(NAT-TREEP
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 (849 12 (:DEFINITION ALWAYS$))
 (745 2 (:REWRITE PLAIN-UQI-INTEGER-LISTP))
 (650 12 (:DEFINITION APPLY$-BADGEP))
 (503 7 (:DEFINITION INTEGER-LISTP))
 (479 4 (:REWRITE INTEGER-LISTP-IMPLIES-ALWAYS$-INTEGERP))
 (459 14 (:REWRITE APPLY$-BADGEP-PROPERTIES . 1))
 (407 12 (:REWRITE APPLY$-SYMBOL-ARITY-1))
 (347 12 (:REWRITE APPLY$-PRIMITIVE))
 (323 12 (:META APPLY$-PRIM-META-FN-CORRECT))
 (321 12 (:DEFINITION SUBSETP-EQUAL))
 (300 33 (:REWRITE SUBSETP-REFLEXIVE-LEMMA))
 (224 9 (:REWRITE APPLY$-BADGEP-PROPERTIES . 2))
 (161 161 (:REWRITE DEFAULT-CDR))
 (151 2 (:REWRITE PLAIN-UQI-TRUE-LIST-LISTP))
 (116 1 (:REWRITE O<=-O-FINP-DEF))
 (99 99 (:TYPE-PRESCRIPTION SUBSETP-EQUAL))
 (97 97 (:REWRITE DEFAULT-CAR))
 (97 2 (:REWRITE PLAIN-UQI-ACL2-NUMBER-LISTP))
 (85 2 (:REWRITE PLAIN-UQI-SYMBOL-LISTP))
 (85 2 (:REWRITE PLAIN-UQI-RATIONAL-LISTP))
 (84 4 (:REWRITE TRUE-LIST-LISTP-IMPLIES-ALWAYS$-TRUE-LISTP))
 (75 36 (:REWRITE DEFAULT-+-2))
 (72 4 (:DEFINITION TRUE-LIST-LISTP))
 (71 9 (:DEFINITION RATIONAL-LISTP))
 (71 6 (:DEFINITION NATP))
 (69 2 (:REWRITE PLAIN-UQI-RATIONAL-LIST-LISTP))
 (60 12 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-1))
 (56 56 (:TYPE-PRESCRIPTION APPLY$-BADGEP))
 (56 17 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (48 36 (:REWRITE DEFAULT-+-1))
 (48 24 (:REWRITE APPLY$-PRIMP-BADGE))
 (48 4 (:REWRITE ACL2-NUMBER-LISTP-IMPLIES-ALWAYS$-ACL2-NUMBERP))
 (45 45 (:TYPE-PRESCRIPTION RATIONAL-LISTP))
 (44 20 (:REWRITE RATIONAL-LISTP-IMPLIES-RATIONALP))
 (40 4 (:REWRITE SYMBOL-LISTP-IMPLIES-ALWAYS$-SYMBOLP))
 (40 4 (:REWRITE RATIONAL-LISTP-IMPLIES-ALWAYS$-RATIONALP))
 (36 36 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (36 36 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-3))
 (36 6 (:DEFINITION LEN))
 (36 4 (:DEFINITION ACL2-NUMBER-LISTP))
 (35 35 (:TYPE-PRESCRIPTION INTEGER-LISTP))
 (33 33 (:REWRITE TRANSITIVITY-OF-SUBSETP-EQUAL))
 (32 8 (:REWRITE O-P-O-INFP-CAR))
 (30 12 (:DEFINITION WEAK-APPLY$-BADGE-P))
 (30 9 (:LINEAR APPLY$-BADGEP-PROPERTIES . 2))
 (28 4 (:DEFINITION SYMBOL-LISTP))
 (27 3 (:DEFINITION LENGTH))
 (26 26 (:REWRITE SUBSETP-IMPLIES-MEMBER))
 (24 24 (:TYPE-PRESCRIPTION LEN))
 (24 24 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-2))
 (24 6 (:REWRITE COMMUTATIVITY-OF-+))
 (24 6 (:DEFINITION INTEGER-ABS))
 (20 20 (:TYPE-PRESCRIPTION TRUE-LIST-LISTP))
 (20 20 (:TYPE-PRESCRIPTION SYMBOL-LISTP))
 (20 20 (:TYPE-PRESCRIPTION ACL2-NUMBER-LISTP))
 (18 9 (:LINEAR APPLY$-BADGEP-PROPERTIES . 1))
 (18 3 (:DEFINITION ALL-NILS))
 (16 16 (:REWRITE TRUE-LIST-LISTP-IMPLIES-TRUE-LISTP))
 (15 15 (:TYPE-PRESCRIPTION ALL-NILS))
 (14 13 (:REWRITE DEFAULT-<-2))
 (14 13 (:REWRITE DEFAULT-<-1))
 (12 12 (:REWRITE APPLY$-CONSP-ARITY-1))
 (12 12 (:REWRITE ALWAYS$-P-LST-IMPLIES-P-ELEMENT))
 (12 6 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (12 6 (:REWRITE APPLY$-BADGEP-PROPERTIES . 3))
