(ASSOC-EQUAL-LOOP$
 (162 15 (:REWRITE TRUE-LIST-LISTP-IMPLIES-TRUE-LISTP-XXX))
 (135 133 (:REWRITE DEFAULT-CAR))
 (128 30 (:DEFINITION TRUE-LISTP))
 (118 116 (:REWRITE DEFAULT-CDR))
 (53 9 (:REWRITE NOT-MEMBER-LOOP$-AS-RATIONAL-LISTP-1))
 (53 7 (:REWRITE NOT-MEMBER-LOOP$-AS-NATP-1))
 (41 9 (:REWRITE NOT-MEMBER-LOOP$-AS-IDENTITY-1))
 (39 7 (:REWRITE NOT-MEMBER-LOOP$-AS-ACL2-NUMBER-1))
 (37 7 (:REWRITE NOT-MEMBER-LOOP$-AS-TRUE-LIST-1))
 (28 7 (:DEFINITION NATP))
 (26 13 (:REWRITE DEFAULT-+-2))
 (25 7 (:REWRITE NOT-MEMBER-LOOP$-AS-SYMBOL-1))
 (25 7 (:REWRITE NOT-MEMBER-LOOP$-AS-RATIONAL-1))
 (25 7 (:REWRITE NOT-MEMBER-LOOP$-AS-INTEGER-1))
 (20 8 (:DEFINITION RATIONAL-LISTP))
 (18 1 (:DEFINITION ALWAYS$))
 (17 17 (:TYPE-PRESCRIPTION RATIONAL-LISTP))
 (16 16 (:REWRITE RATIONAL-LISTP-IMPLIES-RATIONALP))
 (15 15 (:REWRITE ALWAYS$-P-LST-IMPLIES-P-ELEMENT-COROLLARY))
 (14 14 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (14 7 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (13 13 (:REWRITE DEFAULT-+-1))
 (12 4 (:REWRITE EQUAL-LEN-1))
 (10 1 (:REWRITE APPLY$-SYMBOL-ARITY-1))
 (7 7 (:TYPE-PRESCRIPTION NATP))
 (7 7 (:REWRITE SYMBOL-LISTP-IMPLIES-SYMBOLP))
 (7 7 (:REWRITE NOT-MEMBER-LOOP$-AS-GENERAL))
 (7 7 (:REWRITE NOT-MEMBER-LOOP$-AS-ALWAYS$-1))
 (7 7 (:REWRITE DEFAULT-<-2))
 (7 7 (:REWRITE DEFAULT-<-1))
 (7 7 (:REWRITE ACL2-NUMBER-LISTP-IMPLIES-ACL2-NUMBERP))
 (5 5 (:TYPE-PRESCRIPTION ALWAYS$))
 (5 1 (:REWRITE APPLY$-PRIMITIVE))
 (4 2 (:REWRITE APPLY$-PRIMP-BADGE))
 (3 3 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (3 1 (:META APPLY$-PRIM-META-FN-CORRECT))
 (1 1 (:REWRITE APPLY$-CONSP-ARITY-1))
 (1 1 (:REWRITE ALWAYS$-P-LST-IMPLIES-P-ELEMENT))
 (1 1 (:DEFINITION IDENTITY))
 )
(LP12-1
 (167 164 (:REWRITE DEFAULT-CAR))
 (67 66 (:REWRITE DEFAULT-CDR))
 (14 9 (:REWRITE APPLY$-PRIMITIVE))
 (10 10 (:META RELINK-FANCY-SCION-CORRECT))
 (5 5 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 )
(COUNTEREXAMPLE-TO-STRONGER-LP12-1)
(LP12-2
 (119 118 (:REWRITE DEFAULT-CAR))
 (103 102 (:REWRITE DEFAULT-CDR))
 (23 12 (:REWRITE DEFAULT-+-2))
 (14 7 (:REWRITE DEFAULT-<-1))
 (12 12 (:REWRITE LEN-MEMBER-EQUAL-LOOP$-AS))
 (12 12 (:REWRITE DEFAULT-+-1))
 (12 7 (:REWRITE DEFAULT-<-2))
 (10 10 (:META RELINK-FANCY-SCION-CORRECT))
 (10 5 (:REWRITE APPLY$-PRIMITIVE))
 (5 5 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 )
(LP12-2-EQUIVALENCE
 (274 271 (:REWRITE DEFAULT-CDR))
 (255 252 (:REWRITE DEFAULT-CAR))
 (44 22 (:REWRITE DEFAULT-+-2))
 (27 14 (:REWRITE DEFAULT-<-2))
 (26 14 (:REWRITE DEFAULT-<-1))
 (24 24 (:REWRITE LEN-MEMBER-EQUAL-LOOP$-AS))
 (22 22 (:REWRITE DEFAULT-+-1))
 (18 18 (:META RELINK-FANCY-SCION-CORRECT))
