/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) DependencyPairsProof [EQUIVALENT, 310 ms] (2) QDP (3) DependencyGraphProof [EQUIVALENT, 2 ms] (4) AND (5) QDP (6) UsableRulesProof [EQUIVALENT, 0 ms] (7) QDP (8) QDPSizeChangeProof [EQUIVALENT, 1 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, 1 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) UsableRulesProof [EQUIVALENT, 0 ms] (82) QDP (83) QDPSizeChangeProof [EQUIVALENT, 0 ms] (84) YES (85) QDP (86) UsableRulesProof [EQUIVALENT, 0 ms] (87) QDP (88) QDPSizeChangeProof [EQUIVALENT, 0 ms] (89) YES (90) QDP (91) UsableRulesProof [EQUIVALENT, 0 ms] (92) QDP (93) QDPSizeChangeProof [EQUIVALENT, 0 ms] (94) YES (95) QDP (96) UsableRulesProof [EQUIVALENT, 0 ms] (97) QDP (98) QDPSizeChangeProof [EQUIVALENT, 0 ms] (99) YES (100) QDP (101) UsableRulesProof [EQUIVALENT, 0 ms] (102) QDP (103) QDPSizeChangeProof [EQUIVALENT, 0 ms] (104) YES (105) QDP (106) UsableRulesProof [EQUIVALENT, 0 ms] (107) QDP (108) QDPSizeChangeProof [EQUIVALENT, 0 ms] (109) YES (110) QDP (111) UsableRulesProof [EQUIVALENT, 0 ms] (112) QDP (113) QDPSizeChangeProof [EQUIVALENT, 0 ms] (114) YES (115) QDP (116) UsableRulesProof [EQUIVALENT, 0 ms] (117) QDP (118) QDPSizeChangeProof [EQUIVALENT, 0 ms] (119) YES (120) QDP (121) UsableRulesProof [EQUIVALENT, 0 ms] (122) QDP (123) QDPSizeChangeProof [EQUIVALENT, 0 ms] (124) YES (125) QDP (126) UsableRulesProof [EQUIVALENT, 0 ms] (127) QDP (128) QDPSizeChangeProof [EQUIVALENT, 0 ms] (129) YES (130) QDP (131) UsableRulesProof [EQUIVALENT, 0 ms] (132) QDP (133) QDPSizeChangeProof [EQUIVALENT, 0 ms] (134) YES (135) QDP (136) UsableRulesProof [EQUIVALENT, 0 ms] (137) QDP (138) QDPSizeChangeProof [EQUIVALENT, 0 ms] (139) YES (140) QDP (141) UsableRulesProof [EQUIVALENT, 0 ms] (142) QDP (143) QDPSizeChangeProof [EQUIVALENT, 0 ms] (144) YES (145) QDP (146) UsableRulesProof [EQUIVALENT, 0 ms] (147) QDP (148) QDPSizeChangeProof [EQUIVALENT, 0 ms] (149) YES (150) QDP (151) UsableRulesProof [EQUIVALENT, 0 ms] (152) QDP (153) QDPSizeChangeProof [EQUIVALENT, 0 ms] (154) YES (155) QDP (156) UsableRulesProof [EQUIVALENT, 0 ms] (157) QDP (158) QDPSizeChangeProof [EQUIVALENT, 0 ms] (159) YES (160) QDP (161) UsableRulesProof [EQUIVALENT, 0 ms] (162) QDP (163) QDPSizeChangeProof [EQUIVALENT, 0 ms] (164) YES (165) QDP (166) UsableRulesProof [EQUIVALENT, 0 ms] (167) QDP (168) QDPSizeChangeProof [EQUIVALENT, 0 ms] (169) YES (170) QDP (171) UsableRulesProof [EQUIVALENT, 0 ms] (172) QDP (173) QDPSizeChangeProof [EQUIVALENT, 0 ms] (174) YES (175) QDP (176) UsableRulesProof [EQUIVALENT, 0 ms] (177) QDP (178) QDPSizeChangeProof [EQUIVALENT, 0 ms] (179) YES (180) QDP (181) UsableRulesProof [EQUIVALENT, 0 ms] (182) QDP (183) QDPSizeChangeProof [EQUIVALENT, 0 ms] (184) YES (185) QDP (186) UsableRulesProof [EQUIVALENT, 1 ms] (187) QDP (188) QDPSizeChangeProof [EQUIVALENT, 0 ms] (189) YES (190) QDP (191) QDPOrderProof [EQUIVALENT, 1852 ms] (192) QDP (193) QDPOrderProof [EQUIVALENT, 3734 ms] (194) QDP (195) QDPOrderProof [EQUIVALENT, 1395 ms] (196) QDP (197) QDPOrderProof [EQUIVALENT, 1294 ms] (198) QDP (199) QDPOrderProof [EQUIVALENT, 1381 ms] (200) QDP (201) QDPOrderProof [EQUIVALENT, 1262 ms] (202) QDP (203) QDPOrderProof [EQUIVALENT, 1385 ms] (204) QDP (205) QDPOrderProof [EQUIVALENT, 579 ms] (206) QDP (207) QDPOrderProof [EQUIVALENT, 1637 ms] (208) QDP (209) QDPOrderProof [EQUIVALENT, 1058 ms] (210) QDP (211) QDPOrderProof [EQUIVALENT, 1179 ms] (212) QDP (213) QDPOrderProof [EQUIVALENT, 1053 ms] (214) QDP (215) QDPOrderProof [EQUIVALENT, 2548 ms] (216) QDP (217) QDPOrderProof [EQUIVALENT, 2946 ms] (218) QDP (219) QDPOrderProof [EQUIVALENT, 1538 ms] (220) QDP (221) QDPOrderProof [EQUIVALENT, 2834 ms] (222) QDP (223) QDPOrderProof [EQUIVALENT, 2262 ms] (224) QDP (225) QDPOrderProof [EQUIVALENT, 1124 ms] (226) QDP (227) QDPOrderProof [EQUIVALENT, 1094 ms] (228) QDP (229) QDPOrderProof [EQUIVALENT, 2315 ms] (230) QDP (231) QDPOrderProof [EQUIVALENT, 2548 ms] (232) QDP (233) QDPOrderProof [EQUIVALENT, 2254 ms] (234) QDP (235) QDPOrderProof [EQUIVALENT, 1028 ms] (236) QDP (237) QDPOrderProof [EQUIVALENT, 854 ms] (238) QDP (239) QDPOrderProof [EQUIVALENT, 1044 ms] (240) QDP (241) QDPOrderProof [EQUIVALENT, 1003 ms] (242) QDP (243) QDPOrderProof [EQUIVALENT, 1012 ms] (244) QDP (245) QDPOrderProof [EQUIVALENT, 842 ms] (246) QDP (247) QDPOrderProof [EQUIVALENT, 898 ms] (248) QDP (249) QDPOrderProof [EQUIVALENT, 1086 ms] (250) QDP (251) QDPOrderProof [EQUIVALENT, 2151 ms] (252) QDP (253) QDPOrderProof [EQUIVALENT, 1086 ms] (254) QDP (255) QDPOrderProof [EQUIVALENT, 813 ms] (256) QDP (257) QDPOrderProof [EQUIVALENT, 1053 ms] (258) QDP (259) QDPOrderProof [EQUIVALENT, 1984 ms] (260) QDP (261) QDPOrderProof [EQUIVALENT, 2330 ms] (262) QDP (263) QDPOrderProof [EQUIVALENT, 1978 ms] (264) QDP (265) QDPOrderProof [EQUIVALENT, 2062 ms] (266) QDP (267) QDPOrderProof [EQUIVALENT, 2088 ms] (268) QDP (269) QDPOrderProof [EQUIVALENT, 959 ms] (270) QDP (271) QDPOrderProof [EQUIVALENT, 979 ms] (272) QDP (273) QDPOrderProof [EQUIVALENT, 1879 ms] (274) QDP (275) QDPOrderProof [EQUIVALENT, 2313 ms] (276) QDP (277) QDPOrderProof [EQUIVALENT, 2214 ms] (278) QDP (279) QDPOrderProof [EQUIVALENT, 866 ms] (280) QDP (281) QDPOrderProof [EQUIVALENT, 1303 ms] (282) QDP (283) QDPOrderProof [EQUIVALENT, 841 ms] (284) QDP (285) QDPOrderProof [EQUIVALENT, 1442 ms] (286) QDP (287) QDPOrderProof [EQUIVALENT, 928 ms] (288) QDP (289) QDPOrderProof [EQUIVALENT, 909 ms] (290) QDP (291) QDPOrderProof [EQUIVALENT, 847 ms] (292) QDP (293) QDPOrderProof [EQUIVALENT, 861 ms] (294) QDP (295) QDPOrderProof [EQUIVALENT, 1322 ms] (296) QDP (297) QDPOrderProof [EQUIVALENT, 739 ms] (298) QDP (299) QDPOrderProof [EQUIVALENT, 446 ms] (300) QDP (301) QDPOrderProof [EQUIVALENT, 213 ms] (302) QDP (303) QDPOrderProof [EQUIVALENT, 197 ms] (304) QDP (305) QDPOrderProof [EQUIVALENT, 800 ms] (306) QDP (307) UsableRulesProof [EQUIVALENT, 0 ms] (308) QDP (309) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] (310) QDP (311) DependencyGraphProof [EQUIVALENT, 0 ms] (312) QDP (313) QDPSizeChangeProof [EQUIVALENT, 0 ms] (314) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) 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(U101(tt, M, N)) -> MARK(U102(isNatKind(M), M, N)) ACTIVE(U101(tt, M, N)) -> U102^1(isNatKind(M), M, N) ACTIVE(U101(tt, M, N)) -> ISNATKIND(M) ACTIVE(U102(tt, M, N)) -> MARK(U103(isNat(N), M, N)) ACTIVE(U102(tt, M, N)) -> U103^1(isNat(N), M, N) ACTIVE(U102(tt, M, N)) -> ISNAT(N) ACTIVE(U103(tt, M, N)) -> MARK(U104(isNatKind(N), M, N)) ACTIVE(U103(tt, M, N)) -> U104^1(isNatKind(N), M, N) ACTIVE(U103(tt, M, N)) -> ISNATKIND(N) ACTIVE(U104(tt, M, N)) -> MARK(plus(x(N, M), N)) ACTIVE(U104(tt, M, N)) -> PLUS(x(N, M), N) ACTIVE(U104(tt, M, N)) -> X(N, M) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U11(tt, V1, V2)) -> U12^1(isNatKind(V1), V1, V2) ACTIVE(U11(tt, V1, V2)) -> ISNATKIND(V1) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> U13^1(isNatKind(V2), V1, V2) ACTIVE(U12(tt, V1, V2)) -> ISNATKIND(V2) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> U14^1(isNatKind(V2), V1, V2) ACTIVE(U13(tt, V1, V2)) -> ISNATKIND(V2) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) ACTIVE(U14(tt, V1, V2)) -> U15^1(isNat(V1), V2) ACTIVE(U14(tt, V1, V2)) -> ISNAT(V1) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) ACTIVE(U15(tt, V2)) -> U16^1(isNat(V2)) ACTIVE(U15(tt, V2)) -> ISNAT(V2) ACTIVE(U16(tt)) -> MARK(tt) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) ACTIVE(U21(tt, V1)) -> U22^1(isNatKind(V1), V1) ACTIVE(U21(tt, V1)) -> ISNATKIND(V1) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) ACTIVE(U22(tt, V1)) -> U23^1(isNat(V1)) ACTIVE(U22(tt, V1)) -> ISNAT(V1) ACTIVE(U23(tt)) -> MARK(tt) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) ACTIVE(U31(tt, V1, V2)) -> U32^1(isNatKind(V1), V1, V2) ACTIVE(U31(tt, V1, V2)) -> ISNATKIND(V1) ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2)) ACTIVE(U32(tt, V1, V2)) -> U33^1(isNatKind(V2), V1, V2) ACTIVE(U32(tt, V1, V2)) -> ISNATKIND(V2) ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2)) ACTIVE(U33(tt, V1, V2)) -> U34^1(isNatKind(V2), V1, V2) ACTIVE(U33(tt, V1, V2)) -> ISNATKIND(V2) ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) ACTIVE(U34(tt, V1, V2)) -> U35^1(isNat(V1), V2) ACTIVE(U34(tt, V1, V2)) -> ISNAT(V1) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) ACTIVE(U35(tt, V2)) -> U36^1(isNat(V2)) ACTIVE(U35(tt, V2)) -> ISNAT(V2) ACTIVE(U36(tt)) -> MARK(tt) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) ACTIVE(U41(tt, V2)) -> U42^1(isNatKind(V2)) ACTIVE(U41(tt, V2)) -> ISNATKIND(V2) ACTIVE(U42(tt)) -> MARK(tt) ACTIVE(U51(tt)) -> MARK(tt) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2))) ACTIVE(U61(tt, V2)) -> U62^1(isNatKind(V2)) ACTIVE(U61(tt, V2)) -> ISNATKIND(V2) ACTIVE(U62(tt)) -> MARK(tt) ACTIVE(U71(tt, N)) -> MARK(U72(isNatKind(N), N)) ACTIVE(U71(tt, N)) -> U72^1(isNatKind(N), N) ACTIVE(U71(tt, N)) -> ISNATKIND(N) ACTIVE(U72(tt, N)) -> MARK(N) ACTIVE(U81(tt, M, N)) -> MARK(U82(isNatKind(M), M, N)) ACTIVE(U81(tt, M, N)) -> U82^1(isNatKind(M), M, N) ACTIVE(U81(tt, M, N)) -> ISNATKIND(M) ACTIVE(U82(tt, M, N)) -> MARK(U83(isNat(N), M, N)) ACTIVE(U82(tt, M, N)) -> U83^1(isNat(N), M, N) ACTIVE(U82(tt, M, N)) -> ISNAT(N) ACTIVE(U83(tt, M, N)) -> MARK(U84(isNatKind(N), M, N)) ACTIVE(U83(tt, M, N)) -> U84^1(isNatKind(N), M, N) ACTIVE(U83(tt, M, N)) -> ISNATKIND(N) ACTIVE(U84(tt, M, N)) -> MARK(s(plus(N, M))) ACTIVE(U84(tt, M, N)) -> S(plus(N, M)) ACTIVE(U84(tt, M, N)) -> PLUS(N, M) ACTIVE(U91(tt, N)) -> MARK(U92(isNatKind(N))) ACTIVE(U91(tt, N)) -> U92^1(isNatKind(N)) ACTIVE(U91(tt, N)) -> ISNATKIND(N) ACTIVE(U92(tt)) -> MARK(0) ACTIVE(isNat(0)) -> MARK(tt) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) ACTIVE(isNat(plus(V1, V2))) -> U11^1(isNatKind(V1), V1, V2) ACTIVE(isNat(plus(V1, V2))) -> ISNATKIND(V1) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNat(s(V1))) -> U21^1(isNatKind(V1), V1) ACTIVE(isNat(s(V1))) -> ISNATKIND(V1) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) ACTIVE(isNat(x(V1, V2))) -> U31^1(isNatKind(V1), V1, V2) ACTIVE(isNat(x(V1, V2))) -> ISNATKIND(V1) ACTIVE(isNatKind(0)) -> MARK(tt) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(plus(V1, V2))) -> U41^1(isNatKind(V1), V2) ACTIVE(isNatKind(plus(V1, V2))) -> ISNATKIND(V1) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(s(V1))) -> U51^1(isNatKind(V1)) ACTIVE(isNatKind(s(V1))) -> ISNATKIND(V1) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) ACTIVE(isNatKind(x(V1, V2))) -> U61^1(isNatKind(V1), V2) ACTIVE(isNatKind(x(V1, V2))) -> ISNATKIND(V1) ACTIVE(plus(N, 0)) -> MARK(U71(isNat(N), N)) ACTIVE(plus(N, 0)) -> U71^1(isNat(N), N) ACTIVE(plus(N, 0)) -> ISNAT(N) ACTIVE(plus(N, s(M))) -> MARK(U81(isNat(M), M, N)) ACTIVE(plus(N, s(M))) -> U81^1(isNat(M), M, N) ACTIVE(plus(N, s(M))) -> ISNAT(M) ACTIVE(x(N, 0)) -> MARK(U91(isNat(N), N)) ACTIVE(x(N, 0)) -> U91^1(isNat(N), N) ACTIVE(x(N, 0)) -> ISNAT(N) ACTIVE(x(N, s(M))) -> MARK(U101(isNat(M), M, N)) ACTIVE(x(N, s(M))) -> U101^1(isNat(M), M, N) ACTIVE(x(N, s(M))) -> ISNAT(M) MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3)) MARK(U101(X1, X2, X3)) -> U101^1(mark(X1), X2, X3) MARK(U101(X1, X2, X3)) -> MARK(X1) MARK(tt) -> ACTIVE(tt) MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3)) MARK(U102(X1, X2, X3)) -> U102^1(mark(X1), X2, X3) MARK(U102(X1, X2, X3)) -> MARK(X1) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3)) MARK(U103(X1, X2, X3)) -> U103^1(mark(X1), X2, X3) MARK(U103(X1, X2, X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3)) MARK(U104(X1, X2, X3)) -> U104^1(mark(X1), X2, X3) MARK(U104(X1, X2, X3)) -> MARK(X1) 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(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) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) MARK(U11(X1, X2, X3)) -> U11^1(mark(X1), X2, X3) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) MARK(U12(X1, X2, X3)) -> U12^1(mark(X1), X2, X3) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) MARK(U13(X1, X2, X3)) -> U13^1(mark(X1), X2, X3) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) MARK(U14(X1, X2, X3)) -> U14^1(mark(X1), X2, X3) MARK(U14(X1, X2, X3)) -> MARK(X1) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) MARK(U15(X1, X2)) -> U15^1(mark(X1), X2) MARK(U15(X1, X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> U16^1(mark(X)) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) MARK(U21(X1, X2)) -> U21^1(mark(X1), X2) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) MARK(U22(X1, X2)) -> U22^1(mark(X1), X2) MARK(U22(X1, X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> U23^1(mark(X)) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) MARK(U31(X1, X2, X3)) -> U31^1(mark(X1), X2, X3) MARK(U31(X1, X2, X3)) -> MARK(X1) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> U32^1(mark(X1), X2, X3) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3)) MARK(U33(X1, X2, X3)) -> U33^1(mark(X1), X2, X3) MARK(U33(X1, X2, X3)) -> MARK(X1) MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3)) MARK(U34(X1, X2, X3)) -> U34^1(mark(X1), X2, X3) MARK(U34(X1, X2, X3)) -> MARK(X1) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) MARK(U35(X1, X2)) -> U35^1(mark(X1), X2) MARK(U35(X1, X2)) -> MARK(X1) MARK(U36(X)) -> ACTIVE(U36(mark(X))) MARK(U36(X)) -> U36^1(mark(X)) MARK(U36(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(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> U42^1(mark(X)) MARK(U42(X)) -> MARK(X) MARK(U51(X)) -> ACTIVE(U51(mark(X))) MARK(U51(X)) -> U51^1(mark(X)) MARK(U51(X)) -> MARK(X) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) MARK(U61(X1, X2)) -> U61^1(mark(X1), X2) MARK(U61(X1, X2)) -> MARK(X1) MARK(U62(X)) -> ACTIVE(U62(mark(X))) MARK(U62(X)) -> U62^1(mark(X)) MARK(U62(X)) -> MARK(X) MARK(U71(X1, X2)) -> ACTIVE(U71(mark(X1), X2)) MARK(U71(X1, X2)) -> U71^1(mark(X1), X2) MARK(U71(X1, X2)) -> MARK(X1) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) MARK(U72(X1, X2)) -> U72^1(mark(X1), X2) MARK(U72(X1, X2)) -> MARK(X1) MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3)) MARK(U81(X1, X2, X3)) -> U81^1(mark(X1), X2, X3) MARK(U81(X1, X2, X3)) -> MARK(X1) MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3)) MARK(U82(X1, X2, X3)) -> U82^1(mark(X1), X2, X3) MARK(U82(X1, X2, X3)) -> MARK(X1) MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3)) MARK(U83(X1, X2, X3)) -> U83^1(mark(X1), X2, X3) MARK(U83(X1, X2, X3)) -> MARK(X1) MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3)) MARK(U84(X1, X2, X3)) -> U84^1(mark(X1), X2, X3) MARK(U84(X1, X2, X3)) -> MARK(X1) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> S(mark(X)) MARK(s(X)) -> MARK(X) MARK(U91(X1, X2)) -> ACTIVE(U91(mark(X1), X2)) MARK(U91(X1, X2)) -> U91^1(mark(X1), X2) MARK(U91(X1, X2)) -> MARK(X1) MARK(U92(X)) -> ACTIVE(U92(mark(X))) MARK(U92(X)) -> U92^1(mark(X)) MARK(U92(X)) -> MARK(X) MARK(0) -> ACTIVE(0) U101^1(mark(X1), X2, X3) -> U101^1(X1, X2, X3) U101^1(X1, mark(X2), X3) -> U101^1(X1, X2, X3) U101^1(X1, X2, mark(X3)) -> U101^1(X1, X2, X3) U101^1(active(X1), X2, X3) -> U101^1(X1, X2, X3) U101^1(X1, active(X2), X3) -> U101^1(X1, X2, X3) U101^1(X1, X2, active(X3)) -> U101^1(X1, X2, X3) U102^1(mark(X1), X2, X3) -> U102^1(X1, X2, X3) U102^1(X1, mark(X2), X3) -> U102^1(X1, X2, X3) U102^1(X1, X2, mark(X3)) -> U102^1(X1, X2, X3) U102^1(active(X1), X2, X3) -> U102^1(X1, X2, X3) U102^1(X1, active(X2), X3) -> U102^1(X1, X2, X3) U102^1(X1, X2, active(X3)) -> U102^1(X1, X2, X3) ISNATKIND(mark(X)) -> ISNATKIND(X) ISNATKIND(active(X)) -> ISNATKIND(X) U103^1(mark(X1), X2, X3) -> U103^1(X1, X2, X3) U103^1(X1, mark(X2), X3) -> U103^1(X1, X2, X3) U103^1(X1, X2, mark(X3)) -> U103^1(X1, X2, X3) U103^1(active(X1), X2, X3) -> U103^1(X1, X2, X3) U103^1(X1, active(X2), X3) -> U103^1(X1, X2, X3) U103^1(X1, X2, active(X3)) -> U103^1(X1, X2, X3) ISNAT(mark(X)) -> ISNAT(X) ISNAT(active(X)) -> ISNAT(X) U104^1(mark(X1), X2, X3) -> U104^1(X1, X2, X3) U104^1(X1, mark(X2), X3) -> U104^1(X1, X2, X3) U104^1(X1, X2, mark(X3)) -> U104^1(X1, X2, X3) U104^1(active(X1), X2, X3) -> U104^1(X1, X2, X3) U104^1(X1, active(X2), X3) -> U104^1(X1, X2, X3) U104^1(X1, X2, active(X3)) -> U104^1(X1, X2, X3) 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) 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) U11^1(mark(X1), X2, X3) -> U11^1(X1, X2, X3) U11^1(X1, mark(X2), X3) -> U11^1(X1, X2, X3) U11^1(X1, X2, mark(X3)) -> U11^1(X1, X2, X3) U11^1(active(X1), X2, X3) -> U11^1(X1, X2, X3) U11^1(X1, active(X2), X3) -> U11^1(X1, X2, X3) U11^1(X1, X2, active(X3)) -> U11^1(X1, X2, X3) U12^1(mark(X1), X2, X3) -> U12^1(X1, X2, X3) U12^1(X1, mark(X2), X3) -> U12^1(X1, X2, X3) U12^1(X1, X2, mark(X3)) -> U12^1(X1, X2, X3) U12^1(active(X1), X2, X3) -> U12^1(X1, X2, X3) U12^1(X1, active(X2), X3) -> U12^1(X1, X2, X3) U12^1(X1, X2, active(X3)) -> U12^1(X1, X2, X3) U13^1(mark(X1), X2, X3) -> U13^1(X1, X2, X3) U13^1(X1, mark(X2), X3) -> U13^1(X1, X2, X3) U13^1(X1, X2, mark(X3)) -> U13^1(X1, X2, X3) U13^1(active(X1), X2, X3) -> U13^1(X1, X2, X3) U13^1(X1, active(X2), X3) -> U13^1(X1, X2, X3) U13^1(X1, X2, active(X3)) -> U13^1(X1, X2, X3) U14^1(mark(X1), X2, X3) -> U14^1(X1, X2, X3) U14^1(X1, mark(X2), X3) -> U14^1(X1, X2, X3) U14^1(X1, X2, mark(X3)) -> U14^1(X1, X2, X3) U14^1(active(X1), X2, X3) -> U14^1(X1, X2, X3) U14^1(X1, active(X2), X3) -> U14^1(X1, X2, X3) U14^1(X1, X2, active(X3)) -> U14^1(X1, X2, X3) U15^1(mark(X1), X2) -> U15^1(X1, X2) U15^1(X1, mark(X2)) -> U15^1(X1, X2) U15^1(active(X1), X2) -> U15^1(X1, X2) U15^1(X1, active(X2)) -> U15^1(X1, X2) U16^1(mark(X)) -> U16^1(X) U16^1(active(X)) -> U16^1(X) U21^1(mark(X1), X2) -> U21^1(X1, X2) U21^1(X1, mark(X2)) -> U21^1(X1, X2) U21^1(active(X1), X2) -> U21^1(X1, X2) U21^1(X1, active(X2)) -> U21^1(X1, X2) U22^1(mark(X1), X2) -> U22^1(X1, X2) U22^1(X1, mark(X2)) -> U22^1(X1, X2) U22^1(active(X1), X2) -> U22^1(X1, X2) U22^1(X1, active(X2)) -> U22^1(X1, X2) U23^1(mark(X)) -> U23^1(X) U23^1(active(X)) -> U23^1(X) U31^1(mark(X1), X2, X3) -> U31^1(X1, X2, X3) U31^1(X1, mark(X2), X3) -> U31^1(X1, X2, X3) U31^1(X1, X2, mark(X3)) -> U31^1(X1, X2, X3) U31^1(active(X1), X2, X3) -> U31^1(X1, X2, X3) U31^1(X1, active(X2), X3) -> U31^1(X1, X2, X3) U31^1(X1, X2, active(X3)) -> U31^1(X1, X2, X3) U32^1(mark(X1), X2, X3) -> U32^1(X1, X2, X3) U32^1(X1, mark(X2), X3) -> U32^1(X1, X2, X3) U32^1(X1, X2, mark(X3)) -> U32^1(X1, X2, X3) U32^1(active(X1), X2, X3) -> U32^1(X1, X2, X3) U32^1(X1, active(X2), X3) -> U32^1(X1, X2, X3) U32^1(X1, X2, active(X3)) -> U32^1(X1, X2, X3) U33^1(mark(X1), X2, X3) -> U33^1(X1, X2, X3) U33^1(X1, mark(X2), X3) -> U33^1(X1, X2, X3) U33^1(X1, X2, mark(X3)) -> U33^1(X1, X2, X3) U33^1(active(X1), X2, X3) -> U33^1(X1, X2, X3) U33^1(X1, active(X2), X3) -> U33^1(X1, X2, X3) U33^1(X1, X2, active(X3)) -> U33^1(X1, X2, X3) U34^1(mark(X1), X2, X3) -> U34^1(X1, X2, X3) U34^1(X1, mark(X2), X3) -> U34^1(X1, X2, X3) U34^1(X1, X2, mark(X3)) -> U34^1(X1, X2, X3) U34^1(active(X1), X2, X3) -> U34^1(X1, X2, X3) U34^1(X1, active(X2), X3) -> U34^1(X1, X2, X3) U34^1(X1, X2, active(X3)) -> U34^1(X1, X2, X3) U35^1(mark(X1), X2) -> U35^1(X1, X2) U35^1(X1, mark(X2)) -> U35^1(X1, X2) U35^1(active(X1), X2) -> U35^1(X1, X2) U35^1(X1, active(X2)) -> U35^1(X1, X2) U36^1(mark(X)) -> U36^1(X) U36^1(active(X)) -> U36^1(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) U42^1(mark(X)) -> U42^1(X) U42^1(active(X)) -> U42^1(X) U51^1(mark(X)) -> U51^1(X) U51^1(active(X)) -> U51^1(X) U61^1(mark(X1), X2) -> U61^1(X1, X2) U61^1(X1, mark(X2)) -> U61^1(X1, X2) U61^1(active(X1), X2) -> U61^1(X1, X2) U61^1(X1, active(X2)) -> U61^1(X1, X2) U62^1(mark(X)) -> U62^1(X) U62^1(active(X)) -> U62^1(X) U71^1(mark(X1), X2) -> U71^1(X1, X2) U71^1(X1, mark(X2)) -> U71^1(X1, X2) U71^1(active(X1), X2) -> U71^1(X1, X2) U71^1(X1, active(X2)) -> U71^1(X1, X2) U72^1(mark(X1), X2) -> U72^1(X1, X2) U72^1(X1, mark(X2)) -> U72^1(X1, X2) U72^1(active(X1), X2) -> U72^1(X1, X2) U72^1(X1, active(X2)) -> U72^1(X1, X2) U81^1(mark(X1), X2, X3) -> U81^1(X1, X2, X3) U81^1(X1, mark(X2), X3) -> U81^1(X1, X2, X3) U81^1(X1, X2, mark(X3)) -> U81^1(X1, X2, X3) U81^1(active(X1), X2, X3) -> U81^1(X1, X2, X3) U81^1(X1, active(X2), X3) -> U81^1(X1, X2, X3) U81^1(X1, X2, active(X3)) -> U81^1(X1, X2, X3) U82^1(mark(X1), X2, X3) -> U82^1(X1, X2, X3) U82^1(X1, mark(X2), X3) -> U82^1(X1, X2, X3) U82^1(X1, X2, mark(X3)) -> U82^1(X1, X2, X3) U82^1(active(X1), X2, X3) -> U82^1(X1, X2, X3) U82^1(X1, active(X2), X3) -> U82^1(X1, X2, X3) U82^1(X1, X2, active(X3)) -> U82^1(X1, X2, X3) U83^1(mark(X1), X2, X3) -> U83^1(X1, X2, X3) U83^1(X1, mark(X2), X3) -> U83^1(X1, X2, X3) U83^1(X1, X2, mark(X3)) -> U83^1(X1, X2, X3) U83^1(active(X1), X2, X3) -> U83^1(X1, X2, X3) U83^1(X1, active(X2), X3) -> U83^1(X1, X2, X3) U83^1(X1, X2, active(X3)) -> U83^1(X1, X2, X3) U84^1(mark(X1), X2, X3) -> U84^1(X1, X2, X3) U84^1(X1, mark(X2), X3) -> U84^1(X1, X2, X3) U84^1(X1, X2, mark(X3)) -> U84^1(X1, X2, X3) U84^1(active(X1), X2, X3) -> U84^1(X1, X2, X3) U84^1(X1, active(X2), X3) -> U84^1(X1, X2, X3) U84^1(X1, X2, active(X3)) -> U84^1(X1, X2, X3) S(mark(X)) -> S(X) S(active(X)) -> S(X) U91^1(mark(X1), X2) -> U91^1(X1, X2) U91^1(X1, mark(X2)) -> U91^1(X1, X2) U91^1(active(X1), X2) -> U91^1(X1, X2) U91^1(X1, active(X2)) -> U91^1(X1, X2) U92^1(mark(X)) -> U92^1(X) U92^1(active(X)) -> U92^1(X) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) 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 38 SCCs with 114 less nodes. ---------------------------------------- (4) Complex Obligation (AND) ---------------------------------------- (5) Obligation: Q DP problem: The TRS P consists of the following rules: U92^1(active(X)) -> U92^1(X) U92^1(mark(X)) -> U92^1(X) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) 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: U92^1(active(X)) -> U92^1(X) U92^1(mark(X)) -> U92^1(X) 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: *U92^1(active(X)) -> U92^1(X) The graph contains the following edges 1 > 1 *U92^1(mark(X)) -> U92^1(X) The graph contains the following edges 1 > 1 ---------------------------------------- (9) YES ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: U91^1(X1, mark(X2)) -> U91^1(X1, X2) U91^1(mark(X1), X2) -> U91^1(X1, X2) U91^1(active(X1), X2) -> U91^1(X1, X2) U91^1(X1, active(X2)) -> U91^1(X1, X2) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) 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: U91^1(X1, mark(X2)) -> U91^1(X1, X2) U91^1(mark(X1), X2) -> U91^1(X1, X2) U91^1(active(X1), X2) -> U91^1(X1, X2) U91^1(X1, active(X2)) -> U91^1(X1, X2) 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: *U91^1(X1, mark(X2)) -> U91^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 *U91^1(mark(X1), X2) -> U91^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U91^1(active(X1), X2) -> U91^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U91^1(X1, active(X2)) -> U91^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (14) YES ---------------------------------------- (15) 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(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) 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: 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. ---------------------------------------- (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: *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 ---------------------------------------- (19) YES ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: U84^1(X1, mark(X2), X3) -> U84^1(X1, X2, X3) U84^1(mark(X1), X2, X3) -> U84^1(X1, X2, X3) U84^1(X1, X2, mark(X3)) -> U84^1(X1, X2, X3) U84^1(active(X1), X2, X3) -> U84^1(X1, X2, X3) U84^1(X1, active(X2), X3) -> U84^1(X1, X2, X3) U84^1(X1, X2, active(X3)) -> U84^1(X1, X2, X3) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) 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: U84^1(X1, mark(X2), X3) -> U84^1(X1, X2, X3) U84^1(mark(X1), X2, X3) -> U84^1(X1, X2, X3) U84^1(X1, X2, mark(X3)) -> U84^1(X1, X2, X3) U84^1(active(X1), X2, X3) -> U84^1(X1, X2, X3) U84^1(X1, active(X2), X3) -> U84^1(X1, X2, X3) U84^1(X1, X2, active(X3)) -> U84^1(X1, X2, X3) 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: *U84^1(X1, mark(X2), X3) -> U84^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U84^1(mark(X1), X2, X3) -> U84^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U84^1(X1, X2, mark(X3)) -> U84^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 *U84^1(active(X1), X2, X3) -> U84^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U84^1(X1, active(X2), X3) -> U84^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U84^1(X1, X2, active(X3)) -> U84^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 ---------------------------------------- (24) YES ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: U83^1(X1, mark(X2), X3) -> U83^1(X1, X2, X3) U83^1(mark(X1), X2, X3) -> U83^1(X1, X2, X3) U83^1(X1, X2, mark(X3)) -> U83^1(X1, X2, X3) U83^1(active(X1), X2, X3) -> U83^1(X1, X2, X3) U83^1(X1, active(X2), X3) -> U83^1(X1, X2, X3) U83^1(X1, X2, active(X3)) -> U83^1(X1, X2, X3) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) 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: U83^1(X1, mark(X2), X3) -> U83^1(X1, X2, X3) U83^1(mark(X1), X2, X3) -> U83^1(X1, X2, X3) U83^1(X1, X2, mark(X3)) -> U83^1(X1, X2, X3) U83^1(active(X1), X2, X3) -> U83^1(X1, X2, X3) U83^1(X1, active(X2), X3) -> U83^1(X1, X2, X3) U83^1(X1, X2, active(X3)) -> U83^1(X1, X2, X3) 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: *U83^1(X1, mark(X2), X3) -> U83^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U83^1(mark(X1), X2, X3) -> U83^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U83^1(X1, X2, mark(X3)) -> U83^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 *U83^1(active(X1), X2, X3) -> U83^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U83^1(X1, active(X2), X3) -> U83^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U83^1(X1, X2, active(X3)) -> U83^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 ---------------------------------------- (29) YES ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: U82^1(X1, mark(X2), X3) -> U82^1(X1, X2, X3) U82^1(mark(X1), X2, X3) -> U82^1(X1, X2, X3) U82^1(X1, X2, mark(X3)) -> U82^1(X1, X2, X3) U82^1(active(X1), X2, X3) -> U82^1(X1, X2, X3) U82^1(X1, active(X2), X3) -> U82^1(X1, X2, X3) U82^1(X1, X2, active(X3)) -> U82^1(X1, X2, X3) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) 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: U82^1(X1, mark(X2), X3) -> U82^1(X1, X2, X3) U82^1(mark(X1), X2, X3) -> U82^1(X1, X2, X3) U82^1(X1, X2, mark(X3)) -> U82^1(X1, X2, X3) U82^1(active(X1), X2, X3) -> U82^1(X1, X2, X3) U82^1(X1, active(X2), X3) -> U82^1(X1, X2, X3) U82^1(X1, X2, active(X3)) -> U82^1(X1, X2, X3) 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: *U82^1(X1, mark(X2), X3) -> U82^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U82^1(mark(X1), X2, X3) -> U82^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U82^1(X1, X2, mark(X3)) -> U82^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 *U82^1(active(X1), X2, X3) -> U82^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U82^1(X1, active(X2), X3) -> U82^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U82^1(X1, X2, active(X3)) -> U82^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 ---------------------------------------- (34) YES ---------------------------------------- (35) Obligation: Q DP problem: The TRS P consists of the following rules: U81^1(X1, mark(X2), X3) -> U81^1(X1, X2, X3) U81^1(mark(X1), X2, X3) -> U81^1(X1, X2, X3) U81^1(X1, X2, mark(X3)) -> U81^1(X1, X2, X3) U81^1(active(X1), X2, X3) -> U81^1(X1, X2, X3) U81^1(X1, active(X2), X3) -> U81^1(X1, X2, X3) U81^1(X1, X2, active(X3)) -> U81^1(X1, X2, X3) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) 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: U81^1(X1, mark(X2), X3) -> U81^1(X1, X2, X3) U81^1(mark(X1), X2, X3) -> U81^1(X1, X2, X3) U81^1(X1, X2, mark(X3)) -> U81^1(X1, X2, X3) U81^1(active(X1), X2, X3) -> U81^1(X1, X2, X3) U81^1(X1, active(X2), X3) -> U81^1(X1, X2, X3) U81^1(X1, X2, active(X3)) -> U81^1(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: *U81^1(X1, mark(X2), X3) -> U81^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U81^1(mark(X1), X2, X3) -> U81^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U81^1(X1, X2, mark(X3)) -> U81^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 *U81^1(active(X1), X2, X3) -> U81^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U81^1(X1, active(X2), X3) -> U81^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U81^1(X1, X2, active(X3)) -> U81^1(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: U72^1(X1, mark(X2)) -> U72^1(X1, X2) U72^1(mark(X1), X2) -> U72^1(X1, X2) U72^1(active(X1), X2) -> U72^1(X1, X2) U72^1(X1, active(X2)) -> U72^1(X1, X2) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) 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: U72^1(X1, mark(X2)) -> U72^1(X1, X2) U72^1(mark(X1), X2) -> U72^1(X1, X2) U72^1(active(X1), X2) -> U72^1(X1, X2) U72^1(X1, active(X2)) -> U72^1(X1, X2) 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: *U72^1(X1, mark(X2)) -> U72^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 *U72^1(mark(X1), X2) -> U72^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U72^1(active(X1), X2) -> U72^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U72^1(X1, active(X2)) -> U72^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (44) YES ---------------------------------------- (45) Obligation: Q DP problem: The TRS P consists of the following rules: U71^1(X1, mark(X2)) -> U71^1(X1, X2) U71^1(mark(X1), X2) -> U71^1(X1, X2) U71^1(active(X1), X2) -> U71^1(X1, X2) U71^1(X1, active(X2)) -> U71^1(X1, X2) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) 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: U71^1(X1, mark(X2)) -> U71^1(X1, X2) U71^1(mark(X1), X2) -> U71^1(X1, X2) U71^1(active(X1), X2) -> U71^1(X1, X2) U71^1(X1, active(X2)) -> U71^1(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: *U71^1(X1, mark(X2)) -> U71^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 *U71^1(mark(X1), X2) -> U71^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U71^1(active(X1), X2) -> U71^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U71^1(X1, active(X2)) -> U71^1(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: U62^1(active(X)) -> U62^1(X) U62^1(mark(X)) -> U62^1(X) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) 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: U62^1(active(X)) -> U62^1(X) U62^1(mark(X)) -> U62^1(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: *U62^1(active(X)) -> U62^1(X) The graph contains the following edges 1 > 1 *U62^1(mark(X)) -> U62^1(X) The graph contains the following edges 1 > 1 ---------------------------------------- (54) YES ---------------------------------------- (55) Obligation: Q DP problem: The TRS P consists of the following rules: U61^1(X1, mark(X2)) -> U61^1(X1, X2) U61^1(mark(X1), X2) -> U61^1(X1, X2) U61^1(active(X1), X2) -> U61^1(X1, X2) U61^1(X1, active(X2)) -> U61^1(X1, X2) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) 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: U61^1(X1, mark(X2)) -> U61^1(X1, X2) U61^1(mark(X1), X2) -> U61^1(X1, X2) U61^1(active(X1), X2) -> U61^1(X1, X2) U61^1(X1, active(X2)) -> U61^1(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: *U61^1(X1, mark(X2)) -> U61^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 *U61^1(mark(X1), X2) -> U61^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U61^1(active(X1), X2) -> U61^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U61^1(X1, active(X2)) -> U61^1(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: U51^1(active(X)) -> U51^1(X) U51^1(mark(X)) -> U51^1(X) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) 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: U51^1(active(X)) -> U51^1(X) U51^1(mark(X)) -> U51^1(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: *U51^1(active(X)) -> U51^1(X) The graph contains the following edges 1 > 1 *U51^1(mark(X)) -> U51^1(X) The graph contains the following edges 1 > 1 ---------------------------------------- (64) YES ---------------------------------------- (65) Obligation: Q DP problem: The TRS P consists of the following rules: U42^1(active(X)) -> U42^1(X) U42^1(mark(X)) -> U42^1(X) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) 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: U42^1(active(X)) -> U42^1(X) U42^1(mark(X)) -> U42^1(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: *U42^1(active(X)) -> U42^1(X) The graph contains the following edges 1 > 1 *U42^1(mark(X)) -> U42^1(X) The graph contains the following edges 1 > 1 ---------------------------------------- (69) YES ---------------------------------------- (70) 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(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) 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: 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. ---------------------------------------- (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: *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 ---------------------------------------- (74) YES ---------------------------------------- (75) Obligation: Q DP problem: The TRS P consists of the following rules: U36^1(active(X)) -> U36^1(X) U36^1(mark(X)) -> U36^1(X) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) 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: U36^1(active(X)) -> U36^1(X) U36^1(mark(X)) -> U36^1(X) 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: *U36^1(active(X)) -> U36^1(X) The graph contains the following edges 1 > 1 *U36^1(mark(X)) -> U36^1(X) The graph contains the following edges 1 > 1 ---------------------------------------- (79) YES ---------------------------------------- (80) Obligation: Q DP problem: The TRS P consists of the following rules: U35^1(X1, mark(X2)) -> U35^1(X1, X2) U35^1(mark(X1), X2) -> U35^1(X1, X2) U35^1(active(X1), X2) -> U35^1(X1, X2) U35^1(X1, active(X2)) -> U35^1(X1, X2) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (81) 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. ---------------------------------------- (82) Obligation: Q DP problem: The TRS P consists of the following rules: U35^1(X1, mark(X2)) -> U35^1(X1, X2) U35^1(mark(X1), X2) -> U35^1(X1, X2) U35^1(active(X1), X2) -> U35^1(X1, X2) U35^1(X1, active(X2)) -> U35^1(X1, X2) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (83) 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: *U35^1(X1, mark(X2)) -> U35^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 *U35^1(mark(X1), X2) -> U35^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U35^1(active(X1), X2) -> U35^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U35^1(X1, active(X2)) -> U35^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (84) YES ---------------------------------------- (85) Obligation: Q DP problem: The TRS P consists of the following rules: U34^1(X1, mark(X2), X3) -> U34^1(X1, X2, X3) U34^1(mark(X1), X2, X3) -> U34^1(X1, X2, X3) U34^1(X1, X2, mark(X3)) -> U34^1(X1, X2, X3) U34^1(active(X1), X2, X3) -> U34^1(X1, X2, X3) U34^1(X1, active(X2), X3) -> U34^1(X1, X2, X3) U34^1(X1, X2, active(X3)) -> U34^1(X1, X2, X3) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (86) 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. ---------------------------------------- (87) Obligation: Q DP problem: The TRS P consists of the following rules: U34^1(X1, mark(X2), X3) -> U34^1(X1, X2, X3) U34^1(mark(X1), X2, X3) -> U34^1(X1, X2, X3) U34^1(X1, X2, mark(X3)) -> U34^1(X1, X2, X3) U34^1(active(X1), X2, X3) -> U34^1(X1, X2, X3) U34^1(X1, active(X2), X3) -> U34^1(X1, X2, X3) U34^1(X1, X2, active(X3)) -> U34^1(X1, X2, X3) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (88) 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: *U34^1(X1, mark(X2), X3) -> U34^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U34^1(mark(X1), X2, X3) -> U34^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U34^1(X1, X2, mark(X3)) -> U34^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 *U34^1(active(X1), X2, X3) -> U34^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U34^1(X1, active(X2), X3) -> U34^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U34^1(X1, X2, active(X3)) -> U34^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 ---------------------------------------- (89) YES ---------------------------------------- (90) Obligation: Q DP problem: The TRS P consists of the following rules: U33^1(X1, mark(X2), X3) -> U33^1(X1, X2, X3) U33^1(mark(X1), X2, X3) -> U33^1(X1, X2, X3) U33^1(X1, X2, mark(X3)) -> U33^1(X1, X2, X3) U33^1(active(X1), X2, X3) -> U33^1(X1, X2, X3) U33^1(X1, active(X2), X3) -> U33^1(X1, X2, X3) U33^1(X1, X2, active(X3)) -> U33^1(X1, X2, X3) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (91) 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. ---------------------------------------- (92) Obligation: Q DP problem: The TRS P consists of the following rules: U33^1(X1, mark(X2), X3) -> U33^1(X1, X2, X3) U33^1(mark(X1), X2, X3) -> U33^1(X1, X2, X3) U33^1(X1, X2, mark(X3)) -> U33^1(X1, X2, X3) U33^1(active(X1), X2, X3) -> U33^1(X1, X2, X3) U33^1(X1, active(X2), X3) -> U33^1(X1, X2, X3) U33^1(X1, X2, active(X3)) -> U33^1(X1, X2, X3) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (93) 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: *U33^1(X1, mark(X2), X3) -> U33^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U33^1(mark(X1), X2, X3) -> U33^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U33^1(X1, X2, mark(X3)) -> U33^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 *U33^1(active(X1), X2, X3) -> U33^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U33^1(X1, active(X2), X3) -> U33^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U33^1(X1, X2, active(X3)) -> U33^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 ---------------------------------------- (94) YES ---------------------------------------- (95) Obligation: Q DP problem: The TRS P consists of the following rules: U32^1(X1, mark(X2), X3) -> U32^1(X1, X2, X3) U32^1(mark(X1), X2, X3) -> U32^1(X1, X2, X3) U32^1(X1, X2, mark(X3)) -> U32^1(X1, X2, X3) U32^1(active(X1), X2, X3) -> U32^1(X1, X2, X3) U32^1(X1, active(X2), X3) -> U32^1(X1, X2, X3) U32^1(X1, X2, active(X3)) -> U32^1(X1, X2, X3) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (96) 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. ---------------------------------------- (97) Obligation: Q DP problem: The TRS P consists of the following rules: U32^1(X1, mark(X2), X3) -> U32^1(X1, X2, X3) U32^1(mark(X1), X2, X3) -> U32^1(X1, X2, X3) U32^1(X1, X2, mark(X3)) -> U32^1(X1, X2, X3) U32^1(active(X1), X2, X3) -> U32^1(X1, X2, X3) U32^1(X1, active(X2), X3) -> U32^1(X1, X2, X3) U32^1(X1, X2, active(X3)) -> U32^1(X1, X2, X3) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (98) 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(X1, mark(X2), X3) -> U32^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U32^1(mark(X1), X2, X3) -> U32^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U32^1(X1, X2, mark(X3)) -> U32^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 *U32^1(active(X1), X2, X3) -> U32^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U32^1(X1, active(X2), X3) -> U32^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U32^1(X1, X2, active(X3)) -> U32^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 ---------------------------------------- (99) YES ---------------------------------------- (100) Obligation: Q DP problem: The TRS P consists of the following rules: U31^1(X1, mark(X2), X3) -> U31^1(X1, X2, X3) U31^1(mark(X1), X2, X3) -> U31^1(X1, X2, X3) U31^1(X1, X2, mark(X3)) -> U31^1(X1, X2, X3) U31^1(active(X1), X2, X3) -> U31^1(X1, X2, X3) U31^1(X1, active(X2), X3) -> U31^1(X1, X2, X3) U31^1(X1, X2, active(X3)) -> U31^1(X1, X2, X3) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (101) 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. ---------------------------------------- (102) Obligation: Q DP problem: The TRS P consists of the following rules: U31^1(X1, mark(X2), X3) -> U31^1(X1, X2, X3) U31^1(mark(X1), X2, X3) -> U31^1(X1, X2, X3) U31^1(X1, X2, mark(X3)) -> U31^1(X1, X2, X3) U31^1(active(X1), X2, X3) -> U31^1(X1, X2, X3) U31^1(X1, active(X2), X3) -> U31^1(X1, X2, X3) U31^1(X1, X2, active(X3)) -> U31^1(X1, X2, X3) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (103) 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), X3) -> U31^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U31^1(mark(X1), X2, X3) -> U31^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U31^1(X1, X2, mark(X3)) -> U31^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 *U31^1(active(X1), X2, X3) -> U31^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U31^1(X1, active(X2), X3) -> U31^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U31^1(X1, X2, active(X3)) -> U31^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 ---------------------------------------- (104) YES ---------------------------------------- (105) Obligation: Q DP problem: The TRS P consists of the following rules: U23^1(active(X)) -> U23^1(X) U23^1(mark(X)) -> U23^1(X) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (106) 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. ---------------------------------------- (107) Obligation: Q DP problem: The TRS P consists of the following rules: U23^1(active(X)) -> U23^1(X) U23^1(mark(X)) -> U23^1(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (108) 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: *U23^1(active(X)) -> U23^1(X) The graph contains the following edges 1 > 1 *U23^1(mark(X)) -> U23^1(X) The graph contains the following edges 1 > 1 ---------------------------------------- (109) YES ---------------------------------------- (110) Obligation: Q DP problem: The TRS P consists of the following rules: U22^1(X1, mark(X2)) -> U22^1(X1, X2) U22^1(mark(X1), X2) -> U22^1(X1, X2) U22^1(active(X1), X2) -> U22^1(X1, X2) U22^1(X1, active(X2)) -> U22^1(X1, X2) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (111) 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. ---------------------------------------- (112) Obligation: Q DP problem: The TRS P consists of the following rules: U22^1(X1, mark(X2)) -> U22^1(X1, X2) U22^1(mark(X1), X2) -> U22^1(X1, X2) U22^1(active(X1), X2) -> U22^1(X1, X2) U22^1(X1, active(X2)) -> U22^1(X1, X2) 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: *U22^1(X1, mark(X2)) -> U22^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 *U22^1(mark(X1), X2) -> U22^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U22^1(active(X1), X2) -> U22^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U22^1(X1, active(X2)) -> U22^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (114) YES ---------------------------------------- (115) Obligation: Q DP problem: The TRS P consists of the following rules: U21^1(X1, mark(X2)) -> U21^1(X1, X2) U21^1(mark(X1), X2) -> U21^1(X1, X2) U21^1(active(X1), X2) -> U21^1(X1, X2) U21^1(X1, active(X2)) -> U21^1(X1, X2) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (116) 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. ---------------------------------------- (117) Obligation: Q DP problem: The TRS P consists of the following rules: U21^1(X1, mark(X2)) -> U21^1(X1, X2) U21^1(mark(X1), X2) -> U21^1(X1, X2) U21^1(active(X1), X2) -> U21^1(X1, X2) U21^1(X1, active(X2)) -> U21^1(X1, X2) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (118) 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(X1, mark(X2)) -> U21^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 *U21^1(mark(X1), X2) -> U21^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U21^1(active(X1), X2) -> U21^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U21^1(X1, active(X2)) -> U21^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (119) YES ---------------------------------------- (120) Obligation: Q DP problem: The TRS P consists of the following rules: U16^1(active(X)) -> U16^1(X) U16^1(mark(X)) -> U16^1(X) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (121) 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. ---------------------------------------- (122) Obligation: Q DP problem: The TRS P consists of the following rules: U16^1(active(X)) -> U16^1(X) U16^1(mark(X)) -> U16^1(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (123) 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: *U16^1(active(X)) -> U16^1(X) The graph contains the following edges 1 > 1 *U16^1(mark(X)) -> U16^1(X) The graph contains the following edges 1 > 1 ---------------------------------------- (124) YES ---------------------------------------- (125) Obligation: Q DP problem: The TRS P consists of the following rules: U15^1(X1, mark(X2)) -> U15^1(X1, X2) U15^1(mark(X1), X2) -> U15^1(X1, X2) U15^1(active(X1), X2) -> U15^1(X1, X2) U15^1(X1, active(X2)) -> U15^1(X1, X2) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (126) 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. ---------------------------------------- (127) Obligation: Q DP problem: The TRS P consists of the following rules: U15^1(X1, mark(X2)) -> U15^1(X1, X2) U15^1(mark(X1), X2) -> U15^1(X1, X2) U15^1(active(X1), X2) -> U15^1(X1, X2) U15^1(X1, active(X2)) -> U15^1(X1, X2) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (128) 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: *U15^1(X1, mark(X2)) -> U15^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 *U15^1(mark(X1), X2) -> U15^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U15^1(active(X1), X2) -> U15^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U15^1(X1, active(X2)) -> U15^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (129) YES ---------------------------------------- (130) Obligation: Q DP problem: The TRS P consists of the following rules: U14^1(X1, mark(X2), X3) -> U14^1(X1, X2, X3) U14^1(mark(X1), X2, X3) -> U14^1(X1, X2, X3) U14^1(X1, X2, mark(X3)) -> U14^1(X1, X2, X3) U14^1(active(X1), X2, X3) -> U14^1(X1, X2, X3) U14^1(X1, active(X2), X3) -> U14^1(X1, X2, X3) U14^1(X1, X2, active(X3)) -> U14^1(X1, X2, X3) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (131) 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. ---------------------------------------- (132) Obligation: Q DP problem: The TRS P consists of the following rules: U14^1(X1, mark(X2), X3) -> U14^1(X1, X2, X3) U14^1(mark(X1), X2, X3) -> U14^1(X1, X2, X3) U14^1(X1, X2, mark(X3)) -> U14^1(X1, X2, X3) U14^1(active(X1), X2, X3) -> U14^1(X1, X2, X3) U14^1(X1, active(X2), X3) -> U14^1(X1, X2, X3) U14^1(X1, X2, active(X3)) -> U14^1(X1, X2, X3) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (133) 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: *U14^1(X1, mark(X2), X3) -> U14^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U14^1(mark(X1), X2, X3) -> U14^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U14^1(X1, X2, mark(X3)) -> U14^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 *U14^1(active(X1), X2, X3) -> U14^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U14^1(X1, active(X2), X3) -> U14^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U14^1(X1, X2, active(X3)) -> U14^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 ---------------------------------------- (134) YES ---------------------------------------- (135) Obligation: Q DP problem: The TRS P consists of the following rules: U13^1(X1, mark(X2), X3) -> U13^1(X1, X2, X3) U13^1(mark(X1), X2, X3) -> U13^1(X1, X2, X3) U13^1(X1, X2, mark(X3)) -> U13^1(X1, X2, X3) U13^1(active(X1), X2, X3) -> U13^1(X1, X2, X3) U13^1(X1, active(X2), X3) -> U13^1(X1, X2, X3) U13^1(X1, X2, active(X3)) -> U13^1(X1, X2, X3) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (136) 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. ---------------------------------------- (137) Obligation: Q DP problem: The TRS P consists of the following rules: U13^1(X1, mark(X2), X3) -> U13^1(X1, X2, X3) U13^1(mark(X1), X2, X3) -> U13^1(X1, X2, X3) U13^1(X1, X2, mark(X3)) -> U13^1(X1, X2, X3) U13^1(active(X1), X2, X3) -> U13^1(X1, X2, X3) U13^1(X1, active(X2), X3) -> U13^1(X1, X2, X3) U13^1(X1, X2, active(X3)) -> U13^1(X1, X2, X3) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (138) 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: *U13^1(X1, mark(X2), X3) -> U13^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U13^1(mark(X1), X2, X3) -> U13^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U13^1(X1, X2, mark(X3)) -> U13^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 *U13^1(active(X1), X2, X3) -> U13^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U13^1(X1, active(X2), X3) -> U13^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U13^1(X1, X2, active(X3)) -> U13^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 ---------------------------------------- (139) YES ---------------------------------------- (140) Obligation: Q DP problem: The TRS P consists of the following rules: U12^1(X1, mark(X2), X3) -> U12^1(X1, X2, X3) U12^1(mark(X1), X2, X3) -> U12^1(X1, X2, X3) U12^1(X1, X2, mark(X3)) -> U12^1(X1, X2, X3) U12^1(active(X1), X2, X3) -> U12^1(X1, X2, X3) U12^1(X1, active(X2), X3) -> U12^1(X1, X2, X3) U12^1(X1, X2, active(X3)) -> U12^1(X1, X2, X3) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (141) 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. ---------------------------------------- (142) Obligation: Q DP problem: The TRS P consists of the following rules: U12^1(X1, mark(X2), X3) -> U12^1(X1, X2, X3) U12^1(mark(X1), X2, X3) -> U12^1(X1, X2, X3) U12^1(X1, X2, mark(X3)) -> U12^1(X1, X2, X3) U12^1(active(X1), X2, X3) -> U12^1(X1, X2, X3) U12^1(X1, active(X2), X3) -> U12^1(X1, X2, X3) U12^1(X1, X2, active(X3)) -> U12^1(X1, X2, X3) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (143) 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(X1, mark(X2), X3) -> U12^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U12^1(mark(X1), X2, X3) -> U12^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U12^1(X1, X2, mark(X3)) -> U12^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 *U12^1(active(X1), X2, X3) -> U12^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U12^1(X1, active(X2), X3) -> U12^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U12^1(X1, X2, active(X3)) -> U12^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 ---------------------------------------- (144) YES ---------------------------------------- (145) Obligation: Q DP problem: The TRS P consists of the following rules: U11^1(X1, mark(X2), X3) -> U11^1(X1, X2, X3) U11^1(mark(X1), X2, X3) -> U11^1(X1, X2, X3) U11^1(X1, X2, mark(X3)) -> U11^1(X1, X2, X3) U11^1(active(X1), X2, X3) -> U11^1(X1, X2, X3) U11^1(X1, active(X2), X3) -> U11^1(X1, X2, X3) U11^1(X1, X2, active(X3)) -> U11^1(X1, X2, X3) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (146) 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. ---------------------------------------- (147) Obligation: Q DP problem: The TRS P consists of the following rules: U11^1(X1, mark(X2), X3) -> U11^1(X1, X2, X3) U11^1(mark(X1), X2, X3) -> U11^1(X1, X2, X3) U11^1(X1, X2, mark(X3)) -> U11^1(X1, X2, X3) U11^1(active(X1), X2, X3) -> U11^1(X1, X2, X3) U11^1(X1, active(X2), X3) -> U11^1(X1, X2, X3) U11^1(X1, X2, active(X3)) -> U11^1(X1, X2, X3) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (148) 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), X3) -> U11^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U11^1(mark(X1), X2, X3) -> U11^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U11^1(X1, X2, mark(X3)) -> U11^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 *U11^1(active(X1), X2, X3) -> U11^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U11^1(X1, active(X2), X3) -> U11^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U11^1(X1, X2, active(X3)) -> U11^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 ---------------------------------------- (149) YES ---------------------------------------- (150) 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(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (151) 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. ---------------------------------------- (152) 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. ---------------------------------------- (153) 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 ---------------------------------------- (154) YES ---------------------------------------- (155) 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(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (156) 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. ---------------------------------------- (157) 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. ---------------------------------------- (158) 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 ---------------------------------------- (159) YES ---------------------------------------- (160) Obligation: Q DP problem: The TRS P consists of the following rules: U104^1(X1, mark(X2), X3) -> U104^1(X1, X2, X3) U104^1(mark(X1), X2, X3) -> U104^1(X1, X2, X3) U104^1(X1, X2, mark(X3)) -> U104^1(X1, X2, X3) U104^1(active(X1), X2, X3) -> U104^1(X1, X2, X3) U104^1(X1, active(X2), X3) -> U104^1(X1, X2, X3) U104^1(X1, X2, active(X3)) -> U104^1(X1, X2, X3) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (161) 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. ---------------------------------------- (162) Obligation: Q DP problem: The TRS P consists of the following rules: U104^1(X1, mark(X2), X3) -> U104^1(X1, X2, X3) U104^1(mark(X1), X2, X3) -> U104^1(X1, X2, X3) U104^1(X1, X2, mark(X3)) -> U104^1(X1, X2, X3) U104^1(active(X1), X2, X3) -> U104^1(X1, X2, X3) U104^1(X1, active(X2), X3) -> U104^1(X1, X2, X3) U104^1(X1, X2, active(X3)) -> U104^1(X1, X2, X3) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (163) 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: *U104^1(X1, mark(X2), X3) -> U104^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U104^1(mark(X1), X2, X3) -> U104^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U104^1(X1, X2, mark(X3)) -> U104^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 *U104^1(active(X1), X2, X3) -> U104^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U104^1(X1, active(X2), X3) -> U104^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U104^1(X1, X2, active(X3)) -> U104^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 ---------------------------------------- (164) YES ---------------------------------------- (165) 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(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (166) 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. ---------------------------------------- (167) 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. ---------------------------------------- (168) 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 ---------------------------------------- (169) YES ---------------------------------------- (170) Obligation: Q DP problem: The TRS P consists of the following rules: U103^1(X1, mark(X2), X3) -> U103^1(X1, X2, X3) U103^1(mark(X1), X2, X3) -> U103^1(X1, X2, X3) U103^1(X1, X2, mark(X3)) -> U103^1(X1, X2, X3) U103^1(active(X1), X2, X3) -> U103^1(X1, X2, X3) U103^1(X1, active(X2), X3) -> U103^1(X1, X2, X3) U103^1(X1, X2, active(X3)) -> U103^1(X1, X2, X3) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (171) 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. ---------------------------------------- (172) Obligation: Q DP problem: The TRS P consists of the following rules: U103^1(X1, mark(X2), X3) -> U103^1(X1, X2, X3) U103^1(mark(X1), X2, X3) -> U103^1(X1, X2, X3) U103^1(X1, X2, mark(X3)) -> U103^1(X1, X2, X3) U103^1(active(X1), X2, X3) -> U103^1(X1, X2, X3) U103^1(X1, active(X2), X3) -> U103^1(X1, X2, X3) U103^1(X1, X2, active(X3)) -> U103^1(X1, X2, X3) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (173) 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: *U103^1(X1, mark(X2), X3) -> U103^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U103^1(mark(X1), X2, X3) -> U103^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U103^1(X1, X2, mark(X3)) -> U103^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 *U103^1(active(X1), X2, X3) -> U103^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U103^1(X1, active(X2), X3) -> U103^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U103^1(X1, X2, active(X3)) -> U103^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 ---------------------------------------- (174) YES ---------------------------------------- (175) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATKIND(active(X)) -> ISNATKIND(X) ISNATKIND(mark(X)) -> ISNATKIND(X) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (176) 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. ---------------------------------------- (177) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATKIND(active(X)) -> ISNATKIND(X) ISNATKIND(mark(X)) -> ISNATKIND(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (178) 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: *ISNATKIND(active(X)) -> ISNATKIND(X) The graph contains the following edges 1 > 1 *ISNATKIND(mark(X)) -> ISNATKIND(X) The graph contains the following edges 1 > 1 ---------------------------------------- (179) YES ---------------------------------------- (180) Obligation: Q DP problem: The TRS P consists of the following rules: U102^1(X1, mark(X2), X3) -> U102^1(X1, X2, X3) U102^1(mark(X1), X2, X3) -> U102^1(X1, X2, X3) U102^1(X1, X2, mark(X3)) -> U102^1(X1, X2, X3) U102^1(active(X1), X2, X3) -> U102^1(X1, X2, X3) U102^1(X1, active(X2), X3) -> U102^1(X1, X2, X3) U102^1(X1, X2, active(X3)) -> U102^1(X1, X2, X3) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (181) 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. ---------------------------------------- (182) Obligation: Q DP problem: The TRS P consists of the following rules: U102^1(X1, mark(X2), X3) -> U102^1(X1, X2, X3) U102^1(mark(X1), X2, X3) -> U102^1(X1, X2, X3) U102^1(X1, X2, mark(X3)) -> U102^1(X1, X2, X3) U102^1(active(X1), X2, X3) -> U102^1(X1, X2, X3) U102^1(X1, active(X2), X3) -> U102^1(X1, X2, X3) U102^1(X1, X2, active(X3)) -> U102^1(X1, X2, X3) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (183) 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: *U102^1(X1, mark(X2), X3) -> U102^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U102^1(mark(X1), X2, X3) -> U102^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U102^1(X1, X2, mark(X3)) -> U102^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 *U102^1(active(X1), X2, X3) -> U102^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U102^1(X1, active(X2), X3) -> U102^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U102^1(X1, X2, active(X3)) -> U102^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 ---------------------------------------- (184) YES ---------------------------------------- (185) Obligation: Q DP problem: The TRS P consists of the following rules: U101^1(X1, mark(X2), X3) -> U101^1(X1, X2, X3) U101^1(mark(X1), X2, X3) -> U101^1(X1, X2, X3) U101^1(X1, X2, mark(X3)) -> U101^1(X1, X2, X3) U101^1(active(X1), X2, X3) -> U101^1(X1, X2, X3) U101^1(X1, active(X2), X3) -> U101^1(X1, X2, X3) U101^1(X1, X2, active(X3)) -> U101^1(X1, X2, X3) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (186) 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. ---------------------------------------- (187) Obligation: Q DP problem: The TRS P consists of the following rules: U101^1(X1, mark(X2), X3) -> U101^1(X1, X2, X3) U101^1(mark(X1), X2, X3) -> U101^1(X1, X2, X3) U101^1(X1, X2, mark(X3)) -> U101^1(X1, X2, X3) U101^1(active(X1), X2, X3) -> U101^1(X1, X2, X3) U101^1(X1, active(X2), X3) -> U101^1(X1, X2, X3) U101^1(X1, X2, active(X3)) -> U101^1(X1, X2, X3) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (188) 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: *U101^1(X1, mark(X2), X3) -> U101^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U101^1(mark(X1), X2, X3) -> U101^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U101^1(X1, X2, mark(X3)) -> U101^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 *U101^1(active(X1), X2, X3) -> U101^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U101^1(X1, active(X2), X3) -> U101^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U101^1(X1, X2, active(X3)) -> U101^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 ---------------------------------------- (189) YES ---------------------------------------- (190) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3)) ACTIVE(U101(tt, M, N)) -> MARK(U102(isNatKind(M), M, N)) MARK(U101(X1, X2, X3)) -> MARK(X1) MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3)) ACTIVE(U102(tt, M, N)) -> MARK(U103(isNat(N), M, N)) MARK(U102(X1, X2, X3)) -> MARK(X1) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) ACTIVE(U103(tt, M, N)) -> MARK(U104(isNatKind(N), M, N)) MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3)) ACTIVE(U104(tt, M, N)) -> MARK(plus(x(N, M), N)) MARK(U103(X1, X2, X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) MARK(U104(X1, X2, X3)) -> MARK(X1) MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2))) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) MARK(plus(X1, X2)) -> MARK(X1) MARK(plus(X1, X2)) -> MARK(X2) MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), mark(X2))) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(x(X1, X2)) -> MARK(X1) MARK(x(X1, X2)) -> MARK(X2) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U14(X1, X2, X3)) -> MARK(X1) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2)) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2))) MARK(U31(X1, X2, X3)) -> MARK(X1) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) ACTIVE(U71(tt, N)) -> MARK(U72(isNatKind(N), N)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3)) ACTIVE(U72(tt, N)) -> MARK(N) MARK(U33(X1, X2, X3)) -> MARK(X1) MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3)) ACTIVE(U81(tt, M, N)) -> MARK(U82(isNatKind(M), M, N)) MARK(U34(X1, X2, X3)) -> MARK(X1) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) ACTIVE(U82(tt, M, N)) -> MARK(U83(isNat(N), M, N)) MARK(U35(X1, X2)) -> MARK(X1) MARK(U36(X)) -> ACTIVE(U36(mark(X))) ACTIVE(U83(tt, M, N)) -> MARK(U84(isNatKind(N), M, N)) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U84(tt, M, N)) -> MARK(s(plus(N, M))) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) ACTIVE(U91(tt, N)) -> MARK(U92(isNatKind(N))) MARK(U42(X)) -> MARK(X) MARK(U51(X)) -> ACTIVE(U51(mark(X))) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) MARK(U61(X1, X2)) -> MARK(X1) MARK(U62(X)) -> ACTIVE(U62(mark(X))) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) MARK(U62(X)) -> MARK(X) MARK(U71(X1, X2)) -> ACTIVE(U71(mark(X1), X2)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) MARK(U71(X1, X2)) -> MARK(X1) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) MARK(U72(X1, X2)) -> MARK(X1) MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3)) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) MARK(U81(X1, X2, X3)) -> MARK(X1) MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3)) ACTIVE(plus(N, 0)) -> MARK(U71(isNat(N), N)) MARK(U82(X1, X2, X3)) -> MARK(X1) MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3)) ACTIVE(plus(N, s(M))) -> MARK(U81(isNat(M), M, N)) MARK(U83(X1, X2, X3)) -> MARK(X1) MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3)) ACTIVE(x(N, 0)) -> MARK(U91(isNat(N), N)) MARK(U84(X1, X2, X3)) -> MARK(X1) MARK(s(X)) -> ACTIVE(s(mark(X))) ACTIVE(x(N, s(M))) -> MARK(U101(isNat(M), M, N)) MARK(s(X)) -> MARK(X) MARK(U91(X1, X2)) -> ACTIVE(U91(mark(X1), X2)) MARK(U91(X1, X2)) -> MARK(X1) MARK(U92(X)) -> ACTIVE(U92(mark(X))) MARK(U92(X)) -> MARK(X) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (191) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U36(X)) -> ACTIVE(U36(mark(X))) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U51(X)) -> ACTIVE(U51(mark(X))) MARK(U62(X)) -> ACTIVE(U62(mark(X))) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(U92(X)) -> ACTIVE(U92(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( U101_3(x_1, ..., x_3) ) = 2 POL( U102_3(x_1, ..., x_3) ) = 2 POL( U103_3(x_1, ..., x_3) ) = 2 POL( U104_3(x_1, ..., x_3) ) = 2 POL( U11_3(x_1, ..., x_3) ) = 2 POL( U12_3(x_1, ..., x_3) ) = 2 POL( U13_3(x_1, ..., x_3) ) = 2 POL( U14_3(x_1, ..., x_3) ) = 2 POL( U15_2(x_1, x_2) ) = 2 POL( U16_1(x_1) ) = 0 POL( U21_2(x_1, x_2) ) = 2 POL( U22_2(x_1, x_2) ) = 2 POL( U23_1(x_1) ) = max{0, -2} POL( U31_3(x_1, ..., x_3) ) = 2 POL( U32_3(x_1, ..., x_3) ) = 2 POL( U33_3(x_1, ..., x_3) ) = 2 POL( U34_3(x_1, ..., x_3) ) = 2 POL( U35_2(x_1, x_2) ) = 2 POL( U36_1(x_1) ) = max{0, -2} POL( U41_2(x_1, x_2) ) = 2 POL( U42_1(x_1) ) = max{0, -2} POL( U51_1(x_1) ) = max{0, -2} POL( U61_2(x_1, x_2) ) = 2 POL( U62_1(x_1) ) = max{0, -2} POL( U71_2(x_1, x_2) ) = 2 POL( U72_2(x_1, x_2) ) = 2 POL( U81_3(x_1, ..., x_3) ) = 2 POL( U82_3(x_1, ..., x_3) ) = 2 POL( U83_3(x_1, ..., x_3) ) = 2 POL( U84_3(x_1, ..., x_3) ) = 2 POL( U91_2(x_1, x_2) ) = 2 POL( U92_1(x_1) ) = max{0, -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) ) = 2 POL( active_1(x_1) ) = max{0, -2} POL( tt ) = 0 POL( isNatKind_1(x_1) ) = 2 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: U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) plus(X1, mark(X2)) -> plus(X1, X2) plus(mark(X1), X2) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(active(X)) -> U36(X) U36(mark(X)) -> U36(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(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) s(active(X)) -> s(X) s(mark(X)) -> s(X) U92(active(X)) -> U92(X) U92(mark(X)) -> U92(X) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(mark(X1), X2) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U91(X1, mark(X2)) -> U91(X1, X2) U91(mark(X1), X2) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) ---------------------------------------- (192) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3)) ACTIVE(U101(tt, M, N)) -> MARK(U102(isNatKind(M), M, N)) MARK(U101(X1, X2, X3)) -> MARK(X1) MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3)) ACTIVE(U102(tt, M, N)) -> MARK(U103(isNat(N), M, N)) MARK(U102(X1, X2, X3)) -> MARK(X1) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) ACTIVE(U103(tt, M, N)) -> MARK(U104(isNatKind(N), M, N)) MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3)) ACTIVE(U104(tt, M, N)) -> MARK(plus(x(N, M), N)) MARK(U103(X1, X2, X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) MARK(U104(X1, X2, X3)) -> MARK(X1) MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2))) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) MARK(plus(X1, X2)) -> MARK(X1) MARK(plus(X1, X2)) -> MARK(X2) MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), mark(X2))) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(x(X1, X2)) -> MARK(X1) MARK(x(X1, X2)) -> MARK(X2) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U14(X1, X2, X3)) -> MARK(X1) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2)) MARK(U15(X1, X2)) -> MARK(X1) ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2)) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2))) MARK(U31(X1, X2, X3)) -> MARK(X1) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) ACTIVE(U71(tt, N)) -> MARK(U72(isNatKind(N), N)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3)) ACTIVE(U72(tt, N)) -> MARK(N) MARK(U33(X1, X2, X3)) -> MARK(X1) MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3)) ACTIVE(U81(tt, M, N)) -> MARK(U82(isNatKind(M), M, N)) MARK(U34(X1, X2, X3)) -> MARK(X1) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) ACTIVE(U82(tt, M, N)) -> MARK(U83(isNat(N), M, N)) MARK(U35(X1, X2)) -> MARK(X1) ACTIVE(U83(tt, M, N)) -> MARK(U84(isNatKind(N), M, N)) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U84(tt, M, N)) -> MARK(s(plus(N, M))) MARK(U41(X1, X2)) -> MARK(X1) ACTIVE(U91(tt, N)) -> MARK(U92(isNatKind(N))) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) MARK(U61(X1, X2)) -> MARK(X1) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) MARK(U62(X)) -> MARK(X) MARK(U71(X1, X2)) -> ACTIVE(U71(mark(X1), X2)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) MARK(U71(X1, X2)) -> MARK(X1) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) MARK(U72(X1, X2)) -> MARK(X1) MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3)) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) MARK(U81(X1, X2, X3)) -> MARK(X1) MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3)) ACTIVE(plus(N, 0)) -> MARK(U71(isNat(N), N)) MARK(U82(X1, X2, X3)) -> MARK(X1) MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3)) ACTIVE(plus(N, s(M))) -> MARK(U81(isNat(M), M, N)) MARK(U83(X1, X2, X3)) -> MARK(X1) MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3)) ACTIVE(x(N, 0)) -> MARK(U91(isNat(N), N)) MARK(U84(X1, X2, X3)) -> MARK(X1) ACTIVE(x(N, s(M))) -> MARK(U101(isNat(M), M, N)) MARK(s(X)) -> MARK(X) MARK(U91(X1, X2)) -> ACTIVE(U91(mark(X1), X2)) MARK(U91(X1, X2)) -> MARK(X1) MARK(U92(X)) -> MARK(X) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (193) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U101(X1, X2, X3)) -> MARK(X1) MARK(U102(X1, X2, X3)) -> MARK(X1) ACTIVE(U104(tt, M, N)) -> MARK(plus(x(N, M), N)) MARK(U103(X1, X2, X3)) -> MARK(X1) MARK(U104(X1, X2, X3)) -> MARK(X1) MARK(plus(X1, X2)) -> MARK(X1) MARK(plus(X1, X2)) -> MARK(X2) MARK(x(X1, X2)) -> MARK(X1) MARK(x(X1, X2)) -> MARK(X2) ACTIVE(U71(tt, N)) -> MARK(U72(isNatKind(N), N)) ACTIVE(U72(tt, N)) -> MARK(N) ACTIVE(U84(tt, M, N)) -> MARK(s(plus(N, M))) ACTIVE(U91(tt, N)) -> MARK(U92(isNatKind(N))) MARK(U71(X1, X2)) -> MARK(X1) MARK(U72(X1, X2)) -> MARK(X1) MARK(U81(X1, X2, X3)) -> MARK(X1) ACTIVE(plus(N, 