YES proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) DependencyPairsProof [EQUIVALENT, 33 ms] (2) QDP (3) DependencyGraphProof [EQUIVALENT, 0 ms] (4) AND (5) QDP (6) UsableRulesProof [EQUIVALENT, 0 ms] (7) QDP (8) QReductionProof [EQUIVALENT, 0 ms] (9) QDP (10) QDPSizeChangeProof [EQUIVALENT, 0 ms] (11) YES (12) QDP (13) UsableRulesProof [EQUIVALENT, 0 ms] (14) QDP (15) QReductionProof [EQUIVALENT, 0 ms] (16) QDP (17) QDPSizeChangeProof [EQUIVALENT, 0 ms] (18) YES (19) QDP (20) UsableRulesProof [EQUIVALENT, 0 ms] (21) QDP (22) QReductionProof [EQUIVALENT, 0 ms] (23) QDP (24) QDPSizeChangeProof [EQUIVALENT, 0 ms] (25) YES (26) QDP (27) UsableRulesProof [EQUIVALENT, 0 ms] (28) QDP (29) QReductionProof [EQUIVALENT, 0 ms] (30) QDP (31) QDPSizeChangeProof [EQUIVALENT, 0 ms] (32) YES (33) QDP (34) UsableRulesProof [EQUIVALENT, 0 ms] (35) QDP (36) QReductionProof [EQUIVALENT, 0 ms] (37) QDP (38) QDPSizeChangeProof [EQUIVALENT, 0 ms] (39) YES (40) QDP (41) UsableRulesProof [EQUIVALENT, 0 ms] (42) QDP (43) QReductionProof [EQUIVALENT, 0 ms] (44) QDP (45) QDPSizeChangeProof [EQUIVALENT, 0 ms] (46) YES (47) QDP (48) UsableRulesProof [EQUIVALENT, 0 ms] (49) QDP (50) QReductionProof [EQUIVALENT, 0 ms] (51) QDP (52) QDPSizeChangeProof [EQUIVALENT, 0 ms] (53) YES (54) QDP (55) UsableRulesProof [EQUIVALENT, 0 ms] (56) QDP (57) QReductionProof [EQUIVALENT, 0 ms] (58) QDP (59) QDPSizeChangeProof [EQUIVALENT, 0 ms] (60) YES (61) QDP (62) UsableRulesProof [EQUIVALENT, 0 ms] (63) QDP (64) QReductionProof [EQUIVALENT, 0 ms] (65) QDP (66) QDPSizeChangeProof [EQUIVALENT, 0 ms] (67) YES (68) QDP (69) UsableRulesProof [EQUIVALENT, 0 ms] (70) QDP (71) QDPOrderProof [EQUIVALENT, 39 ms] (72) QDP (73) QDPOrderProof [EQUIVALENT, 26 ms] (74) QDP (75) QDPOrderProof [EQUIVALENT, 35 ms] (76) QDP (77) QDPOrderProof [EQUIVALENT, 17 ms] (78) QDP (79) QDPOrderProof [EQUIVALENT, 7 ms] (80) QDP (81) DependencyGraphProof [EQUIVALENT, 0 ms] (82) QDP (83) QDPQMonotonicMRRProof [EQUIVALENT, 102 ms] (84) QDP (85) DependencyGraphProof [EQUIVALENT, 0 ms] (86) QDP (87) QDPQMonotonicMRRProof [EQUIVALENT, 115 ms] (88) QDP (89) DependencyGraphProof [EQUIVALENT, 0 ms] (90) QDP (91) QDPQMonotonicMRRProof [EQUIVALENT, 51 ms] (92) QDP (93) QDPQMonotonicMRRProof [EQUIVALENT, 91 ms] (94) QDP (95) UsableRulesProof [EQUIVALENT, 0 ms] (96) QDP (97) QReductionProof [EQUIVALENT, 0 ms] (98) QDP (99) QDPSizeChangeProof [EQUIVALENT, 0 ms] (100) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) mark(tt) -> active(tt) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(0) -> active(0) U11(mark(X1), X2) -> U11(X1, X2) U11(X1, mark(X2)) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1), X2) -> U31(X1, X2) U31(X1, mark(X2)) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1), X2) -> plus(X1, X2) plus(X1, mark(X2)) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) ---------------------------------------- (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(U11(tt, V2)) -> MARK(U12(isNat(V2))) ACTIVE(U11(tt, V2)) -> U12^1(isNat(V2)) ACTIVE(U11(tt, V2)) -> ISNAT(V2) ACTIVE(U12(tt)) -> MARK(tt) ACTIVE(U21(tt)) -> MARK(tt) ACTIVE(U31(tt, N)) -> MARK(N) ACTIVE(U41(tt, M, N)) -> MARK(U42(isNat(N), M, N)) ACTIVE(U41(tt, M, N)) -> U42^1(isNat(N), M, N) ACTIVE(U41(tt, M, N)) -> ISNAT(N) ACTIVE(U42(tt, M, N)) -> MARK(s(plus(N, M))) ACTIVE(U42(tt, M, N)) -> S(plus(N, M)) ACTIVE(U42(tt, M, N)) -> PLUS(N, M) ACTIVE(isNat(0)) -> MARK(tt) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) ACTIVE(isNat(plus(V1, V2))) -> U11^1(isNat(V1), V2) ACTIVE(isNat(plus(V1, V2))) -> ISNAT(V1) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNat(s(V1))) -> U21^1(isNat(V1)) ACTIVE(isNat(s(V1))) -> ISNAT(V1) ACTIVE(plus(N, 0)) -> MARK(U31(isNat(N), N)) ACTIVE(plus(N, 0)) -> U31^1(isNat(N), N) ACTIVE(plus(N, 0)) -> ISNAT(N) ACTIVE(plus(N, s(M))) -> MARK(U41(isNat(M), M, N)) ACTIVE(plus(N, s(M))) -> U41^1(isNat(M), M, N) ACTIVE(plus(N, s(M))) -> ISNAT(M) MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2)) MARK(U11(X1, X2)) -> U11^1(mark(X1), X2) MARK(U11(X1, X2)) -> MARK(X1) MARK(tt) -> ACTIVE(tt) MARK(U12(X)) -> ACTIVE(U12(mark(X))) MARK(U12(X)) -> U12^1(mark(X)) MARK(U12(X)) -> MARK(X) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U21(X)) -> U21^1(mark(X)) MARK(U21(X)) -> MARK(X) MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2)) MARK(U31(X1, X2)) -> U31^1(mark(X1), X2) MARK(U31(X1, X2)) -> MARK(X1) MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3)) MARK(U41(X1, X2, X3)) -> U41^1(mark(X1), X2, X3) MARK(U41(X1, X2, X3)) -> MARK(X1) MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3)) MARK(U42(X1, X2, X3)) -> U42^1(mark(X1), X2, X3) MARK(U42(X1, X2, X3)) -> MARK(X1) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> S(mark(X)) MARK(s(X)) -> MARK(X) MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2))) MARK(plus(X1, X2)) -> PLUS(mark(X1), mark(X2)) MARK(plus(X1, X2)) -> MARK(X1) MARK(plus(X1, X2)) -> MARK(X2) MARK(0) -> ACTIVE(0) U11^1(mark(X1), X2) -> U11^1(X1, X2) U11^1(X1, mark(X2)) -> U11^1(X1, X2) U11^1(active(X1), X2) -> U11^1(X1, X2) U11^1(X1, active(X2)) -> U11^1(X1, X2) U12^1(mark(X)) -> U12^1(X) U12^1(active(X)) -> U12^1(X) ISNAT(mark(X)) -> ISNAT(X) ISNAT(active(X)) -> ISNAT(X) U21^1(mark(X)) -> U21^1(X) U21^1(active(X)) -> U21^1(X) U31^1(mark(X1), X2) -> U31^1(X1, X2) U31^1(X1, mark(X2)) -> U31^1(X1, X2) U31^1(active(X1), X2) -> U31^1(X1, X2) U31^1(X1, active(X2)) -> U31^1(X1, X2) U41^1(mark(X1), X2, X3) -> U41^1(X1, X2, X3) U41^1(X1, mark(X2), X3) -> U41^1(X1, X2, X3) U41^1(X1, X2, mark(X3)) -> U41^1(X1, X2, X3) U41^1(active(X1), X2, X3) -> U41^1(X1, X2, X3) U41^1(X1, active(X2), X3) -> U41^1(X1, X2, X3) U41^1(X1, X2, active(X3)) -> U41^1(X1, X2, X3) U42^1(mark(X1), X2, X3) -> U42^1(X1, X2, X3) U42^1(X1, mark(X2), X3) -> U42^1(X1, X2, X3) U42^1(X1, X2, mark(X3)) -> U42^1(X1, X2, X3) U42^1(active(X1), X2, X3) -> U42^1(X1, X2, X3) U42^1(X1, active(X2), X3) -> U42^1(X1, X2, X3) U42^1(X1, X2, active(X3)) -> U42^1(X1, X2, X3) S(mark(X)) -> S(X) S(active(X)) -> S(X) PLUS(mark(X1), X2) -> PLUS(X1, X2) PLUS(X1, mark(X2)) -> PLUS(X1, X2) PLUS(active(X1), X2) -> PLUS(X1, X2) PLUS(X1, active(X2)) -> PLUS(X1, X2) The TRS R consists of the following rules: active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) mark(tt) -> active(tt) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(0) -> active(0) U11(mark(X1), X2) -> U11(X1, X2) U11(X1, mark(X2)) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1), X2) -> U31(X1, X2) U31(X1, mark(X2)) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1), X2) -> plus(X1, X2) plus(X1, mark(X2)) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (3) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 10 SCCs with 27 less nodes. ---------------------------------------- (4) Complex Obligation (AND) ---------------------------------------- (5) Obligation: Q DP problem: The TRS P consists of the following rules: PLUS(X1, mark(X2)) -> PLUS(X1, X2) PLUS(mark(X1), X2) -> PLUS(X1, X2) PLUS(active(X1), X2) -> PLUS(X1, X2) PLUS(X1, active(X2)) -> PLUS(X1, X2) The TRS R consists of the following rules: active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) mark(tt) -> active(tt) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(0) -> active(0) U11(mark(X1), X2) -> U11(X1, X2) U11(X1, mark(X2)) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1), X2) -> U31(X1, X2) U31(X1, mark(X2)) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1), X2) -> plus(X1, X2) plus(X1, mark(X2)) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (6) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (7) Obligation: Q DP problem: The TRS P consists of the following rules: PLUS(X1, mark(X2)) -> PLUS(X1, X2) PLUS(mark(X1), X2) -> PLUS(X1, X2) PLUS(active(X1), X2) -> PLUS(X1, X2) PLUS(X1, active(X2)) -> PLUS(X1, X2) R is empty. The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (8) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) ---------------------------------------- (9) Obligation: Q DP problem: The TRS P consists of the following rules: PLUS(X1, mark(X2)) -> PLUS(X1, X2) PLUS(mark(X1), X2) -> PLUS(X1, X2) PLUS(active(X1), X2) -> PLUS(X1, X2) PLUS(X1, active(X2)) -> PLUS(X1, X2) R is empty. The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (10) 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: *PLUS(X1, mark(X2)) -> PLUS(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 *PLUS(mark(X1), X2) -> PLUS(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *PLUS(active(X1), X2) -> PLUS(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *PLUS(X1, active(X2)) -> PLUS(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (11) YES ---------------------------------------- (12) 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(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) mark(tt) -> active(tt) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(0) -> active(0) U11(mark(X1), X2) -> U11(X1, X2) U11(X1, mark(X2)) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1), X2) -> U31(X1, X2) U31(X1, mark(X2)) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1), X2) -> plus(X1, X2) plus(X1, mark(X2)) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (14) 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. The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (15) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) ---------------------------------------- (16) 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. The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (17) 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 ---------------------------------------- (18) YES ---------------------------------------- (19) Obligation: Q DP problem: The TRS P consists of the following rules: U42^1(X1, mark(X2), X3) -> U42^1(X1, X2, X3) U42^1(mark(X1), X2, X3) -> U42^1(X1, X2, X3) U42^1(X1, X2, mark(X3)) -> U42^1(X1, X2, X3) U42^1(active(X1), X2, X3) -> U42^1(X1, X2, X3) U42^1(X1, active(X2), X3) -> U42^1(X1, X2, X3) U42^1(X1, X2, active(X3)) -> U42^1(X1, X2, X3) The TRS R consists of the following rules: active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) mark(tt) -> active(tt) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(0) -> active(0) U11(mark(X1), X2) -> U11(X1, X2) U11(X1, mark(X2)) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1), X2) -> U31(X1, X2) U31(X1, mark(X2)) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1), X2) -> plus(X1, X2) plus(X1, mark(X2)) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (20) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (21) Obligation: Q DP problem: The TRS P consists of the following rules: U42^1(X1, mark(X2), X3) -> U42^1(X1, X2, X3) U42^1(mark(X1), X2, X3) -> U42^1(X1, X2, X3) U42^1(X1, X2, mark(X3)) -> U42^1(X1, X2, X3) U42^1(active(X1), X2, X3) -> U42^1(X1, X2, X3) U42^1(X1, active(X2), X3) -> U42^1(X1, X2, X3) U42^1(X1, X2, active(X3)) -> U42^1(X1, X2, X3) R is empty. The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (22) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) ---------------------------------------- (23) Obligation: Q DP problem: The TRS P consists of the following rules: U42^1(X1, mark(X2), X3) -> U42^1(X1, X2, X3) U42^1(mark(X1), X2, X3) -> U42^1(X1, X2, X3) U42^1(X1, X2, mark(X3)) -> U42^1(X1, X2, X3) U42^1(active(X1), X2, X3) -> U42^1(X1, X2, X3) U42^1(X1, active(X2), X3) -> U42^1(X1, X2, X3) U42^1(X1, X2, active(X3)) -> U42^1(X1, X2, X3) R is empty. The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (24) 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: *U42^1(X1, mark(X2), X3) -> U42^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U42^1(mark(X1), X2, X3) -> U42^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U42^1(X1, X2, mark(X3)) -> U42^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 *U42^1(active(X1), X2, X3) -> U42^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U42^1(X1, active(X2), X3) -> U42^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U42^1(X1, X2, active(X3)) -> U42^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 ---------------------------------------- (25) YES ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: U41^1(X1, mark(X2), X3) -> U41^1(X1, X2, X3) U41^1(mark(X1), X2, X3) -> U41^1(X1, X2, X3) U41^1(X1, X2, mark(X3)) -> U41^1(X1, X2, X3) U41^1(active(X1), X2, X3) -> U41^1(X1, X2, X3) U41^1(X1, active(X2), X3) -> U41^1(X1, X2, X3) U41^1(X1, X2, active(X3)) -> U41^1(X1, X2, X3) The TRS R consists of the following rules: active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) mark(tt) -> active(tt) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(0) -> active(0) U11(mark(X1), X2) -> U11(X1, X2) U11(X1, mark(X2)) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1), X2) -> U31(X1, X2) U31(X1, mark(X2)) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1), X2) -> plus(X1, X2) plus(X1, mark(X2)) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (28) Obligation: Q DP problem: The TRS P consists of the following rules: U41^1(X1, mark(X2), X3) -> U41^1(X1, X2, X3) U41^1(mark(X1), X2, X3) -> U41^1(X1, X2, X3) U41^1(X1, X2, mark(X3)) -> U41^1(X1, X2, X3) U41^1(active(X1), X2, X3) -> U41^1(X1, X2, X3) U41^1(X1, active(X2), X3) -> U41^1(X1, X2, X3) U41^1(X1, X2, active(X3)) -> U41^1(X1, X2, X3) R is empty. The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: U41^1(X1, mark(X2), X3) -> U41^1(X1, X2, X3) U41^1(mark(X1), X2, X3) -> U41^1(X1, X2, X3) U41^1(X1, X2, mark(X3)) -> U41^1(X1, X2, X3) U41^1(active(X1), X2, X3) -> U41^1(X1, X2, X3) U41^1(X1, active(X2), X3) -> U41^1(X1, X2, X3) U41^1(X1, X2, active(X3)) -> U41^1(X1, X2, X3) R is empty. The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) 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: *U41^1(X1, mark(X2), X3) -> U41^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U41^1(mark(X1), X2, X3) -> U41^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U41^1(X1, X2, mark(X3)) -> U41^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 *U41^1(active(X1), X2, X3) -> U41^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U41^1(X1, active(X2), X3) -> U41^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U41^1(X1, X2, active(X3)) -> U41^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 ---------------------------------------- (32) YES ---------------------------------------- (33) Obligation: Q DP problem: The TRS P consists of the following rules: U31^1(X1, mark(X2)) -> U31^1(X1, X2) U31^1(mark(X1), X2) -> U31^1(X1, X2) U31^1(active(X1), X2) -> U31^1(X1, X2) U31^1(X1, active(X2)) -> U31^1(X1, X2) The TRS R consists of the following rules: active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) mark(tt) -> active(tt) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(0) -> active(0) U11(mark(X1), X2) -> U11(X1, X2) U11(X1, mark(X2)) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1), X2) -> U31(X1, X2) U31(X1, mark(X2)) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1), X2) -> plus(X1, X2) plus(X1, mark(X2)) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (34) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (35) Obligation: Q DP problem: The TRS P consists of the following rules: U31^1(X1, mark(X2)) -> U31^1(X1, X2) U31^1(mark(X1), X2) -> U31^1(X1, X2) U31^1(active(X1), X2) -> U31^1(X1, X2) U31^1(X1, active(X2)) -> U31^1(X1, X2) R is empty. The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (36) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) ---------------------------------------- (37) Obligation: Q DP problem: The TRS P consists of the following rules: U31^1(X1, mark(X2)) -> U31^1(X1, X2) U31^1(mark(X1), X2) -> U31^1(X1, X2) U31^1(active(X1), X2) -> U31^1(X1, X2) U31^1(X1, active(X2)) -> U31^1(X1, X2) R is empty. The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) 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: *U31^1(X1, mark(X2)) -> U31^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 *U31^1(mark(X1), X2) -> U31^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U31^1(active(X1), X2) -> U31^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U31^1(X1, active(X2)) -> U31^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (39) YES ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: U21^1(active(X)) -> U21^1(X) U21^1(mark(X)) -> U21^1(X) The TRS R consists of the following rules: active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) mark(tt) -> active(tt) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(0) -> active(0) U11(mark(X1), X2) -> U11(X1, X2) U11(X1, mark(X2)) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1), X2) -> U31(X1, X2) U31(X1, mark(X2)) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1), X2) -> plus(X1, X2) plus(X1, mark(X2)) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (41) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (42) Obligation: Q DP problem: The TRS P consists of the following rules: U21^1(active(X)) -> U21^1(X) U21^1(mark(X)) -> U21^1(X) R is empty. The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (43) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) ---------------------------------------- (44) Obligation: Q DP problem: The TRS P consists of the following rules: U21^1(active(X)) -> U21^1(X) U21^1(mark(X)) -> U21^1(X) R is empty. The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (45) 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: *U21^1(active(X)) -> U21^1(X) The graph contains the following edges 1 > 1 *U21^1(mark(X)) -> U21^1(X) The graph contains the following edges 1 > 1 ---------------------------------------- (46) YES ---------------------------------------- (47) 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(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) mark(tt) -> active(tt) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(0) -> active(0) U11(mark(X1), X2) -> U11(X1, X2) U11(X1, mark(X2)) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1), X2) -> U31(X1, X2) U31(X1, mark(X2)) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1), X2) -> plus(X1, X2) plus(X1, mark(X2)) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (48) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (49) 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. The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (50) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) ---------------------------------------- (51) 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. The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (52) 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 ---------------------------------------- (53) YES ---------------------------------------- (54) Obligation: Q DP problem: The TRS P consists of the following rules: U12^1(active(X)) -> U12^1(X) U12^1(mark(X)) -> U12^1(X) The TRS R consists of the following rules: active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) mark(tt) -> active(tt) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(0) -> active(0) U11(mark(X1), X2) -> U11(X1, X2) U11(X1, mark(X2)) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1), X2) -> U31(X1, X2) U31(X1, mark(X2)) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1), X2) -> plus(X1, X2) plus(X1, mark(X2)) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (55) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (56) Obligation: Q DP problem: The TRS P consists of the following rules: U12^1(active(X)) -> U12^1(X) U12^1(mark(X)) -> U12^1(X) R is empty. The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (57) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) ---------------------------------------- (58) Obligation: Q DP problem: The TRS P consists of the following rules: U12^1(active(X)) -> U12^1(X) U12^1(mark(X)) -> U12^1(X) R is empty. The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (59) 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: *U12^1(active(X)) -> U12^1(X) The graph contains the following edges 1 > 1 *U12^1(mark(X)) -> U12^1(X) The graph contains the following edges 1 > 1 ---------------------------------------- (60) YES ---------------------------------------- (61) Obligation: Q DP problem: The TRS P consists of the following rules: U11^1(X1, mark(X2)) -> U11^1(X1, X2) U11^1(mark(X1), X2) -> U11^1(X1, X2) U11^1(active(X1), X2) -> U11^1(X1, X2) U11^1(X1, active(X2)) -> U11^1(X1, X2) The TRS R consists of the following rules: active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) mark(tt) -> active(tt) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(0) -> active(0) U11(mark(X1), X2) -> U11(X1, X2) U11(X1, mark(X2)) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1), X2) -> U31(X1, X2) U31(X1, mark(X2)) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1), X2) -> plus(X1, X2) plus(X1, mark(X2)) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (62) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (63) Obligation: Q DP problem: The TRS P consists of the following rules: U11^1(X1, mark(X2)) -> U11^1(X1, X2) U11^1(mark(X1), X2) -> U11^1(X1, X2) U11^1(active(X1), X2) -> U11^1(X1, X2) U11^1(X1, active(X2)) -> U11^1(X1, X2) R is empty. The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (64) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) ---------------------------------------- (65) Obligation: Q DP problem: The TRS P consists of the following rules: U11^1(X1, mark(X2)) -> U11^1(X1, X2) U11^1(mark(X1), X2) -> U11^1(X1, X2) U11^1(active(X1), X2) -> U11^1(X1, X2) U11^1(X1, active(X2)) -> U11^1(X1, X2) R is empty. The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (66) 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: *U11^1(X1, mark(X2)) -> U11^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 *U11^1(mark(X1), X2) -> U11^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U11^1(active(X1), X2) -> U11^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U11^1(X1, active(X2)) -> U11^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (67) YES ---------------------------------------- (68) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U12(X)) -> ACTIVE(U12(mark(X))) ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2))) MARK(U12(X)) -> MARK(X) MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2)) ACTIVE(U31(tt, N)) -> MARK(N) MARK(U11(X1, X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U21(X)) -> ACTIVE(U21(mark(X))) ACTIVE(U41(tt, M, N)) -> MARK(U42(isNat(N), M, N)) MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3)) ACTIVE(U42(tt, M, N)) -> MARK(s(plus(N, M))) MARK(s(X)) -> ACTIVE(s(mark(X))) ACTIVE(plus(N, 0)) -> MARK(U31(isNat(N), N)) MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2)) ACTIVE(plus(N, s(M))) -> MARK(U41(isNat(M), M, N)) MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3)) MARK(U41(X1, X2, X3)) -> MARK(X1) MARK(U21(X)) -> MARK(X) MARK(U31(X1, X2)) -> MARK(X1) MARK(U42(X1, X2, X3)) -> MARK(X1) MARK(s(X)) -> MARK(X) MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2))) MARK(plus(X1, X2)) -> MARK(X1) MARK(plus(X1, X2)) -> MARK(X2) The TRS R consists of the following rules: active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt, N)) -> mark(N) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) mark(tt) -> active(tt) mark(U12(X)) -> active(U12(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(0) -> active(0) U11(mark(X1), X2) -> U11(X1, X2) U11(X1, mark(X2)) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1), X2) -> U31(X1, X2) U31(X1, mark(X2)) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) plus(mark(X1), X2) -> plus(X1, X2) plus(X1, mark(X2)) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (69) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (70) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U12(X)) -> ACTIVE(U12(mark(X))) ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2))) MARK(U12(X)) -> MARK(X) MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2)) ACTIVE(U31(tt, N)) -> MARK(N) MARK(U11(X1, X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U21(X)) -> ACTIVE(U21(mark(X))) ACTIVE(U41(tt, M, N)) -> MARK(U42(isNat(N), M, N)) MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3)) ACTIVE(U42(tt, M, N)) -> MARK(s(plus(N, M))) MARK(s(X)) -> ACTIVE(s(mark(X))) ACTIVE(plus(N, 0)) -> MARK(U31(isNat(N), N)) MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2)) ACTIVE(plus(N, s(M))) -> MARK(U41(isNat(M), M, N)) MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3)) MARK(U41(X1, X2, X3)) -> MARK(X1) MARK(U21(X)) -> MARK(X) MARK(U31(X1, X2)) -> MARK(X1) MARK(U42(X1, X2, X3)) -> MARK(X1) MARK(s(X)) -> MARK(X) MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2))) MARK(plus(X1, X2)) -> MARK(X1) MARK(plus(X1, X2)) -> MARK(X2) The TRS R consists of the following rules: mark(U12(X)) -> active(U12(mark(X))) active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U31(tt, N)) -> mark(N) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) mark(s(X)) -> active(s(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(0) -> active(0) plus(X1, mark(X2)) -> plus(X1, X2) plus(mark(X1), X2) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) U12(active(X)) -> U12(X) U12(mark(X)) -> U12(X) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U11(X1, mark(X2)) -> U11(X1, X2) U11(mark(X1), X2) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(X1, mark(X2)) -> U31(X1, X2) U31(mark(X1), X2) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) 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(U12(X)) -> ACTIVE(U12(mark(X))) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(s(X)) -> ACTIVE(s(mark(X))) 