 (9 9 (:REWRITE FOLD-CONSTS-IN-+))
 (8 8 (:REWRITE O-P-DEF-O-FINP-1))
 (6 6 (:REWRITE SYMBOL-LISTP-IMPLIES-SYMBOLP))
 (6 6 (:REWRITE LEN-MEMBER-EQUAL-LOOP$-AS))
 (6 6 (:REWRITE DEFAULT-UNARY-MINUS))
 (6 6 (:REWRITE ACL2-NUMBER-LISTP-IMPLIES-ACL2-NUMBERP))
 (6 6 (:LINEAR MEMBER-EQUAL-ACL2-COUNT-SMALLER-3))
 (6 6 (:LINEAR BOUNDS-POSITION-EQUAL-AC))
 (6 6 (:LINEAR BOUNDS-POSITION-EQUAL))
 (4 4 (:LINEAR MEMBER-EQUAL-ACL2-COUNT-SMALLER-2))
 (3 3 (:REWRITE INTEGERP==>NUMERATOR-=-X))
 (3 3 (:REWRITE INTEGERP==>DENOMINATOR-=-1))
 (3 3 (:REWRITE DEFAULT-REALPART))
 (3 3 (:REWRITE DEFAULT-NUMERATOR))
 (3 3 (:REWRITE DEFAULT-IMAGPART))
 (3 3 (:REWRITE DEFAULT-DENOMINATOR))
 (3 3 (:REWRITE DEFAULT-COERCE-2))
 (3 3 (:REWRITE DEFAULT-COERCE-1))
 (3 1 (:REWRITE AC-<))
 (2 2 (:REWRITE PLAIN-UQI-ALWAYS$))
 (2 2 (:LINEAR MEMBER-EQUAL-ACL2-COUNT-SMALLER-1))
 (2 1 (:REWRITE O-INFP-O-FINP-O<=))
 (1 1 (:REWRITE |a < b & b < c  =>  a < c|))
 )
(APPLY$-WARRANT-NAT-TREEP)
(APPLY$-WARRANT-NAT-TREEP-DEFINITION)
(APPLY$-WARRANT-NAT-TREEP-NECC)
(APPLY$-NAT-TREEP
 (34 34 (:REWRITE DEFAULT-CAR))
 (20 20 (:REWRITE DEFAULT-CDR))
 (20 12 (:REWRITE APPLY$-PRIMP-BADGE))
 (19 10 (:REWRITE APPLY$-PRIMITIVE))
 (17 17 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (6 6 (:REWRITE TAMEP-TRUE-CAR/CDR-NESTP))
 (6 2 (:DEFINITION TRUE-LISTP))
 (6 1 (:REWRITE O-P-O-INFP-CAR))
 (4 4 (:REWRITE TRUE-LIST-LISTP-IMPLIES-TRUE-LISTP))
 (4 4 (:REWRITE ALWAYS$-P-LST-IMPLIES-P-ELEMENT))
 (3 1 (:REWRITE O-P-DEF-O-FINP-1))
 (2 2 (:TYPE-PRESCRIPTION O-P))
 (2 2 (:TYPE-PRESCRIPTION O-FINP))
 (1 1 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (1 1 (:REWRITE DEFAULT-<-2))
 (1 1 (:REWRITE DEFAULT-<-1))
 )
(EXAMPLES-OF-NAT-TREEP
 (1 1 (:REWRITE APPLY$-WARRANT-NAT-TREEP-NECC))
 )
(COPY-NAT-TREE
 (7903 39 (:DEFINITION APPLY$-BADGEP))
 (6896 259 (:DEFINITION MEMBER-EQUAL))
 (6370 6 (:REWRITE PLAIN-UQI-INTEGER-LISTP))
 (5931 21 (:DEFINITION INTEGER-LISTP))
 (5410 67 (:REWRITE APPLY$-BADGEP-PROPERTIES . 1))
 (5227 88 (:DEFINITION SUBSETP-EQUAL))
 (5154 36 (:DEFINITION ALWAYS$))
 (4056 12 (:REWRITE INTEGER-LISTP-IMPLIES-ALWAYS$-INTEGERP))
 (3228 231 (:REWRITE SUBSETP-REFLEXIVE-LEMMA))
 (2727 36 (:REWRITE APPLY$-SYMBOL-ARITY-1))
 (2639 38 (:REWRITE APPLY$-BADGEP-PROPERTIES . 2))
 (2547 36 (:REWRITE APPLY$-PRIMITIVE))
 (2475 36 (:META APPLY$-PRIM-META-FN-CORRECT))
 (2076 208 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-1))
 (1997 89 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (1926 1926 (:REWRITE DEFAULT-CDR))
 (694 694 (:REWRITE DEFAULT-CAR))
 (677 677 (:TYPE-PRESCRIPTION SUBSETP-EQUAL))
 (624 624 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-3))
 (554 554 (:REWRITE SUBSETP-IMPLIES-MEMBER))
 (451 6 (:REWRITE PLAIN-UQI-TRUE-LIST-LISTP))
 (416 416 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-2))
 (380 43 (:DEFINITION TRUE-LISTP))
