/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) DependencyPairsProof [EQUIVALENT, 47 ms] (2) QDP (3) DependencyGraphProof [EQUIVALENT, 0 ms] (4) AND (5) QDP (6) UsableRulesProof [EQUIVALENT, 0 ms] (7) QDP (8) QDPSizeChangeProof [EQUIVALENT, 0 ms] (9) YES (10) QDP (11) UsableRulesProof [EQUIVALENT, 0 ms] (12) QDP (13) QDPSizeChangeProof [EQUIVALENT, 0 ms] (14) YES (15) QDP (16) UsableRulesProof [EQUIVALENT, 0 ms] (17) QDP (18) QDPSizeChangeProof [EQUIVALENT, 0 ms] (19) YES (20) QDP (21) UsableRulesProof [EQUIVALENT, 0 ms] (22) QDP (23) QDPSizeChangeProof [EQUIVALENT, 0 ms] (24) YES (25) QDP (26) UsableRulesProof [EQUIVALENT, 0 ms] (27) QDP (28) QDPSizeChangeProof [EQUIVALENT, 0 ms] (29) YES (30) QDP (31) UsableRulesProof [EQUIVALENT, 0 ms] (32) QDP (33) QDPSizeChangeProof [EQUIVALENT, 0 ms] (34) YES (35) QDP (36) UsableRulesProof [EQUIVALENT, 0 ms] (37) QDP (38) QDPSizeChangeProof [EQUIVALENT, 0 ms] (39) YES (40) QDP (41) UsableRulesProof [EQUIVALENT, 0 ms] (42) QDP (43) QDPSizeChangeProof [EQUIVALENT, 0 ms] (44) YES (45) QDP (46) UsableRulesProof [EQUIVALENT, 0 ms] (47) QDP (48) QDPSizeChangeProof [EQUIVALENT, 0 ms] (49) YES (50) QDP (51) UsableRulesProof [EQUIVALENT, 0 ms] (52) QDP (53) QDPSizeChangeProof [EQUIVALENT, 0 ms] (54) YES (55) QDP (56) UsableRulesProof [EQUIVALENT, 0 ms] (57) QDP (58) QDPSizeChangeProof [EQUIVALENT, 0 ms] (59) YES (60) QDP (61) UsableRulesProof [EQUIVALENT, 0 ms] (62) QDP (63) QDPSizeChangeProof [EQUIVALENT, 0 ms] (64) YES (65) QDP (66) UsableRulesProof [EQUIVALENT, 0 ms] (67) QDP (68) QDPSizeChangeProof [EQUIVALENT, 0 ms] (69) YES (70) QDP (71) UsableRulesProof [EQUIVALENT, 0 ms] (72) QDP (73) QDPSizeChangeProof [EQUIVALENT, 0 ms] (74) YES (75) QDP (76) UsableRulesProof [EQUIVALENT, 0 ms] (77) QDP (78) QDPSizeChangeProof [EQUIVALENT, 0 ms] (79) YES (80) QDP (81) QDPOrderProof [EQUIVALENT, 616 ms] (82) QDP (83) QDPOrderProof [EQUIVALENT, 88 ms] (84) QDP (85) QDPOrderProof [EQUIVALENT, 951 ms] (86) QDP (87) QDPOrderProof [EQUIVALENT, 348 ms] (88) QDP (89) QDPOrderProof [EQUIVALENT, 335 ms] (90) QDP (91) QDPOrderProof [EQUIVALENT, 296 ms] (92) QDP (93) QDPOrderProof [EQUIVALENT, 303 ms] (94) QDP (95) QDPOrderProof [EQUIVALENT, 630 ms] (96) QDP (97) QDPOrderProof [EQUIVALENT, 311 ms] (98) QDP (99) QDPOrderProof [EQUIVALENT, 307 ms] (100) QDP (101) QDPOrderProof [EQUIVALENT, 549 ms] (102) QDP (103) QDPOrderProof [EQUIVALENT, 229 ms] (104) QDP (105) QDPOrderProof [EQUIVALENT, 216 ms] (106) QDP (107) UsableRulesProof [EQUIVALENT, 0 ms] (108) QDP (109) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] (110) QDP (111) DependencyGraphProof [EQUIVALENT, 0 ms] (112) QDP (113) QDPSizeChangeProof [EQUIVALENT, 0 ms] (114) 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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. ---------------------------------------- (1) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (2) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(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, V2)) -> MARK(U32(isNat(V2))) ACTIVE(U31(tt, V2)) -> U32^1(isNat(V2)) ACTIVE(U31(tt, V2)) -> ISNAT(V2) ACTIVE(U32(tt)) -> MARK(tt) ACTIVE(U41(tt, N)) -> MARK(N) ACTIVE(U51(tt, M, N)) -> MARK(U52(isNat(N), M, N)) ACTIVE(U51(tt, M, N)) -> U52^1(isNat(N), M, N) ACTIVE(U51(tt, M, N)) -> ISNAT(N) ACTIVE(U52(tt, M, N)) -> MARK(s(plus(N, M))) ACTIVE(U52(tt, M, N)) -> S(plus(N, M)) ACTIVE(U52(tt, M, N)) -> PLUS(N, M) ACTIVE(U61(tt)) -> MARK(0) ACTIVE(U71(tt, M, N)) -> MARK(U72(isNat(N), M, N)) ACTIVE(U71(tt, M, N)) -> U72^1(isNat(N), M, N) ACTIVE(U71(tt, M, N)) -> ISNAT(N) ACTIVE(U72(tt, M, N)) -> MARK(plus(x(N, M), N)) ACTIVE(U72(tt, M, N)) -> PLUS(x(N, M), N) ACTIVE(U72(tt, M, N)) -> X(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(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2)) ACTIVE(isNat(x(V1, V2))) -> U31^1(isNat(V1), V2) ACTIVE(isNat(x(V1, V2))) -> ISNAT(V1) ACTIVE(plus(N, 0)) -> MARK(U41(isNat(N), N)) ACTIVE(plus(N, 0)) -> U41^1(isNat(N), N) ACTIVE(plus(N, 0)) -> ISNAT(N) ACTIVE(plus(N, s(M))) -> MARK(U51(isNat(M), M, N)) ACTIVE(plus(N, s(M))) -> U51^1(isNat(M), M, N) ACTIVE(plus(N, s(M))) -> ISNAT(M) ACTIVE(x(N, 0)) -> MARK(U61(isNat(N))) ACTIVE(x(N, 0)) -> U61^1(isNat(N)) ACTIVE(x(N, 0)) -> ISNAT(N) ACTIVE(x(N, s(M))) -> MARK(U71(isNat(M), M, N)) ACTIVE(x(N, s(M))) -> U71^1(isNat(M), M, N) ACTIVE(x(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(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> U32^1(mark(X)) MARK(U32(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> U41^1(mark(X1), X2) MARK(U41(X1, X2)) -> MARK(X1) MARK(U51(X1, X2, X3)) -> ACTIVE(U51(mark(X1), X2, X3)) MARK(U51(X1, X2, X3)) -> U51^1(mark(X1), X2, X3) MARK(U51(X1, X2, X3)) -> MARK(X1) MARK(U52(X1, X2, X3)) -> ACTIVE(U52(mark(X1), X2, X3)) MARK(U52(X1, X2, X3)) -> U52^1(mark(X1), X2, X3) MARK(U52(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(U61(X)) -> ACTIVE(U61(mark(X))) MARK(U61(X)) -> U61^1(mark(X)) MARK(U61(X)) -> MARK(X) MARK(0) -> ACTIVE(0) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U71(X1, X2, X3)) -> U71^1(mark(X1), X2, X3) MARK(U71(X1, X2, X3)) -> MARK(X1) MARK(U72(X1, X2, X3)) -> ACTIVE(U72(mark(X1), X2, X3)) MARK(U72(X1, X2, X3)) -> U72^1(mark(X1), X2, X3) MARK(U72(X1, X2, X3)) -> MARK(X1) MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), mark(X2))) MARK(x(X1, X2)) -> X(mark(X1), mark(X2)) MARK(x(X1, X2)) -> MARK(X1) MARK(x(X1, X2)) -> MARK(X2) 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) U32^1(mark(X)) -> U32^1(X) U32^1(active(X)) -> U32^1(X) U41^1(mark(X1), X2) -> U41^1(X1, X2) U41^1(X1, mark(X2)) -> U41^1(X1, X2) U41^1(active(X1), X2) -> U41^1(X1, X2) U41^1(X1, active(X2)) -> U41^1(X1, X2) U51^1(mark(X1), X2, X3) -> U51^1(X1, X2, X3) U51^1(X1, mark(X2), X3) -> U51^1(X1, X2, X3) U51^1(X1, X2, mark(X3)) -> U51^1(X1, X2, X3) U51^1(active(X1), X2, X3) -> U51^1(X1, X2, X3) U51^1(X1, active(X2), X3) -> U51^1(X1, X2, X3) U51^1(X1, X2, active(X3)) -> U51^1(X1, X2, X3) U52^1(mark(X1), X2, X3) -> U52^1(X1, X2, X3) U52^1(X1, mark(X2), X3) -> U52^1(X1, X2, X3) U52^1(X1, X2, mark(X3)) -> U52^1(X1, X2, X3) U52^1(active(X1), X2, X3) -> U52^1(X1, X2, X3) U52^1(X1, active(X2), X3) -> U52^1(X1, X2, X3) U52^1(X1, X2, active(X3)) -> U52^1(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) U61^1(mark(X)) -> U61^1(X) U61^1(active(X)) -> U61^1(X) U71^1(mark(X1), X2, X3) -> U71^1(X1, X2, X3) U71^1(X1, mark(X2), X3) -> U71^1(X1, X2, X3) U71^1(X1, X2, mark(X3)) -> U71^1(X1, X2, X3) U71^1(active(X1), X2, X3) -> U71^1(X1, X2, X3) U71^1(X1, active(X2), X3) -> U71^1(X1, X2, X3) U71^1(X1, X2, active(X3)) -> U71^1(X1, X2, X3) U72^1(mark(X1), X2, X3) -> U72^1(X1, X2, X3) U72^1(X1, mark(X2), X3) -> U72^1(X1, X2, X3) U72^1(X1, X2, mark(X3)) -> U72^1(X1, X2, X3) U72^1(active(X1), X2, X3) -> U72^1(X1, X2, X3) U72^1(X1, active(X2), X3) -> U72^1(X1, X2, X3) U72^1(X1, X2, active(X3)) -> U72^1(X1, X2, X3) X(mark(X1), X2) -> X(X1, X2) X(X1, mark(X2)) -> X(X1, X2) X(active(X1), X2) -> X(X1, X2) X(X1, active(X2)) -> X(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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (3) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 