/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) DependencyPairsProof [EQUIVALENT, 79 ms] (2) QDP (3) DependencyGraphProof [EQUIVALENT, 0 ms] (4) AND (5) QDP (6) UsableRulesProof [EQUIVALENT, 0 ms] (7) QDP (8) QDPSizeChangeProof [EQUIVALENT, 0 ms] (9) YES (10) QDP (11) UsableRulesProof [EQUIVALENT, 0 ms] (12) QDP (13) QDPSizeChangeProof [EQUIVALENT, 0 ms] (14) YES (15) QDP (16) UsableRulesProof [EQUIVALENT, 0 ms] (17) QDP (18) QDPSizeChangeProof [EQUIVALENT, 0 ms] (19) YES (20) QDP (21) UsableRulesProof [EQUIVALENT, 0 ms] (22) QDP (23) QDPSizeChangeProof [EQUIVALENT, 0 ms] (24) YES (25) QDP (26) UsableRulesProof [EQUIVALENT, 0 ms] (27) QDP (28) QDPSizeChangeProof [EQUIVALENT, 0 ms] (29) YES (30) QDP (31) UsableRulesProof [EQUIVALENT, 0 ms] (32) QDP (33) QDPSizeChangeProof [EQUIVALENT, 0 ms] (34) YES (35) QDP (36) UsableRulesProof [EQUIVALENT, 0 ms] (37) QDP (38) QDPSizeChangeProof [EQUIVALENT, 0 ms] (39) YES (40) QDP (41) UsableRulesProof [EQUIVALENT, 0 ms] (42) QDP (43) QDPSizeChangeProof [EQUIVALENT, 0 ms] (44) YES (45) QDP (46) UsableRulesProof [EQUIVALENT, 0 ms] (47) QDP (48) QDPSizeChangeProof [EQUIVALENT, 0 ms] (49) YES (50) QDP (51) UsableRulesProof [EQUIVALENT, 0 ms] (52) QDP (53) QDPSizeChangeProof [EQUIVALENT, 0 ms] (54) YES (55) QDP (56) UsableRulesProof [EQUIVALENT, 0 ms] (57) QDP (58) QDPSizeChangeProof [EQUIVALENT, 0 ms] (59) YES (60) QDP (61) QDPOrderProof [EQUIVALENT, 403 ms] (62) QDP (63) QDPOrderProof [EQUIVALENT, 72 ms] (64) QDP (65) QDPOrderProof [EQUIVALENT, 419 ms] (66) QDP (67) QDPOrderProof [EQUIVALENT, 378 ms] (68) QDP (69) DependencyGraphProof [EQUIVALENT, 0 ms] (70) QDP (71) QDPOrderProof [EQUIVALENT, 275 ms] (72) QDP (73) QDPOrderProof [EQUIVALENT, 337 ms] (74) QDP (75) QDPOrderProof [EQUIVALENT, 264 ms] (76) QDP (77) QDPOrderProof [EQUIVALENT, 345 ms] (78) QDP (79) QDPOrderProof [EQUIVALENT, 437 ms] (80) QDP (81) QDPOrderProof [EQUIVALENT, 375 ms] (82) QDP (83) QDPOrderProof [EQUIVALENT, 615 ms] (84) QDP (85) QDPOrderProof [EQUIVALENT, 765 ms] (86) QDP (87) QDPOrderProof [EQUIVALENT, 240 ms] (88) QDP (89) QDPOrderProof [EQUIVALENT, 249 ms] (90) QDP (91) QDPOrderProof [EQUIVALENT, 237 ms] (92) QDP (93) QDPOrderProof [EQUIVALENT, 225 ms] (94) QDP (95) QDPOrderProof [EQUIVALENT, 587 ms] (96) QDP (97) QDPOrderProof [EQUIVALENT, 295 ms] (98) QDP (99) QDPOrderProof [EQUIVALENT, 382 ms] (100) QDP (101) DependencyGraphProof [EQUIVALENT, 0 ms] (102) TRUE ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. ---------------------------------------- (1) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (2) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(and(tt, T)) -> MARK(T) ACTIVE(isNatIList(IL)) -> MARK(isNatList(IL)) ACTIVE(isNatIList(IL)) -> ISNATLIST(IL) ACTIVE(isNat(0)) -> MARK(tt) ACTIVE(isNat(s(N))) -> MARK(isNat(N)) ACTIVE(isNat(s(N))) -> ISNAT(N) ACTIVE(isNat(length(L))) -> MARK(isNatList(L)) ACTIVE(isNat(length(L))) -> ISNATLIST(L) ACTIVE(isNatIList(zeros)) -> MARK(tt) ACTIVE(isNatIList(cons(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) ACTIVE(isNatIList(cons(N, IL))) -> AND(isNat(N), isNatIList(IL)) ACTIVE(isNatIList(cons(N, IL))) -> ISNAT(N) ACTIVE(isNatIList(cons(N, IL))) -> ISNATILIST(IL) ACTIVE(isNatList(nil)) -> MARK(tt) ACTIVE(isNatList(cons(N, L))) -> MARK(and(isNat(N), isNatList(L))) ACTIVE(isNatList(cons(N, L))) -> AND(isNat(N), isNatList(L)) ACTIVE(isNatList(cons(N, L))) -> ISNAT(N) ACTIVE(isNatList(cons(N, L))) -> ISNATLIST(L) ACTIVE(isNatList(take(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) ACTIVE(isNatList(take(N, IL))) -> AND(isNat(N), isNatIList(IL)) ACTIVE(isNatList(take(N, IL))) -> ISNAT(N) ACTIVE(isNatList(take(N, IL))) -> ISNATILIST(IL) ACTIVE(zeros) -> MARK(cons(0, zeros)) ACTIVE(zeros) -> CONS(0, zeros) ACTIVE(take(0, IL)) -> MARK(uTake1(isNatIList(IL))) ACTIVE(take(0, IL)) -> UTAKE1(isNatIList(IL)) ACTIVE(take(0, IL)) -> ISNATILIST(IL) ACTIVE(uTake1(tt)) -> MARK(nil) ACTIVE(take(s(M), cons(N, IL))) -> MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) ACTIVE(take(s(M), cons(N, IL))) -> UTAKE2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL) ACTIVE(take(s(M), cons(N, IL))) -> AND(isNat(M), and(isNat(N), isNatIList(IL))) ACTIVE(take(s(M), cons(N, IL))) -> ISNAT(M) ACTIVE(take(s(M), cons(N, IL))) -> AND(isNat(N), isNatIList(IL)) ACTIVE(take(s(M), cons(N, IL))) -> ISNAT(N) ACTIVE(take(s(M), cons(N, IL))) -> ISNATILIST(IL) ACTIVE(uTake2(tt, M, N, IL)) -> MARK(cons(N, take(M, IL))) ACTIVE(uTake2(tt, M, N, IL)) -> CONS(N, take(M, IL)) ACTIVE(uTake2(tt, M, N, IL)) -> TAKE(M, IL) ACTIVE(length(cons(N, L))) -> MARK(uLength(and(isNat(N), isNatList(L)), L)) ACTIVE(length(cons(N, L))) -> ULENGTH(and(isNat(N), isNatList(L)), L) ACTIVE(length(cons(N, L))) -> AND(isNat(N), isNatList(L)) ACTIVE(length(cons(N, L))) -> ISNAT(N) ACTIVE(length(cons(N, L))) -> ISNATLIST(L) ACTIVE(uLength(tt, L)) -> MARK(s(length(L))) ACTIVE(uLength(tt, L)) -> S(length(L)) ACTIVE(uLength(tt, L)) -> LENGTH(L) MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), mark(X2))) MARK(and(X1, X2)) -> AND(mark(X1), mark(X2)) MARK(and(X1, X2)) -> MARK(X1) MARK(and(X1, X2)) -> MARK(X2) MARK(tt) -> ACTIVE(tt) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(0) -> ACTIVE(0) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> S(mark(X)) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> LENGTH(mark(X)) MARK(length(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2)) MARK(cons(X1, X2)) -> CONS(mark(X1), X2) MARK(cons(X1, X2)) -> MARK(X1) MARK(nil) -> ACTIVE(nil) MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2))) MARK(take(X1, X2)) -> TAKE(mark(X1), mark(X2)) MARK(take(X1, X2)) -> MARK(X1) MARK(take(X1, X2)) -> MARK(X2) MARK(uTake1(X)) -> ACTIVE(uTake1(mark(X))) MARK(uTake1(X)) -> UTAKE1(mark(X)) MARK(uTake1(X)) -> MARK(X) MARK(uTake2(X1, X2, X3, X4)) -> ACTIVE(uTake2(mark(X1), X2, X3, X4)) MARK(uTake2(X1, X2, X3, X4)) -> UTAKE2(mark(X1), X2, X3, X4) MARK(uTake2(X1, X2, X3, X4)) -> MARK(X1) MARK(uLength(X1, X2)) -> ACTIVE(uLength(mark(X1), X2)) MARK(uLength(X1, X2)) -> ULENGTH(mark(X1), X2) MARK(uLength(X1, X2)) -> MARK(X1) AND(mark(X1), X2) -> AND(X1, X2) AND(X1, mark(X2)) -> AND(X1, X2) AND(active(X1), X2) -> AND(X1, X2) AND(X1, active(X2)) -> AND(X1, X2) ISNATILIST(mark(X)) -> ISNATILIST(X) ISNATILIST(active(X)) -> ISNATILIST(X) ISNATLIST(mark(X)) -> ISNATLIST(X) ISNATLIST(active(X)) -> ISNATLIST(X) ISNAT(mark(X)) -> ISNAT(X) ISNAT(active(X)) -> ISNAT(X) S(mark(X)) -> S(X) S(active(X)) -> S(X) LENGTH(mark(X)) -> LENGTH(X) LENGTH(active(X)) -> LENGTH(X) CONS(mark(X1), X2) -> CONS(X1, X2) CONS(X1, mark(X2)) -> CONS(X1, X2) CONS(active(X1), X2) -> CONS(X1, X2) CONS(X1, active(X2)) -> CONS(X1, X2) TAKE(mark(X1), X2) -> TAKE(X1, X2) TAKE(X1, mark(X2)) -> TAKE(X1, X2) TAKE(active(X1), X2) -> TAKE(X1, X2) TAKE(X1, active(X2)) -> TAKE(X1, X2) UTAKE1(mark(X)) -> UTAKE1(X) UTAKE1(active(X)) -> UTAKE1(X) UTAKE2(mark(X1), X2, X3, X4) -> UTAKE2(X1, X2, X3, X4) UTAKE2(X1, mark(X2), X3, X4) -> UTAKE2(X1, X2, X3, X4) UTAKE2(X1, X2, mark(X3), X4) -> UTAKE2(X1, X2, X3, X4) UTAKE2(X1, X2, X3, mark(X4)) -> UTAKE2(X1, X2, X3, X4) UTAKE2(active(X1), X2, X3, X4) -> UTAKE2(X1, X2, X3, X4) UTAKE2(X1, active(X2), X3, X4) -> UTAKE2(X1, X2, X3, X4) UTAKE2(X1, X2, active(X3), X4) -> UTAKE2(X1, X2, X3, X4) UTAKE2(X1, X2, X3, active(X4)) -> UTAKE2(X1, X2, X3, X4) ULENGTH(mark(X1), X2) -> ULENGTH(X1, X2) ULENGTH(X1, mark(X2)) -> ULENGTH(X1, X2) ULENGTH(active(X1), X2) -> ULENGTH(X1, X2) ULENGTH(X1, active(X2)) -> ULENGTH(X1, X2) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (3) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 12 SCCs with 44 less nodes. ---------------------------------------- (4) Complex Obligation (AND) ---------------------------------------- (5) Obligation: Q DP problem: The TRS P consists of the following rules: ULENGTH(X1, mark(X2)) -> ULENGTH(X1, X2) ULENGTH(mark(X1), X2) -> ULENGTH(X1, X2) ULENGTH(active(X1), X2) -> ULENGTH(X1, X2) ULENGTH(X1, active(X2)) -> ULENGTH(X1, X2) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (6) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (7) Obligation: Q DP problem: The TRS P consists of the following rules: ULENGTH(X1, mark(X2)) -> ULENGTH(X1, X2) ULENGTH(mark(X1), X2) -> ULENGTH(X1, X2) ULENGTH(active(X1), X2) -> ULENGTH(X1, X2) ULENGTH(X1, active(X2)) -> ULENGTH(X1, X2) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (8) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *ULENGTH(X1, mark(X2)) -> ULENGTH(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 *ULENGTH(mark(X1), X2) -> ULENGTH(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *ULENGTH(active(X1), X2) -> ULENGTH(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *ULENGTH(X1, active(X2)) -> ULENGTH(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (9) YES ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: UTAKE2(X1, mark(X2), X3, X4) -> UTAKE2(X1, X2, X3, X4) UTAKE2(mark(X1), X2, X3, X4) -> UTAKE2(X1, X2, X3, X4) UTAKE2(X1, X2, mark(X3), X4) -> UTAKE2(X1, X2, X3, X4) UTAKE2(X1, X2, X3, mark(X4)) -> UTAKE2(X1, X2, X3, X4) UTAKE2(active(X1), X2, X3, X4) -> UTAKE2(X1, X2, X3, X4) UTAKE2(X1, active(X2), X3, X4) -> UTAKE2(X1, X2, X3, X4) UTAKE2(X1, X2, active(X3), X4) -> UTAKE2(X1, X2, X3, X4) UTAKE2(X1, X2, X3, active(X4)) -> UTAKE2(X1, X2, X3, X4) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: UTAKE2(X1, mark(X2), X3, X4) -> UTAKE2(X1, X2, X3, X4) UTAKE2(mark(X1), X2, X3, X4) -> UTAKE2(X1, X2, X3, X4) UTAKE2(X1, X2, mark(X3), X4) -> UTAKE2(X1, X2, X3, X4) UTAKE2(X1, X2, X3, mark(X4)) -> UTAKE2(X1, X2, X3, X4) UTAKE2(active(X1), X2, X3, X4) -> UTAKE2(X1, X2, X3, X4) UTAKE2(X1, active(X2), X3, X4) -> UTAKE2(X1, X2, X3, X4) UTAKE2(X1, X2, active(X3), X4) -> UTAKE2(X1, X2, X3, X4) UTAKE2(X1, X2, X3, active(X4)) -> UTAKE2(X1, X2, X3, X4) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *UTAKE2(X1, mark(X2), X3, X4) -> UTAKE2(X1, X2, X3, X4) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4 *UTAKE2(mark(X1), X2, X3, X4) -> UTAKE2(X1, X2, X3, X4) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4 *UTAKE2(X1, X2, mark(X3), X4) -> UTAKE2(X1, X2, X3, X4) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4 *UTAKE2(X1, X2, X3, mark(X4)) -> UTAKE2(X1, X2, X3, X4) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4 *UTAKE2(active(X1), X2, X3, X4) -> UTAKE2(X1, X2, X3, X4) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4 *UTAKE2(X1, active(X2), X3, X4) -> UTAKE2(X1, X2, X3, X4) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4 *UTAKE2(X1, X2, active(X3), X4) -> UTAKE2(X1, X2, X3, X4) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4 *UTAKE2(X1, X2, X3, active(X4)) -> UTAKE2(X1, X2, X3, X4) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4 ---------------------------------------- (14) YES ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: UTAKE1(active(X)) -> UTAKE1(X) UTAKE1(mark(X)) -> UTAKE1(X) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (17) Obligation: Q DP problem: The TRS P consists of the following rules: UTAKE1(active(X)) -> UTAKE1(X) UTAKE1(mark(X)) -> UTAKE1(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (18) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *UTAKE1(active(X)) -> UTAKE1(X) The graph contains the following edges 1 > 1 *UTAKE1(mark(X)) -> UTAKE1(X) The graph contains the following edges 1 > 1 ---------------------------------------- (19) YES ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: TAKE(X1, mark(X2)) -> TAKE(X1, X2) TAKE(mark(X1), X2) -> TAKE(X1, X2) TAKE(active(X1), X2) -> TAKE(X1, X2) TAKE(X1, active(X2)) -> TAKE(X1, X2) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: TAKE(X1, mark(X2)) -> TAKE(X1, X2) TAKE(mark(X1), X2) -> TAKE(X1, X2) TAKE(active(X1), X2) -> TAKE(X1, X2) TAKE(X1, active(X2)) -> TAKE(X1, X2) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *TAKE(X1, mark(X2)) -> TAKE(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 *TAKE(mark(X1), X2) -> TAKE(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *TAKE(active(X1), X2) -> TAKE(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *TAKE(X1, active(X2)) -> TAKE(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (24) YES ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: CONS(X1, mark(X2)) -> CONS(X1, X2) CONS(mark(X1), X2) -> CONS(X1, X2) CONS(active(X1), X2) -> CONS(X1, X2) CONS(X1, active(X2)) -> CONS(X1, X2) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (26) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (27) Obligation: Q DP problem: The TRS P consists of the following rules: CONS(X1, mark(X2)) -> CONS(X1, X2) CONS(mark(X1), X2) -> CONS(X1, X2) CONS(active(X1), X2) -> CONS(X1, X2) CONS(X1, active(X2)) -> CONS(X1, X2) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (28) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *CONS(X1, mark(X2)) -> CONS(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 *CONS(mark(X1), X2) -> CONS(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *CONS(active(X1), X2) -> CONS(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *CONS(X1, active(X2)) -> CONS(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (29) YES ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: LENGTH(active(X)) -> LENGTH(X) LENGTH(mark(X)) -> LENGTH(X) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: LENGTH(active(X)) -> LENGTH(X) LENGTH(mark(X)) -> LENGTH(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *LENGTH(active(X)) -> LENGTH(X) The graph contains the following edges 1 > 1 *LENGTH(mark(X)) -> LENGTH(X) The graph contains the following edges 1 > 1 ---------------------------------------- (34) YES ---------------------------------------- (35) Obligation: Q DP problem: The TRS P consists of the following rules: S(active(X)) -> S(X) S(mark(X)) -> S(X) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (36) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (37) Obligation: Q DP problem: The TRS P consists of the following rules: S(active(X)) -> S(X) S(mark(X)) -> S(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (38) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *S(active(X)) -> S(X) The graph contains the following edges 1 > 1 *S(mark(X)) -> S(X) The graph contains the following edges 1 > 1 ---------------------------------------- (39) YES ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(active(X)) -> ISNAT(X) ISNAT(mark(X)) -> ISNAT(X) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (41) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (42) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(active(X)) -> ISNAT(X) ISNAT(mark(X)) -> ISNAT(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (43) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *ISNAT(active(X)) -> ISNAT(X) The graph contains the following edges 1 > 1 *ISNAT(mark(X)) -> ISNAT(X) The graph contains the following edges 1 > 1 ---------------------------------------- (44) YES ---------------------------------------- (45) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATLIST(active(X)) -> ISNATLIST(X) ISNATLIST(mark(X)) -> ISNATLIST(X) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (46) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (47) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATLIST(active(X)) -> ISNATLIST(X) ISNATLIST(mark(X)) -> ISNATLIST(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (48) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *ISNATLIST(active(X)) -> ISNATLIST(X) The graph contains the following edges 1 > 1 *ISNATLIST(mark(X)) -> ISNATLIST(X) The graph contains the following edges 1 > 1 ---------------------------------------- (49) YES ---------------------------------------- (50) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(active(X)) -> ISNATILIST(X) ISNATILIST(mark(X)) -> ISNATILIST(X) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (51) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (52) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(active(X)) -> ISNATILIST(X) ISNATILIST(mark(X)) -> ISNATILIST(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (53) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *ISNATILIST(active(X)) -> ISNATILIST(X) The graph contains the following edges 1 > 1 *ISNATILIST(mark(X)) -> ISNATILIST(X) The graph contains the following edges 1 > 1 ---------------------------------------- (54) YES ---------------------------------------- (55) Obligation: Q DP problem: The TRS P consists of the following rules: AND(X1, mark(X2)) -> AND(X1, X2) AND(mark(X1), X2) -> AND(X1, X2) AND(active(X1), X2) -> AND(X1, X2) AND(X1, active(X2)) -> AND(X1, X2) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (56) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (57) Obligation: Q DP problem: The TRS P consists of the following rules: AND(X1, mark(X2)) -> AND(X1, X2) AND(mark(X1), X2) -> AND(X1, X2) AND(active(X1), X2) -> AND(X1, X2) AND(X1, active(X2)) -> AND(X1, X2) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (58) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *AND(X1, mark(X2)) -> AND(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 *AND(mark(X1), X2) -> AND(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *AND(active(X1), X2) -> AND(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *AND(X1, active(X2)) -> AND(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (59) YES ---------------------------------------- (60) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), mark(X2))) ACTIVE(and(tt, T)) -> MARK(T) MARK(and(X1, X2)) -> MARK(X1) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(isNatIList(IL)) -> MARK(isNatList(IL)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNat(s(N))) -> MARK(isNat(N)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(length(L))) -> MARK(isNatList(L)) MARK(s(X)) -> ACTIVE(s(mark(X))) ACTIVE(isNatIList(cons(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNatList(cons(N, L))) -> MARK(and(isNat(N), isNatList(L))) MARK(length(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ACTIVE(zeros) -> MARK(cons(0, zeros)) MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2)) ACTIVE(isNatList(take(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) MARK(cons(X1, X2)) -> MARK(X1) MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2))) ACTIVE(take(0, IL)) -> MARK(uTake1(isNatIList(IL))) MARK(take(X1, X2)) -> MARK(X1) MARK(take(X1, X2)) -> MARK(X2) MARK(uTake1(X)) -> ACTIVE(uTake1(mark(X))) ACTIVE(take(s(M), cons(N, IL))) -> MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) MARK(uTake1(X)) -> MARK(X) MARK(uTake2(X1, X2, X3, X4)) -> ACTIVE(uTake2(mark(X1), X2, X3, X4)) ACTIVE(uTake2(tt, M, N, IL)) -> MARK(cons(N, take(M, IL))) MARK(uTake2(X1, X2, X3, X4)) -> MARK(X1) MARK(uLength(X1, X2)) -> ACTIVE(uLength(mark(X1), X2)) ACTIVE(length(cons(N, L))) -> MARK(uLength(and(isNat(N), isNatList(L)), L)) MARK(uLength(X1, X2)) -> MARK(X1) ACTIVE(uLength(tt, L)) -> MARK(s(length(L))) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (61) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2)) MARK(uTake1(X)) -> ACTIVE(uTake1(mark(X))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2} POL( and_2(x_1, x_2) ) = 2 POL( cons_2(x_1, x_2) ) = 0 POL( length_1(x_1) ) = 2 POL( s_1(x_1) ) = 0 POL( take_2(x_1, x_2) ) = 2 POL( uLength_2(x_1, x_2) ) = 2 POL( uTake1_1(x_1) ) = max{0, -2} POL( uTake2_4(x_1, ..., x_4) ) = 2 POL( mark_1(x_1) ) = max{0, -2} POL( active_1(x_1) ) = 2 POL( tt ) = 0 POL( isNatIList_1(x_1) ) = 2 POL( isNatList_1(x_1) ) = 2 POL( isNat_1(x_1) ) = 2 POL( zeros ) = 2 POL( 0 ) = 0 POL( nil ) = 0 POL( MARK_1(x_1) ) = 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: and(X1, mark(X2)) -> and(X1, X2) and(mark(X1), X2) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) cons(X1, mark(X2)) -> cons(X1, X2) cons(mark(X1), X2) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(active(X)) -> uTake1(X) uTake1(mark(X)) -> uTake1(X) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) ---------------------------------------- (62) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), mark(X2))) ACTIVE(and(tt, T)) -> MARK(T) MARK(and(X1, X2)) -> MARK(X1) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(isNatIList(IL)) -> MARK(isNatList(IL)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNat(s(N))) -> MARK(isNat(N)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(length(L))) -> MARK(isNatList(L)) ACTIVE(isNatIList(cons(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNatList(cons(N, L))) -> MARK(and(isNat(N), isNatList(L))) MARK(length(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ACTIVE(zeros) -> MARK(cons(0, zeros)) ACTIVE(isNatList(take(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) MARK(cons(X1, X2)) -> MARK(X1) MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2))) ACTIVE(take(0, IL)) -> MARK(uTake1(isNatIList(IL))) MARK(take(X1, X2)) -> MARK(X1) MARK(take(X1, X2)) -> MARK(X2) ACTIVE(take(s(M), cons(N, IL))) -> MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) MARK(uTake1(X)) -> MARK(X) MARK(uTake2(X1, X2, X3, X4)) -> ACTIVE(uTake2(mark(X1), X2, X3, X4)) ACTIVE(uTake2(tt, M, N, IL)) -> MARK(cons(N, take(M, IL))) MARK(uTake2(X1, X2, X3, X4)) -> MARK(X1) MARK(uLength(X1, X2)) -> ACTIVE(uLength(mark(X1), X2)) ACTIVE(length(cons(N, L))) -> MARK(uLength(and(isNat(N), isNatList(L)), L)) MARK(uLength(X1, X2)) -> MARK(X1) ACTIVE(uLength(tt, L)) -> MARK(s(length(L))) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (63) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(take(0, IL)) -> MARK(uTake1(isNatIList(IL))) MARK(take(X1, X2)) -> MARK(X1) MARK(take(X1, X2)) -> MARK(X2) MARK(uTake2(X1, X2, X3, X4)) -> MARK(X1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVE(x_1)) = x_1 POL(MARK(x_1)) = x_1 POL(active(x_1)) = x_1 POL(and(x_1, x_2)) = x_1 + x_2 POL(cons(x_1, x_2)) = x_1 + x_2 POL(isNat(x_1)) = 0 POL(isNatIList(x_1)) = 0 POL(isNatList(x_1)) = 0 POL(length(x_1)) = x_1 POL(mark(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = x_1 POL(take(x_1, x_2)) = 1 + x_1 + x_2 POL(tt) = 0 POL(uLength(x_1, x_2)) = x_1 + x_2 POL(uTake1(x_1)) = x_1 POL(uTake2(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 POL(zeros) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) active(and(tt, T)) -> mark(T) mark(isNatIList(X)) -> active(isNatIList(X)) active(isNatIList(IL)) -> mark(isNatList(IL)) mark(isNatList(X)) -> active(isNatList(X)) active(isNat(s(N))) -> mark(isNat(N)) mark(isNat(X)) -> active(isNat(X)) active(isNat(length(L))) -> mark(isNatList(L)) mark(s(X)) -> active(s(mark(X))) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) mark(length(X)) -> active(length(mark(X))) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) mark(zeros) -> active(zeros) active(zeros) -> mark(cons(0, zeros)) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) mark(uTake1(X)) -> active(uTake1(mark(X))) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(tt) -> active(tt) mark(0) -> active(0) mark(nil) -> active(nil) and(X1, mark(X2)) -> and(X1, X2) and(mark(X1), X2) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) take(X1, mark(X2)) -> take(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(active(X)) -> uTake1(X) uTake1(mark(X)) -> uTake1(X) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) cons(X1, mark(X2)) -> cons(X1, X2) cons(mark(X1), X2) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) active(isNat(0)) -> mark(tt) active(isNatIList(zeros)) -> mark(tt) active(isNatList(nil)) -> mark(tt) active(uTake1(tt)) -> mark(nil) ---------------------------------------- (64) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), mark(X2))) ACTIVE(and(tt, T)) -> MARK(T) MARK(and(X1, X2)) -> MARK(X1) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(isNatIList(IL)) -> MARK(isNatList(IL)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNat(s(N))) -> MARK(isNat(N)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(length(L))) -> MARK(isNatList(L)) ACTIVE(isNatIList(cons(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNatList(cons(N, L))) -> MARK(and(isNat(N), isNatList(L))) MARK(length(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ACTIVE(zeros) -> MARK(cons(0, zeros)) ACTIVE(isNatList(take(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) MARK(cons(X1, X2)) -> MARK(X1) MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2))) ACTIVE(take(s(M), cons(N, IL))) -> MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) MARK(uTake1(X)) -> MARK(X) MARK(uTake2(X1, X2, X3, X4)) -> ACTIVE(uTake2(mark(X1), X2, X3, X4)) ACTIVE(uTake2(tt, M, N, IL)) -> MARK(cons(N, take(M, IL))) MARK(uLength(X1, X2)) -> ACTIVE(uLength(mark(X1), X2)) ACTIVE(length(cons(N, L))) -> MARK(uLength(and(isNat(N), isNatList(L)), L)) MARK(uLength(X1, X2)) -> MARK(X1) ACTIVE(uLength(tt, L)) -> MARK(s(length(L))) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (65) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(length(X)) -> MARK(X) MARK(uLength(X1, X2)) -> MARK(X1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = x_1 POL( and_2(x_1, x_2) ) = x_1 + 2x_2 POL( length_1(x_1) ) = 2x_1 + 2 POL( take_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( uLength_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 POL( uTake2_4(x_1, ..., x_4) ) = x_1 + 2x_2 + 2x_3 + 2x_4 POL( mark_1(x_1) ) = x_1 POL( active_1(x_1) ) = x_1 POL( tt ) = 0 POL( isNatIList_1(x_1) ) = 0 POL( isNatList_1(x_1) ) = 0 POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = x_1 POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( zeros ) = 0 POL( 0 ) = 0 POL( uTake1_1(x_1) ) = x_1 POL( nil ) = 0 