0)) -> MARK(U71(isNat(N), N)) MARK(U82(X1, X2, X3)) -> MARK(X1) ACTIVE(plus(N, s(M))) -> MARK(U81(isNat(M), M, N)) MARK(U83(X1, X2, X3)) -> MARK(X1) ACTIVE(x(N, 0)) -> MARK(U91(isNat(N), N)) MARK(U84(X1, X2, X3)) -> MARK(X1) ACTIVE(x(N, s(M))) -> MARK(U101(isNat(M), M, N)) MARK(s(X)) -> MARK(X) MARK(U91(X1, X2)) -> MARK(X1) The remaining pairs can at least be oriented weakly. Used ordering: Combined order from the following AFS and order. MARK(x1) = x1 U101(x1, x2, x3) = U101(x1, x2, x3) ACTIVE(x1) = x1 mark(x1) = x1 tt = tt U102(x1, x2, x3) = U102(x1, x2, x3) isNatKind(x1) = isNatKind U103(x1, x2, x3) = U103(x1, x2, x3) isNat(x1) = isNat U104(x1, x2, x3) = U104(x1, x2, x3) plus(x1, x2) = plus(x1, x2) x(x1, x2) = x(x1, x2) U11(x1, x2, x3) = x1 U12(x1, x2, x3) = x1 U13(x1, x2, x3) = x1 U14(x1, x2, x3) = x1 U15(x1, x2) = x1 U16(x1) = x1 U21(x1, x2) = x1 U22(x1, x2) = x1 U23(x1) = x1 U31(x1, x2, x3) = x1 U32(x1, x2, x3) = x1 U33(x1, x2, x3) = x1 U34(x1, x2, x3) = x1 U35(x1, x2) = x1 U36(x1) = x1 U41(x1, x2) = x1 U42(x1) = x1 U61(x1, x2) = x1 U62(x1) = x1 U71(x1, x2) = U71(x1, x2) U72(x1, x2) = U72(x1, x2) U81(x1, x2, x3) = U81(x1, x2, x3) U82(x1, x2, x3) = U82(x1, x2, x3) U83(x1, x2, x3) = U83(x1, x2, x3) U84(x1, x2, x3) = U84(x1, x2, x3) s(x1) = s(x1) U91(x1, x2) = U91(x1, x2) U92(x1) = x1 U51(x1) = x1 0 = 0 active(x1) = x1 Recursive path order with status [RPO]. Quasi-Precedence: [U101_3, U102_3, U103_3, U104_3, x_2] > [plus_2, U81_3, U82_3, U83_3, U84_3] > U71_2 > [tt, isNatKind, isNat] > U72_2 [U101_3, U102_3, U103_3, U104_3, x_2] > [plus_2, U81_3, U82_3, U83_3, U84_3] > U71_2 > [tt, isNatKind, isNat] > s_1 [U101_3, U102_3, U103_3, U104_3, x_2] > [plus_2, U81_3, U82_3, U83_3, U84_3] > U71_2 > [tt, isNatKind, isNat] > 0 [U101_3, U102_3, U103_3, U104_3, x_2] > U91_2 > [tt, isNatKind, isNat] > U72_2 [U101_3, U102_3, U103_3, U104_3, x_2] > U91_2 > [tt, isNatKind, isNat] > s_1 [U101_3, U102_3, U103_3, U104_3, x_2] > U91_2 > [tt, isNatKind, isNat] > 0 Status: U101_3: [2,3,1] tt: multiset status U102_3: [2,3,1] isNatKind: multiset status U103_3: [2,3,1] isNat: multiset status U104_3: [2,3,1] plus_2: [2,1] x_2: [2,1] U71_2: [1,2] U72_2: multiset status U81_3: [2,3,1] U82_3: [2,3,1] U83_3: [2,3,1] U84_3: [2,3,1] s_1: [1] U91_2: [1,2] 0: multiset status The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) mark(isNatKind(X)) -> active(isNatKind(X)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) mark(isNat(X)) -> active(isNat(X)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) mark(U16(X)) -> active(U16(mark(X))) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) mark(U23(X)) -> active(U23(mark(X))) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) active(U72(tt, N)) -> mark(N) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) mark(U36(X)) -> active(U36(mark(X))) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) mark(U42(X)) -> active(U42(mark(X))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) mark(U51(X)) -> active(U51(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) mark(U62(X)) -> active(U62(mark(X))) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) active(x(N, 0)) -> mark(U91(isNat(N), N)) mark(s(X)) -> active(s(mark(X))) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(tt) -> active(tt) mark(0) -> active(0) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) plus(X1, mark(X2)) -> plus(X1, X2) plus(mark(X1), X2) -> plus(X1, X2) plus(active(X1), X2) -> plus(X1, X2) plus(X1, active(X2)) -> plus(X1, X2) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(active(X)) -> U36(X) U36(mark(X)) -> U36(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(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) s(active(X)) -> s(X) s(mark(X)) -> s(X) U92(active(X)) -> U92(X) U92(mark(X)) -> U92(X) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(mark(X1), X2) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U91(X1, mark(X2)) -> U91(X1, X2) U91(mark(X1), X2) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) active(U16(tt)) -> mark(tt) active(U23(tt)) -> mark(tt) active(U36(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U62(tt)) -> mark(tt) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNatKind(0)) -> mark(tt) ---------------------------------------- (194) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3)) ACTIVE(U101(tt, M, N)) -> MARK(U102(isNatKind(M), M, N)) MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3)) ACTIVE(U102(tt, M, N)) -> MARK(U103(isNat(N), M, N)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) ACTIVE(U103(tt, M, N)) -> MARK(U104(isNatKind(N), M, N)) MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2))) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), mark(X2))) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U14(X1, X2, X3)) -> MARK(X1) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2)) MARK(U15(X1, X2)) -> MARK(X1) ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2)) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2))) MARK(U31(X1, X2, X3)) -> MARK(X1) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3)) MARK(U33(X1, X2, X3)) -> MARK(X1) MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3)) ACTIVE(U81(tt, M, N)) -> MARK(U82(isNatKind(M), M, N)) MARK(U34(X1, X2, X3)) -> MARK(X1) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) ACTIVE(U82(tt, M, N)) -> MARK(U83(isNat(N), M, N)) MARK(U35(X1, X2)) -> MARK(X1) ACTIVE(U83(tt, M, N)) -> MARK(U84(isNatKind(N), M, N)) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) MARK(U61(X1, X2)) -> MARK(X1) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) MARK(U62(X)) -> MARK(X) MARK(U71(X1, X2)) -> ACTIVE(U71(mark(X1), X2)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3)) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3)) MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3)) MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3)) MARK(U91(X1, X2)) -> ACTIVE(U91(mark(X1), X2)) MARK(U92(X)) -> MARK(X) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (195) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2))) MARK(x(X1, X2)) -> ACTIVE(x(mark(X1), mark(X2))) MARK(U71(X1, X2)) -> ACTIVE(U71(mark(X1), X2)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) MARK(U84(X1, X2, X3)) -> ACTIVE(U84(mark(X1), X2, X3)) MARK(U91(X1, X2)) -> ACTIVE(U91(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, x_1 - 1} POL( U101_3(x_1, ..., x_3) ) = 2 POL( U102_3(x_1, ..., x_3) ) = 2 POL( U103_3(x_1, ..., x_3) ) = 2 POL( U104_3(x_1, ..., x_3) ) = 2 POL( U11_3(x_1, ..., x_3) ) = 2 POL( U12_3(x_1, ..., x_3) ) = 2 POL( U13_3(x_1, ..., x_3) ) = 2 POL( U14_3(x_1, ..., x_3) ) = 2 POL( U15_2(x_1, x_2) ) = 2 POL( U21_2(x_1, x_2) ) = 2 POL( U22_2(x_1, x_2) ) = 2 POL( U31_3(x_1, ..., x_3) ) = 2 POL( U32_3(x_1, ..., x_3) ) = 2 POL( U33_3(x_1, ..., x_3) ) = 2 POL( U34_3(x_1, ..., x_3) ) = 2 POL( U35_2(x_1, x_2) ) = 2 POL( U41_2(x_1, x_2) ) = 2 POL( U61_2(x_1, x_2) ) = 2 POL( U71_2(x_1, x_2) ) = 0 POL( U72_2(x_1, x_2) ) = max{0, -2} POL( U81_3(x_1, ..., x_3) ) = 2 POL( U82_3(x_1, ..., x_3) ) = 2 POL( U83_3(x_1, ..., x_3) ) = 2 POL( U84_3(x_1, ..., x_3) ) = 0 POL( U91_2(x_1, x_2) ) = 0 POL( plus_2(x_1, x_2) ) = 0 POL( x_2(x_1, x_2) ) = 0 POL( mark_1(x_1) ) = max{0, -2} POL( active_1(x_1) ) = 2 POL( tt ) = 0 POL( isNatKind_1(x_1) ) = 2 POL( isNat_1(x_1) ) = 2 POL( U16_1(x_1) ) = 2x_1 POL( U23_1(x_1) ) = max{0, -2} POL( U36_1(x_1) ) = x_1 + 2 POL( U42_1(x_1) ) = 0 POL( U62_1(x_1) ) = 2 POL( s_1(x_1) ) = max{0, x_1 - 2} POL( U92_1(x_1) ) = max{0, x_1 - 2} POL( U51_1(x_1) ) = 2x_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: U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(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) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) 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) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(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) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(mark(X1), X2) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U91(X1, mark(X2)) -> U91(X1, X2) U91(mark(X1), X2) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) ---------------------------------------- (196) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3)) ACTIVE(U101(tt, M, N)) -> MARK(U102(isNatKind(M), M, N)) MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3)) ACTIVE(U102(tt, M, N)) -> MARK(U103(isNat(N), M, N)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) ACTIVE(U103(tt, M, N)) -> MARK(U104(isNatKind(N), M, N)) MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U14(X1, X2, X3)) -> MARK(X1) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2)) MARK(U15(X1, X2)) -> MARK(X1) ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2)) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2))) MARK(U31(X1, X2, X3)) -> MARK(X1) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3)) MARK(U33(X1, X2, X3)) -> MARK(X1) MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3)) ACTIVE(U81(tt, M, N)) -> MARK(U82(isNatKind(M), M, N)) MARK(U34(X1, X2, X3)) -> MARK(X1) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) ACTIVE(U82(tt, M, N)) -> MARK(U83(isNat(N), M, N)) MARK(U35(X1, X2)) -> MARK(X1) ACTIVE(U83(tt, M, N)) -> MARK(U84(isNatKind(N), M, N)) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) MARK(U61(X1, X2)) -> MARK(X1) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) MARK(U62(X)) -> MARK(X) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3)) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3)) MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3)) MARK(U92(X)) -> MARK(X) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (197) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U104(X1, X2, X3)) -> ACTIVE(U104(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) ) = max{0, 2x_1 - 1} POL( U101_3(x_1, ..., x_3) ) = 1 POL( U102_3(x_1, ..., x_3) ) = 1 POL( U103_3(x_1, ..., x_3) ) = 1 POL( U104_3(x_1, ..., x_3) ) = 0 POL( U11_3(x_1, ..., x_3) ) = 1 POL( U12_3(x_1, ..., x_3) ) = 1 POL( U13_3(x_1, ..., x_3) ) = 1 POL( U14_3(x_1, ..., x_3) ) = 1 POL( U15_2(x_1, x_2) ) = 1 POL( U21_2(x_1, x_2) ) = 1 POL( U22_2(x_1, x_2) ) = 1 POL( U31_3(x_1, ..., x_3) ) = 1 POL( U32_3(x_1, ..., x_3) ) = 1 POL( U33_3(x_1, ..., x_3) ) = 1 POL( U34_3(x_1, ..., x_3) ) = 1 POL( U35_2(x_1, x_2) ) = 1 POL( U41_2(x_1, x_2) ) = 1 POL( U61_2(x_1, x_2) ) = 1 POL( U81_3(x_1, ..., x_3) ) = 1 POL( U82_3(x_1, ..., x_3) ) = 1 POL( U83_3(x_1, ..., x_3) ) = 1 POL( mark_1(x_1) ) = 0 POL( active_1(x_1) ) = max{0, 2x_1 - 2} POL( tt ) = 0 POL( isNatKind_1(x_1) ) = 1 POL( isNat_1(x_1) ) = 1 POL( plus_2(x_1, x_2) ) = max{0, x_1 - 2} POL( x_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U16_1(x_1) ) = 2 POL( U23_1(x_1) ) = max{0, 2x_1 - 2} POL( U36_1(x_1) ) = max{0, x_1 - 2} POL( U42_1(x_1) ) = max{0, 2x_1 - 2} POL( U62_1(x_1) ) = max{0, 2x_1 - 2} POL( U71_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( U72_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2} POL( U84_3(x_1, ..., x_3) ) = max{0, x_2 - 2} POL( s_1(x_1) ) = max{0, x_1 - 2} POL( U91_2(x_1, x_2) ) = x_2 + 2 POL( U92_1(x_1) ) = max{0, x_1 - 2} POL( U51_1(x_1) ) = 0 POL( 0 ) = 2 POL( MARK_1(x_1) ) = 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(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) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) ---------------------------------------- (198) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3)) ACTIVE(U101(tt, M, N)) -> MARK(U102(isNatKind(M), M, N)) MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3)) ACTIVE(U102(tt, M, N)) -> MARK(U103(isNat(N), M, N)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) ACTIVE(U103(tt, M, N)) -> MARK(U104(isNatKind(N), M, N)) MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U14(X1, X2, X3)) -> MARK(X1) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2)) MARK(U15(X1, X2)) -> MARK(X1) ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2)) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2))) MARK(U31(X1, X2, X3)) -> MARK(X1) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3)) MARK(U33(X1, X2, X3)) -> MARK(X1) MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3)) ACTIVE(U81(tt, M, N)) -> MARK(U82(isNatKind(M), M, N)) MARK(U34(X1, X2, X3)) -> MARK(X1) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) ACTIVE(U82(tt, M, N)) -> MARK(U83(isNat(N), M, N)) MARK(U35(X1, X2)) -> MARK(X1) ACTIVE(U83(tt, M, N)) -> MARK(U84(isNatKind(N), M, N)) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) MARK(U61(X1, X2)) -> MARK(X1) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) MARK(U62(X)) -> MARK(X) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3)) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3)) MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3)) MARK(U92(X)) -> MARK(X) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (199) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3)) MARK(U92(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( U101_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + x_3 + 2 POL( U102_3(x_1, ..., x_3) ) = max{0, -2} POL( U103_3(x_1, ..., x_3) ) = max{0, -2} POL( U11_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U12_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U13_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U14_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U15_2(x_1, x_2) ) = 2x_1 + 1 POL( U21_2(x_1, x_2) ) = 2x_1 + 1 POL( U22_2(x_1, x_2) ) = 2x_1 + 1 POL( U31_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U32_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U33_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U34_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U35_2(x_1, x_2) ) = 2x_1 + 1 POL( U41_2(x_1, x_2) ) = 2x_1 + 1 POL( U61_2(x_1, x_2) ) = x_1 + 1 POL( U81_3(x_1, ..., x_3) ) = max{0, -2} POL( U82_3(x_1, ..., x_3) ) = 1 POL( U83_3(x_1, ..., x_3) ) = max{0, -2} POL( mark_1(x_1) ) = 2x_1 POL( active_1(x_1) ) = x_1 + 2 POL( tt ) = 0 POL( isNatKind_1(x_1) ) = 0 POL( isNat_1(x_1) ) = 0 POL( U104_3(x_1, ..., x_3) ) = max{0, -2} POL( plus_2(x_1, x_2) ) = max{0, -2} POL( x_2(x_1, x_2) ) = max{0, x_2 - 2} POL( U16_1(x_1) ) = x_1 + 1 POL( U23_1(x_1) ) = x_1 + 1 POL( U36_1(x_1) ) = x_1 + 1 POL( U42_1(x_1) ) = x_1 + 1 POL( U62_1(x_1) ) = x_1 + 1 POL( U71_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2} POL( U72_2(x_1, x_2) ) = max{0, 2x_1 - 1} POL( U84_3(x_1, ..., x_3) ) = max{0, -2} POL( s_1(x_1) ) = max{0, 2x_1 - 2} POL( U91_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U92_1(x_1) ) = x_1 + 2 POL( U51_1(x_1) ) = 2x_1 + 1 POL( 0 ) = 0 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: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(active(X)) -> U36(X) U36(mark(X)) -> U36(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(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) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) ---------------------------------------- (200) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(U101(tt, M, N)) -> MARK(U102(isNatKind(M), M, N)) MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3)) ACTIVE(U102(tt, M, N)) -> MARK(U103(isNat(N), M, N)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) ACTIVE(U103(tt, M, N)) -> MARK(U104(isNatKind(N), M, N)) MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U14(X1, X2, X3)) -> MARK(X1) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2)) MARK(U15(X1, X2)) -> MARK(X1) ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2)) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2))) MARK(U31(X1, X2, X3)) -> MARK(X1) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3)) MARK(U33(X1, X2, X3)) -> MARK(X1) MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3)) ACTIVE(U81(tt, M, N)) -> MARK(U82(isNatKind(M), M, N)) MARK(U34(X1, X2, X3)) -> MARK(X1) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) ACTIVE(U82(tt, M, N)) -> MARK(U83(isNat(N), M, N)) MARK(U35(X1, X2)) -> MARK(X1) ACTIVE(U83(tt, M, N)) -> MARK(U84(isNatKind(N), M, N)) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) MARK(U61(X1, X2)) -> MARK(X1) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) MARK(U62(X)) -> MARK(X) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3)) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3)) MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (201) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U101(tt, M, N)) -> MARK(U102(isNatKind(M), M, N)) 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 - 2} POL( MARK_1(x_1) ) = max{0, -2} POL( U102_3(x_1, ..., x_3) ) = max{0, -2} POL( U104_3(x_1, ..., x_3) ) = 2x_2 + 2 POL( U11_3(x_1, ..., x_3) ) = 1 POL( U12_3(x_1, ..., x_3) ) = max{0, -2} POL( U13_3(x_1, ..., x_3) ) = 2 POL( U14_3(x_1, ..., x_3) ) = 0 POL( U21_2(x_1, x_2) ) = 2 POL( U22_2(x_1, x_2) ) = max{0, -2} POL( U31_3(x_1, ..., x_3) ) = max{0, -2} POL( U32_3(x_1, ..., x_3) ) = 2 POL( U33_3(x_1, ..., x_3) ) = 2 POL( U34_3(x_1, ..., x_3) ) = 2 POL( U41_2(x_1, x_2) ) = 2 POL( U42_1(x_1) ) = max{0, -2} POL( U51_1(x_1) ) = 0 POL( U61_2(x_1, x_2) ) = max{0, -2} POL( U62_1(x_1) ) = max{0, x_1 - 2} POL( U82_3(x_1, ..., x_3) ) = max{0, -2} POL( U84_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 - 2} POL( isNatKind_1(x_1) ) = 2 POL( active_1(x_1) ) = max{0, -2} POL( mark_1(x_1) ) = max{0, -2} POL( U103_3(x_1, ..., x_3) ) = 2 POL( U15_2(x_1, x_2) ) = max{0, -2} POL( U35_2(x_1, x_2) ) = max{0, -2} POL( U81_3(x_1, ..., x_3) ) = 2 POL( U83_3(x_1, ..., x_3) ) = 0 POL( U101_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( tt ) = 2 POL( isNat_1(x_1) ) = 0 POL( plus_2(x_1, x_2) ) = 2x_2 + 2 POL( x_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U16_1(x_1) ) = max{0, x_1 - 2} POL( U23_1(x_1) ) = max{0, 2x_1 - 2} POL( U36_1(x_1) ) = max{0, x_1 - 2} POL( U71_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 POL( U72_2(x_1, x_2) ) = max{0, x_1 - 2} POL( s_1(x_1) ) = 2 POL( U91_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U92_1(x_1) ) = max{0, -2} POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(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) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) ---------------------------------------- (202) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3)) ACTIVE(U102(tt, M, N)) -> MARK(U103(isNat(N), M, N)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) ACTIVE(U103(tt, M, N)) -> MARK(U104(isNatKind(N), M, N)) MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U14(X1, X2, X3)) -> MARK(X1) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2)) MARK(U15(X1, X2)) -> MARK(X1) ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2)) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2))) MARK(U31(X1, X2, X3)) -> MARK(X1) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3)) MARK(U33(X1, X2, X3)) -> MARK(X1) MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3)) ACTIVE(U81(tt, M, N)) -> MARK(U82(isNatKind(M), M, N)) MARK(U34(X1, X2, X3)) -> MARK(X1) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) ACTIVE(U82(tt, M, N)) -> MARK(U83(isNat(N), M, N)) MARK(U35(X1, X2)) -> MARK(X1) ACTIVE(U83(tt, M, N)) -> MARK(U84(isNatKind(N), M, N)) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) MARK(U61(X1, X2)) -> MARK(X1) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) MARK(U62(X)) -> MARK(X) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) MARK(U81(X1, X2, X3)) -> ACTIVE(U81(mark(X1), X2, X3)) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3)) MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (203) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3)) MARK(U81(X1, X2, X3)) -> ACTIVE(U81(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) ) = max{0, -2} POL( U102_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2 POL( U103_3(x_1, ..., x_3) ) = max{0, -2} POL( U11_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U12_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U13_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U14_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U15_2(x_1, x_2) ) = 2x_1 + 1 POL( U21_2(x_1, x_2) ) = 2x_1 + 1 POL( U22_2(x_1, x_2) ) = 2x_1 + 1 POL( U31_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U32_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U33_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U34_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U35_2(x_1, x_2) ) = 2x_1 + 1 POL( U41_2(x_1, x_2) ) = 2x_1 + 1 POL( U61_2(x_1, x_2) ) = 2x_1 + 1 POL( U81_3(x_1, ..., x_3) ) = x_2 + x_3 + 2 POL( U82_3(x_1, ..., x_3) ) = max{0, -2} POL( U83_3(x_1, ..., x_3) ) = 0 POL( mark_1(x_1) ) = 2x_1 POL( U101_3(x_1, ..., x_3) ) = 2 POL( active_1(x_1) ) = x_1 + 2 POL( tt ) = 0 POL( isNatKind_1(x_1) ) = 0 POL( isNat_1(x_1) ) = 0 POL( U104_3(x_1, ..., x_3) ) = max{0, -2} POL( plus_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( x_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U16_1(x_1) ) = x_1 + 1 POL( U23_1(x_1) ) = x_1 + 1 POL( U36_1(x_1) ) = x_1 + 1 POL( U42_1(x_1) ) = x_1 + 1 POL( U62_1(x_1) ) = x_1 + 1 POL( U71_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2} POL( U72_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U84_3(x_1, ..., x_3) ) = max{0, -2} POL( s_1(x_1) ) = 2 POL( U91_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U92_1(x_1) ) = x_1 + 1 POL( U51_1(x_1) ) = x_1 + 1 POL( 0 ) = 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: isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(active(X)) -> U36(X) U36(mark(X)) -> U36(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(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) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) ---------------------------------------- (204) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(U102(tt, M, N)) -> MARK(U103(isNat(N), M, N)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) ACTIVE(U103(tt, M, N)) -> MARK(U104(isNatKind(N), M, N)) MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U14(X1, X2, X3)) -> MARK(X1) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2)) MARK(U15(X1, X2)) -> MARK(X1) ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2)) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2))) MARK(U31(X1, X2, X3)) -> MARK(X1) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3)) MARK(U33(X1, X2, X3)) -> MARK(X1) MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3)) ACTIVE(U81(tt, M, N)) -> MARK(U82(isNatKind(M), M, N)) MARK(U34(X1, X2, X3)) -> MARK(X1) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) ACTIVE(U82(tt, M, N)) -> MARK(U83(isNat(N), M, N)) MARK(U35(X1, X2)) -> MARK(X1) ACTIVE(U83(tt, M, N)) -> MARK(U84(isNatKind(N), M, N)) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) MARK(U61(X1, X2)) -> MARK(X1) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) MARK(U62(X)) -> MARK(X) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3)) MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (205) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U102(tt, M, N)) -> MARK(U103(isNat(N), M, N)) ACTIVE(U81(tt, M, N)) -> MARK(U82(isNatKind(M), M, N)) 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( MARK_1(x_1) ) = max{0, -2} POL( U103_3(x_1, ..., x_3) ) = 