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)) = 1 POL(U11(x_1, x_2)) = 1 POL(U12(x_1)) = 0 POL(U21(x_1)) = 0 POL(U31(x_1, x_2)) = 1 POL(U41(x_1, x_2, x_3)) = 1 POL(U42(x_1, x_2, x_3)) = 1 POL(active(x_1)) = 0 POL(isNat(x_1)) = 1 POL(mark(x_1)) = 0 POL(plus(x_1, x_2)) = 1 POL(s(x_1)) = 0 POL(tt) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U12(active(X)) -> U12(X) U12(mark(X)) -> U12(X) U11(X1, mark(X2)) -> U11(X1, X2) U11(mark(X1), X2) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U31(X1, mark(X2)) -> U31(X1, X2) U31(mark(X1), X2) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) plus(X1, mark(X2)) -> plus(X1, X2) plus(mark(X1), X2) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) ---------------------------------------- (72) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2))) MARK(U12(X)) -> MARK(X) MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2)) ACTIVE(U31(tt, N)) -> MARK(N) MARK(U11(X1, X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(U41(tt, M, N)) -> MARK(U42(isNat(N), M, N)) MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3)) ACTIVE(U42(tt, M, N)) -> MARK(s(plus(N, M))) ACTIVE(plus(N, 0)) -> MARK(U31(isNat(N), N)) MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2)) ACTIVE(plus(N, s(M))) -> MARK(U41(isNat(M), M, N)) MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3)) MARK(U41(X1, X2, X3)) -> MARK(X1) MARK(U21(X)) -> MARK(X) MARK(U31(X1, X2)) -> MARK(X1) MARK(U42(X1, X2, X3)) -> MARK(X1) MARK(s(X)) -> MARK(X) MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2))) MARK(plus(X1, X2)) -> MARK(X1) MARK(plus(X1, X2)) -> MARK(X2) The TRS R consists of the following rules: mark(U12(X)) -> active(U12(mark(X))) active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U31(tt, N)) -> mark(N) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) mark(s(X)) -> active(s(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(0) -> active(0) plus(X1, mark(X2)) -> plus(X1, X2) plus(mark(X1), X2) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) U12(active(X)) -> U12(X) U12(mark(X)) -> U12(X) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U11(X1, mark(X2)) -> U11(X1, X2) U11(mark(X1), X2) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(X1, mark(X2)) -> U31(X1, X2) U31(mark(X1), X2) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) 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. ACTIVE(plus(N, 0)) -> MARK(U31(isNat(N), N)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0) = 1 POL(ACTIVE(x_1)) = x_1 POL(MARK(x_1)) = x_1 POL(U11(x_1, x_2)) = x_1 POL(U12(x_1)) = x_1 POL(U21(x_1)) = x_1 POL(U31(x_1, x_2)) = x_1 + x_2 POL(U41(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U42(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(active(x_1)) = x_1 POL(isNat(x_1)) = 0 POL(mark(x_1)) = x_1 POL(plus(x_1, x_2)) = x_1 + x_2 POL(s(x_1)) = x_1 POL(tt) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(U12(X)) -> active(U12(mark(X))) active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U31(tt, N)) -> mark(N) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) mark(s(X)) -> active(s(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(0) -> active(0) U11(X1, mark(X2)) -> U11(X1, X2) U11(mark(X1), X2) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) U31(X1, mark(X2)) -> U31(X1, X2) U31(mark(X1), X2) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) plus(X1, mark(X2)) -> plus(X1, X2) plus(mark(X1), X2) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) U12(active(X)) -> U12(X) U12(mark(X)) -> U12(X) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) ---------------------------------------- (74) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2))) MARK(U12(X)) -> MARK(X) MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2)) ACTIVE(U31(tt, N)) -> MARK(N) MARK(U11(X1, X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(U41(tt, M, N)) -> MARK(U42(isNat(N), M, N)) MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3)) ACTIVE(U42(tt, M, N)) -> MARK(s(plus(N, M))) MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2)) ACTIVE(plus(N, s(M))) -> MARK(U41(isNat(M), M, N)) MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3)) MARK(U41(X1, X2, X3)) -> MARK(X1) MARK(U21(X)) -> MARK(X) MARK(U31(X1, X2)) -> MARK(X1) MARK(U42(X1, X2, X3)) -> MARK(X1) MARK(s(X)) -> MARK(X) MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2))) MARK(plus(X1, X2)) -> MARK(X1) MARK(plus(X1, X2)) -> MARK(X2) The TRS R consists of the following rules: mark(U12(X)) -> active(U12(mark(X))) active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U31(tt, N)) -> mark(N) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) mark(s(X)) -> active(s(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(0) -> active(0) plus(X1, mark(X2)) -> plus(X1, X2) plus(mark(X1), X2) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) U12(active(X)) -> U12(X) U12(mark(X)) -> U12(X) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U11(X1, mark(X2)) -> U11(X1, X2) U11(mark(X1), X2) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(X1, mark(X2)) -> U31(X1, X2) U31(mark(X1), X2) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) 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(U31(tt, N)) -> MARK(N) MARK(U31(X1, X2)) -> MARK(X1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0) = 1 POL(ACTIVE(x_1)) = x_1 POL(MARK(x_1)) = x_1 POL(U11(x_1, x_2)) = x_1 POL(U12(x_1)) = x_1 POL(U21(x_1)) = x_1 POL(U31(x_1, x_2)) = 1 + x_1 + x_2 POL(U41(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U42(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(active(x_1)) = x_1 POL(isNat(x_1)) = 0 POL(mark(x_1)) = x_1 POL(plus(x_1, x_2)) = x_1 + x_2 POL(s(x_1)) = x_1 