 (342 169 (:REWRITE DEFAULT-+-2))
 (333 333 (:TYPE-PRESCRIPTION APPLY$-BADGEP))
 (289 6 (:REWRITE PLAIN-UQI-ACL2-NUMBER-LISTP))
 (260 42 (:DEFINITION LEN))
 (253 6 (:REWRITE PLAIN-UQI-SYMBOL-LISTP))
 (253 6 (:REWRITE PLAIN-UQI-RATIONAL-LISTP))
 (252 12 (:REWRITE TRUE-LIST-LISTP-IMPLIES-ALWAYS$-TRUE-LISTP))
 (231 231 (:REWRITE TRANSITIVITY-OF-SUBSETP-EQUAL))
 (224 169 (:REWRITE DEFAULT-+-1))
 (216 12 (:DEFINITION TRUE-LIST-LISTP))
 (213 27 (:DEFINITION RATIONAL-LISTP))
 (205 6 (:REWRITE PLAIN-UQI-RATIONAL-LIST-LISTP))
 (204 3 (:REWRITE O<=-O-FINP-DEF))
 (200 100 (:REWRITE APPLY$-BADGEP-PROPERTIES . 3))
 (192 192 (:TYPE-PRESCRIPTION LEN))
 (180 180 (:TYPE-PRESCRIPTION ALWAYS$))
 (172 172 (:TYPE-PRESCRIPTION TRUE-LISTP))
 (153 17 (:DEFINITION LENGTH))
 (150 75 (:LINEAR APPLY$-BADGEP-PROPERTIES . 1))
 (150 25 (:DEFINITION ALL-NILS))
 (144 72 (:REWRITE APPLY$-PRIMP-BADGE))
 (144 39 (:DEFINITION WEAK-APPLY$-BADGE-P))
 (144 12 (:REWRITE ACL2-NUMBER-LISTP-IMPLIES-ALWAYS$-ACL2-NUMBERP))
 (140 68 (:REWRITE RATIONAL-LISTP-IMPLIES-RATIONALP))
 (136 34 (:REWRITE COMMUTATIVITY-OF-+))
 (136 34 (:DEFINITION INTEGER-ABS))
 (135 135 (:TYPE-PRESCRIPTION RATIONAL-LISTP))
 (128 37 (:LINEAR APPLY$-BADGEP-PROPERTIES . 2))
 (125 125 (:TYPE-PRESCRIPTION ALL-NILS))
 (120 12 (:REWRITE SYMBOL-LISTP-IMPLIES-ALWAYS$-SYMBOLP))
 (120 12 (:REWRITE RATIONAL-LISTP-IMPLIES-ALWAYS$-RATIONALP))
 (108 108 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (108 12 (:DEFINITION ACL2-NUMBER-LISTP))
 (105 105 (:TYPE-PRESCRIPTION INTEGER-LISTP))
 (96 24 (:REWRITE O-P-O-INFP-CAR))
 (95 89 (:REWRITE DEFAULT-<-2))
 (94 89 (:REWRITE DEFAULT-<-1))
 (86 86 (:REWRITE TRUE-LIST-LISTP-IMPLIES-TRUE-LISTP))
 (84 12 (:DEFINITION SYMBOL-LISTP))
 (60 60 (:TYPE-PRESCRIPTION TRUE-LIST-LISTP))
 (60 60 (:TYPE-PRESCRIPTION SYMBOL-LISTP))
 (60 60 (:TYPE-PRESCRIPTION ACL2-NUMBER-LISTP))
 (50 50 (:LINEAR BOUNDS-POSITION-EQUAL-AC))
 (50 50 (:LINEAR BOUNDS-POSITION-EQUAL))
 (42 42 (:REWRITE LEN-MEMBER-EQUAL-LOOP$-AS))
 (38 38 (:REWRITE FOLD-CONSTS-IN-+))
 (36 36 (:REWRITE APPLY$-CONSP-ARITY-1))
 (36 36 (:REWRITE ALWAYS$-P-LST-IMPLIES-P-ELEMENT))
 (36 18 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (34 34 (:REWRITE DEFAULT-UNARY-MINUS))
 (30 30 (:LINEAR MEMBER-EQUAL-ACL2-COUNT-SMALLER-3))
 (24 24 (:REWRITE O-P-DEF-O-FINP-1))
 (20 20 (:LINEAR MEMBER-EQUAL-ACL2-COUNT-SMALLER-2))
 (18 18 (:REWRITE SYMBOL-LISTP-IMPLIES-SYMBOLP))
 (18 18 (:REWRITE ACL2-NUMBER-LISTP-IMPLIES-ACL2-NUMBERP))
 (17 17 (:REWRITE INTEGERP==>NUMERATOR-=-X))
 (17 17 (:REWRITE INTEGERP==>DENOMINATOR-=-1))
 (17 17 (:REWRITE DEFAULT-REALPART))
 (17 17 (:REWRITE DEFAULT-NUMERATOR))
 (17 17 (:REWRITE DEFAULT-IMAGPART))
 (17 17 (:REWRITE DEFAULT-DENOMINATOR))
 (17 17 (:REWRITE DEFAULT-COERCE-2))
 (17 17 (:REWRITE DEFAULT-COERCE-1))
 (12 2 (:REWRITE O-FIRST-EXPT-<))
 (10 10 (:LINEAR MEMBER-EQUAL-ACL2-COUNT-SMALLER-1))