16 SCCs with 47 less nodes. ---------------------------------------- (4) Complex Obligation (AND) ---------------------------------------- (5) Obligation: Q DP problem: The TRS P consists of the following rules: X(X1, mark(X2)) -> X(X1, X2) X(mark(X1), X2) -> X(X1, X2) X(active(X1), X2) -> X(X1, X2) X(X1, active(X2)) -> X(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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (6) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (7) Obligation: Q DP problem: The TRS P consists of the following rules: X(X1, mark(X2)) -> X(X1, X2) X(mark(X1), X2) -> X(X1, X2) X(active(X1), X2) -> X(X1, X2) X(X1, active(X2)) -> X(X1, X2) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (8) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *X(X1, mark(X2)) -> X(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 *X(mark(X1), X2) -> X(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *X(active(X1), X2) -> X(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *X(X1, active(X2)) -> X(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (9) YES ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: U72^1(X1, mark(X2), X3) -> U72^1(X1, X2, X3) U72^1(mark(X1), X2, X3) -> U72^1(X1, X2, X3) U72^1(X1, X2, mark(X3)) -> U72^1(X1, X2, X3) U72^1(active(X1), X2, X3) -> U72^1(X1, X2, X3) U72^1(X1, active(X2), X3) -> U72^1(X1, X2, X3) U72^1(X1, X2, active(X3)) -> U72^1(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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: U72^1(X1, mark(X2), X3) -> U72^1(X1, X2, X3) U72^1(mark(X1), X2, X3) -> U72^1(X1, X2, X3) U72^1(X1, X2, mark(X3)) -> U72^1(X1, X2, X3) U72^1(active(X1), X2, X3) -> U72^1(X1, X2, X3) U72^1(X1, active(X2), X3) -> U72^1(X1, X2, X3) U72^1(X1, X2, active(X3)) -> U72^1(X1, X2, X3) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *U72^1(X1, mark(X2), X3) -> U72^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U72^1(mark(X1), X2, X3) -> U72^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U72^1(X1, X2, mark(X3)) -> U72^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 *U72^1(active(X1), X2, X3) -> U72^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U72^1(X1, active(X2), X3) -> U72^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U72^1(X1, X2, active(X3)) -> U72^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 ---------------------------------------- (14) YES ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: U71^1(X1, mark(X2), X3) -> U71^1(X1, X2, X3) U71^1(mark(X1), X2, X3) -> U71^1(X1, X2, X3) U71^1(X1, X2, mark(X3)) -> U71^1(X1, X2, X3) U71^1(active(X1), X2, X3) -> U71^1(X1, X2, X3) U71^1(X1, active(X2), X3) -> U71^1(X1, X2, X3) U71^1(X1, X2, active(X3)) -> U71^1(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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (17) Obligation: Q DP problem: The TRS P consists of the following rules: U71^1(X1, mark(X2), X3) -> U71^1(X1, X2, X3) U71^1(mark(X1), X2, X3) -> U71^1(X1, X2, X3) U71^1(X1, X2, mark(X3)) -> U71^1(X1, X2, X3) U71^1(active(X1), X2, X3) -> U71^1(X1, X2, X3) U71^1(X1, active(X2), X3) -> U71^1(X1, X2, X3) U71^1(X1, X2, active(X3)) -> U71^1(X1, X2, X3) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (18) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *U71^1(X1, mark(X2), X3) -> U71^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U71^1(mark(X1), X2, X3) -> U71^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U71^1(X1, X2, mark(X3)) -> U71^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 *U71^1(active(X1), X2, X3) -> U71^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U71^1(X1, active(X2), X3) -> U71^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U71^1(X1, X2, active(X3)) -> U71^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 ---------------------------------------- (19) YES ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: U61^1(active(X)) -> U61^1(X) U61^1(mark(X)) -> U61^1(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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: U61^1(active(X)) -> U61^1(X) U61^1(mark(X)) -> U61^1(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *U61^1(active(X)) -> U61^1(X) The graph contains the following edges 1 > 1 *U61^1(mark(X)) -> U61^1(X) The graph contains the following edges 1 > 1 ---------------------------------------- (24) YES ---------------------------------------- (25) 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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (26) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (27) Obligation: Q DP problem: The TRS P consists of the following rules: 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. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (28) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *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 ---------------------------------------- (29) YES ---------------------------------------- (30) 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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: S(active(X)) -> S(X) S(mark(X)) -> S(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *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 ---------------------------------------- (34) YES ---------------------------------------- (35) Obligation: Q DP problem: The TRS P consists of the following rules: U52^1(X1, mark(X2), X3) -> U52^1(X1, X2, X3) U52^1(mark(X1), X2, X3) -> U52^1(X1, X2, X3) U52^1(X1, X2, mark(X3)) -> U52^1(X1, X2, X3) U52^1(active(X1), X2, X3) -> U52^1(X1, X2, X3) U52^1(X1, active(X2), X3) -> U52^1(X1, X2, X3) U52^1(X1, X2, active(X3)) -> U52^1(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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (36) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (37) Obligation: Q DP problem: The TRS P consists of the following rules: U52^1(X1, mark(X2), X3) -> U52^1(X1, X2, X3) U52^1(mark(X1), X2, X3) -> U52^1(X1, X2, X3) U52^1(X1, X2, mark(X3)) -> U52^1(X1, X2, X3) U52^1(active(X1), X2, X3) -> U52^1(X1, X2, X3) U52^1(X1, active(X2), X3) -> U52^1(X1, X2, X3) U52^1(X1, X2, active(X3)) -> U52^1(X1, X2, X3) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (38) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *U52^1(X1, mark(X2), X3) -> U52^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U52^1(mark(X1), X2, X3) -> U52^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U52^1(X1, X2, mark(X3)) -> U52^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 *U52^1(active(X1), X2, X3) -> U52^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U52^1(X1, active(X2), X3) -> U52^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U52^1(X1, X2, active(X3)) -> U52^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 ---------------------------------------- (39) YES ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(X1, mark(X2), X3) -> U51^1(X1, X2, X3) U51^1(mark(X1), X2, X3) -> U51^1(X1, X2, X3) U51^1(X1, X2, mark(X3)) -> U51^1(X1, X2, X3) U51^1(active(X1), X2, X3) -> U51^1(X1, X2, X3) U51^1(X1, active(X2), X3) -> U51^1(X1, X2, X3) U51^1(X1, X2, active(X3)) -> U51^1(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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (41) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (42) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(X1, mark(X2), X3) -> U51^1(X1, X2, X3) U51^1(mark(X1), X2, X3) -> U51^1(X1, X2, X3) U51^1(X1, X2, mark(X3)) -> U51^1(X1, X2, X3) U51^1(active(X1), X2, X3) -> U51^1(X1, X2, X3) U51^1(X1, active(X2), X3) -> U51^1(X1, X2, X3) U51^1(X1, X2, active(X3)) -> U51^1(X1, X2, X3) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (43) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *U51^1(X1, mark(X2), X3) -> U51^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U51^1(mark(X1), X2, X3) -> U51^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U51^1(X1, X2, mark(X3)) -> U51^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 *U51^1(active(X1), X2, X3) -> U51^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U51^1(X1, active(X2), X3) -> U51^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U51^1(X1, X2, active(X3)) -> U51^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 ---------------------------------------- (44) YES ---------------------------------------- (45) Obligation: Q DP problem: The TRS P consists of the following rules: U41^1(X1, mark(X2)) -> U41^1(X1, X2) U41^1(mark(X1), X2) -> U41^1(X1, X2) U41^1(active(X1), X2) -> U41^1(X1, X2) U41^1(X1, active(X2)) -> U41^1(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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (46) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (47) Obligation: Q DP problem: The TRS P consists of the following rules: U41^1(X1, mark(X2)) -> U41^1(X1, X2) U41^1(mark(X1), X2) -> U41^1(X1, X2) U41^1(active(X1), X2) -> U41^1(X1, X2) U41^1(X1, active(X2)) -> U41^1(X1, X2) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (48) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *U41^1(X1, mark(X2)) -> U41^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 *U41^1(mark(X1), X2) -> U41^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U41^1(active(X1), X2) -> U41^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U41^1(X1, active(X2)) -> U41^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (49) YES ---------------------------------------- (50) Obligation: Q DP problem: The TRS P consists of the following rules: U32^1(active(X)) -> U32^1(X) U32^1(mark(X)) -> U32^1(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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (51) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (52) Obligation: Q DP problem: The TRS P consists of the following rules: U32^1(active(X)) -> U32^1(X) U32^1(mark(X)) -> U32^1(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (53) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *U32^1(active(X)) -> U32^1(X) The graph contains the following edges 1 > 1 *U32^1(mark(X)) -> U32^1(X) The graph contains the following edges 1 > 1 ---------------------------------------- (54) YES ---------------------------------------- (55) 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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (56) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (57) Obligation: Q DP problem: The TRS P consists of the following rules: 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. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (58) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *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 ---------------------------------------- (59) YES ---------------------------------------- (60) 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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (61) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (62) 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. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (63) 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 ---------------------------------------- (64) YES ---------------------------------------- (65) 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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (66) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (67) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(active(X)) -> ISNAT(X) ISNAT(mark(X)) -> ISNAT(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (68) 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 ---------------------------------------- (69) YES ---------------------------------------- (70) 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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (71) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (72) 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. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (73) 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 ---------------------------------------- (74) YES ---------------------------------------- (75) 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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (76) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (77) 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. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (78) 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 ---------------------------------------- (79) YES ---------------------------------------- (80) 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(U11(X1, X2)) -> MARK(X1) MARK(U12(X)) -> ACTIVE(U12(mark(X))) ACTIVE(U31(tt, V2)) -> MARK(U32(isNat(V2))) MARK(U12(X)) -> MARK(X) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U41(tt, N)) -> MARK(N) MARK(U21(X)) -> ACTIVE(U21(mark(X))) ACTIVE(U51(tt, M, N)) -> MARK(U52(isNat(N), M, N)) MARK(U21(X)) -> MARK(X) MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2)) ACTIVE(U52(tt, M, N)) -> MARK(s(plus(N, M))) MARK(U31(X1, X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) ACTIVE(U71(tt, M, N)) -> MARK(U72(isNat(N), M, N)) MARK(U32(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U72(tt, M, N)) -> MARK(plus(x(N, M), N)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U51(X1, X2, X3)) -> ACTIVE(U51(mark(X1), X2, X3)) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) MARK(U51(X1, X2, X3)) -> MARK(X1) MARK(U52(X1, X2, X3)) -> ACTIVE(U52(mark(X1), X2, X3)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U52(X1, X2, X3)) -> MARK(X1) MARK(s(X)) -> ACTIVE(s(mark(X))) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2)) MARK(s(X)) -> MARK(X) MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2))) ACTIVE(plus(N, 0)) -> MARK(U41(isNat(N), N)) MARK(plus(X1, X2)) -> MARK(X1) MARK(plus(X1, X2)) -> MARK(X2) MARK(U61(X)) -> ACTIVE(U61(mark(X))) ACTIVE(plus(N, s(M))) -> MARK(U51(isNat(M), M, N)) MARK(U61(X)) -> MARK(X) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) ACTIVE(x(N, 0)) -> MARK(U61(isNat(N))) MARK(U71(X1, X2, X3)) -> MARK(X1) MARK(U72(X1, X2, X3)) -> ACTIVE(U72(mark(X1), X2, X3)) ACTIVE(x(N, s(M))) -> MARK(U71(isNat(M), M, N)) MARK(U72(X1, X2, X3)) -> MARK(X1) MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), mark(X2))) MARK(x(X1, X2)) -> MARK(X1) MARK(x(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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (81) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U12(X)) -> ACTIVE(U12(mark(X))) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(s(X)) -> ACTIVE(s(mark(X))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1} POL( U11_2(x_1, x_2) ) = 2 POL( U12_1(x_1) ) = max{0, -2} POL( U21_1(x_1) ) = 0 POL( U31_2(x_1, x_2) ) = 2 POL( U32_1(x_1) ) = max{0, -2} POL( U41_2(x_1, x_2) ) = 2 POL( U51_3(x_1, ..., x_3) ) = 2 POL( U52_3(x_1, ..., x_3) ) = 2 POL( U61_1(x_1) ) = 2 POL( U71_3(x_1, ..., x_3) ) = 2 POL( U72_3(x_1, ..., x_3) ) = 2 POL( plus_2(x_1, x_2) ) = 2 POL( s_1(x_1) ) = max{0, -2} POL( x_2(x_1, x_2) ) = 2 POL( mark_1(x_1) ) = max{0, -2} POL( active_1(x_1) ) = max{0, -2} POL( tt ) = 0 POL( isNat_1(x_1) ) = 2 POL( 0 ) = 0 POL( MARK_1(x_1) ) = 1 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) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(active(X)) -> U12(X) U12(mark(X)) -> U12(X) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) 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) s(active(X)) -> s(X) s(mark(X)) -> s(X) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(mark(X1), X2) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U61(active(X)) -> U61(X) U61(mark(X)) -> U61(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) ---------------------------------------- (82) 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(U11(X1, X2)) -> MARK(X1) ACTIVE(U31(tt, V2)) -> MARK(U32(isNat(V2))) MARK(U12(X)) -> MARK(X) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U41(tt, N)) -> MARK(N) ACTIVE(U51(tt, M, N)) -> MARK(U52(isNat(N), M, N)) MARK(U21(X)) -> MARK(X) MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2)) ACTIVE(U52(tt, M, N)) -> MARK(s(plus(N, M))) MARK(U31(X1, X2)) -> MARK(X1) ACTIVE(U71(tt, M, N)) -> MARK(U72(isNat(N), M, N)) MARK(U32(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U72(tt, M, N)) -> MARK(plus(x(N, M), N)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U51(X1, X2, X3)) -> ACTIVE(U51(mark(X1), X2, X3)) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) MARK(U51(X1, X2, X3)) -> MARK(X1) MARK(U52(X1, X2, X3)) -> ACTIVE(U52(mark(X1), X2, X3)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U52(X1, X2, X3)) -> MARK(X1) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2)) MARK(s(X)) -> MARK(X) MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2))) ACTIVE(plus(N, 0)) -> MARK(U41(isNat(N), N)) MARK(plus(X1, X2)) -> MARK(X1) MARK(plus(X1, X2)) -> MARK(X2) MARK(U61(X)) -> ACTIVE(U61(mark(X))) ACTIVE(plus(N, s(M))) -> MARK(U51(isNat(M), M, N)) MARK(U61(X)) -> MARK(X) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) ACTIVE(x(N, 0)) -> MARK(U61(isNat(N))) MARK(U71(X1, X2, X3)) -> MARK(X1) MARK(U72(X1, X2, X3)) -> ACTIVE(U72(mark(X1), X2, X3)) ACTIVE(x(N, s(M))) -> MARK(U71(isNat(M), M, N)) MARK(U72(X1, X2, X3)) -> MARK(X1) MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), mark(X2))) MARK(x(X1, X2)) -> MARK(X1) MARK(x(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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (83) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U61(X)) -> ACTIVE(U61(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(U32(x_1)) = 0 POL(U41(x_1, x_2)) = 1 POL(U51(x_1, x_2, x_3)) = 1 POL(U52(x_1, x_2, x_3)) = 1 POL(U61(x_1)) = 0 POL(U71(x_1, x_2, x_3)) = 1 POL(U72(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 POL(x(x_1, x_2)) = 1 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) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) 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) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(mark(X1), X2) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U61(active(X)) -> U61(X) U61(mark(X)) -> U61(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) ---------------------------------------- (84) 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(U11(X1, X2)) -> MARK(X1) ACTIVE(U31(tt, V2)) -> MARK(U32(isNat(V2))) MARK(U12(X)) -> MARK(X) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U41(tt, N)) -> MARK(N) ACTIVE(U51(tt, M, N)) -> MARK(U52(isNat(N), M, N)) MARK(U21(X)) -> MARK(X) MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2)) ACTIVE(U52(tt, M, N)) -> MARK(s(plus(N, M))) MARK(U31(X1, X2)) -> MARK(X1) ACTIVE(U71(tt, M, N)) -> MARK(U72(isNat(N), M, N)) MARK(U32(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U72(tt, M, N)) -> MARK(plus(x(N, M), N)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U51(X1, X2, X3)) -> ACTIVE(U51(mark(X1), X2, X3)) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) MARK(U51(X1, X2, X3)) -> MARK(X1) MARK(U52(X1, X2, X3)) -> ACTIVE(U52(mark(X1), X2, X3)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U52(X1, X2, X3)) -> MARK(X1) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2)) MARK(s(X)) -> MARK(X) MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2))) ACTIVE(plus(N, 0)) -> MARK(U41(isNat(N), N)) MARK(plus(X1, X2)) -> MARK(X1) MARK(plus(X1, X2)) -> MARK(X2) ACTIVE(plus(N, s(M))) -> MARK(U51(isNat(M), M, N)) MARK(U61(X)) -> MARK(X) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) ACTIVE(x(N, 