 (18 9 (:REWRITE APPLY$-PRIMITIVE))
 (9 9 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 )
(LP12-2-OBSERVATION)
(LP12-3-LEMMA
 (48 24 (:REWRITE DEFAULT-+-2))
 (31 24 (:REWRITE DEFAULT-+-1))
 (20 10 (:TYPE-PRESCRIPTION TRUE-LISTP-APPEND))
 (18 15 (:REWRITE DEFAULT-CDR))
 (15 15 (:REWRITE LEN-MEMBER-EQUAL-LOOP$-AS))
 (10 10 (:TYPE-PRESCRIPTION TRUE-LISTP))
 (10 10 (:TYPE-PRESCRIPTION BINARY-APPEND))
 (3 3 (:REWRITE DEFAULT-CAR))
 (1 1 (:REWRITE FOLD-CONSTS-IN-+))
 )
(LP12-3
 (1209 22 (:REWRITE TRUE-LISTP-TRUE-LIST-FIX-ID))
 (1144 23 (:DEFINITION TRUE-LISTP))
 (1028 16 (:REWRITE APPLY$-BADGEP-PROPERTIES . 3))
 (1020 4 (:DEFINITION APPLY$-BADGEP))
 (708 4 (:DEFINITION SUBSETP-EQUAL))
 (656 56 (:DEFINITION MEMBER-EQUAL))
 (458 452 (:REWRITE DEFAULT-CDR))
 (408 28 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-1))
 (142 140 (:REWRITE DEFAULT-CAR))
 (124 124 (:TYPE-PRESCRIPTION MEMBER-EQUAL))
 (120 120 (:TYPE-PRESCRIPTION TRUE-LISTP))
 (110 46 (:REWRITE DEFAULT-+-2))
 (84 84 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-3))
 (80 8 (:DEFINITION NATP))
 (78 6 (:REWRITE APPLY$-CONSP-ARITY-1))
 (62 46 (:REWRITE DEFAULT-+-1))
 (56 56 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-2))
 (47 24 (:REWRITE DEFAULT-*-2))
 (44 44 (:TYPE-PRESCRIPTION APPLY$-BADGEP))
 (36 24 (:REWRITE DEFAULT-*-1))
 (24 24 (:REWRITE LEN-MEMBER-EQUAL-LOOP$-AS))
 (24 12 (:LINEAR APPLY$-BADGEP-PROPERTIES . 1))
 (24 4 (:DEFINITION ALL-NILS))
 (20 20 (:TYPE-PRESCRIPTION ALL-NILS))
 (16 16 (:TYPE-PRESCRIPTION SUBSETP-EQUAL))
 (16 8 (:REWRITE APPLY$-BADGEP-PROPERTIES . 1))
 (16 4 (:DEFINITION WEAK-APPLY$-BADGE-P))
 (14 7 (:REWRITE APPLY$-PRIMITIVE))
 (8 8 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (8 8 (:REWRITE DEFAULT-<-2))
 (8 8 (:REWRITE DEFAULT-<-1))
 (8 4 (:REWRITE APPLY$-BADGEP-PROPERTIES . 2))
 (8 4 (:LINEAR APPLY$-BADGEP-PROPERTIES . 2))
 (7 7 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 )
(NATS)
(LP12-4
 (482 12 (:REWRITE APPLY$-CONSP-ARITY-1))
 (147 105 (:REWRITE DEFAULT-CDR))
 (147 105 (:REWRITE DEFAULT-CAR))
 (89 89 (:REWRITE DEFAULT-+-2))
 (89 89 (:REWRITE DEFAULT-+-1))
 (76 76 (:REWRITE DEFAULT-<-2))
 (76 76 (:REWRITE DEFAULT-<-1))
 (75 25 (:REWRITE FOLD-CONSTS-IN-+))
 (28 14 (:REWRITE APPLY$-PRIMITIVE))
 (23 11 (:REWRITE ZP-OPEN))
 (14 14 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (12 12 (:TYPE-PRESCRIPTION LEN))
 (3 3 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (1 1 (:INDUCTION COLLECT$-FTB-RULE))
 )
(NATS-UP
 (56 52 (:REWRITE DEFAULT-<-2))
 (56 52 (:REWRITE DEFAULT-<-1))
 (38 38 (:REWRITE DEFAULT-+-2))
 (38 38 (:REWRITE DEFAULT-+-1))