 (10 3 (:REWRITE AC-<))
 (8 4 (:REWRITE O-FIRST-EXPT-DEF-O-FINP))
 (6 6 (:REWRITE PLAIN-UQI-ALWAYS$))
 (6 3 (:REWRITE O-INFP-O-FINP-O<=))
 (4 2 (:REWRITE O-FIRST-COEFF-<))
 (3 3 (:REWRITE |a < b & b < c  =>  a < c|))
 )
(APPLY$-WARRANT-COPY-NAT-TREE)
(APPLY$-WARRANT-COPY-NAT-TREE-DEFINITION)
(APPLY$-WARRANT-COPY-NAT-TREE-NECC)
(APPLY$-COPY-NAT-TREE
 (127 2 (:DEFINITION COLLECT$))
 (121 4 (:REWRITE APPLY$-CONSP-ARITY-1))
 (113 2 (:REWRITE BETA-REDUCTION))
 (91 4 (:REWRITE EV$-OPENER))
 (36 4 (:DEFINITION EV$-LIST-DEF))
 (31 4 (:DEFINITION TAMEP))
 (26 26 (:REWRITE DEFAULT-CAR))
 (22 4 (:REWRITE DEFAULT-+-2))
 (20 4 (:DEFINITION SUITABLY-TAMEP-LISTP))
 (20 4 (:DEFINITION PAIRLIS$))
 (18 18 (:REWRITE DEFAULT-CDR))
 (16 2 (:REWRITE APPLY$-SYMBOL-ARITY-1))
 (14 2 (:DEFINITION ASSOC-EQUAL))
 (12 2 (:REWRITE O-P-O-INFP-CAR))
 (10 7 (:REWRITE APPLY$-PRIMP-BADGE))
 (10 2 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (9 5 (:REWRITE APPLY$-PRIMITIVE))
 (8 8 (:TYPE-PRESCRIPTION TAMEP))
 (8 8 (:TYPE-PRESCRIPTION SUITABLY-TAMEP-LISTP))
 (7 7 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (6 6 (:REWRITE TAMEP-TRUE-CAR/CDR-NESTP))
 (6 6 (:REWRITE CAR-CONS))
 (6 2 (:REWRITE O-P-DEF-O-FINP-1))
 (4 4 (:TYPE-PRESCRIPTION O-P))
 (4 4 (:TYPE-PRESCRIPTION O-FINP))
 (4 4 (:REWRITE DEFAULT-+-1))
 (2 2 (:TYPE-PRESCRIPTION LEN))
 (2 2 (:REWRITE RATIONAL-LISTP-IMPLIES-RATIONALP))
 (2 2 (:REWRITE O-INFP->NEQ-0))
 (2 2 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (2 2 (:REWRITE DEFAULT-<-2))
 (2 2 (:REWRITE DEFAULT-<-1))
 (2 2 (:REWRITE CDR-CONS))
 (2 2 (:REWRITE ACL2-NUMBER-LISTP-IMPLIES-ACL2-NUMBERP))
 )
(FAILURE-0
 (807 12 (:REWRITE APPLY$-CONSP-ARITY-1))
 (759 12 (:REWRITE BETA-REDUCTION))
 (659 24 (:REWRITE EV$-OPENER))
 (444 6 (:DEFINITION COLLECT$))
 (411 6 (:DEFINITION ALWAYS$))
 (184 24 (:DEFINITION EV$-LIST-DEF))
 (168 24 (:DEFINITION TAMEP))
 (164 12 (:REWRITE APPLY$-COPY-NAT-TREE))
 (138 133 (:REWRITE DEFAULT-CAR))
 (128 123 (:REWRITE DEFAULT-CDR))
 (120 24 (:DEFINITION SUITABLY-TAMEP-LISTP))
 (119 12 (:REWRITE APPLY$-NAT-TREEP))
 (88 24 (:DEFINITION PAIRLIS$))
 (88 16 (:REWRITE DEFAULT-+-2))
 (84 12 (:DEFINITION ASSOC-EQUAL))
 (48 48 (:TYPE-PRESCRIPTION TAMEP))
 (48 48 (:TYPE-PRESCRIPTION SUITABLY-TAMEP-LISTP))
 (40 10 (:REWRITE O-P-O-INFP-CAR))
 (40 8 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (36 36 (:REWRITE TAMEP-TRUE-CAR/CDR-NESTP))
 (30 6 (:REWRITE APPLY$-BADGEP-PROPERTIES . 1))
 (24 24 (:REWRITE TRUE-LIST-LISTP-IMPLIES-TRUE-LISTP))
 (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))
 (10 10 (:REWRITE O-P-DEF-O-FINP-1))
 (8 8 (:REWRITE RATIONAL-LISTP-IMPLIES-RATIONALP))
 (8 8 (:REWRITE O-INFP->NEQ-0))
 (8 8 (:REWRITE ACL2-NUMBER-LISTP-IMPLIES-ACL2-NUMBERP))
 (6 6 (:TYPE-PRESCRIPTION APPLY$-BADGEP))
 (6 6 (:REWRITE ALWAYS$-P-LST-IMPLIES-P-ELEMENT))
 )
(FAILURE-1
 (807 12 (:REWRITE APPLY$-CONSP-ARITY-1))