0)) -> MARK(U61(isNat(N))) MARK(U71(X1, X2, X3)) -> MARK(X1) MARK(U72(X1, X2, X3)) -> ACTIVE(U72(mark(X1), X2, X3)) ACTIVE(x(N, s(M))) -> MARK(U71(isNat(M), M, N)) MARK(U72(X1, X2, X3)) -> MARK(X1) MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), mark(X2))) MARK(x(X1, X2)) -> MARK(X1) MARK(x(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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (85) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U41(tt, N)) -> MARK(N) ACTIVE(U52(tt, M, N)) -> MARK(s(plus(N, M))) ACTIVE(U72(tt, M, N)) -> MARK(plus(x(N, M), N)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U51(X1, X2, X3)) -> MARK(X1) MARK(U52(X1, X2, X3)) -> MARK(X1) MARK(s(X)) -> MARK(X) ACTIVE(plus(N, 0)) -> MARK(U41(isNat(N), N)) MARK(plus(X1, X2)) -> MARK(X1) MARK(plus(X1, X2)) -> MARK(X2) ACTIVE(plus(N, s(M))) -> MARK(U51(isNat(M), M, N)) MARK(U61(X)) -> MARK(X) ACTIVE(x(N, 0)) -> MARK(U61(isNat(N))) MARK(U71(X1, X2, X3)) -> MARK(X1) ACTIVE(x(N, s(M))) -> MARK(U71(isNat(M), M, N)) MARK(U72(X1, X2, X3)) -> MARK(X1) MARK(x(X1, X2)) -> MARK(X1) MARK(x(X1, X2)) -> MARK(X2) The remaining pairs can at least be oriented weakly. Used ordering: Combined order from the following AFS and order. MARK(x1) = MARK(x1) U11(x1, x2) = x1 ACTIVE(x1) = ACTIVE(x1) mark(x1) = x1 tt = tt U12(x1) = x1 isNat(x1) = isNat U31(x1, x2) = x1 U32(x1) = x1 U41(x1, x2) = U41(x1, x2) U51(x1, x2, x3) = U51(x1, x2, x3) U52(x1, x2, x3) = U52(x1, x2, x3) U21(x1) = x1 s(x1) = s(x1) plus(x1, x2) = plus(x1, x2) U71(x1, x2, x3) = U71(x1, x2, x3) U72(x1, x2, x3) = U72(x1, x2, x3) x(x1, x2) = x(x1, x2) 0 = 0 U61(x1) = U61(x1) active(x1) = x1 Recursive path order with status [RPO]. Quasi-Precedence: [MARK_1, ACTIVE_1, U71_3, U72_3, x_2, 0, U61_1] > [U41_2, U51_3, U52_3, plus_2] > [tt, isNat, s_1] Status: MARK_1: [1] ACTIVE_1: [1] tt: multiset status isNat: multiset status U41_2: multiset status U51_3: multiset status U52_3: multiset status s_1: multiset status plus_2: multiset status U71_3: multiset status U72_3: multiset status x_2: multiset status 0: multiset status U61_1: multiset status The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) active(U11(tt, V2)) -> mark(U12(isNat(V2))) mark(U12(X)) -> active(U12(mark(X))) active(U31(tt, V2)) -> mark(U32(isNat(V2))) mark(isNat(X)) -> active(isNat(X)) active(U41(tt, N)) -> mark(N) mark(U21(X)) -> active(U21(mark(X))) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) mark(U32(X)) -> active(U32(mark(X))) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(s(X)) -> active(s(mark(X))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) active(plus(N, 0)) -> mark(U41(isNat(N), N)) mark(U61(X)) -> active(U61(mark(X))) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) active(x(N, 0)) -> mark(U61(isNat(N))) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) active(x(N, s(M))) -> mark(U71(isNat(M), M, N)) mark(x(X1, X2)) -> active(x(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) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(active(X)) -> U12(X) U12(mark(X)) -> U12(X) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) 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) s(active(X)) -> s(X) s(mark(X)) -> s(X) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(mark(X1), X2) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U61(active(X)) -> U61(X) U61(mark(X)) -> U61(X) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U32(tt)) -> mark(tt) active(U61(tt)) -> mark(0) active(isNat(0)) -> mark(tt) ---------------------------------------- (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(U11(X1, X2)) -> MARK(X1) ACTIVE(U31(tt, V2)) -> MARK(U32(isNat(V2))) MARK(U12(X)) -> MARK(X) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U51(tt, M, N)) -> MARK(U52(isNat(N), M, N)) MARK(U21(X)) -> MARK(X) MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2)) MARK(U31(X1, X2)) -> MARK(X1) ACTIVE(U71(tt, M, N)) -> MARK(U72(isNat(N), M, N)) MARK(U32(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U51(X1, X2, X3)) -> ACTIVE(U51(mark(X1), X2, X3)) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) MARK(U52(X1, X2, X3)) -> ACTIVE(U52(mark(X1), X2, X3)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2)) MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2))) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2, X3)) -> ACTIVE(U72(mark(X1), X2, X3)) MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), 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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (87) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U51(X1, X2, X3)) -> ACTIVE(U51(mark(X1), X2, X3)) MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2))) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), mark(X2))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, -2} POL( U11_2(x_1, x_2) ) = 2x_1 + 1 POL( U31_2(x_1, x_2) ) = 2x_1 + 1 POL( U41_2(x_1, x_2) ) = 2 POL( U51_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 2 POL( U52_3(x_1, ..., x_3) ) = max{0, -2} POL( U71_3(x_1, ..., x_3) ) = x_2 + x_3 + 2 POL( U72_3(x_1, ..., x_3) ) = 1 POL( plus_2(x_1, x_2) ) = 2 POL( x_2(x_1, x_2) ) = 2 POL( mark_1(x_1) ) = 2x_1 POL( active_1(x_1) ) = 2x_1 + 2 POL( tt ) = 0 POL( U12_1(x_1) ) = 2x_1 + 1 POL( isNat_1(x_1) ) = 0 POL( U32_1(x_1) ) = x_1 + 1 POL( U21_1(x_1) ) = 2x_1 + 1 POL( s_1(x_1) ) = max{0, -2} POL( 0 ) = 0 POL( U61_1(x_1) ) = x_1 + 2 POL( MARK_1(x_1) ) = max{0, 2x_1 - 2} 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) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(active(X)) -> U12(X) U12(mark(X)) -> U12(X) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) ---------------------------------------- (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(U11(X1, X2)) -> MARK(X1) ACTIVE(U31(tt, V2)) -> MARK(U32(isNat(V2))) MARK(U12(X)) -> MARK(X) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U51(tt, M, N)) -> MARK(U52(isNat(N), M, N)) MARK(U21(X)) -> MARK(X) MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2)) MARK(U31(X1, X2)) -> MARK(X1) ACTIVE(U71(tt, M, N)) -> MARK(U72(isNat(N), M, N)) MARK(U32(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) MARK(U52(X1, X2, X3)) -> ACTIVE(U52(mark(X1), X2, X3)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2)) MARK(U72(X1, X2, X3)) -> ACTIVE(U72(mark(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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (89) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U71(tt, M, N)) -> MARK(U72(isNat(N), M, N)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = x_1 + 2 POL( U11_2(x_1, x_2) ) = 0 POL( U31_2(x_1, x_2) ) = max{0, -2} POL( U52_3(x_1, ..., x_3) ) = max{0, -2} POL( U72_3(x_1, ..., x_3) ) = max{0, -2} POL( mark_1(x_1) ) = 2 POL( active_1(x_1) ) = max{0, x_1 - 2} POL( tt ) = 2 POL( U12_1(x_1) ) = 2 POL( isNat_1(x_1) ) = 0 POL( U32_1(x_1) ) = 2 POL( U41_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( U21_1(x_1) ) = 2 POL( U51_3(x_1, ..., x_3) ) = max{0, x_1 - 2} POL( s_1(x_1) ) = x_1 + 2 