 (23 23 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (6 6 (:REWRITE DEFAULT-UNARY-MINUS))
 (4 2 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (2 2 (:REWRITE RATIONAL-LISTP-IMPLIES-RATIONALP))
 (2 2 (:REWRITE ACL2-NUMBER-LISTP-IMPLIES-ACL2-NUMBERP))
 )
(LP12-5-LEMMA
 (312 6 (:REWRITE APPLY$-CONSP-ARITY-1))
 (100 20 (:DEFINITION BINARY-APPEND))
 (97 97 (:REWRITE DEFAULT-+-2))
 (97 97 (:REWRITE DEFAULT-+-1))
 (91 91 (:REWRITE DEFAULT-<-2))
 (91 91 (:REWRITE DEFAULT-<-1))
 (41 40 (:REWRITE DEFAULT-CDR))
 (41 40 (:REWRITE DEFAULT-CAR))
 (30 10 (:REWRITE FOLD-CONSTS-IN-+))
 (14 7 (:REWRITE APPLY$-PRIMITIVE))
 (12 12 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (7 7 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (6 6 (:TYPE-PRESCRIPTION LEN))
 )
(LP12-5
 (21 2 (:DEFINITION FROM-TO-BY-DOWN-OPENER))
 (11 11 (:REWRITE DEFAULT-<-2))
 (11 11 (:REWRITE DEFAULT-<-1))
 (10 2 (:DEFINITION BINARY-APPEND))
 (4 3 (:REWRITE DEFAULT-CDR))
 (4 3 (:REWRITE DEFAULT-CAR))
 (2 2 (:REWRITE DEFAULT-+-2))
 (2 2 (:REWRITE DEFAULT-+-1))
 (2 1 (:REWRITE APPLY$-PRIMITIVE))
 (1 1 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (1 1 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 )
(LP12-6
 (28 28 (:REWRITE DEFAULT-CAR))
 (17 17 (:REWRITE DEFAULT-CDR))
 (8 4 (:REWRITE APPLY$-PRIMITIVE))
 (4 4 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 )
(LP12-7-LEMMA
 (1772 4 (:REWRITE APPLY$-CONSP-ARITY-1))
 (390 78 (:DEFINITION BINARY-APPEND))
 (386 383 (:REWRITE DEFAULT-CAR))
 (372 369 (:REWRITE DEFAULT-CDR))
 (282 262 (:REWRITE DEFAULT-+-2))
 (274 14 (:REWRITE LP12-2))
 (262 262 (:REWRITE DEFAULT-+-1))
 (179 169 (:REWRITE DEFAULT-<-2))
 (179 169 (:REWRITE DEFAULT-<-1))
 (124 124 (:TYPE-PRESCRIPTION LEN))
 (120 20 (:DEFINITION LEN))
 (38 19 (:REWRITE APPLY$-PRIMITIVE))
 (28 28 (:META RELINK-FANCY-SCION-CORRECT))
 (23 23 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (20 20 (:REWRITE LEN-MEMBER-EQUAL-LOOP$-AS))
 (19 19 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (17 5 (:REWRITE ALWAYS$-P-LST-IMPLIES-P-ELEMENT))
 (12 4 (:REWRITE FOLD-CONSTS-IN-+))
 (10 1 (:DEFINITION MEMBER-EQUAL))
 (5 5 (:TYPE-PRESCRIPTION MEMBER-EQUAL))
 (1 1 (:INDUCTION LOOP$-AS-CDR-FBT-RULE . 1))
 (1 1 (:INDUCTION LOOP$-AS-CDR-CDR-RULE . 1))
 )
(LP12-7
 (314 2 (:DEFINITION LOOP$-AS))
 (219 1 (:DEFINITION COLLECT$+))
 (211 1 (:REWRITE BETA-REDUCTION))
 (207 2 (:DEFINITION PAIRLIS$))
 (176 8 (:DEFINITION FROM-TO-BY-DOWN-OPENER))
 (104 4 (:DEFINITION CDR-LOOP$-AS-TUPLE))
 (104 4 (:DEFINITION CAR-LOOP$-AS-TUPLE))
 (102 4 (:DEFINITION EMPTY-LOOP$-AS-TUPLEP))
 (64 8 (:DEFINITION FROM-TO-BY))
 (53 35 (:REWRITE DEFAULT-+-2))
 (52 10 (:DEFINITION LEN))
 (42 42 (:REWRITE DEFAULT-CDR))