1 POL( U15_2(x_1, x_2) ) = max{0, -2} POL( U16_1(x_1) ) = max{0, x_1 - 2} POL( U23_1(x_1) ) = max{0, x_1 - 2} POL( U35_2(x_1, x_2) ) = 1 POL( U36_1(x_1) ) = max{0, x_1 - 1} POL( U83_3(x_1, ..., x_3) ) = max{0, -2} POL( isNat_1(x_1) ) = 0 POL( active_1(x_1) ) = x_1 + 2 POL( mark_1(x_1) ) = max{0, -2} POL( U104_3(x_1, ..., x_3) ) = x_2 + 2 POL( U11_3(x_1, ..., x_3) ) = max{0, -2} POL( U12_3(x_1, ..., x_3) ) = max{0, -2} POL( U13_3(x_1, ..., x_3) ) = max{0, -2} POL( U14_3(x_1, ..., x_3) ) = 0 POL( U21_2(x_1, x_2) ) = max{0, -2} POL( U22_2(x_1, x_2) ) = max{0, -2} POL( U31_3(x_1, ..., x_3) ) = max{0, -2} POL( U32_3(x_1, ..., x_3) ) = 1 POL( U33_3(x_1, ..., x_3) ) = max{0, -2} POL( U34_3(x_1, ..., x_3) ) = max{0, -2} POL( U41_2(x_1, x_2) ) = 1 POL( U42_1(x_1) ) = max{0, x_1 - 2} POL( U51_1(x_1) ) = max{0, -2} POL( U61_2(x_1, x_2) ) = 1 POL( U62_1(x_1) ) = max{0, 2x_1 - 2} POL( U82_3(x_1, ..., x_3) ) = 1 POL( U84_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 - 2} POL( isNatKind_1(x_1) ) = 1 POL( U101_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2} POL( tt ) = 0 POL( U102_3(x_1, ..., x_3) ) = 2x_1 + 2x_3 + 2 POL( plus_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( x_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2} POL( U71_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U72_2(x_1, x_2) ) = 2 POL( U81_3(x_1, ..., x_3) ) = 2 POL( s_1(x_1) ) = max{0, 2x_1 - 2} POL( U91_2(x_1, x_2) ) = max{0, x_2 - 2} POL( U92_1(x_1) ) = max{0, x_1 - 2} POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(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) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) ---------------------------------------- (206) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) ACTIVE(U103(tt, M, N)) -> MARK(U104(isNatKind(N), M, N)) MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U14(X1, X2, X3)) -> MARK(X1) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2)) MARK(U15(X1, X2)) -> MARK(X1) ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2)) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2))) MARK(U31(X1, X2, X3)) -> MARK(X1) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3)) MARK(U33(X1, X2, X3)) -> MARK(X1) MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3)) MARK(U34(X1, X2, X3)) -> MARK(X1) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) ACTIVE(U82(tt, M, N)) -> MARK(U83(isNat(N), M, N)) MARK(U35(X1, X2)) -> MARK(X1) ACTIVE(U83(tt, M, N)) -> MARK(U84(isNatKind(N), M, N)) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) MARK(U61(X1, X2)) -> MARK(X1) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) MARK(U62(X)) -> MARK(X) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) MARK(U82(X1, X2, X3)) -> ACTIVE(U82(mark(X1), X2, X3)) MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (207) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U82(X1, X2, X3)) -> ACTIVE(U82(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) ) = max{0, -2} POL( MARK_1(x_1) ) = max{0, x_1 - 1} POL( U104_3(x_1, ..., x_3) ) = max{0, -2} POL( U11_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U12_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U13_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U14_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U21_2(x_1, x_2) ) = 2x_1 + 1 POL( U22_2(x_1, x_2) ) = 2x_1 + 1 POL( U31_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U32_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U33_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U34_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U41_2(x_1, x_2) ) = 2x_1 + 1 POL( U42_1(x_1) ) = 2x_1 + 1 POL( U51_1(x_1) ) = 2x_1 + 1 POL( U61_2(x_1, x_2) ) = 2x_1 + 1 POL( U62_1(x_1) ) = x_1 + 1 POL( U84_3(x_1, ..., x_3) ) = max{0, -2} POL( isNatKind_1(x_1) ) = 0 POL( active_1(x_1) ) = 2x_1 POL( mark_1(x_1) ) = 2x_1 POL( U103_3(x_1, ..., x_3) ) = 2x_2 + x_3 + 1 POL( U15_2(x_1, x_2) ) = 2x_1 + 1 POL( U35_2(x_1, x_2) ) = 2x_1 + 1 POL( U82_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 2 POL( U83_3(x_1, ..., x_3) ) = max{0, -2} POL( U101_3(x_1, ..., x_3) ) = x_1 + 2 POL( tt ) = 0 POL( U102_3(x_1, ..., x_3) ) = 2 POL( isNat_1(x_1) ) = 0 POL( plus_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 1} POL( x_2(x_1, x_2) ) = 2 POL( U16_1(x_1) ) = x_1 + 1 POL( U23_1(x_1) ) = x_1 + 1 POL( U36_1(x_1) ) = x_1 + 1 POL( U71_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U72_2(x_1, x_2) ) = x_2 + 2 POL( U81_3(x_1, ..., x_3) ) = 2 POL( s_1(x_1) ) = max{0, x_1 - 2} POL( U91_2(x_1, x_2) ) = max{0, x_2 - 2} POL( U92_1(x_1) ) = 2 POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(active(X)) -> U36(X) U36(mark(X)) -> U36(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(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) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) ---------------------------------------- (208) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) ACTIVE(U103(tt, M, N)) -> MARK(U104(isNatKind(N), M, N)) MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U14(X1, X2, X3)) -> MARK(X1) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2)) MARK(U15(X1, X2)) -> MARK(X1) ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2)) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2))) MARK(U31(X1, X2, X3)) -> MARK(X1) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3)) MARK(U33(X1, X2, X3)) -> MARK(X1) MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3)) MARK(U34(X1, X2, X3)) -> MARK(X1) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) ACTIVE(U82(tt, M, N)) -> MARK(U83(isNat(N), M, N)) MARK(U35(X1, X2)) -> MARK(X1) ACTIVE(U83(tt, M, N)) -> MARK(U84(isNatKind(N), M, N)) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) MARK(U61(X1, X2)) -> MARK(X1) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) MARK(U62(X)) -> MARK(X) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (209) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U82(tt, M, N)) -> MARK(U83(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) ) = max{0, x_1 - 2} POL( MARK_1(x_1) ) = max{0, -2} POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_2 + x_3 - 2} POL( U11_3(x_1, ..., x_3) ) = max{0, -2} POL( U12_3(x_1, ..., x_3) ) = 2 POL( U13_3(x_1, ..., x_3) ) = max{0, -2} POL( U14_3(x_1, ..., x_3) ) = max{0, -2} POL( U21_2(x_1, x_2) ) = 0 POL( U22_2(x_1, x_2) ) = 2 POL( U31_3(x_1, ..., x_3) ) = 2 POL( U32_3(x_1, ..., x_3) ) = 0 POL( U33_3(x_1, ..., x_3) ) = max{0, -2} POL( U34_3(x_1, ..., x_3) ) = 2 POL( U41_2(x_1, x_2) ) = 0 POL( U42_1(x_1) ) = max{0, x_1 - 2} POL( U51_1(x_1) ) = 2 POL( U61_2(x_1, x_2) ) = 2 POL( U62_1(x_1) ) = 2 POL( U84_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 + x_3 - 2} POL( isNatKind_1(x_1) ) = 0 POL( active_1(x_1) ) = max{0, x_1 - 2} POL( mark_1(x_1) ) = max{0, -2} POL( U103_3(x_1, ..., x_3) ) = 2 POL( U15_2(x_1, x_2) ) = 2 POL( U35_2(x_1, x_2) ) = 2 POL( U83_3(x_1, ..., x_3) ) = max{0, -2} POL( U101_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2} POL( tt ) = 2 POL( U102_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2} POL( isNat_1(x_1) ) = 0 POL( plus_2(x_1, x_2) ) = max{0, x_1 - 2} POL( x_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U16_1(x_1) ) = 2 POL( U23_1(x_1) ) = max{0, -2} POL( U36_1(x_1) ) = 2 POL( U71_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2} POL( U72_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U81_3(x_1, ..., x_3) ) = 2x_1 + 2 POL( U82_3(x_1, ..., x_3) ) = x_1 + 2 POL( s_1(x_1) ) = max{0, x_1 - 2} POL( U91_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2} POL( U92_1(x_1) ) = 2 POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(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) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) ---------------------------------------- (210) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) ACTIVE(U103(tt, M, N)) -> MARK(U104(isNatKind(N), M, N)) MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U14(X1, X2, X3)) -> MARK(X1) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2)) MARK(U15(X1, X2)) -> MARK(X1) ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2)) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2))) MARK(U31(X1, X2, X3)) -> MARK(X1) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3)) MARK(U33(X1, X2, X3)) -> MARK(X1) MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3)) MARK(U34(X1, X2, X3)) -> MARK(X1) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) MARK(U35(X1, X2)) -> MARK(X1) ACTIVE(U83(tt, M, N)) -> MARK(U84(isNatKind(N), M, N)) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) MARK(U61(X1, X2)) -> MARK(X1) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) MARK(U62(X)) -> MARK(X) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) MARK(U83(X1, X2, X3)) -> ACTIVE(U83(mark(X1), X2, X3)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (211) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3)) MARK(U83(X1, X2, X3)) -> ACTIVE(U83(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) ) = 2 POL( MARK_1(x_1) ) = x_1 + 2 POL( U104_3(x_1, ..., x_3) ) = max{0, -2} POL( U11_3(x_1, ..., x_3) ) = 2x_1 POL( U12_3(x_1, ..., x_3) ) = 2x_1 POL( U13_3(x_1, ..., x_3) ) = 2x_1 POL( U14_3(x_1, ..., x_3) ) = 2x_1 POL( U21_2(x_1, x_2) ) = 2x_1 POL( U22_2(x_1, x_2) ) = x_1 POL( U31_3(x_1, ..., x_3) ) = 2x_1 POL( U32_3(x_1, ..., x_3) ) = 2x_1 POL( U33_3(x_1, ..., x_3) ) = x_1 POL( U34_3(x_1, ..., x_3) ) = 2x_1 POL( U41_2(x_1, x_2) ) = x_1 POL( U42_1(x_1) ) = 2x_1 POL( U51_1(x_1) ) = 2x_1 POL( U61_2(x_1, x_2) ) = 2x_1 POL( U62_1(x_1) ) = x_1 POL( U84_3(x_1, ..., x_3) ) = max{0, -2} POL( isNatKind_1(x_1) ) = 0 POL( active_1(x_1) ) = x_1 + 2 POL( mark_1(x_1) ) = x_1 + 2 POL( U103_3(x_1, ..., x_3) ) = 1 POL( U15_2(x_1, x_2) ) = 2x_1 POL( U35_2(x_1, x_2) ) = 2x_1 POL( U83_3(x_1, ..., x_3) ) = x_1 + 2x_2 + x_3 + 1 POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 - 2} POL( tt ) = 0 POL( U102_3(x_1, ..., x_3) ) = 2 POL( isNat_1(x_1) ) = 0 POL( plus_2(x_1, x_2) ) = x_1 + 1 POL( x_2(x_1, x_2) ) = x_1 + 1 POL( U16_1(x_1) ) = x_1 POL( U23_1(x_1) ) = x_1 POL( U36_1(x_1) ) = 2x_1 POL( U71_2(x_1, x_2) ) = 2x_2 + 2 POL( U72_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( U81_3(x_1, ..., x_3) ) = max{0, x_1 + x_3 - 2} POL( U82_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 1} POL( s_1(x_1) ) = 0 POL( U91_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U92_1(x_1) ) = max{0, -2} POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(active(X)) -> U36(X) U36(mark(X)) -> U36(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(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) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) ---------------------------------------- (212) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) ACTIVE(U103(tt, M, N)) -> MARK(U104(isNatKind(N), M, N)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U14(X1, X2, X3)) -> MARK(X1) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2)) MARK(U15(X1, X2)) -> MARK(X1) ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2)) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2))) MARK(U31(X1, X2, X3)) -> MARK(X1) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3)) MARK(U33(X1, X2, X3)) -> MARK(X1) MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3)) MARK(U34(X1, X2, X3)) -> MARK(X1) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) MARK(U35(X1, X2)) -> MARK(X1) ACTIVE(U83(tt, M, N)) -> MARK(U84(isNatKind(N), M, N)) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) MARK(U61(X1, X2)) -> MARK(X1) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) MARK(U62(X)) -> MARK(X) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (213) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U103(tt, M, N)) -> MARK(U104(isNatKind(N), M, N)) ACTIVE(U83(tt, M, N)) -> MARK(U84(isNatKind(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( MARK_1(x_1) ) = 2 POL( U104_3(x_1, ..., x_3) ) = x_1 + x_2 + 2 POL( U11_3(x_1, ..., x_3) ) = max{0, -2} POL( U12_3(x_1, ..., x_3) ) = max{0, -2} POL( U13_3(x_1, ..., x_3) ) = max{0, -2} POL( U14_3(x_1, ..., x_3) ) = max{0, -2} POL( U21_2(x_1, x_2) ) = max{0, -2} POL( U22_2(x_1, x_2) ) = max{0, -2} POL( U31_3(x_1, ..., x_3) ) = max{0, -2} POL( U32_3(x_1, ..., x_3) ) = 0 POL( U33_3(x_1, ..., x_3) ) = max{0, -2} POL( U34_3(x_1, ..., x_3) ) = max{0, -2} POL( U41_2(x_1, x_2) ) = max{0, -2} POL( U42_1(x_1) ) = max{0, 2x_1 - 2} POL( U51_1(x_1) ) = max{0, -2} POL( U61_2(x_1, x_2) ) = max{0, -2} POL( U62_1(x_1) ) = max{0, 2x_1 - 2} POL( U84_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + 2x_3 - 2} POL( isNatKind_1(x_1) ) = 0 POL( active_1(x_1) ) = max{0, 2x_1 - 2} POL( mark_1(x_1) ) = max{0, x_1 - 2} POL( U15_2(x_1, x_2) ) = max{0, -2} POL( U35_2(x_1, x_2) ) = 0 POL( U16_1(x_1) ) = max{0, x_1 - 2} POL( U23_1(x_1) ) = 2 POL( U36_1(x_1) ) = max{0, x_1 - 2} POL( isNat_1(x_1) ) = 0 POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( tt ) = 2 POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 - 2} POL( U103_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 2x_3 + 1 POL( plus_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2} POL( x_2(x_1, x_2) ) = x_1 + 2 POL( U71_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U72_2(x_1, x_2) ) = max{0, x_2 - 2} POL( U81_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 + x_3 - 2} POL( U82_3(x_1, ..., x_3) ) = x_1 + 2 POL( U83_3(x_1, ..., x_3) ) = 2x_1 + 2 POL( s_1(x_1) ) = max{0, 2x_1 - 2} POL( U91_2(x_1, x_2) ) = x_1 + 2 POL( U92_1(x_1) ) = max{0, 2x_1 - 2} POL( 0 ) = 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(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) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) ---------------------------------------- (214) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U14(X1, X2, X3)) -> MARK(X1) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2)) MARK(U15(X1, X2)) -> MARK(X1) ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2)) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2))) MARK(U31(X1, X2, X3)) -> MARK(X1) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3)) MARK(U33(X1, X2, X3)) -> MARK(X1) MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3)) MARK(U34(X1, X2, X3)) -> MARK(X1) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) MARK(U35(X1, X2)) -> MARK(X1) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) MARK(U61(X1, X2)) -> MARK(X1) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) MARK(U62(X)) -> MARK(X) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (215) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U34(X1, X2, X3)) -> MARK(X1) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(MARK(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(isNatKind(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(ACTIVE(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(isNat(x_1)) = [[0A]] + [[1A]] * x_1 >>> <<< POL(U11(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(tt) = [[0A]] >>> <<< POL(U12(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U13(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U14(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U15(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(mark(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U16(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U21(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(U22(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(U23(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U31(x_1, x_2, x_3)) = [[2A]] + [[0A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U32(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U33(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U34(x_1, x_2, x_3)) = [[0A]] + [[1A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U35(x_1, x_2)) = [[1A]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(U36(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(U41(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U42(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U61(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U62(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(plus(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U51(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(s(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(x(x_1, x_2)) = [[1A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(active(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U101(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U102(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U103(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U104(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U71(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U72(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U81(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U82(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U83(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U84(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U91(x_1, x_2)) = [[1A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U92(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(0) = [[0A]] >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) mark(isNatKind(X)) -> active(isNatKind(X)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) mark(isNat(X)) -> active(isNat(X)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) mark(U16(X)) -> active(U16(mark(X))) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) mark(U23(X)) -> active(U23(mark(X))) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) active(U72(tt, N)) -> mark(N) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) mark(U36(X)) -> active(U36(mark(X))) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) mark(U42(X)) -> active(U42(mark(X))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) mark(U51(X)) -> active(U51(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) mark(U62(X)) -> active(U62(mark(X))) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) active(x(N, 0)) -> mark(U91(isNat(N), N)) mark(s(X)) -> active(s(mark(X))) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(tt) -> active(tt) mark(0) -> active(0) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(active(X)) -> U36(X) U36(mark(X)) -> U36(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) 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) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) active(U16(tt)) -> mark(tt) active(U23(tt)) -> mark(tt) active(U36(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U62(tt)) -> mark(tt) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNatKind(0)) -> mark(tt) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U92(active(X)) -> U92(X) U92(mark(X)) -> U92(X) U71(X1, mark(X2)) -> U71(X1, X2) U71(mark(X1), X2) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U91(X1, mark(X2)) -> U91(X1, X2) U91(mark(X1), X2) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) ---------------------------------------- (216) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U14(X1, X2, X3)) -> MARK(X1) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2)) MARK(U15(X1, X2)) -> MARK(X1) ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2)) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2))) MARK(U31(X1, X2, X3)) -> MARK(X1) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3)) MARK(U33(X1, X2, X3)) -> MARK(X1) MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3)) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) MARK(U35(X1, X2)) -> MARK(X1) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) MARK(U61(X1, X2)) -> MARK(X1) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) MARK(U62(X)) -> MARK(X) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (217) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U61(X1, X2)) -> MARK(X1) 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(isNatKind(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(ACTIVE(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(isNat(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U11(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(tt) = [[0A]] >>> <<< POL(U12(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U13(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U14(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U15(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(mark(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U16(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U21(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U22(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U23(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U31(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U32(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U33(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U34(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U35(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U36(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U41(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U42(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U61(x_1, x_2)) = [[1A]] + [[1A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U62(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(plus(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U51(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(s(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(x(x_1, x_2)) = [[1A]] + [[1A]] * x_1 + [[0A]] * x_2 >>> <<< POL(active(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U101(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U102(x_1, x_2, x_3)) = [[1A]] + [[0A]] * x_1 + [[0A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U103(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U104(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U71(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U72(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U81(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U82(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U83(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U84(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U91(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(U92(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(0) = [[0A]] >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) mark(isNatKind(X)) -> active(isNatKind(X)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) mark(isNat(X)) -> active(isNat(X)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) mark(U16(X)) -> active(U16(mark(X))) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) mark(U23(X)) -> active(U23(mark(X))) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) active(U72(tt, N)) -> mark(N) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) mark(U36(X)) -> active(U36(mark(X))) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) mark(U42(X)) -> active(U42(mark(X))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) mark(U51(X)) -> active(U51(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) mark(U62(X)) -> active(U62(mark(X))) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) active(x(N, 