POL( plus_2(x_1, x_2) ) = max{0, -2} POL( U71_3(x_1, ..., x_3) ) = 2x_1 + 2 POL( x_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2} POL( 0 ) = 0 POL( U61_1(x_1) ) = max{0, -2} POL( MARK_1(x_1) ) = 2 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) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) ---------------------------------------- (90) 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(U11(X1, X2)) -> MARK(X1) ACTIVE(U31(tt, V2)) -> MARK(U32(isNat(V2))) MARK(U12(X)) -> MARK(X) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U51(tt, M, N)) -> MARK(U52(isNat(N), M, N)) MARK(U21(X)) -> MARK(X) MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2)) MARK(U31(X1, X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) MARK(U52(X1, X2, X3)) -> ACTIVE(U52(mark(X1), X2, X3)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2)) MARK(U72(X1, X2, X3)) -> ACTIVE(U72(mark(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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (91) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U51(tt, M, N)) -> MARK(U52(isNat(N), M, N)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = x_1 POL( U11_2(x_1, x_2) ) = max{0, -2} POL( U31_2(x_1, x_2) ) = max{0, -2} POL( U52_3(x_1, ..., x_3) ) = max{0, -2} POL( U72_3(x_1, ..., x_3) ) = 0 POL( mark_1(x_1) ) = 1 POL( active_1(x_1) ) = max{0, x_1 - 2} POL( tt ) = 1 POL( U12_1(x_1) ) = 0 POL( isNat_1(x_1) ) = 0 POL( U32_1(x_1) ) = 2 POL( U41_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 1} POL( U21_1(x_1) ) = max{0, -2} POL( U51_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 1 POL( s_1(x_1) ) = x_1 + 2 POL( plus_2(x_1, x_2) ) = x_2 + 2 POL( U71_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2} POL( x_2(x_1, x_2) ) = 2x_2 + 2 POL( 0 ) = 0 POL( U61_1(x_1) ) = max{0, -2} POL( MARK_1(x_1) ) = max{0, -2} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 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) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) ---------------------------------------- (92) 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(U11(X1, X2)) -> MARK(X1) ACTIVE(U31(tt, V2)) -> MARK(U32(isNat(V2))) MARK(U12(X)) -> MARK(X) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(U21(X)) -> MARK(X) MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2)) MARK(U31(X1, X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) MARK(U52(X1, X2, X3)) -> ACTIVE(U52(mark(X1), X2, X3)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2)) MARK(U72(X1, X2, X3)) -> ACTIVE(U72(mark(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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (93) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U52(X1, X2, X3)) -> ACTIVE(U52(mark(X1), X2, X3)) MARK(U72(X1, X2, X3)) -> ACTIVE(U72(mark(X1), X2, X3)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = x_1 POL( U11_2(x_1, x_2) ) = 2 POL( U31_2(x_1, x_2) ) = 2 POL( U52_3(x_1, ..., x_3) ) = 1 POL( U72_3(x_1, ..., x_3) ) = 1 POL( mark_1(x_1) ) = max{0, -2} POL( active_1(x_1) ) = max{0, x_1 - 2} POL( tt ) = 0 POL( U12_1(x_1) ) = 2 POL( isNat_1(x_1) ) = 2 POL( U32_1(x_1) ) = max{0, -2} POL( U41_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( U21_1(x_1) ) = max{0, -2} POL( U51_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( s_1(x_1) ) = max{0, 2x_1 - 2} POL( plus_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U71_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2} POL( x_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2} POL( 0 ) = 0 POL( U61_1(x_1) ) = 2 POL( MARK_1(x_1) ) = 2 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) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(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) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) ---------------------------------------- (94) 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(U11(X1, X2)) -> MARK(X1) ACTIVE(U31(tt, V2)) -> MARK(U32(isNat(V2))) MARK(U12(X)) -> MARK(X) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(U21(X)) -> MARK(X) MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2)) MARK(U31(X1, X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2)) 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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (95) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U31(tt, V2)) -> MARK(U32(isNat(V2))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(MARK(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U11(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(ACTIVE(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(mark(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(tt) = [[0A]] >>> <<< POL(U12(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(isNat(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U31(x_1, x_2)) = [[-I]] + [[1A]] * x_1 + [[5A]] * x_2 >>> <<< POL(U32(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U21(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(plus(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(s(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(x(x_1, x_2)) = [[-I]] + [[1A]] * x_1 + [[5A]] * x_2 >>> <<< POL(active(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U41(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U51(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U52(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U71(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[5A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U72(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[5A]] * x_2 + [[1A]] * x_3 >>> <<< POL(0) = [[5A]] >>> <<< POL(U61(x_1)) = [[5A]] + [[0A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) active(U11(tt, V2)) -> mark(U12(isNat(V2))) mark(U12(X)) -> active(U12(mark(X))) active(U31(tt, V2)) -> mark(U32(isNat(V2))) mark(isNat(X)) -> active(isNat(X)) active(U41(tt, N)) -> mark(N) mark(U21(X)) -> active(U21(mark(X))) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) mark(U32(X)) -> active(U32(mark(X))) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(s(X)) -> active(s(mark(X))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) active(plus(N, 0)) -> mark(U41(isNat(N), N)) mark(U61(X)) -> active(U61(mark(X))) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) active(x(N, 0)) -> mark(U61(isNat(N))) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) active(x(N, s(M))) -> mark(U71(isNat(M), M, N)) mark(x(X1, X2)) -> active(x(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) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(active(X)) -> U12(X) U12(mark(X)) -> U12(X) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(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) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U32(tt)) -> mark(tt) active(U61(tt)) -> mark(0) active(isNat(0)) -> mark(tt) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) s(active(X)) -> s(X) s(mark(X)) -> s(X) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(mark(X1), X2) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U61(active(X)) -> U61(X) U61(mark(X)) -> U61(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) ---------------------------------------- (96) 