 (40 8 (:DEFINITION BINARY-APPEND))
 (36 35 (:REWRITE DEFAULT-+-1))
 (34 34 (:REWRITE DEFAULT-CAR))
 (34 17 (:REWRITE DEFAULT-<-1))
 (32 8 (:DEFINITION NOT))
 (27 1 (:REWRITE LP12-2))
 (24 24 (:TYPE-PRESCRIPTION FROM-TO-BY))
 (24 8 (:REWRITE FOLD-CONSTS-IN-+))
 (18 17 (:REWRITE DEFAULT-<-2))
 (14 14 (:REWRITE CDR-CONS))
 (14 14 (:REWRITE CAR-CONS))
 (10 10 (:REWRITE LEN-MEMBER-EQUAL-LOOP$-AS))
 (2 2 (:TYPE-PRESCRIPTION LOOP$-AS))
 (2 2 (:META RELINK-FANCY-SCION-CORRECT))
 (2 1 (:REWRITE APPLY$-PRIMITIVE))
 (1 1 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (1 1 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 )
(LP12-8-LEMMA-LEMMA
 (153 151 (:REWRITE DEFAULT-CAR))
 (150 30 (:DEFINITION BINARY-APPEND))
 (123 121 (:REWRITE DEFAULT-CDR))
 (98 98 (:REWRITE DEFAULT-+-2))
 (98 98 (:REWRITE DEFAULT-+-1))
 (87 87 (:REWRITE DEFAULT-<-2))
 (87 87 (:REWRITE DEFAULT-<-1))
 (18 18 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (12 12 (:META RELINK-FANCY-SCION-CORRECT))
 (12 4 (:REWRITE FOLD-CONSTS-IN-+))
 (10 5 (:REWRITE APPLY$-PRIMITIVE))
 (5 5 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (1 1 (:INDUCTION LOOP$-AS-CDR-FBT-RULE . 1))
 (1 1 (:INDUCTION LOOP$-AS-CDR-CDR-RULE . 1))
 )
(LP12-8-LEMMA
 (3623 192 (:DEFINITION FROM-TO-BY-DOWN-OPENER))
 (1716 1305 (:REWRITE DEFAULT-+-2))
 (1306 1305 (:REWRITE DEFAULT-+-1))
 (1036 1014 (:REWRITE DEFAULT-CDR))
 (960 192 (:DEFINITION BINARY-APPEND))
 (944 919 (:REWRITE DEFAULT-CAR))
 (805 473 (:REWRITE DEFAULT-<-1))
 (473 473 (:REWRITE DEFAULT-<-2))
 (135 135 (:REWRITE LEN-MEMBER-EQUAL-LOOP$-AS))
 (50 33 (:REWRITE APPLY$-PRIMITIVE))
 (32 32 (:META RELINK-FANCY-SCION-CORRECT))
 (22 22 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (17 17 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (15 15 (:REWRITE DEFAULT-UNARY-MINUS))
 (3 3 (:TYPE-PRESCRIPTION NATP))
 (1 1 (:INDUCTION LOOP$-AS-CDR-FBT-RULE . 5))
 (1 1 (:INDUCTION LOOP$-AS-CDR-CDR-RULE . 1))
 (1 1 (:INDUCTION FROM-TO-BY-DOWN-INDUCTION))
 )
(LP12-8-PROOF-1
 (942 6 (:DEFINITION LOOP$-AS))
 (528 24 (:DEFINITION FROM-TO-BY-DOWN-OPENER))
 (312 12 (:DEFINITION CDR-LOOP$-AS-TUPLE))
 (312 12 (:DEFINITION CAR-LOOP$-AS-TUPLE))
 (306 12 (:DEFINITION EMPTY-LOOP$-AS-TUPLEP))
 (192 24 (:DEFINITION FROM-TO-BY))
 (150 102 (:REWRITE DEFAULT-+-2))
 (120 120 (:REWRITE DEFAULT-CDR))
 (120 24 (:DEFINITION LEN))
 (120 24 (:DEFINITION BINARY-APPEND))
 (114 114 (:REWRITE DEFAULT-CAR))
 (104 102 (:REWRITE DEFAULT-+-1))
 (102 52 (:REWRITE DEFAULT-<-1))
 (72 24 (:REWRITE FOLD-CONSTS-IN-+))
 (66 66 (:TYPE-PRESCRIPTION FROM-TO-BY))
 (52 52 (:REWRITE DEFAULT-<-2))
 (30 2 (:REWRITE LP12-8-LEMMA-LEMMA))
 (24 24 (:REWRITE LEN-MEMBER-EQUAL-LOOP$-AS))