 (759 12 (:REWRITE BETA-REDUCTION))
 (659 24 (:REWRITE EV$-OPENER))
 (444 6 (:DEFINITION COLLECT$))
 (411 6 (:DEFINITION ALWAYS$))
 (184 24 (:DEFINITION EV$-LIST-DEF))
 (168 24 (:DEFINITION TAMEP))
 (164 12 (:REWRITE APPLY$-COPY-NAT-TREE))
 (138 133 (:REWRITE DEFAULT-CAR))
 (128 123 (:REWRITE DEFAULT-CDR))
 (120 24 (:DEFINITION SUITABLY-TAMEP-LISTP))
 (119 12 (:REWRITE APPLY$-NAT-TREEP))
 (88 24 (:DEFINITION PAIRLIS$))
 (88 16 (:REWRITE DEFAULT-+-2))
 (84 12 (:DEFINITION ASSOC-EQUAL))
 (48 48 (:TYPE-PRESCRIPTION TAMEP))
 (48 48 (:TYPE-PRESCRIPTION SUITABLY-TAMEP-LISTP))
 (40 10 (:REWRITE O-P-O-INFP-CAR))
 (40 8 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (36 36 (:REWRITE TAMEP-TRUE-CAR/CDR-NESTP))
 (30 6 (:REWRITE APPLY$-BADGEP-PROPERTIES . 1))
 (24 24 (:REWRITE TRUE-LIST-LISTP-IMPLIES-TRUE-LISTP))
 (24 12 (:REWRITE APPLY$-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$-BADGE-P))
 (10 10 (:REWRITE O-P-DEF-O-FINP-1))
 (8 8 (:REWRITE RATIONAL-LISTP-IMPLIES-RATIONALP))
 (8 8 (:REWRITE O-INFP->NEQ-0))
 (8 8 (:REWRITE APPLY$-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$-P-LST-IMPLIES-P-ELEMENT))
 )
(INDUCTION-HINT)
(FAILURE-2
 (4446 169 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-1))
 (4111 3867 (:REWRITE DEFAULT-CAR))
 (3758 3712 (:REWRITE DEFAULT-CDR))
 (2828 59 (:REWRITE APPLY$-BADGEP-PROPERTIES . 1))
 (2722 53 (:DEFINITION APPLY$-BADGEP))
 (2150 20 (:DEFINITION SUBSETP-EQUAL))
 (2104 112 (:DEFINITION LOOP$-AS))
 (1722 322 (:REWRITE DEFAULT-+-2))
 (1195 50 (:REWRITE SUBSETP-REFLEXIVE-LEMMA))
 (984 984 (:REWRITE TAMEP-TRUE-CAR/CDR-NESTP))
 (834 181 (:REWRITE O-P-O-INFP-CAR))
 (775 155 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (673 110 (:DEFINITION CDR-LOOP$-AS-TUPLE))
 (673 110 (:DEFINITION CAR-LOOP$-AS-TUPLE))
 (567 112 (:DEFINITION EMPTY-LOOP$-AS-TUPLEP))
 (551 276 (:REWRITE APPLY$-PRIMITIVE))
 (526 526 (:REWRITE SUBSETP-IMPLIES-MEMBER))
 (397 397 (:REWRITE TRUE-LIST-LISTP-IMPLIES-TRUE-LISTP))
 (385 385 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-3))
 (322 322 (:REWRITE DEFAULT-+-1))
 (308 308 (:TYPE-PRESCRIPTION EV$-LIST))
 (303 303 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-2))
 (291 181 (:REWRITE O-P-DEF-O-FINP-1))
 (275 275 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (267 267 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (267 267 (:REWRITE DEFAULT-<-2))
 (267 267 (:REWRITE DEFAULT-<-1))
 (220 220 (:TYPE-PRESCRIPTION LOOP$-AS))
 (156 156 (:REWRITE O-INFP->NEQ-0))
 (155 155 (:REWRITE RATIONAL-LISTP-IMPLIES-RATIONALP))
 (155 155 (:REWRITE ACL2-NUMBER-LISTP-IMPLIES-ACL2-NUMBERP))
 (140 140 (:TYPE-PRESCRIPTION SUBSETP-EQUAL))
 (126 126 (:REWRITE APPLY$-WARRANT-NAT-TREEP-NECC))
 (126 126 (:REWRITE APPLY$-WARRANT-COPY-NAT-TREE-NECC))
 (116 53 (:DEFINITION WEAK-APPLY$-BADGE-P))
 (86 86 (:TYPE-PRESCRIPTION APPLY$-BADGEP))
 (50 50 (:REWRITE TRANSITIVITY-OF-SUBSETP-EQUAL))