0)) -> mark(U91(isNat(N), N)) mark(s(X)) -> active(s(mark(X))) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(tt) -> active(tt) mark(0) -> active(0) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(active(X)) -> U36(X) U36(mark(X)) -> U36(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) 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) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) active(U16(tt)) -> mark(tt) active(U23(tt)) -> mark(tt) active(U36(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U62(tt)) -> mark(tt) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNatKind(0)) -> mark(tt) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U92(active(X)) -> U92(X) U92(mark(X)) -> U92(X) U71(X1, mark(X2)) -> U71(X1, X2) U71(mark(X1), X2) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U91(X1, mark(X2)) -> U91(X1, X2) U91(mark(X1), X2) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) ---------------------------------------- (218) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U14(X1, X2, X3)) -> MARK(X1) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2)) MARK(U15(X1, X2)) -> MARK(X1) ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2)) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2))) MARK(U31(X1, X2, X3)) -> MARK(X1) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3)) MARK(U33(X1, X2, X3)) -> MARK(X1) MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3)) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) MARK(U35(X1, X2)) -> MARK(X1) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) MARK(U62(X)) -> MARK(X) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (219) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U14(X1, X2, X3)) -> MARK(X1) 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(isNatKind(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(ACTIVE(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(isNat(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(U11(x_1, x_2, x_3)) = [[1A]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(tt) = [[0A]] >>> <<< POL(U12(x_1, x_2, x_3)) = [[1A]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U13(x_1, x_2, x_3)) = [[1A]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U14(x_1, x_2, x_3)) = [[1A]] + [[1A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U15(x_1, x_2)) = [[1A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(mark(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U16(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U21(x_1, x_2)) = [[1A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U22(x_1, x_2)) = [[1A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U23(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U31(x_1, x_2, x_3)) = [[1A]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U32(x_1, x_2, x_3)) = [[1A]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U33(x_1, x_2, x_3)) = [[1A]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U34(x_1, x_2, x_3)) = [[1A]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U35(x_1, x_2)) = [[1A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U36(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U41(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(U42(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U61(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U62(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(plus(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U51(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(s(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(x(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(active(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U101(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U102(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U103(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U104(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U71(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U72(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U81(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U82(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U83(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U84(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U91(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U92(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(0) = [[0A]] >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) mark(isNatKind(X)) -> active(isNatKind(X)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) mark(isNat(X)) -> active(isNat(X)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) mark(U16(X)) -> active(U16(mark(X))) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) mark(U23(X)) -> active(U23(mark(X))) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) active(U72(tt, N)) -> mark(N) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) mark(U36(X)) -> active(U36(mark(X))) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) mark(U42(X)) -> active(U42(mark(X))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) mark(U51(X)) -> active(U51(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) mark(U62(X)) -> active(U62(mark(X))) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) active(x(N, 0)) -> mark(U91(isNat(N), N)) mark(s(X)) -> active(s(mark(X))) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(tt) -> active(tt) mark(0) -> active(0) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(active(X)) -> U36(X) U36(mark(X)) -> U36(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) 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) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) active(U16(tt)) -> mark(tt) active(U23(tt)) -> mark(tt) active(U36(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U62(tt)) -> mark(tt) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNatKind(0)) -> mark(tt) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U92(active(X)) -> U92(X) U92(mark(X)) -> U92(X) U71(X1, mark(X2)) -> U71(X1, X2) U71(mark(X1), X2) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U91(X1, mark(X2)) -> U91(X1, X2) U91(mark(X1), X2) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) ---------------------------------------- (220) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2)) MARK(U15(X1, X2)) -> MARK(X1) ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2)) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2))) MARK(U31(X1, X2, X3)) -> MARK(X1) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3)) MARK(U33(X1, X2, X3)) -> MARK(X1) MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3)) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) MARK(U35(X1, X2)) -> MARK(X1) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) MARK(U62(X)) -> MARK(X) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (221) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U31(X1, X2, X3)) -> MARK(X1) 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(isNatKind(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(ACTIVE(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(isNat(x_1)) = [[0A]] + [[1A]] * x_1 >>> <<< POL(U11(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(tt) = [[0A]] >>> <<< POL(U12(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U13(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U14(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U15(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(mark(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U16(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U21(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(U22(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(U23(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U31(x_1, x_2, x_3)) = [[1A]] + [[1A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U32(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U33(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U34(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U35(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(U36(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U41(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U42(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U61(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U62(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(plus(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U51(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(s(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(x(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(active(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U101(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U102(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U103(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U104(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U71(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U72(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U81(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U82(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U83(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U84(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U91(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U92(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(0) = [[0A]] >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) mark(isNatKind(X)) -> active(isNatKind(X)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) mark(isNat(X)) -> active(isNat(X)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) mark(U16(X)) -> active(U16(mark(X))) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) mark(U23(X)) -> active(U23(mark(X))) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) active(U72(tt, N)) -> mark(N) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) mark(U36(X)) -> active(U36(mark(X))) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) mark(U42(X)) -> active(U42(mark(X))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) mark(U51(X)) -> active(U51(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) mark(U62(X)) -> active(U62(mark(X))) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) active(x(N, 0)) -> mark(U91(isNat(N), N)) mark(s(X)) -> active(s(mark(X))) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(tt) -> active(tt) mark(0) -> active(0) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(active(X)) -> U36(X) U36(mark(X)) -> U36(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) 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) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) active(U16(tt)) -> mark(tt) active(U23(tt)) -> mark(tt) active(U36(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U62(tt)) -> mark(tt) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNatKind(0)) -> mark(tt) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U92(active(X)) -> U92(X) U92(mark(X)) -> U92(X) U71(X1, mark(X2)) -> U71(X1, X2) U71(mark(X1), X2) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U91(X1, mark(X2)) -> U91(X1, X2) U91(mark(X1), X2) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) ---------------------------------------- (222) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2)) MARK(U15(X1, X2)) -> MARK(X1) ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2)) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(V2))) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3)) MARK(U33(X1, X2, X3)) -> MARK(X1) MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3)) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) MARK(U35(X1, X2)) -> MARK(X1) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) MARK(U62(X)) -> MARK(X) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (223) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U61(tt, V2)) -> MARK(U62(isNatKind(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(isNatKind(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(ACTIVE(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(isNat(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U11(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(tt) = [[0A]] >>> <<< POL(U12(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U13(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U14(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U15(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(mark(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U16(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U21(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U22(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U23(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U31(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U32(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U33(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U34(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U35(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(U36(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U41(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U42(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U61(x_1, x_2)) = [[1A]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(U62(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(plus(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U51(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(s(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(x(x_1, x_2)) = [[1A]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(active(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U101(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U102(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U103(x_1, x_2, x_3)) = [[1A]] + [[0A]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U104(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U71(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U72(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U81(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U82(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U83(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U84(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U91(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(U92(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(0) = [[0A]] >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) mark(isNatKind(X)) -> active(isNatKind(X)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) mark(isNat(X)) -> active(isNat(X)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) mark(U16(X)) -> active(U16(mark(X))) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) mark(U23(X)) -> active(U23(mark(X))) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) active(U72(tt, N)) -> mark(N) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) mark(U36(X)) -> active(U36(mark(X))) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) mark(U42(X)) -> active(U42(mark(X))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) mark(U51(X)) -> active(U51(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) mark(U62(X)) -> active(U62(mark(X))) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) active(x(N, 0)) -> mark(U91(isNat(N), N)) mark(s(X)) -> active(s(mark(X))) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(tt) -> active(tt) mark(0) -> active(0) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(active(X)) -> U36(X) U36(mark(X)) -> U36(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) 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) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) active(U16(tt)) -> mark(tt) active(U23(tt)) -> mark(tt) active(U36(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U62(tt)) -> mark(tt) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNatKind(0)) -> mark(tt) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U92(active(X)) -> U92(X) U92(mark(X)) -> U92(X) U71(X1, mark(X2)) -> U71(X1, X2) U71(mark(X1), X2) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U91(X1, mark(X2)) -> U91(X1, X2) U91(mark(X1), X2) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) ---------------------------------------- (224) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2)) MARK(U15(X1, X2)) -> MARK(X1) ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2)) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3)) MARK(U33(X1, X2, X3)) -> MARK(X1) MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3)) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) MARK(U35(X1, X2)) -> MARK(X1) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) MARK(U62(X)) -> MARK(X) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (225) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U62(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( MARK_1(x_1) ) = max{0, x_1 - 1} POL( U11_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U12_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U13_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U14_3(x_1, ..., x_3) ) = 0 POL( U21_2(x_1, x_2) ) = 2x_1 + 1 POL( U22_2(x_1, x_2) ) = x_1 + 1 POL( U31_3(x_1, ..., x_3) ) = x_1 + 1 POL( U32_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U33_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U34_3(x_1, ..., x_3) ) = max{0, -2} POL( U41_2(x_1, x_2) ) = 2x_1 + 1 POL( U42_1(x_1) ) = x_1 + 1 POL( U51_1(x_1) ) = 2x_1 + 1 POL( U61_2(x_1, x_2) ) = 0 POL( isNatKind_1(x_1) ) = 0 POL( active_1(x_1) ) = x_1 + 2 POL( mark_1(x_1) ) = x_1 POL( U15_2(x_1, x_2) ) = 2x_1 + 1 POL( U16_1(x_1) ) = x_1 + 1 POL( U23_1(x_1) ) = 2x_1 + 1 POL( U35_2(x_1, x_2) ) = x_1 + 1 POL( U36_1(x_1) ) = x_1 + 1 POL( isNat_1(x_1) ) = 0 POL( U101_3(x_1, ..., x_3) ) = x_2 + 2 POL( tt ) = 0 POL( U102_3(x_1, ..., x_3) ) = 2 POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2} POL( U104_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( plus_2(x_1, x_2) ) = max{0, x_1 - 2} POL( x_2(x_1, x_2) ) = x_1 + 2 POL( U62_1(x_1) ) = 2x_1 + 2 POL( U71_2(x_1, x_2) ) = x_2 + 2 POL( U72_2(x_1, x_2) ) = 2 POL( U81_3(x_1, ..., x_3) ) = 2 POL( U82_3(x_1, ..., x_3) ) = 2 POL( U83_3(x_1, ..., x_3) ) = max{0, x_3 - 2} POL( U84_3(x_1, ..., x_3) ) = 2 POL( s_1(x_1) ) = x_1 + 2 POL( U91_2(x_1, x_2) ) = 2x_2 + 2 POL( U92_1(x_1) ) = 2 POL( 0 ) = 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(active(X)) -> U36(X) U36(mark(X)) -> U36(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(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) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) ---------------------------------------- (226) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2)) MARK(U15(X1, X2)) -> MARK(X1) ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2)) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3)) MARK(U33(X1, X2, X3)) -> MARK(X1) MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3)) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) MARK(U35(X1, X2)) -> MARK(X1) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (227) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U61(X1, X2)) -> ACTIVE(U61(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, x_1 - 1} POL( MARK_1(x_1) ) = 1 POL( U11_3(x_1, ..., x_3) ) = 2 POL( U12_3(x_1, ..., x_3) ) = 2 POL( U13_3(x_1, ..., x_3) ) = 2 POL( U14_3(x_1, ..., x_3) ) = 2 POL( U21_2(x_1, x_2) ) = 2 POL( U22_2(x_1, x_2) ) = 2 POL( U31_3(x_1, ..., x_3) ) = 2 POL( U32_3(x_1, ..., x_3) ) = 2 POL( U33_3(x_1, ..., x_3) ) = 2 POL( U34_3(x_1, ..., x_3) ) = 2 POL( U41_2(x_1, x_2) ) = 2 POL( U42_1(x_1) ) = max{0, -2} POL( U51_1(x_1) ) = max{0, x_1 - 2} POL( U61_2(x_1, x_2) ) = max{0, -2} POL( isNatKind_1(x_1) ) = 2 POL( active_1(x_1) ) = max{0, x_1 - 2} POL( mark_1(x_1) ) = 1 POL( U15_2(x_1, x_2) ) = 2 POL( U16_1(x_1) ) = x_1 + 2 POL( U23_1(x_1) ) = x_1 + 2 POL( U35_2(x_1, x_2) ) = 2 POL( U36_1(x_1) ) = x_1 + 1 POL( isNat_1(x_1) ) = 2 POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + 2x_3 - 2} POL( tt ) = 0 POL( U102_3(x_1, ..., x_3) ) = max{0, x_1 - 2} POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2} POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + x_3 - 2} POL( plus_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( x_2(x_1, x_2) ) = max{0, x_1 - 1} POL( U62_1(x_1) ) = 2x_1 + 1 POL( U71_2(x_1, x_2) ) = 2x_2 + 2 POL( U72_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( U81_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + 2x_3 - 2} POL( U82_3(x_1, ..., x_3) ) = max{0, x_3 - 1} POL( U83_3(x_1, ..., x_3) ) = x_1 + x_2 + 2x_3 + 1 POL( U84_3(x_1, ..., x_3) ) = max{0, x_3 - 2} POL( s_1(x_1) ) = 2 POL( U91_2(x_1, x_2) ) = 2x_1 + 2 POL( U92_1(x_1) ) = x_1 + 2 POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(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) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) ---------------------------------------- (228) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2)) MARK(U15(X1, X2)) -> MARK(X1) ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2)) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3)) MARK(U33(X1, X2, X3)) -> MARK(X1) MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3)) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) MARK(U35(X1, X2)) -> MARK(X1) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (229) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U21(X1, X2)) -> MARK(X1) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(MARK(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(isNatKind(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(ACTIVE(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(isNat(x_1)) = [[0A]] + [[1A]] * x_1 >>> <<< POL(U11(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(tt) = [[0A]] >>> <<< POL(U12(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U13(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U14(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U15(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(mark(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U16(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U21(x_1, x_2)) = [[1A]] + [[1A]] * x_1 + [[1A]] * x_2 >>> <<< POL(U22(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(U23(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U31(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U32(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U33(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U34(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U35(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(U36(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U41(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U42(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(plus(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U51(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(s(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(x(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U61(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(active(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U101(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U102(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U103(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U104(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U62(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(U71(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U72(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U81(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U82(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U83(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U84(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U91(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U92(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(0) = [[0A]] >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) mark(isNatKind(X)) -> active(isNatKind(X)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) mark(isNat(X)) -> active(isNat(X)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) mark(U16(X)) -> active(U16(mark(X))) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) mark(U23(X)) -> active(U23(mark(X))) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) active(U72(tt, N)) -> mark(N) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) mark(U36(X)) -> active(U36(mark(X))) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) mark(U42(X)) -> active(U42(mark(X))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) mark(U51(X)) -> active(U51(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) mark(U62(X)) -> active(U62(mark(X))) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) active(x(N, 