POL( MARK_1(x_1) ) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) active(and(tt, T)) -> mark(T) mark(isNatIList(X)) -> active(isNatIList(X)) active(isNatIList(IL)) -> mark(isNatList(IL)) mark(isNatList(X)) -> active(isNatList(X)) active(isNat(s(N))) -> mark(isNat(N)) mark(isNat(X)) -> active(isNat(X)) active(isNat(length(L))) -> mark(isNatList(L)) mark(s(X)) -> active(s(mark(X))) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) mark(length(X)) -> active(length(mark(X))) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) mark(zeros) -> active(zeros) active(zeros) -> mark(cons(0, zeros)) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) mark(uTake1(X)) -> active(uTake1(mark(X))) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(tt) -> active(tt) mark(0) -> active(0) mark(nil) -> active(nil) and(X1, mark(X2)) -> and(X1, X2) and(mark(X1), X2) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) take(X1, mark(X2)) -> take(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) cons(X1, mark(X2)) -> cons(X1, X2) cons(mark(X1), X2) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) active(isNat(0)) -> mark(tt) active(isNatIList(zeros)) -> mark(tt) active(isNatList(nil)) -> mark(tt) active(uTake1(tt)) -> mark(nil) uTake1(active(X)) -> uTake1(X) uTake1(mark(X)) -> uTake1(X) ---------------------------------------- (66) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), mark(X2))) ACTIVE(and(tt, T)) -> MARK(T) MARK(and(X1, X2)) -> MARK(X1) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(isNatIList(IL)) -> MARK(isNatList(IL)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNat(s(N))) -> MARK(isNat(N)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(length(L))) -> MARK(isNatList(L)) ACTIVE(isNatIList(cons(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNatList(cons(N, L))) -> MARK(and(isNat(N), isNatList(L))) MARK(zeros) -> ACTIVE(zeros) ACTIVE(zeros) -> MARK(cons(0, zeros)) ACTIVE(isNatList(take(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) MARK(cons(X1, X2)) -> MARK(X1) MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2))) ACTIVE(take(s(M), cons(N, IL))) -> MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) MARK(uTake1(X)) -> MARK(X) MARK(uTake2(X1, X2, X3, X4)) -> ACTIVE(uTake2(mark(X1), X2, X3, X4)) ACTIVE(uTake2(tt, M, N, IL)) -> MARK(cons(N, take(M, IL))) MARK(uLength(X1, X2)) -> ACTIVE(uLength(mark(X1), X2)) ACTIVE(length(cons(N, L))) -> MARK(uLength(and(isNat(N), isNatList(L)), L)) ACTIVE(uLength(tt, L)) -> MARK(s(length(L))) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (67) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(zeros) -> MARK(cons(0, zeros)) ACTIVE(take(s(M), cons(N, IL))) -> MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) MARK(uTake1(X)) -> MARK(X) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = 2x_1 POL( and_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( length_1(x_1) ) = max{0, -2} POL( take_2(x_1, x_2) ) = x_2 + 2 POL( uLength_2(x_1, x_2) ) = max{0, -2} POL( uTake2_4(x_1, ..., x_4) ) = x_3 POL( mark_1(x_1) ) = x_1 POL( active_1(x_1) ) = x_1 POL( tt ) = 0 POL( isNatIList_1(x_1) ) = 0 POL( isNatList_1(x_1) ) = 0 POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = 2x_1 POL( cons_2(x_1, x_2) ) = x_1 POL( zeros ) = 2 POL( 0 ) = 1 POL( uTake1_1(x_1) ) = x_1 + 2 POL( nil ) = 1 POL( MARK_1(x_1) ) = 2x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) active(and(tt, T)) -> mark(T) mark(isNatIList(X)) -> active(isNatIList(X)) active(isNatIList(IL)) -> mark(isNatList(IL)) mark(isNatList(X)) -> active(isNatList(X)) active(isNat(s(N))) -> mark(isNat(N)) mark(isNat(X)) -> active(isNat(X)) active(isNat(length(L))) -> mark(isNatList(L)) mark(s(X)) -> active(s(mark(X))) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) mark(length(X)) -> active(length(mark(X))) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) mark(zeros) -> active(zeros) active(zeros) -> mark(cons(0, zeros)) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) mark(uTake1(X)) -> active(uTake1(mark(X))) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(tt) -> active(tt) mark(0) -> active(0) mark(nil) -> active(nil) and(X1, mark(X2)) -> and(X1, X2) and(mark(X1), X2) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) take(X1, mark(X2)) -> take(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) cons(X1, mark(X2)) -> cons(X1, X2) cons(mark(X1), X2) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) active(isNat(0)) -> mark(tt) active(isNatIList(zeros)) -> mark(tt) active(isNatList(nil)) -> mark(tt) active(uTake1(tt)) -> mark(nil) uTake1(active(X)) -> uTake1(X) uTake1(mark(X)) -> uTake1(X) ---------------------------------------- (68) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), mark(X2))) ACTIVE(and(tt, T)) -> MARK(T) MARK(and(X1, X2)) -> MARK(X1) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(isNatIList(IL)) -> MARK(isNatList(IL)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNat(s(N))) -> MARK(isNat(N)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(length(L))) -> MARK(isNatList(L)) ACTIVE(isNatIList(cons(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNatList(cons(N, L))) -> MARK(and(isNat(N), isNatList(L))) MARK(zeros) -> ACTIVE(zeros) ACTIVE(isNatList(take(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) MARK(cons(X1, X2)) -> MARK(X1) MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2))) MARK(uTake2(X1, X2, X3, X4)) -> ACTIVE(uTake2(mark(X1), X2, X3, X4)) ACTIVE(uTake2(tt, M, N, IL)) -> MARK(cons(N, take(M, IL))) MARK(uLength(X1, X2)) -> ACTIVE(uLength(mark(X1), X2)) ACTIVE(length(cons(N, L))) -> MARK(uLength(and(isNat(N), isNatList(L)), L)) ACTIVE(uLength(tt, L)) -> MARK(s(length(L))) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (69) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (70) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(and(tt, T)) -> MARK(T) MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), mark(X2))) ACTIVE(isNatIList(IL)) -> MARK(isNatList(IL)) MARK(and(X1, X2)) -> MARK(X1) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(isNat(s(N))) -> MARK(isNat(N)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNat(length(L))) -> MARK(isNatList(L)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNatIList(cons(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNatList(cons(N, L))) -> MARK(and(isNat(N), isNatList(L))) MARK(cons(X1, X2)) -> MARK(X1) MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2))) ACTIVE(isNatList(take(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) MARK(uTake2(X1, X2, X3, X4)) -> ACTIVE(uTake2(mark(X1), X2, X3, X4)) ACTIVE(uTake2(tt, M, N, IL)) -> MARK(cons(N, take(M, IL))) MARK(uLength(X1, X2)) -> ACTIVE(uLength(mark(X1), X2)) ACTIVE(length(cons(N, L))) -> MARK(uLength(and(isNat(N), isNatList(L)), L)) ACTIVE(uLength(tt, L)) -> MARK(s(length(L))) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (71) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2} POL( and_2(x_1, x_2) ) = 2 POL( length_1(x_1) ) = 2 POL( take_2(x_1, x_2) ) = max{0, -2} POL( uLength_2(x_1, x_2) ) = 2 POL( uTake2_4(x_1, ..., x_4) ) = 2 POL( mark_1(x_1) ) = 0 POL( active_1(x_1) ) = max{0, 2x_1 - 1} POL( tt ) = 0 POL( isNatIList_1(x_1) ) = 2 POL( isNatList_1(x_1) ) = 2 POL( isNat_1(x_1) ) = 2 POL( s_1(x_1) ) = max{0, x_1 - 2} POL( cons_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 1} POL( zeros ) = 2 POL( 0 ) = 0 POL( uTake1_1(x_1) ) = max{0, x_1 - 2} POL( nil ) = 0 POL( MARK_1(x_1) ) = 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: and(X1, mark(X2)) -> and(X1, X2) and(mark(X1), X2) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) take(X1, mark(X2)) -> take(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) ---------------------------------------- (72) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(and(tt, T)) -> MARK(T) MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), mark(X2))) ACTIVE(isNatIList(IL)) -> MARK(isNatList(IL)) MARK(and(X1, X2)) -> MARK(X1) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(isNat(s(N))) -> MARK(isNat(N)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNat(length(L))) -> MARK(isNatList(L)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNatIList(cons(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNatList(cons(N, L))) -> MARK(and(isNat(N), isNatList(L))) MARK(cons(X1, X2)) -> MARK(X1) ACTIVE(isNatList(take(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) MARK(uTake2(X1, X2, X3, X4)) -> ACTIVE(uTake2(mark(X1), X2, X3, X4)) ACTIVE(uTake2(tt, M, N, IL)) -> MARK(cons(N, take(M, IL))) MARK(uLength(X1, X2)) -> ACTIVE(uLength(mark(X1), X2)) ACTIVE(length(cons(N, L))) -> MARK(uLength(and(isNat(N), isNatList(L)), L)) ACTIVE(uLength(tt, L)) -> MARK(s(length(L))) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (73) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(cons(X1, X2)) -> MARK(X1) MARK(uTake2(X1, X2, X3, X4)) -> ACTIVE(uTake2(mark(X1), X2, X3, X4)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = x_1 