 (44 44 (:TYPE-PRESCRIPTION O-FINP))
 (35 5 (:DEFINITION LEN))
 (30 5 (:DEFINITION ALL-NILS))
 (25 25 (:TYPE-PRESCRIPTION ALL-NILS))
 (22 11 (:LINEAR APPLY$-BADGEP-PROPERTIES . 1))
 (20 10 (:REWRITE APPLY$-BADGEP-PROPERTIES . 3))
 (10 10 (:LINEAR BOUNDS-POSITION-EQUAL-AC))
 (10 10 (:LINEAR BOUNDS-POSITION-EQUAL))
 (6 3 (:REWRITE APPLY$-BADGEP-PROPERTIES . 2))
 (6 3 (:LINEAR APPLY$-BADGEP-PROPERTIES . 2))
 (5 5 (:REWRITE LEN-MEMBER-EQUAL-LOOP$-AS))
 )
(FAILURE-3
 (188794 777 (:DEFINITION APPLY$-BADGEP))
 (155794 1404 (:DEFINITION SUBSETP-EQUAL))
 (141100 746 (:REWRITE APPLY$-BADGEP-PROPERTIES . 1))
 (131538 10662 (:DEFINITION MEMBER-EQUAL))
 (86017 3510 (:REWRITE SUBSETP-REFLEXIVE-LEMMA))
 (84516 5736 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-1))
 (33341 32190 (:REWRITE DEFAULT-CAR))
 (30420 267 (:REWRITE APPLY$-BADGEP-PROPERTIES . 2))
 (21843 21843 (:TYPE-PRESCRIPTION MEMBER-EQUAL))
 (21216 1176 (:REWRITE APPLY$-BADGEP-PROPERTIES . 3))
 (18576 18576 (:REWRITE SUBSETP-IMPLIES-MEMBER))
 (17022 17022 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-3))
 (11406 11406 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-2))
 (9972 2043 (:REWRITE DEFAULT-+-2))
 (9828 9828 (:TYPE-PRESCRIPTION SUBSETP-EQUAL))
 (6148 563 (:REWRITE ALWAYS$-P-LST-IMPLIES-P-ELEMENT))
 (5003 1107 (:REWRITE O-P-O-INFP-CAR))
 (4974 4974 (:REWRITE TAMEP-TRUE-CAR/CDR-NESTP))
 (4264 869 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (3510 3510 (:REWRITE TRANSITIVITY-OF-SUBSETP-EQUAL))
 (3195 3195 (:TYPE-PRESCRIPTION APPLY$-BADGEP))
 (2803 1402 (:REWRITE APPLY$-PRIMITIVE))
 (2683 138 (:DEFINITION LOOP$-AS))
 (2457 351 (:DEFINITION LEN))
 (2440 777 (:DEFINITION WEAK-APPLY$-BADGE-P))
 (2227 2227 (:REWRITE TRUE-LIST-LISTP-IMPLIES-TRUE-LISTP))
 (2106 351 (:DEFINITION ALL-NILS))
 (2043 2043 (:REWRITE DEFAULT-+-1))
 (1755 1755 (:TYPE-PRESCRIPTION ALL-NILS))
 (1682 1107 (:REWRITE O-P-DEF-O-FINP-1))
 (1649 1568 (:REWRITE DEFAULT-<-1))
 (1610 1610 (:TYPE-PRESCRIPTION EV$-LIST))
 (1581 1581 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (1568 1568 (:REWRITE DEFAULT-<-2))
 (1401 1401 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (1174 587 (:LINEAR APPLY$-BADGEP-PROPERTIES . 1))
 (1008 267 (:LINEAR APPLY$-BADGEP-PROPERTIES . 2))
 (869 869 (:REWRITE RATIONAL-LISTP-IMPLIES-RATIONALP))
 (869 869 (:REWRITE ACL2-NUMBER-LISTP-IMPLIES-ACL2-NUMBERP))
 (843 843 (:REWRITE O-INFP->NEQ-0))
 (839 138 (:DEFINITION CDR-LOOP$-AS-TUPLE))
 (839 138 (:DEFINITION CAR-LOOP$-AS-TUPLE))
 (710 138 (:DEFINITION EMPTY-LOOP$-AS-TUPLEP))
 (702 702 (:LINEAR BOUNDS-POSITION-EQUAL-AC))
 (702 702 (:LINEAR BOUNDS-POSITION-EQUAL))
 (433 433 (:REWRITE APPLY$-WARRANT-NAT-TREEP-NECC))
 (433 433 (:REWRITE APPLY$-WARRANT-COPY-NAT-TREE-NECC))
 (351 351 (:REWRITE LEN-MEMBER-EQUAL-LOOP$-AS))
 (276 276 (:TYPE-PRESCRIPTION LOOP$-AS))
 (230 230 (:TYPE-PRESCRIPTION O-FINP))
 )
(FINALLY-SUCCESS!