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(U11(X1, X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(U21(X)) -> MARK(X) MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2)) MARK(U31(X1, X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2)) 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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (97) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = x_1 POL( U11_2(x_1, x_2) ) = 2 POL( U31_2(x_1, x_2) ) = max{0, -2} POL( mark_1(x_1) ) = max{0, x_1 - 2} POL( active_1(x_1) ) = max{0, x_1 - 2} POL( tt ) = 2 POL( U12_1(x_1) ) = max{0, -2} POL( isNat_1(x_1) ) = 2 POL( U32_1(x_1) ) = 0 POL( U41_2(x_1, x_2) ) = 2 POL( U21_1(x_1) ) = 2x_1 POL( U51_3(x_1, ..., x_3) ) = 2 POL( U52_3(x_1, ..., x_3) ) = max{0, x_1 + x_3 - 2} POL( s_1(x_1) ) = max{0, x_1 - 2} POL( plus_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( U71_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 - 2} POL( U72_3(x_1, ..., x_3) ) = 2x_2 + 2 POL( x_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( 0 ) = 0 POL( U61_1(x_1) ) = max{0, -2} POL( MARK_1(x_1) ) = 2 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) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(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) ---------------------------------------- (98) 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(U11(X1, X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(U21(X)) -> MARK(X) MARK(U31(X1, X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2)) 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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (99) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U32(X)) -> MARK(X) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = 0 POL( U11_2(x_1, x_2) ) = x_1 + 1 POL( mark_1(x_1) ) = 2x_1 + 2 POL( active_1(x_1) ) = x_1 + 1 POL( tt ) = 1 POL( U12_1(x_1) ) = 2x_1 + 1 POL( isNat_1(x_1) ) = 0 POL( U31_2(x_1, x_2) ) = 2x_1 + 1 POL( U32_1(x_1) ) = x_1 + 2 POL( U41_2(x_1, x_2) ) = x_1 + x_2 + 1 POL( U21_1(x_1) ) = x_1 + 1 POL( U51_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( U52_3(x_1, ..., x_3) ) = max{0, 2x_1 - 1} POL( s_1(x_1) ) = x_1 POL( plus_2(x_1, x_2) ) = max{0, x_2 - 2} POL( U71_3(x_1, ..., x_3) ) = 2x_1 + x_3 + 2 POL( U72_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2} POL( x_2(x_1, x_2) ) = 2x_1 + 2 POL( 0 ) = 2 POL( U61_1(x_1) ) = 2 POL( MARK_1(x_1) ) = max{0, 2x_1 - 2} 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) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(active(X)) -> U12(X) U12(mark(X)) -> U12(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) ---------------------------------------- (100) 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(U11(X1, X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(U21(X)) -> MARK(X) MARK(U31(X1, X2)) -> MARK(X1) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2)) 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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (101) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U11(tt, V2)) -> MARK(U12(isNat(V2))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(MARK(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U11(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[4A]] * x_2 >>> <<< POL(ACTIVE(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(mark(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(tt) = [[4A]] >>> <<< POL(U12(x_1)) = [[-I]] + [[3A]] * x_1 >>> <<< POL(isNat(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U21(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U31(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(plus(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[4A]] * x_2 >>> <<< POL(s(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(x(x_1, x_2)) = [[0A]] + [[4A]] * x_1 + [[1A]] * x_2 >>> <<< POL(active(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U32(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U41(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U51(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[4A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U52(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[4A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U71(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[1A]] * x_2 + [[4A]] * x_3 >>> <<< POL(U72(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[1A]] * x_2 + [[4A]] * x_3 >>> <<< POL(0) = [[4A]] >>> <<< POL(U61(x_1)) = [[-I]] + [[4A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) active(U11(tt, V2)) -> mark(U12(isNat(V2))) mark(U12(X)) -> active(U12(mark(X))) active(U31(tt, V2)) -> mark(U32(isNat(V2))) mark(isNat(X)) -> active(isNat(X)) active(U41(tt, N)) -> mark(N) mark(U21(X)) -> active(U21(mark(X))) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) mark(U32(X)) -> active(U32(mark(X))) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(s(X)) -> active(s(mark(X))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) active(plus(N, 0)) -> mark(U41(isNat(N), N)) mark(U61(X)) -> active(U61(mark(X))) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) active(x(N, 0)) -> mark(U61(isNat(N))) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) active(x(N, s(M))) -> mark(U71(isNat(M), M, N)) mark(x(X1, X2)) -> active(x(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) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(active(X)) -> U12(X) U12(mark(X)) -> U12(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) active(U12(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U32(tt)) -> mark(tt) active(U61(tt)) -> mark(0) active(isNat(0)) -> mark(tt) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) s(active(X)) -> s(X) s(mark(X)) -> s(X) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(mark(X1), X2) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U61(active(X)) -> U61(X) U61(mark(X)) -> U61(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) ---------------------------------------- (102) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2)) MARK(U11(X1, X2)) -> MARK(X1) MARK(U12(X)) -> MARK(X) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(U21(X)) -> MARK(X) MARK(U31(X1, X2)) -> MARK(X1) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2)) 