POL( and_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( length_1(x_1) ) = 0 POL( uLength_2(x_1, x_2) ) = max{0, -2} POL( uTake2_4(x_1, ..., x_4) ) = 2x_3 + 2 POL( mark_1(x_1) ) = x_1 POL( active_1(x_1) ) = x_1 POL( tt ) = 0 POL( isNatIList_1(x_1) ) = 0 POL( isNatList_1(x_1) ) = 0 POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = x_1 POL( cons_2(x_1, x_2) ) = x_1 + 1 POL( zeros ) = 1 POL( 0 ) = 0 POL( take_2(x_1, x_2) ) = 2x_2 POL( uTake1_1(x_1) ) = 0 POL( nil ) = 0 POL( MARK_1(x_1) ) = 2x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) active(and(tt, T)) -> mark(T) mark(isNatIList(X)) -> active(isNatIList(X)) active(isNatIList(IL)) -> mark(isNatList(IL)) mark(isNatList(X)) -> active(isNatList(X)) active(isNat(s(N))) -> mark(isNat(N)) mark(isNat(X)) -> active(isNat(X)) active(isNat(length(L))) -> mark(isNatList(L)) mark(s(X)) -> active(s(mark(X))) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) mark(length(X)) -> active(length(mark(X))) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) mark(zeros) -> active(zeros) active(zeros) -> mark(cons(0, zeros)) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) mark(uTake1(X)) -> active(uTake1(mark(X))) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(tt) -> active(tt) mark(0) -> active(0) mark(nil) -> active(nil) and(X1, mark(X2)) -> and(X1, X2) and(mark(X1), X2) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) take(X1, mark(X2)) -> take(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(mark(X1), X2) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) active(isNat(0)) -> mark(tt) active(isNatIList(zeros)) -> mark(tt) active(isNatList(nil)) -> mark(tt) active(uTake1(tt)) -> mark(nil) uTake1(active(X)) -> uTake1(X) uTake1(mark(X)) -> uTake1(X) ---------------------------------------- (74) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(and(tt, T)) -> MARK(T) MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), mark(X2))) ACTIVE(isNatIList(IL)) -> MARK(isNatList(IL)) MARK(and(X1, X2)) -> MARK(X1) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(isNat(s(N))) -> MARK(isNat(N)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNat(length(L))) -> MARK(isNatList(L)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNatIList(cons(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNatList(cons(N, L))) -> MARK(and(isNat(N), isNatList(L))) ACTIVE(isNatList(take(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) ACTIVE(uTake2(tt, M, N, IL)) -> MARK(cons(N, take(M, IL))) MARK(uLength(X1, X2)) -> ACTIVE(uLength(mark(X1), X2)) ACTIVE(length(cons(N, L))) -> MARK(uLength(and(isNat(N), isNatList(L)), L)) ACTIVE(uLength(tt, L)) -> MARK(s(length(L))) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (75) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(uTake2(tt, M, N, IL)) -> MARK(cons(N, take(M, IL))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2} POL( and_2(x_1, x_2) ) = max{0, -2} POL( length_1(x_1) ) = 0 POL( uLength_2(x_1, x_2) ) = 1 POL( mark_1(x_1) ) = 1 POL( active_1(x_1) ) = max{0, 2x_1 - 2} POL( tt ) = 2 POL( isNatIList_1(x_1) ) = 0 POL( isNatList_1(x_1) ) = 0 POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = max{0, -2} POL( cons_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2} POL( zeros ) = 2 POL( 0 ) = 2 POL( take_2(x_1, x_2) ) = max{0, 2x_2 - 1} POL( uTake1_1(x_1) ) = x_1 + 1 POL( uTake2_4(x_1, ..., x_4) ) = 2x_1 + x_2 + 2x_3 + 2x_4 + 2 POL( nil ) = 0 POL( MARK_1(x_1) ) = max{0, -2} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: and(X1, mark(X2)) -> and(X1, X2) and(mark(X1), X2) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) ---------------------------------------- (76) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(and(tt, T)) -> MARK(T) MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), mark(X2))) ACTIVE(isNatIList(IL)) -> MARK(isNatList(IL)) MARK(and(X1, X2)) -> MARK(X1) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(isNat(s(N))) -> MARK(isNat(N)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNat(length(L))) -> MARK(isNatList(L)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNatIList(cons(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNatList(cons(N, L))) -> MARK(and(isNat(N), isNatList(L))) ACTIVE(isNatList(take(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) MARK(uLength(X1, X2)) -> ACTIVE(uLength(mark(X1), X2)) ACTIVE(length(cons(N, L))) -> MARK(uLength(and(isNat(N), isNatList(L)), L)) ACTIVE(uLength(tt, L)) -> MARK(s(length(L))) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (77) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(isNatList(take(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = 2x_1 POL( and_2(x_1, x_2) ) = x_1 + x_2 POL( length_1(x_1) ) = 2x_1 POL( uLength_2(x_1, x_2) ) = 2x_2 POL( mark_1(x_1) ) = x_1 POL( active_1(x_1) ) = x_1 POL( tt ) = 0 POL( isNatIList_1(x_1) ) = 2x_1 POL( isNatList_1(x_1) ) = 2x_1 POL( isNat_1(x_1) ) = 2x_1 POL( s_1(x_1) ) = x_1 POL( cons_2(x_1, x_2) ) = x_1 + x_2 POL( zeros ) = 0 POL( 0 ) = 0 POL( take_2(x_1, x_2) ) = x_1 + x_2 + 2 POL( uTake1_1(x_1) ) = 2 POL( uTake2_4(x_1, ..., x_4) ) = x_2 + x_3 + x_4 + 2 POL( nil ) = 0 POL( MARK_1(x_1) ) = 2x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) active(and(tt, T)) -> mark(T) mark(isNatIList(X)) -> active(isNatIList(X)) active(isNatIList(IL)) -> mark(isNatList(IL)) mark(isNatList(X)) -> active(isNatList(X)) active(isNat(s(N))) -> mark(isNat(N)) mark(isNat(X)) -> active(isNat(X)) active(isNat(length(L))) -> mark(isNatList(L)) mark(s(X)) -> active(s(mark(X))) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) mark(length(X)) -> active(length(mark(X))) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) mark(zeros) -> active(zeros) active(zeros) -> mark(cons(0, zeros)) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) mark(uTake1(X)) -> active(uTake1(mark(X))) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(tt) -> active(tt) mark(0) -> active(0) mark(nil) -> active(nil) and(X1, mark(X2)) -> and(X1, X2) and(mark(X1), X2) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) active(isNat(0)) -> mark(tt) active(isNatIList(zeros)) -> mark(tt) active(isNatList(nil)) -> mark(tt) active(uTake1(tt)) -> mark(nil) cons(X1, mark(X2)) -> cons(X1, X2) cons(mark(X1), X2) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(active(X)) -> uTake1(X) uTake1(mark(X)) -> uTake1(X) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) ---------------------------------------- (78) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(and(tt, T)) -> MARK(T) MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), mark(X2))) ACTIVE(isNatIList(IL)) -> MARK(isNatList(IL)) MARK(and(X1, X2)) -> MARK(X1) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(isNat(s(N))) -> MARK(isNat(N)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNat(length(L))) -> MARK(isNatList(L)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNatIList(cons(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNatList(cons(N, L))) -> MARK(and(isNat(N), isNatList(L))) MARK(uLength(X1, X2)) -> ACTIVE(uLength(mark(X1), X2)) ACTIVE(length(cons(N, L))) -> MARK(uLength(and(isNat(N), isNatList(L)), L)) ACTIVE(uLength(tt, L)) -> MARK(s(length(L))) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (79) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(isNat(length(L))) -> MARK(isNatList(L)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = x_1 + 2 POL( and_2(x_1, x_2) ) = 2x_1 + x_2 POL( length_1(x_1) ) = 2x_1 + 1 POL( uLength_2(x_1, x_2) ) = 2x_2 + 1 POL( mark_1(x_1) ) = x_1 POL( active_1(x_1) ) = x_1 POL( tt ) = 0 POL( isNatIList_1(x_1) ) = 2x_1 POL( isNatList_1(x_1) ) = 2x_1 POL( isNat_1(x_1) ) = 2x_1 POL( s_1(x_1) ) = x_1 POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( zeros ) = 0 POL( 0 ) = 0 POL( take_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( uTake1_1(x_1) ) = max{0, -2} POL( uTake2_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 POL( nil ) = 0 POL( MARK_1(x_1) ) = x_1 + 