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 (6321 45 (:REWRITE APPLY$-BADGEP-PROPERTIES . 1))
 (6231 45 (:DEFINITION APPLY$-BADGEP))
 (5808 308 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-1))
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 (4970 4965 (:REWRITE DEFAULT-CDR))
 (4460 230 (:REWRITE ALWAYS$-P-LST-IMPLIES-P-ELEMENT))
 (3939 3934 (:REWRITE DEFAULT-CAR))
 (2868 120 (:REWRITE SUBSETP-REFLEXIVE-LEMMA))
 (2032 116 (:DEFINITION LOOP$-AS))
 (1558 1558 (:TYPE-PRESCRIPTION MEMBER-EQUAL))
 (985 985 (:REWRITE TRUE-LIST-LISTP-IMPLIES-TRUE-LISTP))
 (954 954 (:REWRITE SUBSETP-IMPLIES-MEMBER))
 (798 798 (:REWRITE TAMEP-TRUE-CAR/CDR-NESTP))
 (720 720 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-3))
 (672 112 (:DEFINITION CDR-LOOP$-AS-TUPLE))
 (672 112 (:DEFINITION CAR-LOOP$-AS-TUPLE))
 (610 305 (:REWRITE APPLY$-PRIMITIVE))
 (604 124 (:REWRITE DEFAULT-+-2))
 (572 116 (:DEFINITION EMPTY-LOOP$-AS-TUPLEP))
 (524 524 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-2))
 (402 108 (:REWRITE O-P-O-INFP-CAR))
 (336 336 (:TYPE-PRESCRIPTION SUBSETP-EQUAL))
 (305 305 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (260 52 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (248 248 (:REWRITE APPLY$-WARRANT-NAT-TREEP-NECC))
 (248 248 (:REWRITE APPLY$-WARRANT-COPY-NAT-TREE-NECC))
 (224 224 (:TYPE-PRESCRIPTION LOOP$-AS))
 (126 126 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (126 126 (:REWRITE DEFAULT-<-2))
 (126 126 (:REWRITE DEFAULT-<-1))
 (124 124 (:REWRITE DEFAULT-+-1))
 (120 120 (:REWRITE TRANSITIVITY-OF-SUBSETP-EQUAL))
 (114 45 (:DEFINITION WEAK-APPLY$-BADGE-P))
 (98 98 (:REWRITE O-P-DEF-O-FINP-1))
 (91 91 (:TYPE-PRESCRIPTION APPLY$-BADGEP))
 (84 12 (:DEFINITION LEN))
 (72 12 (:DEFINITION ALL-NILS))
 (68 34 (:REWRITE APPLY$-BADGEP-PROPERTIES . 3))
 (60 60 (:TYPE-PRESCRIPTION ALL-NILS))
 (53 53 (:REWRITE O-INFP->NEQ-0))
 (52 52 (:REWRITE RATIONAL-LISTP-IMPLIES-RATIONALP))
 (52 52 (:REWRITE ACL2-NUMBER-LISTP-IMPLIES-ACL2-NUMBERP))
 (24 24 (:LINEAR BOUNDS-POSITION-EQUAL-AC))
 (24 24 (:LINEAR BOUNDS-POSITION-EQUAL))
 (24 12 (:LINEAR APPLY$-BADGEP-PROPERTIES . 1))
 (20 20 (:TYPE-PRESCRIPTION TRUE-LISTP))
 (12 12 (:REWRITE LEN-MEMBER-EQUAL-LOOP$-AS))
 )
(MR-COPY-NAT-TREE)
(MR-COPY-NAT-TREE-COPIES
 (4940 348 (:DEFINITION MEMBER-EQUAL))
 (4596 224 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-1))
 (4306 224 (:REWRITE ALWAYS$-P-LST-IMPLIES-P-ELEMENT))
 (3657 3657 (:REWRITE DEFAULT-CDR))
 (3636 23 (:REWRITE APPLY$-BADGEP-PROPERTIES . 1))
 (3590 23 (:DEFINITION APPLY$-BADGEP))
 (3010 28 (:DEFINITION SUBSETP-EQUAL))
 (2968 2968 (:REWRITE DEFAULT-CAR))
 (1960 112 (:DEFINITION LOOP$-AS))
 (1673 70 (:REWRITE SUBSETP-REFLEXIVE-LEMMA))
 (1232 1232 (:TYPE-PRESCRIPTION MEMBER-EQUAL))
 (963 963 (:REWRITE TRUE-LIST-LISTP-IMPLIES-TRUE-LISTP))
 (684 684 (:REWRITE SUBSETP-IMPLIES-MEMBER))
 (648 108 (:DEFINITION CDR-LOOP$-AS-TUPLE))
 (648 108 (:DEFINITION CAR-LOOP$-AS-TUPLE))
 (570 570 (:REWRITE TAMEP-TRUE-CAR/CDR-NESTP))
 (552 112 (:DEFINITION EMPTY-LOOP$-AS-TUPLEP))
 (472 472 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-3))
 (448 224 (:REWRITE APPLY$-PRIMITIVE))
 (356 356 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-2))
 (286 79 (:REWRITE O-P-O-INFP-CAR))
 (245 245 (:REWRITE APPLY$-WARRANT-NAT-TREEP-NECC))
 (224 224 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (216 216 (:TYPE-PRESCRIPTION LOOP$-AS))
 (209 49 (:REWRITE DEFAULT-+-2))
 (196 196 (:TYPE-PRESCRIPTION SUBSETP-EQUAL))
 (97 97 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (97 97 (:REWRITE DEFAULT-<-2))
 (97 97 (:REWRITE DEFAULT-<-1))
 (85 17 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (70 70 (:REWRITE TRANSITIVITY-OF-SUBSETP-EQUAL))