POL(tt) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(U12(X)) -> active(U12(mark(X))) active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U31(tt, N)) -> mark(N) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) mark(s(X)) -> active(s(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(0) -> active(0) U11(X1, mark(X2)) -> U11(X1, X2) U11(mark(X1), X2) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) U31(X1, mark(X2)) -> U31(X1, X2) U31(mark(X1), X2) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) plus(X1, mark(X2)) -> plus(X1, X2) plus(mark(X1), X2) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) U12(active(X)) -> U12(X) U12(mark(X)) -> U12(X) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) ---------------------------------------- (76) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2))) MARK(U12(X)) -> MARK(X) MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2)) MARK(U11(X1, X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(U41(tt, M, N)) -> MARK(U42(isNat(N), M, N)) MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3)) ACTIVE(U42(tt, M, N)) -> MARK(s(plus(N, M))) MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2)) ACTIVE(plus(N, s(M))) -> MARK(U41(isNat(M), M, N)) MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3)) MARK(U41(X1, X2, X3)) -> MARK(X1) MARK(U21(X)) -> MARK(X) MARK(U42(X1, X2, X3)) -> MARK(X1) MARK(s(X)) -> MARK(X) MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2))) MARK(plus(X1, X2)) -> MARK(X1) MARK(plus(X1, X2)) -> MARK(X2) The TRS R consists of the following rules: mark(U12(X)) -> active(U12(mark(X))) active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U31(tt, N)) -> mark(N) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) mark(s(X)) -> active(s(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(0) -> active(0) plus(X1, mark(X2)) -> plus(X1, X2) plus(mark(X1), X2) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) U12(active(X)) -> U12(X) U12(mark(X)) -> U12(X) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U11(X1, mark(X2)) -> U11(X1, X2) U11(mark(X1), X2) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(X1, mark(X2)) -> U31(X1, X2) U31(mark(X1), X2) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) 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. MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2)) 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)) = 1 POL(U11(x_1, x_2)) = 1 POL(U12(x_1)) = 0 POL(U21(x_1)) = 0 POL(U31(x_1, x_2)) = 0 POL(U41(x_1, x_2, x_3)) = 1 POL(U42(x_1, x_2, x_3)) = 1 POL(active(x_1)) = 0 POL(isNat(x_1)) = 1 POL(mark(x_1)) = 0 POL(plus(x_1, x_2)) = 1 POL(s(x_1)) = 0 POL(tt) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U11(X1, mark(X2)) -> U11(X1, X2) U11(mark(X1), X2) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) U31(X1, mark(X2)) -> U31(X1, X2) U31(mark(X1), X2) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) plus(X1, mark(X2)) -> plus(X1, X2) plus(mark(X1), X2) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) ---------------------------------------- (78) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2))) MARK(U12(X)) -> MARK(X) MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2)) MARK(U11(X1, X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(U41(tt, M, N)) -> MARK(U42(isNat(N), M, N)) MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3)) ACTIVE(U42(tt, M, N)) -> MARK(s(plus(N, M))) ACTIVE(plus(N, s(M))) -> MARK(U41(isNat(M), M, N)) MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3)) MARK(U41(X1, X2, X3)) -> MARK(X1) MARK(U21(X)) -> MARK(X) MARK(U42(X1, X2, X3)) -> MARK(X1) MARK(s(X)) -> MARK(X) MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2))) MARK(plus(X1, X2)) -> MARK(X1) MARK(plus(X1, X2)) -> MARK(X2) The TRS R consists of the following rules: mark(U12(X)) -> active(U12(mark(X))) active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U31(tt, N)) -> mark(N) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) mark(s(X)) -> active(s(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(0) -> active(0) plus(X1, mark(X2)) -> plus(X1, X2) plus(mark(X1), X2) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) U12(active(X)) -> U12(X) U12(mark(X)) -> U12(X) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U11(X1, mark(X2)) -> U11(X1, X2) U11(mark(X1), X2) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(X1, mark(X2)) -> U31(X1, X2) U31(mark(X1), X2) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) 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. MARK(U41(X1, X2, X3)) -> MARK(X1) MARK(U42(X1, X2, X3)) -> MARK(X1) MARK(s(X)) -> MARK(X) MARK(plus(X1, X2)) -> MARK(X1) MARK(plus(X1, X2)) -> MARK(X2) 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(U11(x_1, x_2)) = x_1 POL(U12(x_1)) = x_1 POL(U21(x_1)) = x_1 POL(U31(x_1, x_2)) = x_2 POL(U41(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 POL(U42(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 POL(active(x_1)) = x_1 POL(isNat(x_1)) = 1 POL(mark(x_1)) = x_1 POL(plus(x_1, x_2)) = 1 + x_1 + x_2 POL(s(x_1)) = 1 + x_1 POL(tt) = 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(U12(X)) -> active(U12(mark(X))) active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U31(tt, N)) -> mark(N) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) mark(s(X)) -> active(s(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(0) -> active(0) U11(X1, mark(X2)) -> U11(X1, X2) U11(mark(X1), X2) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) plus(X1, mark(X2)) -> plus(X1, X2) plus(mark(X1), X2) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) U12(active(X)) -> U12(X) U12(mark(X)) -> U12(X) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(X1, mark(X2)) -> U31(X1, X2) U31(mark(X1), X2) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) ---------------------------------------- (80) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2))) MARK(U12(X)) -> MARK(X) MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2)) MARK(U11(X1, X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(U41(tt, M, N)) -> MARK(U42(isNat(N), M, N)) MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3)) ACTIVE(U42(tt, M, N)) -> MARK(s(plus(N, M))) ACTIVE(plus(N, s(M))) -> MARK(U41(isNat(M), M, N)) MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3)) MARK(U21(X)) -> MARK(X) MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2))) The TRS R consists of the following rules: mark(U12(X)) -> active(U12(mark(X))) active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U31(tt, N)) -> mark(N) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) mark(s(X)) -> active(s(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(0) -> active(0) plus(X1, mark(X2)) -> plus(X1, X2) plus(mark(X1), X2) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) U12(active(X)) -> U12(X) U12(mark(X)) -> U12(X) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U11(X1, mark(X2)) -> U11(X1, X2) U11(mark(X1), X2) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(X1, mark(X2)) -> U31(X1, X2) U31(mark(X1), X2) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (81) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (82) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U12(X)) -> MARK(X) MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2)) ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2))) ACTIVE(U41(tt, M, N)) -> MARK(U42(isNat(N), M, N)) MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3)) ACTIVE(plus(N, s(M))) -> MARK(U41(isNat(M), M, N)) MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3)) MARK(U11(X1, X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U21(X)) -> MARK(X) MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2))) The TRS R consists of the following rules: mark(U12(X)) -> active(U12(mark(X))) active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U31(tt, N)) -> mark(N) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) mark(s(X)) -> active(s(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(0) -> active(0) plus(X1, mark(X2)) -> plus(X1, X2) plus(mark(X1), X2) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) U12(active(X)) -> U12(X) U12(mark(X)) -> U12(X) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U11(X1, mark(X2)) -> U11(X1, X2) U11(mark(X1), X2) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(X1, mark(X2)) -> U31(X1, X2) U31(mark(X1), X2) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (83) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented dependency pairs: MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3)) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVE(x_1)) = x_1 POL(MARK(x_1)) = 1 POL(U11(x_1, x_2)) = 1 POL(U12(x_1)) = 0 POL(U21(x_1)) = 0 POL(U31(x_1, x_2)) = 0 POL(U41(x_1, x_2, x_3)) = 1 POL(U42(x_1, x_2, x_3)) = 0 POL(active(x_1)) = 0 POL(isNat(x_1)) = 1 POL(mark(x_1)) = 0 POL(plus(x_1, x_2)) = 1 POL(s(x_1)) = 0 POL(tt) = 0 ---------------------------------------- (84) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U12(X)) -> MARK(X) MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2)) ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2))) ACTIVE(U41(tt, M, N)) -> MARK(U42(isNat(N), M, N)) ACTIVE(plus(N, s(M))) -> MARK(U41(isNat(M), M, N)) MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3)) MARK(U11(X1, X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U21(X)) -> MARK(X) MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2))) The TRS R consists of the following rules: mark(U12(X)) -> active(U12(mark(X))) active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U31(tt, N)) -> mark(N) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) mark(s(X)) -> active(s(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(0) -> active(0) plus(X1, mark(X2)) -> plus(X1, X2) plus(mark(X1), X2) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) U12(active(X)) -> U12(X) U12(mark(X)) -> U12(X) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U11(X1, mark(X2)) -> U11(X1, X2) U11(mark(X1), X2) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(X1, mark(X2)) -> U31(X1, X2) U31(mark(X1), X2) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (85) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (86) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2)) ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2))) MARK(U12(X)) -> MARK(X) MARK(U11(X1, X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U21(X)) -> MARK(X) MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3)) ACTIVE(plus(N, s(M))) -> MARK(U41(isNat(M), M, N)) MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2))) The TRS R consists of the following rules: mark(U12(X)) -> active(U12(mark(X))) active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U31(tt, N)) -> mark(N) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) mark(s(X)) -> active(s(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(0) -> active(0) plus(X1, mark(X2)) -> plus(X1, X2) plus(mark(X1), X2) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) U12(active(X)) -> U12(X) U12(mark(X)) -> U12(X) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U11(X1, mark(X2)) -> U11(X1, X2) U11(mark(X1), X2) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(X1, mark(X2)) -> U31(X1, X2) U31(mark(X1), X2) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (87) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented dependency pairs: MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3)) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVE(x_1)) = 2*x_1 POL(MARK(x_1)) = 2 POL(U11(x_1, x_2)) = 1 POL(U12(x_1)) = 0 POL(U21(x_1)) = 0 POL(U31(x_1, x_2)) = 0 POL(U41(x_1, x_2, x_3)) = 0 POL(U42(x_1, x_2, x_3)) = 0 POL(active(x_1)) = 0 POL(isNat(x_1)) = 1 POL(mark(x_1)) = 0 POL(plus(x_1, x_2)) = 1 POL(s(x_1)) = 0 POL(tt) = 0 ---------------------------------------- (88) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2)) ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2))) MARK(U12(X)) -> MARK(X) MARK(U11(X1, X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U21(X)) -> MARK(X) ACTIVE(plus(N, s(M))) -> MARK(U41(isNat(M), M, N)) MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2))) The TRS R consists of the following rules: mark(U12(X)) -> active(U12(mark(X))) active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U31(tt, N)) -> mark(N) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) mark(s(X)) -> active(s(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(0) -> active(0) plus(X1, mark(X2)) -> plus(X1, X2) plus(mark(X1), X2) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) U12(active(X)) -> U12(X) U12(mark(X)) -> U12(X) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U11(X1, mark(X2)) -> U11(X1, X2) U11(mark(X1), X2) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(X1, mark(X2)) -> U31(X1, X2) U31(mark(X1), X2) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (89) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (90) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2))) MARK(U12(X)) -> MARK(X) MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2)) MARK(U11(X1, X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U21(X)) -> MARK(X) MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2))) The TRS R consists of the following rules: mark(U12(X)) -> active(U12(mark(X))) active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U31(tt, N)) -> mark(N) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) mark(s(X)) -> active(s(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(0) -> active(0) plus(X1, mark(X2)) -> plus(X1, X2) plus(mark(X1), X2) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) U12(active(X)) -> U12(X) U12(mark(X)) -> U12(X) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U11(X1, mark(X2)) -> U11(X1, X2) U11(mark(X1), X2) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(X1, mark(X2)) -> U31(X1, X2) U31(mark(X1), X2) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (91) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented dependency pairs: MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2))) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVE(x_1)) = 2*x_1 POL(MARK(x_1)) = 2 POL(U11(x_1, x_2)) = 1 POL(U12(x_1)) = 0 POL(U21(x_1)) = 0 POL(U31(x_1, x_2)) = 0 POL(U41(x_1, x_2, x_3)) = 0 POL(U42(x_1, x_2, x_3)) = 0 POL(active(x_1)) = 0 POL(isNat(x_1)) = 1 POL(mark(x_1)) = 0 POL(plus(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(tt) = 0 ---------------------------------------- (92) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2))) MARK(U12(X)) -> MARK(X) MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2)) MARK(U11(X1, X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U21(X)) -> MARK(X) The TRS R consists of the following rules: mark(U12(X)) -> active(U12(mark(X))) active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U31(tt, N)) -> mark(N) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) mark(s(X)) -> active(s(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(0) -> active(0) plus(X1, mark(X2)) -> plus(X1, X2) plus(mark(X1), X2) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) U12(active(X)) -> U12(X) U12(mark(X)) -> U12(X) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U11(X1, mark(X2)) -> U11(X1, X2) U11(mark(X1), X2) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(X1, mark(X2)) -> U31(X1, X2) U31(mark(X1), X2) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (93) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented dependency pairs: ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2))) MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) Used ordering: Polynomial interpretation [POLO]: POL(0) = 2 POL(ACTIVE(x_1)) = 2*x_1 POL(MARK(x_1)) = 2 + 2*x_1 POL(U11(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U12(x_1)) = x_1 POL(U21(x_1)) = x_1 POL(U31(x_1, x_2)) = 2 + 2*x_2 POL(U41(x_1, x_2, x_3)) = 2 + 2*x_2 + 2*x_3 POL(U42(x_1, x_2, x_3)) = 2 + 2*x_2 + 2*x_3 POL(active(x_1)) = x_1 POL(isNat(x_1)) = 2*x_1 POL(mark(x_1)) = x_1 POL(plus(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(s(x_1)) = 1 + x_1 POL(tt) = 1 ---------------------------------------- (94) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U12(X)) -> MARK(X) MARK(U11(X1, X2)) -> MARK(X1) MARK(U21(X)) -> MARK(X) The TRS R consists of the following rules: mark(U12(X)) -> active(U12(mark(X))) active(U11(tt, V2)) -> mark(U12(isNat(V2))) active(U31(tt, N)) -> mark(N) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N)) mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3)) active(U42(tt, M, N)) -> mark(s(plus(N, M))) mark(s(X)) -> active(s(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U21(X)) -> active(U21(mark(X))) active(plus(N, 0)) -> mark(U31(isNat(N), N)) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) active(plus(N, s(M))) -> mark(U41(isNat(M), M, N)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(tt) -> active(tt) mark(0) -> active(0) plus(X1, mark(X2)) -> plus(X1, X2) plus(mark(X1), X2) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) U12(active(X)) -> U12(X) U12(mark(X)) -> U12(X) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(isNat(0)) -> mark(tt) U11(X1, mark(X2)) -> U11(X1, X2) U11(mark(X1), X2) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U42(X1, mark(X2), X3) -> U42(X1, X2, X3) U42(mark(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, X2, mark(X3)) -> U42(X1, X2, X3) U42(active(X1), X2, X3) -> U42(X1, X2, X3) U42(X1, active(X2), X3) -> U42(X1, X2, X3) U42(X1, X2, active(X3)) -> U42(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(X1, mark(X2)) -> U31(X1, X2) U31(mark(X1), X2) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (95) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (96) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U12(X)) -> MARK(X) MARK(U11(X1, X2)) -> MARK(X1) MARK(U21(X)) -> MARK(X) R is empty. The set Q consists of the following terms: active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) isNat(mark(x0)) isNat(active(x0)) U21(mark(x0)) U21(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (97) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. active(U11(tt, x0)) active(U12(tt)) active(U21(tt)) active(U31(tt, x0)) active(U41(tt, x0, x1)) active(U42(tt, x0, x1)) active(isNat(0)) active(isNat(plus(x0, x1))) active(isNat(s(x0))) active(plus(x0, 0)) active(plus(x0, s(x1))) mark(U11(x0, x1)) mark(tt) mark(U12(x0)) mark(isNat(x0)) mark(U21(x0)) mark(U31(x0, x1)) mark(U41(x0, x1, x2)) mark(U42(x0, x1, x2)) mark(s(x0)) mark(plus(x0, x1)) mark(0) isNat(mark(x0)) isNat(active(x0)) U31(mark(x0), x1) U31(x0, mark(x1)) U31(active(x0), x1) U31(x0, active(x1)) U41(mark(x0), x1, x2) U41(x0, mark(x1), x2) U41(x0, x1, mark(x2)) U41(active(x0), x1, x2) U41(x0, active(x1), x2) U41(x0, x1, active(x2)) U42(mark(x0), x1, x2) U42(x0, mark(x1), x2) U42(x0, x1, mark(x2)) U42(active(x0), x1, x2) U42(x0, active(x1), x2) U42(x0, x1, active(x2)) s(mark(x0)) s(active(x0)) plus(mark(x0), x1) plus(x0, mark(x1)) plus(active(x0), x1) plus(x0, active(x1)) ---------------------------------------- (98) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U12(X)) -> MARK(X) MARK(U11(X1, X2)) -> MARK(X1) MARK(U21(X)) -> MARK(X) R is empty. The set Q consists of the following terms: U11(mark(x0), x1) U11(x0, mark(x1)) U11(active(x0), x1) U11(x0, active(x1)) U12(mark(x0)) U12(active(x0)) U21(mark(x0)) U21(active(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (99) 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: *MARK(U12(X)) -> MARK(X) The graph contains the following edges 1 > 1 *MARK(U11(X1, X2)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U21(X)) -> MARK(X) The graph contains the following edges 1 > 1 ---------------------------------------- (100) YES