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) active(and(tt, T)) -> mark(T) mark(isNatIList(X)) -> active(isNatIList(X)) active(isNatIList(IL)) -> mark(isNatList(IL)) mark(isNatList(X)) -> active(isNatList(X)) active(isNat(s(N))) -> mark(isNat(N)) mark(isNat(X)) -> active(isNat(X)) active(isNat(length(L))) -> mark(isNatList(L)) mark(s(X)) -> active(s(mark(X))) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) mark(length(X)) -> active(length(mark(X))) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) mark(zeros) -> active(zeros) active(zeros) -> mark(cons(0, zeros)) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) mark(uTake1(X)) -> active(uTake1(mark(X))) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(tt) -> active(tt) mark(0) -> active(0) mark(nil) -> active(nil) and(X1, mark(X2)) -> and(X1, X2) and(mark(X1), X2) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) active(isNat(0)) -> mark(tt) active(isNatIList(zeros)) -> mark(tt) active(isNatList(nil)) -> mark(tt) active(uTake1(tt)) -> mark(nil) cons(X1, mark(X2)) -> cons(X1, X2) cons(mark(X1), X2) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(active(X)) -> uTake1(X) uTake1(mark(X)) -> uTake1(X) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) ---------------------------------------- (80) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(and(tt, T)) -> MARK(T) MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), mark(X2))) ACTIVE(isNatIList(IL)) -> MARK(isNatList(IL)) MARK(and(X1, X2)) -> MARK(X1) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(isNat(s(N))) -> MARK(isNat(N)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNatIList(cons(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNatList(cons(N, L))) -> MARK(and(isNat(N), isNatList(L))) MARK(uLength(X1, X2)) -> ACTIVE(uLength(mark(X1), X2)) ACTIVE(length(cons(N, L))) -> MARK(uLength(and(isNat(N), isNatList(L)), L)) ACTIVE(uLength(tt, L)) -> MARK(s(length(L))) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (81) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(isNatIList(IL)) -> MARK(isNatList(IL)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = 2x_1 POL( and_2(x_1, x_2) ) = x_1 + x_2 POL( length_1(x_1) ) = 2x_1 POL( uLength_2(x_1, x_2) ) = 2x_2 POL( mark_1(x_1) ) = x_1 POL( active_1(x_1) ) = x_1 POL( tt ) = 0 POL( isNatIList_1(x_1) ) = 2x_1 + 2 POL( isNatList_1(x_1) ) = 2x_1 POL( isNat_1(x_1) ) = 2x_1 POL( s_1(x_1) ) = x_1 POL( cons_2(x_1, x_2) ) = x_1 + x_2 POL( zeros ) = 0 POL( 0 ) = 0 POL( take_2(x_1, x_2) ) = x_1 + x_2 + 2 POL( uTake1_1(x_1) ) = 0 POL( uTake2_4(x_1, ..., x_4) ) = x_2 + x_3 + x_4 + 2 POL( nil ) = 0 POL( MARK_1(x_1) ) = 2x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) active(and(tt, T)) -> mark(T) mark(isNatIList(X)) -> active(isNatIList(X)) active(isNatIList(IL)) -> mark(isNatList(IL)) mark(isNatList(X)) -> active(isNatList(X)) active(isNat(s(N))) -> mark(isNat(N)) mark(isNat(X)) -> active(isNat(X)) active(isNat(length(L))) -> mark(isNatList(L)) mark(s(X)) -> active(s(mark(X))) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) mark(length(X)) -> active(length(mark(X))) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) mark(zeros) -> active(zeros) active(zeros) -> mark(cons(0, zeros)) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) mark(uTake1(X)) -> active(uTake1(mark(X))) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(tt) -> active(tt) mark(0) -> active(0) mark(nil) -> active(nil) and(X1, mark(X2)) -> and(X1, X2) and(mark(X1), X2) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) active(isNat(0)) -> mark(tt) active(isNatIList(zeros)) -> mark(tt) active(isNatList(nil)) -> mark(tt) active(uTake1(tt)) -> mark(nil) cons(X1, mark(X2)) -> cons(X1, X2) cons(mark(X1), X2) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(active(X)) -> uTake1(X) uTake1(mark(X)) -> uTake1(X) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) ---------------------------------------- (82) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(and(tt, T)) -> MARK(T) MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), mark(X2))) MARK(and(X1, X2)) -> MARK(X1) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(isNat(s(N))) -> MARK(isNat(N)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNatIList(cons(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNatList(cons(N, L))) -> MARK(and(isNat(N), isNatList(L))) MARK(uLength(X1, X2)) -> ACTIVE(uLength(mark(X1), X2)) ACTIVE(length(cons(N, L))) -> MARK(uLength(and(isNat(N), isNatList(L)), L)) ACTIVE(uLength(tt, L)) -> MARK(s(length(L))) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (83) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(and(X1, X2)) -> MARK(X1) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(ACTIVE(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(and(x_1, x_2)) = [[-I]] + [[1A]] * x_1 + [[0A]] * x_2 >>> <<< POL(tt) = [[0A]] >>> <<< POL(MARK(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(mark(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(isNatIList(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(isNat(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(s(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(isNatList(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(cons(x_1, x_2)) = [[-I]] + [[1A]] * x_1 + [[0A]] * x_2 >>> <<< POL(length(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(uLength(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(active(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(zeros) = [[2A]] >>> <<< POL(0) = [[0A]] >>> <<< POL(take(x_1, x_2)) = [[-I]] + [[1A]] * x_1 + [[1A]] * x_2 >>> <<< POL(uTake1(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(uTake2(x_1, x_2, x_3, x_4)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 + [[2A]] * x_3 + [[1A]] * x_4 >>> <<< POL(nil) = [[0A]] >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) active(and(tt, T)) -> mark(T) mark(isNatIList(X)) -> active(isNatIList(X)) active(isNatIList(IL)) -> mark(isNatList(IL)) mark(isNatList(X)) -> active(isNatList(X)) active(isNat(s(N))) -> mark(isNat(N)) mark(isNat(X)) -> active(isNat(X)) active(isNat(length(L))) -> mark(isNatList(L)) mark(s(X)) -> active(s(mark(X))) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) mark(length(X)) -> active(length(mark(X))) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) mark(zeros) -> active(zeros) active(zeros) -> mark(cons(0, zeros)) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) mark(uTake1(X)) -> active(uTake1(mark(X))) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(tt) -> active(tt) mark(0) -> active(0) mark(nil) -> active(nil) and(X1, mark(X2)) -> and(X1, X2) and(mark(X1), X2) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) active(isNat(0)) -> mark(tt) active(isNatIList(zeros)) -> mark(tt) active(isNatList(nil)) -> mark(tt) active(uTake1(tt)) -> mark(nil) cons(X1, mark(X2)) -> cons(X1, X2) cons(mark(X1), X2) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(active(X)) -> uTake1(X) uTake1(mark(X)) -> uTake1(X) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) ---------------------------------------- (84) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(and(tt, T)) -> MARK(T) MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), mark(X2))) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(isNat(s(N))) -> MARK(isNat(N)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNatIList(cons(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNatList(cons(N, L))) -> MARK(and(isNat(N), isNatList(L))) MARK(uLength(X1, X2)) -> ACTIVE(uLength(mark(X1), X2)) ACTIVE(length(cons(N, L))) -> MARK(uLength(and(isNat(N), isNatList(L)), L)) ACTIVE(uLength(tt, L)) -> MARK(s(length(L))) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (85) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(isNat(s(N))) -> MARK(isNat(N)) MARK(s(X)) -> MARK(X) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(ACTIVE(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(and(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(tt) = [[2A]] >>> <<< POL(MARK(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(mark(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(isNatIList(x_1)) = [[-I]] + [[2A]] * x_1 >>> <<< POL(isNat(x_1)) = [[-I]] + [[1A]] * x_1 >>> <<< POL(s(x_1)) = [[2A]] + [[1A]] * x_1 >>> <<< POL(isNatList(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(cons(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(length(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(uLength(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(active(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(zeros) = [[0A]] >>> <<< POL(0) = [[1A]] >>> <<< POL(take(x_1, x_2)) = [[3A]] + [[2A]] * x_1 + [[2A]] * x_2 >>> <<< POL(uTake1(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(uTake2(x_1, x_2, x_3, x_4)) = [[4A]] + [[-I]] * x_1 + [[3A]] * x_2 + [[1A]] * x_3 + [[3A]] * x_4 >>> <<< POL(nil) = [[2A]] >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) active(and(tt, T)) -> mark(T) mark(isNatIList(X)) -> active(isNatIList(X)) active(isNatIList(IL)) -> mark(isNatList(IL)) mark(isNatList(X)) -> active(isNatList(X)) active(isNat(s(N))) -> mark(isNat(N)) mark(isNat(X)) -> active(isNat(X)) active(isNat(length(L))) -> mark(isNatList(L)) mark(s(X)) -> active(s(mark(X))) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) mark(length(X)) -> active(length(mark(X))) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) mark(zeros) -> active(zeros) active(zeros) -> mark(cons(0, zeros)) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) mark(uTake1(X)) -> active(uTake1(mark(X))) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(tt) -> active(tt) mark(0) -> active(0) mark(nil) -> active(nil) and(X1, mark(X2)) -> and(X1, X2) and(mark(X1), X2) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) active(isNat(0)) -> mark(tt) active(isNatIList(zeros)) -> mark(tt) active(isNatList(nil)) -> mark(tt) active(uTake1(tt)) -> mark(nil) cons(X1, mark(X2)) -> cons(X1, X2) cons(mark(X1), X2) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(active(X)) -> uTake1(X) uTake1(mark(X)) -> uTake1(X) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) ---------------------------------------- (86) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(and(tt, T)) -> MARK(T) MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), mark(X2))) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNatIList(cons(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNatList(cons(N, L))) -> MARK(and(isNat(N), isNatList(L))) MARK(uLength(X1, X2)) -> ACTIVE(uLength(mark(X1), X2)) ACTIVE(length(cons(N, L))) -> MARK(uLength(and(isNat(N), isNatList(L)), L)) ACTIVE(uLength(tt, L)) -> MARK(s(length(L))) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (87) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(isNat(X)) -> ACTIVE(isNat(X)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1} POL( and_2(x_1, x_2) ) = 2 POL( length_1(x_1) ) = 2 POL( uLength_2(x_1, x_2) ) = 2 POL( mark_1(x_1) ) = max{0, -2} POL( active_1(x_1) ) = 0 POL( tt ) = 0 POL( isNatIList_1(x_1) ) = 2 POL( isNatList_1(x_1) ) = 2 POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = 1 POL( cons_2(x_1, x_2) ) = 2 POL( zeros ) = 0 POL( 0 ) = 0 POL( take_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( uTake1_1(x_1) ) = max{0, 2x_1 - 2} POL( uTake2_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_4 - 2} POL( nil ) = 0 POL( MARK_1(x_1) ) = 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: and(X1, mark(X2)) -> and(X1, X2) and(mark(X1), X2) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) ---------------------------------------- (88) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(and(tt, T)) -> MARK(T) MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), mark(X2))) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNatIList(cons(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNatList(cons(N, L))) -> MARK(and(isNat(N), isNatList(L))) MARK(uLength(X1, X2)) -> ACTIVE(uLength(mark(X1), X2)) ACTIVE(length(cons(N, L))) -> MARK(uLength(and(isNat(N), isNatList(L)), L)) ACTIVE(uLength(tt, L)) -> MARK(s(length(L))) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (89) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(length(cons(N, L))) -> MARK(uLength(and(isNat(N), isNatList(L)), L)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = x_1 + 2 POL( and_2(x_1, x_2) ) = 2x_2 POL( length_1(x_1) ) = 2 POL( uLength_2(x_1, x_2) ) = max{0, -2} POL( mark_1(x_1) ) = 2x_1 POL( active_1(x_1) ) = x_1 POL( tt ) = 0 POL( isNatIList_1(x_1) ) = 0 POL( isNatList_1(x_1) ) = 0 POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = max{0, -2} POL( cons_2(x_1, x_2) ) = max{0, -2} POL( zeros ) = 0 POL( 0 ) = 2 POL( take_2(x_1, x_2) ) = max{0, -2} POL( uTake1_1(x_1) ) = max{0, -2} POL( uTake2_4(x_1, ..., x_4) ) = max{0, -2} POL( nil ) = 0 POL( MARK_1(x_1) ) = 2x_1 + 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) active(and(tt, T)) -> mark(T) mark(isNatIList(X)) -> active(isNatIList(X)) active(isNatIList(IL)) -> mark(isNatList(IL)) mark(isNatList(X)) -> active(isNatList(X)) active(isNat(s(N))) -> mark(isNat(N)) mark(isNat(X)) -> active(isNat(X)) active(isNat(length(L))) -> mark(isNatList(L)) mark(s(X)) -> active(s(mark(X))) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) mark(length(X)) -> active(length(mark(X))) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) mark(zeros) -> active(zeros) active(zeros) -> mark(cons(0, zeros)) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) mark(uTake1(X)) -> active(uTake1(mark(X))) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(tt) -> active(tt) mark(0) -> active(0) mark(nil) -> active(nil) and(X1, mark(X2)) -> and(X1, X2) and(mark(X1), X2) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) active(isNat(0)) -> mark(tt) active(isNatIList(zeros)) -> mark(tt) active(isNatList(nil)) -> mark(tt) active(uTake1(tt)) -> mark(nil) cons(X1, mark(X2)) -> cons(X1, X2) cons(mark(X1), X2) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(active(X)) -> uTake1(X) uTake1(mark(X)) -> uTake1(X) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) ---------------------------------------- (90) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(and(tt, T)) -> MARK(T) MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), mark(X2))) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNatIList(cons(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) ACTIVE(isNatList(cons(N, L))) -> MARK(and(isNat(N), isNatList(L))) MARK(uLength(X1, X2)) -> ACTIVE(uLength(mark(X1), X2)) ACTIVE(uLength(tt, L)) -> MARK(s(length(L))) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (91) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(uLength(tt, L)) -> MARK(s(length(L))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = x_1 + 2 POL( and_2(x_1, x_2) ) = 2x_2 POL( uLength_2(x_1, x_2) ) = 1 POL( mark_1(x_1) ) = x_1 POL( active_1(x_1) ) = x_1 POL( tt ) = 0 POL( isNatIList_1(x_1) ) = 0 POL( isNatList_1(x_1) ) = 0 POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = max{0, -2} POL( length_1(x_1) ) = 1 POL( cons_2(x_1, x_2) ) = 0 POL( zeros ) = 0 POL( 0 ) = 0 POL( take_2(x_1, x_2) ) = 2 POL( uTake1_1(x_1) ) = x_1 + 2 POL( uTake2_4(x_1, ..., x_4) ) = 1 POL( nil ) = 0 POL( MARK_1(x_1) ) = x_1 + 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) active(and(tt, T)) -> mark(T) mark(isNatIList(X)) -> active(isNatIList(X)) active(isNatIList(IL)) -> mark(isNatList(IL)) mark(isNatList(X)) -> active(isNatList(X)) active(isNat(s(N))) -> mark(isNat(N)) mark(isNat(X)) -> active(isNat(X)) active(isNat(length(L))) -> mark(isNatList(L)) mark(s(X)) -> active(s(mark(X))) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) mark(length(X)) -> active(length(mark(X))) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) mark(zeros) -> active(zeros) active(zeros) -> mark(cons(0, zeros)) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) mark(uTake1(X)) -> active(uTake1(mark(X))) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(tt) -> active(tt) mark(0) -> active(0) mark(nil) -> active(nil) and(X1, mark(X2)) -> and(X1, X2) and(mark(X1), X2) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) active(isNat(0)) -> mark(tt) active(isNatIList(zeros)) -> mark(tt) active(isNatList(nil)) -> mark(tt) active(uTake1(tt)) -> mark(nil) cons(X1, mark(X2)) -> cons(X1, X2) cons(mark(X1), X2) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(active(X)) -> uTake1(X) uTake1(mark(X)) -> uTake1(X) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) ---------------------------------------- (92) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(and(tt, T)) -> MARK(T) MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), mark(X2))) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNatIList(cons(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) ACTIVE(isNatList(cons(N, L))) -> MARK(and(isNat(N), isNatList(L))) MARK(uLength(X1, X2)) -> ACTIVE(uLength(mark(X1), X2)) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (93) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(uLength(X1, X2)) -> ACTIVE(uLength(mark(X1), X2)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 1} POL( and_2(x_1, x_2) ) = 1 POL( uLength_2(x_1, x_2) ) = max{0, -2} POL( mark_1(x_1) ) = 2 POL( active_1(x_1) ) = max{0, 2x_1 - 2} POL( tt ) = 0 POL( isNatIList_1(x_1) ) = 1 POL( isNatList_1(x_1) ) = 1 POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = max{0, 2x_1 - 2} POL( length_1(x_1) ) = x_1 + 1 POL( cons_2(x_1, x_2) ) = 2x_1 + 2 POL( zeros ) = 1 POL( 0 ) = 0 POL( take_2(x_1, x_2) ) = 2x_2 + 2 POL( uTake1_1(x_1) ) = max{0, x_1 - 1} POL( uTake2_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_3 - 2} POL( nil ) = 0 POL( MARK_1(x_1) ) = 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: and(X1, mark(X2)) -> and(X1, X2) and(mark(X1), X2) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) ---------------------------------------- (94) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(and(tt, T)) -> MARK(T) MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), mark(X2))) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNatIList(cons(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) ACTIVE(isNatList(cons(N, L))) -> MARK(and(isNat(N), isNatList(L))) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (95) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(isNatIList(cons(N, IL))) -> MARK(and(isNat(N), isNatIList(IL))) ACTIVE(isNatList(cons(N, L))) -> MARK(and(isNat(N), isNatList(L))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(ACTIVE(x_1)) = [[5A]] + [[0A]] * x_1 >>> <<< POL(and(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(tt) = [[0A]] >>> <<< POL(MARK(x_1)) = [[5A]] + [[0A]] * x_1 >>> <<< POL(mark(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(isNatIList(x_1)) = [[-I]] + [[5A]] * x_1 >>> <<< POL(isNatList(x_1)) = [[0A]] + [[5A]] * x_1 >>> <<< POL(cons(x_1, x_2)) = [[1A]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(isNat(x_1)) = [[-I]] + [[5A]] * x_1 >>> <<< POL(active(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(s(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(length(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(zeros) = [[0A]] >>> <<< POL(0) = [[0A]] >>> <<< POL(take(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(uTake1(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(uTake2(x_1, x_2, x_3, x_4)) = [[-I]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 + [[1A]] * x_4 >>> <<< POL(uLength(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(nil) = [[1A]] >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) active(and(tt, T)) -> mark(T) mark(isNatIList(X)) -> active(isNatIList(X)) active(isNatIList(IL)) -> mark(isNatList(IL)) mark(isNatList(X)) -> active(isNatList(X)) active(isNat(s(N))) -> mark(isNat(N)) mark(isNat(X)) -> active(isNat(X)) active(isNat(length(L))) -> mark(isNatList(L)) mark(s(X)) -> active(s(mark(X))) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) mark(length(X)) -> active(length(mark(X))) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) mark(zeros) -> active(zeros) active(zeros) -> mark(cons(0, zeros)) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) mark(uTake1(X)) -> active(uTake1(mark(X))) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(tt) -> active(tt) mark(0) -> active(0) mark(nil) -> active(nil) and(X1, mark(X2)) -> and(X1, X2) and(mark(X1), X2) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) active(isNat(0)) -> mark(tt) active(isNatIList(zeros)) -> mark(tt) active(isNatList(nil)) -> mark(tt) active(uTake1(tt)) -> mark(nil) s(active(X)) -> s(X) s(mark(X)) -> s(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) cons(X1, mark(X2)) -> cons(X1, X2) cons(mark(X1), X2) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(active(X)) -> uTake1(X) uTake1(mark(X)) -> uTake1(X) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) ---------------------------------------- (96) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(and(tt, T)) -> MARK(T) MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), mark(X2))) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (97) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2} POL( and_2(x_1, x_2) ) = 2 POL( mark_1(x_1) ) = max{0, -2} POL( active_1(x_1) ) = max{0, -2} POL( tt ) = 0 POL( isNatIList_1(x_1) ) = 0 POL( isNatList_1(x_1) ) = 0 POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = max{0, -2} POL( length_1(x_1) ) = 2 POL( cons_2(x_1, x_2) ) = max{0, x_1 - 2} POL( zeros ) = 0 POL( 0 ) = 0 POL( take_2(x_1, x_2) ) = max{0, x_2 - 2} POL( uTake1_1(x_1) ) = max{0, -2} POL( uTake2_4(x_1, ..., x_4) ) = max{0, x_2 - 2} POL( uLength_2(x_1, x_2) ) = max{0, x_1 - 2} POL( nil ) = 0 POL( MARK_1(x_1) ) = 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: and(X1, mark(X2)) -> and(X1, X2) and(mark(X1), X2) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) ---------------------------------------- (98) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(and(tt, T)) -> MARK(T) MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), mark(X2))) MARK(and(X1, X2)) -> MARK(X2) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (99) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(and(tt, T)) -> MARK(T) MARK(and(X1, X2)) -> MARK(X2) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(ACTIVE(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(and(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(tt) = [[3A]] >>> <<< POL(MARK(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(mark(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(active(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(isNatIList(x_1)) = [[-I]] + [[3A]] * x_1 >>> <<< POL(isNatList(x_1)) = [[-I]] + [[1A]] * x_1 >>> <<< POL(isNat(x_1)) = [[-I]] + [[1A]] * x_1 >>> <<< POL(s(x_1)) = [[3A]] + [[1A]] * x_1 >>> <<< POL(length(x_1)) = [[2A]] + [[1A]] * x_1 >>> <<< POL(cons(x_1, x_2)) = [[1A]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(zeros) = [[0A]] >>> <<< POL(0) = [[2A]] >>> <<< POL(take(x_1, x_2)) = [[4A]] + [[2A]] * x_1 + [[3A]] * x_2 >>> <<< POL(uTake1(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(uTake2(x_1, x_2, x_3, x_4)) = [[5A]] + [[-I]] * x_1 + [[3A]] * x_2 + [[-I]] * x_3 + [[4A]] * x_4 >>> <<< POL(uLength(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[2A]] * x_2 >>> <<< POL(nil) = [[2A]] >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) active(and(tt, T)) -> mark(T) mark(isNatIList(X)) -> active(isNatIList(X)) active(isNatIList(IL)) -> mark(isNatList(IL)) mark(isNatList(X)) -> active(isNatList(X)) active(isNat(s(N))) -> mark(isNat(N)) mark(isNat(X)) -> active(isNat(X)) active(isNat(length(L))) -> mark(isNatList(L)) mark(s(X)) -> active(s(mark(X))) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) mark(length(X)) -> active(length(mark(X))) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) mark(zeros) -> active(zeros) active(zeros) -> mark(cons(0, zeros)) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) mark(uTake1(X)) -> active(uTake1(mark(X))) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(tt) -> active(tt) mark(0) -> active(0) mark(nil) -> active(nil) and(X1, mark(X2)) -> and(X1, X2) and(mark(X1), X2) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) active(isNat(0)) -> mark(tt) active(isNatIList(zeros)) -> mark(tt) active(isNatList(nil)) -> mark(tt) active(uTake1(tt)) -> mark(nil) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) cons(X1, mark(X2)) -> cons(X1, X2) cons(mark(X1), X2) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(active(X)) -> uTake1(X) uTake1(mark(X)) -> uTake1(X) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) ---------------------------------------- (100) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(and(X1, X2)) -> ACTIVE(and(mark(X1), mark(X2))) The TRS R consists of the following rules: active(and(tt, T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0)) -> mark(tt) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(N, L))) -> mark(and(isNat(N), isNatList(L))) active(isNatList(take(N, IL))) -> mark(and(isNat(N), isNatIList(IL))) active(zeros) -> mark(cons(0, zeros)) active(take(0, IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt)) -> mark(nil) active(take(s(M), cons(N, IL))) -> mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)) active(uTake2(tt, M, N, IL)) -> mark(cons(N, take(M, IL))) active(length(cons(N, L))) -> mark(uLength(and(isNat(N), isNatList(L)), L)) active(uLength(tt, L)) -> mark(s(length(L))) mark(and(X1, X2)) -> active(and(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0) -> active(0) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(nil) -> active(nil) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1, X2, X3, X4)) -> active(uTake2(mark(X1), X2, X3, X4)) mark(uLength(X1, X2)) -> active(uLength(mark(X1), X2)) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, mark(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, mark(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, mark(X4)) -> uTake2(X1, X2, X3, X4) uTake2(active(X1), X2, X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, active(X2), X3, X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, active(X3), X4) -> uTake2(X1, X2, X3, X4) uTake2(X1, X2, X3, active(X4)) -> uTake2(X1, X2, X3, X4) uLength(mark(X1), X2) -> uLength(X1, X2) uLength(X1, mark(X2)) -> uLength(X1, X2) uLength(active(X1), X2) -> uLength(X1, X2) uLength(X1, active(X2)) -> uLength(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (101) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (102) TRUE