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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (103) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U12(X)) -> MARK(X) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, -2} POL( U11_2(x_1, x_2) ) = x_1 + 1 POL( mark_1(x_1) ) = 2x_1 + 1 POL( active_1(x_1) ) = x_1 + 2 POL( tt ) = 0 POL( U12_1(x_1) ) = x_1 + 2 POL( isNat_1(x_1) ) = 0 POL( U31_2(x_1, x_2) ) = x_1 + 1 POL( U32_1(x_1) ) = 2 POL( U41_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U21_1(x_1) ) = x_1 + 1 POL( U51_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 2 POL( U52_3(x_1, ..., x_3) ) = max{0, 2x_1 - 1} POL( s_1(x_1) ) = x_1 + 1 POL( plus_2(x_1, x_2) ) = max{0, -2} POL( U71_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + x_3 - 2} POL( U72_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( x_2(x_1, x_2) ) = 0 POL( 0 ) = 0 POL( U61_1(x_1) ) = max{0, x_1 - 2} POL( MARK_1(x_1) ) = max{0, x_1 - 1} 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) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(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) ---------------------------------------- (104) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2)) MARK(U11(X1, X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(U21(X)) -> MARK(X) MARK(U31(X1, X2)) -> MARK(X1) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2)) 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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (105) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2} POL( U11_2(x_1, x_2) ) = max{0, -2} POL( mark_1(x_1) ) = 2 POL( active_1(x_1) ) = max{0, -2} POL( tt ) = 0 POL( U12_1(x_1) ) = 0 POL( isNat_1(x_1) ) = 2 POL( U31_2(x_1, x_2) ) = max{0, -2} POL( U32_1(x_1) ) = max{0, 2x_1 - 2} POL( U41_2(x_1, x_2) ) = max{0, -2} POL( U21_1(x_1) ) = max{0, -2} POL( U51_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2 POL( U52_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 - 2} POL( s_1(x_1) ) = max{0, x_1 - 2} POL( plus_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2} POL( U71_3(x_1, ..., x_3) ) = 2 POL( U72_3(x_1, ..., x_3) ) = x_1 + x_3 + 1 POL( x_2(x_1, x_2) ) = max{0, x_1 - 2} POL( 0 ) = 0 POL( U61_1(x_1) ) = 2 POL( MARK_1(x_1) ) = 2 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) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) ---------------------------------------- (106) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U11(X1, X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(U21(X)) -> MARK(X) MARK(U31(X1, X2)) -> MARK(X1) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2)) 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, V2)) -> mark(U32(isNat(V2))) active(U32(tt)) -> mark(tt) active(U41(tt, N)) -> mark(N) active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N)) active(U52(tt, M, N)) -> mark(s(plus(N, M))) active(U61(tt)) -> mark(0) active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N)) active(U72(tt, M, N)) -> mark(plus(x(N, M), N)) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2)) active(plus(N, 0)) -> mark(U41(isNat(N), N)) active(plus(N, s(M))) -> mark(U51(isNat(M), M, N)) active(x(N, 0)) -> mark(U61(isNat(N))) active(x(N, s(M))) -> mark(U71(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(U32(X)) -> active(U32(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2, X3)) -> active(U52(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(U61(X)) -> active(U61(mark(X))) mark(0) -> active(0) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2, X3)) -> active(U72(mark(X1), X2, X3)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) 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) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, mark(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, mark(X3)) -> U52(X1, X2, X3) U52(active(X1), X2, X3) -> U52(X1, X2, X3) U52(X1, active(X2), X3) -> U52(X1, X2, X3) U52(X1, X2, active(X3)) -> U52(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) U61(mark(X)) -> U61(X) U61(active(X)) -> U61(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, mark(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, mark(X3)) -> U72(X1, X2, X3) U72(active(X1), X2, X3) -> U72(X1, X2, X3) U72(X1, active(X2), X3) -> U72(X1, X2, X3) U72(X1, X2, active(X3)) -> U72(X1, X2, X3) x(mark(X1), X2) -> x(X1, X2) x(X1, mark(X2)) -> x(X1, X2) x(active(X1), X2) -> x(X1, X2) x(X1, active(X2)) -> x(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (107) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (108) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U11(X1, X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(U21(X)) -> MARK(X) MARK(U31(X1, X2)) -> MARK(X1) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2)) The TRS R consists of the following rules: isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(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) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(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) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (109) UsableRulesReductionPairsProof (EQUIVALENT) By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. The following dependency pairs can be deleted: ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNat(V1), V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNat(V1), V2)) The following rules are removed from R: isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(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) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(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) Used ordering: POLO with Polynomial interpretation [POLO]: POL(ACTIVE(x_1)) = x_1 POL(MARK(x_1)) = 2*x_1 POL(U11(x_1, x_2)) = x_1 + x_2 POL(U21(x_1)) = x_1 POL(U31(x_1, x_2)) = x_1 + x_2 POL(active(x_1)) = 2*x_1 POL(isNat(x_1)) = 2*x_1 POL(mark(x_1)) = 2*x_1 POL(plus(x_1, x_2)) = 1 + 2*x_1 + x_2 POL(s(x_1)) = 2*x_1 POL(x(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 ---------------------------------------- (110) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U11(X1, X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(U21(X)) -> MARK(X) MARK(U31(X1, X2)) -> MARK(X1) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (111) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (112) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U21(X)) -> MARK(X) MARK(U11(X1, X2)) -> MARK(X1) MARK(U31(X1, X2)) -> MARK(X1) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (113) 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(U21(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(U31(X1, X2)) -> MARK(X1) The graph contains the following edges 1 > 1 ---------------------------------------- (114) YES