 (12 2 (:DEFINITION NTH))
 (6 4 (:REWRITE APPLY$-PRIMITIVE))
 (4 4 (:META RELINK-FANCY-SCION-CORRECT))
 (2 2 (:TYPE-PRESCRIPTION NATP))
 (2 2 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (2 2 (:REWRITE ZP-OPEN))
 (1 1 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 )
(LP12-8-LEMMA-BRIDGE)
(LP12-8-PROOF-2
 (133 1 (:DEFINITION LOOP$-AS))
 (88 4 (:DEFINITION FROM-TO-BY-DOWN-OPENER))
 (44 2 (:DEFINITION CDR-LOOP$-AS-TUPLE))
 (44 2 (:DEFINITION CAR-LOOP$-AS-TUPLE))
 (43 2 (:DEFINITION EMPTY-LOOP$-AS-TUPLEP))
 (38 26 (:REWRITE DEFAULT-+-2))
 (28 26 (:REWRITE DEFAULT-+-1))
 (25 5 (:DEFINITION LEN))
 (21 21 (:REWRITE DEFAULT-CDR))
 (20 4 (:DEFINITION BINARY-APPEND))
 (18 18 (:REWRITE DEFAULT-CAR))
 (12 2 (:DEFINITION NTH))
 (11 11 (:TYPE-PRESCRIPTION FROM-TO-BY))
 (9 5 (:REWRITE DEFAULT-<-1))
 (6 6 (:REWRITE CDR-CONS))
 (6 6 (:REWRITE CAR-CONS))
 (5 5 (:REWRITE LEN-MEMBER-EQUAL-LOOP$-AS))
 (5 5 (:REWRITE DEFAULT-<-2))
 (2 2 (:REWRITE ZP-OPEN))
 (1 1 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 )
(LP12-8-LEMMA-DIFFERENT-BRIDGE
 (190 1 (:DEFINITION LOOP$-AS))
 (106 5 (:DEFINITION FROM-TO-BY-DOWN-OPENER))
 (73 50 (:REWRITE DEFAULT-+-2))
 (63 2 (:DEFINITION CDR-LOOP$-AS-TUPLE))
 (63 2 (:DEFINITION CAR-LOOP$-AS-TUPLE))
 (62 2 (:DEFINITION EMPTY-LOOP$-AS-TUPLEP))
 (60 5 (:DEFINITION FROM-TO-BY))
 (52 50 (:REWRITE DEFAULT-+-1))
 (29 21 (:REWRITE DEFAULT-<-1))
 (29 17 (:REWRITE FOLD-CONSTS-IN-+))
 (28 2 (:DEFINITION NATS-UP))
 (25 5 (:DEFINITION LEN))
 (25 5 (:DEFINITION BINARY-APPEND))
 (22 22 (:REWRITE DEFAULT-CDR))
 (21 21 (:REWRITE DEFAULT-<-2))
 (20 2 (:DEFINITION NTH))
 (19 19 (:REWRITE DEFAULT-CAR))
 (12 4 (:DEFINITION NFIX))
 (8 2 (:REWRITE ZP-OPEN))
 (6 6 (:REWRITE CDR-CONS))
 (6 6 (:REWRITE CAR-CONS))
 (5 5 (:REWRITE LEN-MEMBER-EQUAL-LOOP$-AS))
 (2 2 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (2 2 (:REWRITE DEFAULT-UNARY-MINUS))
 )
(LP12-8-PROOF-3
 (157 1 (:DEFINITION LOOP$-AS))
 (110 5 (:DEFINITION FROM-TO-BY-DOWN-OPENER))
 (79 79 (:TYPE-PRESCRIPTION LEN))
 (52 2 (:DEFINITION CDR-LOOP$-AS-TUPLE))
 (52 2 (:DEFINITION CAR-LOOP$-AS-TUPLE))
 (51 2 (:DEFINITION EMPTY-LOOP$-AS-TUPLEP))
 (40 28 (:REWRITE DEFAULT-+-2))
 (40 5 (:DEFINITION FROM-TO-BY))
 (30 28 (:REWRITE DEFAULT-+-1))
 (25 5 (:DEFINITION LEN))
 (25 5 (:DEFINITION BINARY-APPEND))
 (22 22 (:REWRITE DEFAULT-CDR))
 (21 11 (:REWRITE DEFAULT-<-1))
 (19 19 (:REWRITE DEFAULT-CAR))
 (12 2 (:DEFINITION NTH))
 (11 11 (:REWRITE DEFAULT-<-2))
 (6 6 (:REWRITE CDR-CONS))
 (6 6 (:REWRITE CAR-CONS))
 (5 5 (:REWRITE LEN-MEMBER-EQUAL-LOOP$-AS))
 (2 2 (:REWRITE ZP-OPEN))
 (1 1 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 )