 (69 69 (:REWRITE O-P-DEF-O-FINP-1))
 (60 23 (:DEFINITION WEAK-APPLY$-BADGE-P))
 (49 49 (:REWRITE DEFAULT-+-1))
 (49 7 (:DEFINITION LEN))
 (44 44 (:TYPE-PRESCRIPTION APPLY$-BADGEP))
 (42 7 (:DEFINITION ALL-NILS))
 (35 35 (:TYPE-PRESCRIPTION ALL-NILS))
 (28 14 (:REWRITE APPLY$-BADGEP-PROPERTIES . 3))
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 (17 17 (:REWRITE RATIONAL-LISTP-IMPLIES-RATIONALP))
 (17 17 (:REWRITE ACL2-NUMBER-LISTP-IMPLIES-ACL2-NUMBERP))
 (14 14 (:LINEAR BOUNDS-POSITION-EQUAL-AC))
 (14 14 (:LINEAR BOUNDS-POSITION-EQUAL))
 (14 7 (:LINEAR APPLY$-BADGEP-PROPERTIES . 1))
 (7 7 (:REWRITE LEN-MEMBER-EQUAL-LOOP$-AS))
 )
(COPY-NAT-TREE-IS-MR-COPY-NAT-TREE
 (10366 42 (:REWRITE APPLY$-BADGEP-PROPERTIES . 1))
 (10282 42 (:DEFINITION APPLY$-BADGEP))
 (9779 763 (:DEFINITION MEMBER-EQUAL))
 (8600 80 (:DEFINITION SUBSETP-EQUAL))
 (7824 448 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-1))
 (6824 6824 (:REWRITE DEFAULT-CDR))
 (4855 257 (:REWRITE ALWAYS$-P-LST-IMPLIES-P-ELEMENT))
 (4780 200 (:REWRITE SUBSETP-REFLEXIVE-LEMMA))
 (4600 4600 (:REWRITE DEFAULT-CAR))
 (2192 128 (:DEFINITION LOOP$-AS))
 (2115 2115 (:TYPE-PRESCRIPTION MEMBER-EQUAL))
 (1655 1655 (:REWRITE TRUE-LIST-LISTP-IMPLIES-TRUE-LISTP))
 (1406 1406 (:REWRITE SUBSETP-IMPLIES-MEMBER))
 (1116 1116 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-3))
 (1059 211 (:REWRITE DEFAULT-+-2))
 (930 930 (:REWRITE TAMEP-TRUE-CAR/CDR-NESTP))
 (792 792 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-2))
 (720 120 (:DEFINITION CDR-LOOP$-AS-TUPLE))
 (720 120 (:DEFINITION CAR-LOOP$-AS-TUPLE))
 (662 331 (:REWRITE APPLY$-PRIMITIVE))
 (624 128 (:DEFINITION EMPTY-LOOP$-AS-TUPLEP))
 (560 560 (:TYPE-PRESCRIPTION SUBSETP-EQUAL))
 (460 92 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (456 114 (:REWRITE O-P-O-INFP-CAR))
 (331 331 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (277 277 (:REWRITE APPLY$-WARRANT-NAT-TREEP-NECC))
 (277 277 (:REWRITE APPLY$-WARRANT-COPY-NAT-TREE-NECC))
 (240 240 (:TYPE-PRESCRIPTION LOOP$-AS))
 (211 211 (:REWRITE DEFAULT-+-1))
 (200 200 (:REWRITE TRANSITIVITY-OF-SUBSETP-EQUAL))
 (176 176 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (176 176 (:REWRITE DEFAULT-<-2))
 (176 176 (:REWRITE DEFAULT-<-1))
 (140 20 (:DEFINITION LEN))
 (124 42 (:DEFINITION WEAK-APPLY$-BADGE-P))
 (120 20 (:DEFINITION ALL-NILS))
 (118 118 (:TYPE-PRESCRIPTION APPLY$-BADGEP))
 (114 114 (:REWRITE O-P-DEF-O-FINP-1))
 (112 56 (:REWRITE APPLY$-BADGEP-PROPERTIES . 3))
 (100 100 (:TYPE-PRESCRIPTION ALL-NILS))
 (92 92 (:REWRITE RATIONAL-LISTP-IMPLIES-RATIONALP))
 (92 92 (:REWRITE ACL2-NUMBER-LISTP-IMPLIES-ACL2-NUMBERP))
 (90 90 (:REWRITE O-INFP->NEQ-0))
 (40 40 (:LINEAR BOUNDS-POSITION-EQUAL-AC))
 (40 40 (:LINEAR BOUNDS-POSITION-EQUAL))
 (40 20 (:LINEAR APPLY$-BADGEP-PROPERTIES . 1))
 (20 20 (:REWRITE LEN-MEMBER-EQUAL-LOOP$-AS))
 (9 3 (:REWRITE FOLD-CONSTS-IN-+))
 )
(FLAGGED-MR-COPY-NAT-TREE)
(FLAGGED-EQUIVALENCE
 (171 36 (:REWRITE DEFAULT-+-2))
 (90 90 (:REWRITE DEFAULT-CAR))
 (76 17 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (67 67 (:REWRITE DEFAULT-CDR))
 (36 36 (:REWRITE DEFAULT-+-1))
 (27 27 (:REWRITE TAMEP-TRUE-CAR/CDR-NESTP))
 (24 6 (:REWRITE O-P-O-INFP-CAR))
 (18 9 (:REWRITE APPLY$-PRIMITIVE))
 (17 17 (:REWRITE RATIONAL-LISTP-IMPLIES-RATIONALP))
 (17 17 (:REWRITE APPLY$-WARRANT-COPY-NAT-TREE-NECC))
 (17 17 (:REWRITE ACL2-NUMBER-LISTP-IMPLIES-ACL2-NUMBERP))
 (15 15 (:REWRITE O-INFP->NEQ-0))
 (14 14 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (14 14 (:REWRITE DEFAULT-<-2))
 (14 14 (:REWRITE DEFAULT-<-1))
 (9 9 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (6 6 (:REWRITE O-P-DEF-O-FINP-1))
 (6 2 (:REWRITE FOLD-CONSTS-IN-+))
 )
(ELIMINATE-COPY-NAT-TREE)