0)) -> mark(U91(isNat(N), N)) mark(s(X)) -> active(s(mark(X))) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(tt) -> active(tt) mark(0) -> active(0) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(active(X)) -> U36(X) U36(mark(X)) -> U36(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(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) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) active(U16(tt)) -> mark(tt) active(U23(tt)) -> mark(tt) active(U36(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U62(tt)) -> mark(tt) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNatKind(0)) -> mark(tt) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U92(active(X)) -> U92(X) U92(mark(X)) -> U92(X) U71(X1, mark(X2)) -> U71(X1, X2) U71(mark(X1), X2) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U91(X1, mark(X2)) -> U91(X1, X2) U91(mark(X1), X2) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) ---------------------------------------- (230) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2)) MARK(U15(X1, X2)) -> MARK(X1) ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2)) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3)) MARK(U33(X1, X2, X3)) -> MARK(X1) MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3)) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) MARK(U35(X1, X2)) -> MARK(X1) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (231) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U33(X1, X2, X3)) -> MARK(X1) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(MARK(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(isNatKind(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(ACTIVE(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(isNat(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(U11(x_1, x_2, x_3)) = [[2A]] + [[0A]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(tt) = [[0A]] >>> <<< POL(U12(x_1, x_2, x_3)) = [[2A]] + [[0A]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U13(x_1, x_2, x_3)) = [[2A]] + [[0A]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U14(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U15(x_1, x_2)) = [[2A]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(mark(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U16(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U21(x_1, x_2)) = [[2A]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(U22(x_1, x_2)) = [[2A]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(U23(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U31(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U32(x_1, x_2, x_3)) = [[2A]] + [[0A]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U33(x_1, x_2, x_3)) = [[2A]] + [[1A]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U34(x_1, x_2, x_3)) = [[2A]] + [[0A]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U35(x_1, x_2)) = [[2A]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(U36(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U41(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(U42(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(plus(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(U51(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(s(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(x(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(U61(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(active(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U101(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U102(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U103(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U104(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U62(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U71(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U72(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U81(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U82(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U83(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U84(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U91(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U92(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(0) = [[0A]] >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) mark(isNatKind(X)) -> active(isNatKind(X)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) mark(isNat(X)) -> active(isNat(X)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) mark(U16(X)) -> active(U16(mark(X))) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) mark(U23(X)) -> active(U23(mark(X))) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) active(U72(tt, N)) -> mark(N) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) mark(U36(X)) -> active(U36(mark(X))) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) mark(U42(X)) -> active(U42(mark(X))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) mark(U51(X)) -> active(U51(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) mark(U62(X)) -> active(U62(mark(X))) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) active(x(N, 0)) -> mark(U91(isNat(N), N)) mark(s(X)) -> active(s(mark(X))) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(tt) -> active(tt) mark(0) -> active(0) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(active(X)) -> U36(X) U36(mark(X)) -> U36(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(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) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) active(U16(tt)) -> mark(tt) active(U23(tt)) -> mark(tt) active(U36(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U62(tt)) -> mark(tt) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNatKind(0)) -> mark(tt) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U92(active(X)) -> U92(X) U92(mark(X)) -> U92(X) U71(X1, mark(X2)) -> U71(X1, X2) U71(mark(X1), X2) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U91(X1, mark(X2)) -> U91(X1, X2) U91(mark(X1), X2) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) ---------------------------------------- (232) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2)) MARK(U15(X1, X2)) -> MARK(X1) ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2)) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3)) MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3)) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) MARK(U35(X1, X2)) -> MARK(X1) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (233) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U32(tt, V1, V2)) -> MARK(U33(isNatKind(V2), V1, V2)) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(MARK(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(isNatKind(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(ACTIVE(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(isNat(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U11(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(tt) = [[0A]] >>> <<< POL(U12(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U13(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U14(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U15(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(mark(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U16(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U21(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U22(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U23(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U31(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[4A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U32(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[4A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U33(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U34(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U35(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U36(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U41(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U42(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(plus(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U51(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(s(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(x(x_1, x_2)) = [[-I]] + [[4A]] * x_1 + [[1A]] * x_2 >>> <<< POL(U61(x_1, x_2)) = [[-I]] + [[4A]] * x_1 + [[-I]] * x_2 >>> <<< POL(active(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U101(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 + [[4A]] * x_3 >>> <<< POL(U102(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[1A]] * x_2 + [[4A]] * x_3 >>> <<< POL(U103(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[1A]] * x_2 + [[4A]] * x_3 >>> <<< POL(U104(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[1A]] * x_2 + [[4A]] * x_3 >>> <<< POL(U62(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(U71(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U72(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U81(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U82(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U83(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U84(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U91(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(U92(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(0) = [[0A]] >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) mark(isNatKind(X)) -> active(isNatKind(X)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) mark(isNat(X)) -> active(isNat(X)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) mark(U16(X)) -> active(U16(mark(X))) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) mark(U23(X)) -> active(U23(mark(X))) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) active(U72(tt, N)) -> mark(N) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) mark(U36(X)) -> active(U36(mark(X))) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) mark(U42(X)) -> active(U42(mark(X))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) mark(U51(X)) -> active(U51(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) mark(U62(X)) -> active(U62(mark(X))) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) active(x(N, 0)) -> mark(U91(isNat(N), N)) mark(s(X)) -> active(s(mark(X))) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(tt) -> active(tt) mark(0) -> active(0) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(active(X)) -> U36(X) U36(mark(X)) -> U36(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(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) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) active(U16(tt)) -> mark(tt) active(U23(tt)) -> mark(tt) active(U36(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U62(tt)) -> mark(tt) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNatKind(0)) -> mark(tt) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U92(active(X)) -> U92(X) U92(mark(X)) -> U92(X) U71(X1, mark(X2)) -> U71(X1, X2) U71(mark(X1), X2) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U91(X1, mark(X2)) -> U91(X1, X2) U91(mark(X1), X2) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) ---------------------------------------- (234) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) MARK(U15(X1, X2)) -> MARK(X1) ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2)) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U33(X1, X2, X3)) -> ACTIVE(U33(mark(X1), X2, X3)) MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3)) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) MARK(U35(X1, X2)) -> MARK(X1) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (235) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U33(X1, X2, X3)) -> ACTIVE(U33(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) ) = max{0, -2} POL( MARK_1(x_1) ) = max{0, x_1 - 1} POL( U11_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U12_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U13_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U14_3(x_1, ..., x_3) ) = max{0, -2} POL( U21_2(x_1, x_2) ) = 2x_1 + 1 POL( U22_2(x_1, x_2) ) = 2x_1 + 1 POL( U31_3(x_1, ..., x_3) ) = max{0, -2} POL( U32_3(x_1, ..., x_3) ) = x_1 + 1 POL( U34_3(x_1, ..., x_3) ) = 0 POL( U41_2(x_1, x_2) ) = x_1 + 1 POL( U42_1(x_1) ) = x_1 + 1 POL( U51_1(x_1) ) = x_1 + 1 POL( U61_2(x_1, x_2) ) = max{0, -2} POL( isNatKind_1(x_1) ) = 0 POL( active_1(x_1) ) = x_1 + 1 POL( mark_1(x_1) ) = 2x_1 POL( U15_2(x_1, x_2) ) = 2x_1 + 1 POL( U16_1(x_1) ) = x_1 + 1 POL( U23_1(x_1) ) = x_1 + 1 POL( U35_2(x_1, x_2) ) = x_1 + 1 POL( U36_1(x_1) ) = x_1 + 1 POL( isNat_1(x_1) ) = 0 POL( U33_3(x_1, ..., x_3) ) = x_2 + x_3 + 2 POL( U101_3(x_1, ..., x_3) ) = 2 POL( tt ) = 1 POL( U102_3(x_1, ..., x_3) ) = 2 POL( U103_3(x_1, ..., x_3) ) = 2 POL( U104_3(x_1, ..., x_3) ) = 2 POL( plus_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2} POL( x_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U62_1(x_1) ) = 2 POL( U71_2(x_1, x_2) ) = max{0, x_2 - 2} POL( U72_2(x_1, x_2) ) = max{0, x_2 - 2} POL( U81_3(x_1, ..., x_3) ) = max{0, x_3 - 2} POL( U82_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 - 2} POL( U83_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( U84_3(x_1, ..., x_3) ) = 2 POL( s_1(x_1) ) = 0 POL( U91_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( U92_1(x_1) ) = 2 POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(active(X)) -> U36(X) U36(mark(X)) -> U36(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(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) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) ---------------------------------------- (236) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) MARK(U15(X1, X2)) -> MARK(X1) ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2)) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3)) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) MARK(U35(X1, X2)) -> MARK(X1) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (237) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U33(tt, V1, V2)) -> MARK(U34(isNatKind(V2), V1, V2)) 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 - 2} POL( MARK_1(x_1) ) = max{0, -2} POL( U11_3(x_1, ..., x_3) ) = 2 POL( U12_3(x_1, ..., x_3) ) = 0 POL( U13_3(x_1, ..., x_3) ) = max{0, -2} POL( U14_3(x_1, ..., x_3) ) = 2 POL( U21_2(x_1, x_2) ) = 2 POL( U22_2(x_1, x_2) ) = max{0, -2} POL( U31_3(x_1, ..., x_3) ) = 2 POL( U32_3(x_1, ..., x_3) ) = max{0, -2} POL( U34_3(x_1, ..., x_3) ) = 0 POL( U41_2(x_1, x_2) ) = 2 POL( U42_1(x_1) ) = max{0, -2} POL( U51_1(x_1) ) = 2 POL( U61_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( isNatKind_1(x_1) ) = 2 POL( active_1(x_1) ) = x_1 + 1 POL( mark_1(x_1) ) = 2 POL( U15_2(x_1, x_2) ) = 2 POL( U16_1(x_1) ) = 2x_1 + 2 POL( U23_1(x_1) ) = max{0, x_1 - 1} POL( U35_2(x_1, x_2) ) = max{0, -2} POL( U36_1(x_1) ) = max{0, -2} POL( isNat_1(x_1) ) = 0 POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( tt ) = 2 POL( U102_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2 POL( U103_3(x_1, ..., x_3) ) = 2 POL( U104_3(x_1, ..., x_3) ) = max{0, x_2 - 1} POL( plus_2(x_1, x_2) ) = max{0, x_1 - 2} POL( x_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2} POL( U33_3(x_1, ..., x_3) ) = max{0, 2x_1 - 1} POL( U62_1(x_1) ) = 2 POL( U71_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U72_2(x_1, x_2) ) = 2 POL( U81_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2} POL( U82_3(x_1, ..., x_3) ) = 2 POL( U83_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( U84_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2} POL( s_1(x_1) ) = max{0, x_1 - 2} POL( U91_2(x_1, x_2) ) = 2x_1 + 2 POL( U92_1(x_1) ) = 2 POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(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) ---------------------------------------- (238) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U34(X1, X2, X3)) -> ACTIVE(U34(mark(X1), X2, X3)) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) MARK(U35(X1, X2)) -> MARK(X1) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (239) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U34(X1, X2, X3)) -> ACTIVE(U34(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) ) = max{0, -2} POL( MARK_1(x_1) ) = max{0, 2x_1 - 2} POL( U11_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U12_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U13_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U14_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U21_2(x_1, x_2) ) = 1 POL( U22_2(x_1, x_2) ) = 2x_1 + 1 POL( U31_3(x_1, ..., x_3) ) = 1 POL( U32_3(x_1, ..., x_3) ) = x_1 + 1 POL( U41_2(x_1, x_2) ) = 2x_1 + 1 POL( U42_1(x_1) ) = x_1 + 1 POL( U51_1(x_1) ) = 2x_1 + 1 POL( U61_2(x_1, x_2) ) = max{0, -2} POL( isNatKind_1(x_1) ) = 0 POL( active_1(x_1) ) = x_1 + 2 POL( mark_1(x_1) ) = 2x_1 + 2 POL( U15_2(x_1, x_2) ) = 2x_1 + 1 POL( U16_1(x_1) ) = x_1 + 1 POL( U23_1(x_1) ) = x_1 + 1 POL( U35_2(x_1, x_2) ) = x_1 + 1 POL( U36_1(x_1) ) = x_1 + 1 POL( isNat_1(x_1) ) = 0 POL( U34_3(x_1, ..., x_3) ) = x_2 + x_3 + 2 POL( U101_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( tt ) = 2 POL( U102_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_3 - 2} POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2} POL( U104_3(x_1, ..., x_3) ) = max{0, x_2 - 2} POL( plus_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( x_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U33_3(x_1, ..., x_3) ) = 2 POL( U62_1(x_1) ) = 2 POL( U71_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( U72_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U81_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2} POL( U82_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2} POL( U83_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2} POL( U84_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 - 2} POL( s_1(x_1) ) = max{0, -2} POL( U91_2(x_1, x_2) ) = 2x_1 + 2 POL( U92_1(x_1) ) = 2 POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(active(X)) -> U36(X) U36(mark(X)) -> U36(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(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) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) ---------------------------------------- (240) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) MARK(U35(X1, X2)) -> MARK(X1) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (241) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U34(tt, V1, V2)) -> MARK(U35(isNat(V1), V2)) 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 - 2} POL( MARK_1(x_1) ) = max{0, -2} POL( U11_3(x_1, ..., x_3) ) = max{0, -2} POL( U12_3(x_1, ..., x_3) ) = 0 POL( U13_3(x_1, ..., x_3) ) = 0 POL( U14_3(x_1, ..., x_3) ) = max{0, -2} POL( U21_2(x_1, x_2) ) = 2 POL( U22_2(x_1, x_2) ) = 2 POL( U31_3(x_1, ..., x_3) ) = 2 POL( U32_3(x_1, ..., x_3) ) = max{0, -2} POL( U41_2(x_1, x_2) ) = 2 POL( U42_1(x_1) ) = max{0, x_1 - 2} POL( U51_1(x_1) ) = 2 POL( U61_2(x_1, x_2) ) = max{0, x_1 - 2} POL( isNatKind_1(x_1) ) = 0 POL( active_1(x_1) ) = max{0, -2} POL( mark_1(x_1) ) = max{0, -2} POL( U15_2(x_1, x_2) ) = 2 POL( U16_1(x_1) ) = max{0, -2} POL( U23_1(x_1) ) = max{0, 2x_1 - 2} POL( U35_2(x_1, x_2) ) = max{0, -2} POL( U36_1(x_1) ) = max{0, 2x_1 - 2} POL( isNat_1(x_1) ) = 0 POL( U101_3(x_1, ..., x_3) ) = 2 POL( tt ) = 2 POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( U104_3(x_1, ..., x_3) ) = max{0, x_2 - 2} POL( plus_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( x_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( U33_3(x_1, ..., x_3) ) = 0 POL( U34_3(x_1, ..., x_3) ) = 2x_1 POL( U62_1(x_1) ) = 2 POL( U71_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U72_2(x_1, x_2) ) = 2x_1 + 2 POL( U81_3(x_1, ..., x_3) ) = max{0, 2x_2 + x_3 - 2} POL( U82_3(x_1, ..., x_3) ) = x_2 + x_3 POL( U83_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( U84_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( s_1(x_1) ) = max{0, x_1 - 2} POL( U91_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( U92_1(x_1) ) = x_1 POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(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) ---------------------------------------- (242) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) MARK(U35(X1, X2)) -> MARK(X1) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (243) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U35(X1, X2)) -> ACTIVE(U35(mark(X1), X2)) MARK(U35(X1, X2)) -> MARK(X1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, -2} POL( MARK_1(x_1) ) = max{0, 2x_1 - 2} POL( U11_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U12_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U13_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U14_3(x_1, ..., x_3) ) = x_1 + 1 POL( U21_2(x_1, x_2) ) = 1 POL( U22_2(x_1, x_2) ) = x_1 + 1 POL( U31_3(x_1, ..., x_3) ) = max{0, -2} POL( U32_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U41_2(x_1, x_2) ) = 2x_1 + 1 POL( U42_1(x_1) ) = 2x_1 + 1 POL( U51_1(x_1) ) = x_1 + 1 POL( U61_2(x_1, x_2) ) = max{0, -2} POL( isNatKind_1(x_1) ) = 0 POL( active_1(x_1) ) = 2x_1 POL( mark_1(x_1) ) = x_1 POL( U15_2(x_1, x_2) ) = 2x_1 + 1 POL( U16_1(x_1) ) = 2x_1 + 1 POL( U23_1(x_1) ) = x_1 + 1 POL( U36_1(x_1) ) = 2x_1 + 1 POL( isNat_1(x_1) ) = 0 POL( U35_2(x_1, x_2) ) = 2x_1 + x_2 + 2 POL( U101_3(x_1, ..., x_3) ) = 2 POL( tt ) = 1 POL( U102_3(x_1, ..., x_3) ) = 2 POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 - 1} POL( U104_3(x_1, ..., x_3) ) = max{0, x_1 - 2} POL( plus_2(x_1, x_2) ) = 2 POL( x_2(x_1, x_2) ) = 2x_1 + 2 POL( U33_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2} POL( U34_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + 2x_3 - 2} POL( U62_1(x_1) ) = 2 POL( U71_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2} POL( U72_2(x_1, x_2) ) = 2 POL( U81_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 2x_3 + 2 POL( U82_3(x_1, ..., x_3) ) = max{0, x_1 - 2} POL( U83_3(x_1, ..., x_3) ) = 2x_2 + 2 POL( U84_3(x_1, ..., x_3) ) = x_1 + 2 POL( s_1(x_1) ) = max{0, 2x_1 - 2} POL( U91_2(x_1, x_2) ) = 2 POL( U92_1(x_1) ) = max{0, 2x_1 - 1} POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U36(active(X)) -> U36(X) U36(mark(X)) -> U36(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(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) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) ---------------------------------------- (244) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (245) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U35(tt, V2)) -> MARK(U36(isNat(V2))) 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 - 2} POL( MARK_1(x_1) ) = max{0, -2} POL( U11_3(x_1, ..., x_3) ) = max{0, -2} POL( U12_3(x_1, ..., x_3) ) = 2 POL( U13_3(x_1, ..., x_3) ) = 0 POL( U14_3(x_1, ..., x_3) ) = 2 POL( U21_2(x_1, x_2) ) = max{0, -2} POL( U22_2(x_1, x_2) ) = max{0, -2} POL( U31_3(x_1, ..., x_3) ) = 2 POL( U32_3(x_1, ..., x_3) ) = 0 POL( U41_2(x_1, x_2) ) = 2 POL( U42_1(x_1) ) = max{0, 2x_1 - 2} POL( U51_1(x_1) ) = 2 POL( U61_2(x_1, x_2) ) = max{0, x_1 - 2} POL( isNatKind_1(x_1) ) = 0 POL( active_1(x_1) ) = 0 POL( mark_1(x_1) ) = 2 POL( U15_2(x_1, x_2) ) = 0 POL( U16_1(x_1) ) = max{0, 2x_1 - 2} POL( U23_1(x_1) ) = max{0, 2x_1 - 2} POL( U36_1(x_1) ) = 2 POL( isNat_1(x_1) ) = 0 POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( tt ) = 2 POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( U104_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 - 1} POL( plus_2(x_1, x_2) ) = 2 POL( x_2(x_1, x_2) ) = 2 POL( U33_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 - 2} POL( U34_3(x_1, ..., x_3) ) = max{0, x_2 - 2} POL( U35_2(x_1, x_2) ) = 2x_1 + x_2 POL( U62_1(x_1) ) = max{0, 2x_1 - 2} POL( U71_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U72_2(x_1, x_2) ) = 2 POL( U81_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 - 2} POL( U82_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2} POL( U83_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( U84_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( s_1(x_1) ) = 2 POL( U91_2(x_1, x_2) ) = x_2 + 1 POL( U92_1(x_1) ) = 0 POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(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) ---------------------------------------- (246) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> ACTIVE(U32(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (247) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U32(X1, X2, X3)) -> ACTIVE(U32(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) ) = max{0, x_1 - 1} POL( MARK_1(x_1) ) = 1 POL( U11_3(x_1, ..., x_3) ) = 2 POL( U12_3(x_1, ..., x_3) ) = 2 POL( U13_3(x_1, ..., x_3) ) = 2 POL( U14_3(x_1, ..., x_3) ) = 2 POL( U21_2(x_1, x_2) ) = 2 POL( U22_2(x_1, x_2) ) = 2 POL( U31_3(x_1, ..., x_3) ) = 2 POL( U32_3(x_1, ..., x_3) ) = max{0, -2} POL( U41_2(x_1, x_2) ) = 2 POL( U42_1(x_1) ) = max{0, x_1 - 2} POL( U51_1(x_1) ) = 0 POL( U61_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( isNatKind_1(x_1) ) = 2 POL( active_1(x_1) ) = max{0, x_1 - 2} POL( mark_1(x_1) ) = max{0, x_1 - 2} POL( U15_2(x_1, x_2) ) = 2 POL( U16_1(x_1) ) = max{0, -2} POL( U23_1(x_1) ) = 2 POL( isNat_1(x_1) ) = 2 POL( U101_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 - 2} POL( tt ) = 0 POL( U102_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 + x_3 - 2} POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( U104_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + x_3 + 2 POL( plus_2(x_1, x_2) ) = max{0, x_2 - 2} POL( x_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 1} POL( U33_3(x_1, ..., x_3) ) = max{0, -2} POL( U34_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2} POL( U35_2(x_1, x_2) ) = 2 POL( U36_1(x_1) ) = 0 POL( U62_1(x_1) ) = 2 POL( U71_2(x_1, x_2) ) = x_2 + 2 POL( U72_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U81_3(x_1, ..., x_3) ) = x_2 + 2 POL( U82_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2} POL( U83_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + x_3 - 2} POL( U84_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 - 2} POL( s_1(x_1) ) = 2 POL( U91_2(x_1, x_2) ) = max{0, x_2 - 2} POL( U92_1(x_1) ) = 2 POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(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) ---------------------------------------- (248) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U36(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (249) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U36(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) ) = 2 POL( MARK_1(x_1) ) = x_1 + 2 POL( U11_3(x_1, ..., x_3) ) = 2x_1 POL( U12_3(x_1, ..., x_3) ) = 2x_1 POL( U13_3(x_1, ..., x_3) ) = 2x_1 POL( U14_3(x_1, ..., x_3) ) = max{0, -2} POL( U21_2(x_1, x_2) ) = max{0, -2} POL( U22_2(x_1, x_2) ) = 2x_1 POL( U31_3(x_1, ..., x_3) ) = max{0, -2} POL( U32_3(x_1, ..., x_3) ) = x_1 POL( U41_2(x_1, x_2) ) = 2x_1 POL( U42_1(x_1) ) = x_1 POL( U51_1(x_1) ) = 2x_1 POL( U61_2(x_1, x_2) ) = 0 POL( isNatKind_1(x_1) ) = 0 POL( active_1(x_1) ) = x_1 + 2 POL( mark_1(x_1) ) = 2x_1 POL( U15_2(x_1, x_2) ) = 2x_1 POL( U16_1(x_1) ) = x_1 POL( U23_1(x_1) ) = 2x_1 POL( isNat_1(x_1) ) = 0 POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_2 - 1} POL( tt ) = 2 POL( U102_3(x_1, ..., x_3) ) = x_1 + 2 POL( U103_3(x_1, ..., x_3) ) = max{0, x_2 + 2x_3 - 2} POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2} POL( plus_2(x_1, x_2) ) = x_1 POL( x_2(x_1, x_2) ) = 2 POL( U33_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 + 2x_3 - 2} POL( U34_3(x_1, ..., x_3) ) = max{0, x_3 - 2} POL( U35_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U36_1(x_1) ) = 2x_1 + 1 POL( U62_1(x_1) ) = 2 POL( U71_2(x_1, x_2) ) = 2x_2 + 2 POL( U72_2(x_1, x_2) ) = x_1 POL( U81_3(x_1, ..., x_3) ) = 2x_1 + x_3 + 2 POL( U82_3(x_1, ..., x_3) ) = 2 POL( U83_3(x_1, ..., x_3) ) = x_1 + x_2 + x_3 POL( U84_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2} POL( s_1(x_1) ) = max{0, x_1 - 2} POL( U91_2(x_1, x_2) ) = 2x_1 + 2 POL( U92_1(x_1) ) = max{0, -2} POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(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) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) ---------------------------------------- (250) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, V2)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (251) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(isNat(x(V1, V2))) -> MARK(U31(isNatKind(V1), V1, 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(isNatKind(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(ACTIVE(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(isNat(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U11(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(tt) = [[0A]] >>> <<< POL(U12(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U13(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U14(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U15(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(mark(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U16(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U21(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U22(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U23(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U31(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[1A]] * x_2 + [[-I]] * x_3 >>> <<< POL(U32(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 + [[-I]] * x_3 >>> <<< POL(U41(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U42(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(plus(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U51(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(s(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(x(x_1, x_2)) = [[2A]] + [[2A]] * x_1 + [[-I]] * x_2 >>> <<< POL(U61(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(active(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U101(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[2A]] * x_3 >>> <<< POL(U102(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[2A]] * x_3 >>> <<< POL(U103(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[2A]] * x_3 >>> <<< POL(U104(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[2A]] * x_3 >>> <<< POL(U33(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[1A]] * x_2 + [[-I]] * x_3 >>> <<< POL(U34(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[-I]] * x_3 >>> <<< POL(U35(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U36(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(U62(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(U71(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U72(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U81(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U82(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U83(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U84(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U91(x_1, x_2)) = [[1A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U92(x_1)) = [[1A]] + [[-I]] * x_1 >>> <<< POL(0) = [[1A]] >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) mark(isNatKind(X)) -> active(isNatKind(X)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) mark(isNat(X)) -> active(isNat(X)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) mark(U16(X)) -> active(U16(mark(X))) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) mark(U23(X)) -> active(U23(mark(X))) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) active(U72(tt, N)) -> mark(N) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) mark(U36(X)) -> active(U36(mark(X))) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) mark(U42(X)) -> active(U42(mark(X))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) mark(U51(X)) -> active(U51(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) mark(U62(X)) -> active(U62(mark(X))) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) active(x(N, 0)) -> mark(U91(isNat(N), N)) mark(s(X)) -> active(s(mark(X))) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(tt) -> active(tt) mark(0) -> active(0) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(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) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) active(U16(tt)) -> mark(tt) active(U23(tt)) -> mark(tt) active(U36(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U62(tt)) -> mark(tt) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNatKind(0)) -> mark(tt) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(active(X)) -> U36(X) U36(mark(X)) -> U36(X) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U92(active(X)) -> U92(X) U92(mark(X)) -> U92(X) U71(X1, mark(X2)) -> U71(X1, X2) U71(mark(X1), X2) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U91(X1, mark(X2)) -> U91(X1, X2) U91(mark(X1), X2) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) ---------------------------------------- (252) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2, X3)) -> ACTIVE(U31(mark(X1), X2, X3)) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (253) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U31(X1, X2, X3)) -> ACTIVE(U31(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) ) = 0 POL( MARK_1(x_1) ) = max{0, x_1 - 1} POL( U11_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U12_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U13_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U14_3(x_1, ..., x_3) ) = 0 POL( U21_2(x_1, x_2) ) = max{0, -2} POL( U22_2(x_1, x_2) ) = 2x_1 + 1 POL( U32_3(x_1, ..., x_3) ) = x_1 + 1 POL( U41_2(x_1, x_2) ) = 2x_1 + 1 POL( U42_1(x_1) ) = x_1 + 1 POL( U51_1(x_1) ) = x_1 + 1 POL( U61_2(x_1, x_2) ) = max{0, -2} POL( isNatKind_1(x_1) ) = 0 POL( active_1(x_1) ) = 2x_1 + 2 POL( mark_1(x_1) ) = 2x_1 POL( U15_2(x_1, x_2) ) = 2x_1 + 1 POL( U16_1(x_1) ) = x_1 + 1 POL( U23_1(x_1) ) = 2x_1 + 1 POL( isNat_1(x_1) ) = 0 POL( U31_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 2 POL( U101_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 - 2} POL( tt ) = 0 POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2} POL( U103_3(x_1, ..., x_3) ) = x_2 + x_3 POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2} POL( plus_2(x_1, x_2) ) = 2 POL( x_2(x_1, x_2) ) = 2 POL( U33_3(x_1, ..., x_3) ) = max{0, -2} POL( U34_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2} POL( U35_2(x_1, x_2) ) = x_2 + 2 POL( U36_1(x_1) ) = 2 POL( U62_1(x_1) ) = x_1 + 2 POL( U71_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2} POL( U72_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( U81_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 2 POL( U82_3(x_1, ..., x_3) ) = 2 POL( U83_3(x_1, ..., x_3) ) = x_1 + 2 POL( U84_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_3 - 2} POL( s_1(x_1) ) = max{0, x_1 - 2} POL( U91_2(x_1, x_2) ) = x_1 + 2x_2 + 2 POL( U92_1(x_1) ) = x_1 + 2 POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) 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) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) ---------------------------------------- (254) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (255) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U31(tt, V1, V2)) -> MARK(U32(isNatKind(V1), V1, V2)) 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 - 2} POL( MARK_1(x_1) ) = max{0, -2} POL( U11_3(x_1, ..., x_3) ) = 0 POL( U12_3(x_1, ..., x_3) ) = 2 POL( U13_3(x_1, ..., x_3) ) = 2 POL( U14_3(x_1, ..., x_3) ) = 2 POL( U21_2(x_1, x_2) ) = 2 POL( U22_2(x_1, x_2) ) = max{0, -2} POL( U32_3(x_1, ..., x_3) ) = max{0, x_3 - 2} POL( U41_2(x_1, x_2) ) = max{0, -2} POL( U42_1(x_1) ) = 2 POL( U51_1(x_1) ) = 2 POL( U61_2(x_1, x_2) ) = max{0, x_1 - 2} POL( isNatKind_1(x_1) ) = 0 POL( active_1(x_1) ) = max{0, 2x_1 - 2} POL( mark_1(x_1) ) = 2 POL( U15_2(x_1, x_2) ) = max{0, -2} POL( U16_1(x_1) ) = max{0, -2} POL( U23_1(x_1) ) = max{0, -2} POL( isNat_1(x_1) ) = 2 POL( U101_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( tt ) = 2 POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2} POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( U104_3(x_1, ..., x_3) ) = 2 POL( plus_2(x_1, x_2) ) = max{0, x_1 - 1} POL( x_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U31_3(x_1, ..., x_3) ) = max{0, 2x_1 - 1} POL( U33_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2} POL( U34_3(x_1, ..., x_3) ) = x_1 + 2x_3 + 2 POL( U35_2(x_1, x_2) ) = 2 POL( U36_1(x_1) ) = max{0, -2} POL( U62_1(x_1) ) = 2 POL( U71_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U72_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U81_3(x_1, ..., x_3) ) = x_1 + x_2 + x_3 POL( U82_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + x_3 - 2} POL( U83_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( U84_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( s_1(x_1) ) = max{0, x_1 - 2} POL( U91_2(x_1, x_2) ) = max{0, x_2 - 2} POL( U92_1(x_1) ) = 2 POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) 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) ---------------------------------------- (256) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U32(X1, X2, X3)) -> MARK(X1) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (257) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U32(X1, X2, X3)) -> MARK(X1) 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( MARK_1(x_1) ) = max{0, 2x_1 - 2} POL( U11_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U12_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U13_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U14_3(x_1, ..., x_3) ) = max{0, -2} POL( U21_2(x_1, x_2) ) = 1 POL( U22_2(x_1, x_2) ) = 2x_1 + 1 POL( U41_2(x_1, x_2) ) = 2x_1 + 1 POL( U42_1(x_1) ) = 2x_1 + 1 POL( U51_1(x_1) ) = x_1 + 1 POL( U61_2(x_1, x_2) ) = 0 POL( isNatKind_1(x_1) ) = 0 POL( active_1(x_1) ) = x_1 POL( mark_1(x_1) ) = x_1 POL( U15_2(x_1, x_2) ) = 2x_1 + 1 POL( U16_1(x_1) ) = x_1 + 1 POL( U23_1(x_1) ) = 2x_1 + 1 POL( isNat_1(x_1) ) = 0 POL( U101_3(x_1, ..., x_3) ) = 2 POL( tt ) = 0 POL( U102_3(x_1, ..., x_3) ) = 2 POL( U103_3(x_1, ..., x_3) ) = 2 POL( U104_3(x_1, ..., x_3) ) = 2 POL( plus_2(x_1, x_2) ) = 2 POL( x_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U31_3(x_1, ..., x_3) ) = 2x_1 + 2 POL( U32_3(x_1, ..., x_3) ) = 2x_1 + 2 POL( U33_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( U34_3(x_1, ..., x_3) ) = max{0, x_1 - 2} POL( U35_2(x_1, x_2) ) = 2x_2 + 2 POL( U36_1(x_1) ) = max{0, 2x_1 - 2} POL( U62_1(x_1) ) = 2 POL( U71_2(x_1, x_2) ) = x_1 + 2x_2 + 1 POL( U72_2(x_1, x_2) ) = 2 POL( U81_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( U82_3(x_1, ..., x_3) ) = 2 POL( U83_3(x_1, ..., x_3) ) = x_1 + 2x_2 + x_3 + 2 POL( U84_3(x_1, ..., x_3) ) = 2x_2 + 2 POL( s_1(x_1) ) = max{0, -2} POL( U91_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U92_1(x_1) ) = 2x_1 + 2 POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) 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) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) ---------------------------------------- (258) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U13(X1, X2, X3)) -> MARK(X1) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (259) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U13(X1, X2, X3)) -> MARK(X1) 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(isNatKind(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(ACTIVE(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(isNat(x_1)) = [[1A]] + [[-I]] * x_1 >>> <<< POL(U11(x_1, x_2, x_3)) = [[1A]] + [[0A]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(tt) = [[0A]] >>> <<< POL(U12(x_1, x_2, x_3)) = [[1A]] + [[0A]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U13(x_1, x_2, x_3)) = [[1A]] + [[1A]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U14(x_1, x_2, x_3)) = [[1A]] + [[0A]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U15(x_1, x_2)) = [[1A]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(mark(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U16(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U21(x_1, x_2)) = [[1A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U22(x_1, x_2)) = [[1A]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(U23(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U41(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(U42(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(plus(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(U51(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(s(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(x(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(U61(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(active(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U101(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U102(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U103(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U104(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U31(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U32(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U33(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U34(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U35(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U36(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(U62(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U71(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U72(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U81(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U82(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U83(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U84(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U91(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U92(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(0) = [[0A]] >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) mark(isNatKind(X)) -> active(isNatKind(X)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) mark(isNat(X)) -> active(isNat(X)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) mark(U16(X)) -> active(U16(mark(X))) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) mark(U23(X)) -> active(U23(mark(X))) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) active(U72(tt, N)) -> mark(N) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) mark(U36(X)) -> active(U36(mark(X))) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) mark(U42(X)) -> active(U42(mark(X))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) mark(U51(X)) -> active(U51(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) mark(U62(X)) -> active(U62(mark(X))) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) active(x(N, 0)) -> mark(U91(isNat(N), N)) mark(s(X)) -> active(s(mark(X))) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(tt) -> active(tt) mark(0) -> active(0) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) 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) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) active(U16(tt)) -> mark(tt) active(U23(tt)) -> mark(tt) active(U36(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U62(tt)) -> mark(tt) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNatKind(0)) -> mark(tt) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(active(X)) -> U36(X) U36(mark(X)) -> U36(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U92(active(X)) -> U92(X) U92(mark(X)) -> U92(X) U71(X1, mark(X2)) -> U71(X1, X2) U71(mark(X1), X2) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U91(X1, mark(X2)) -> U91(X1, X2) U91(mark(X1), X2) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) ---------------------------------------- (260) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) MARK(U22(X1, X2)) -> MARK(X1) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (261) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U22(X1, X2)) -> MARK(X1) 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(isNatKind(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(ACTIVE(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(isNat(x_1)) = [[-I]] + [[5A]] * x_1 >>> <<< POL(U11(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[5A]] * x_2 + [[5A]] * x_3 >>> <<< POL(tt) = [[0A]] >>> <<< POL(U12(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[5A]] * x_2 + [[5A]] * x_3 >>> <<< POL(U13(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[5A]] * x_2 + [[5A]] * x_3 >>> <<< POL(U14(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[5A]] * x_2 + [[5A]] * x_3 >>> <<< POL(U15(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[5A]] * x_2 >>> <<< POL(mark(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U16(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U21(x_1, x_2)) = [[1A]] + [[0A]] * x_1 + [[5A]] * x_2 >>> <<< POL(U22(x_1, x_2)) = [[1A]] + [[1A]] * x_1 + [[5A]] * x_2 >>> <<< POL(U23(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U41(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(U42(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(plus(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U51(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(s(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(x(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U61(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(active(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U101(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U102(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U103(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U104(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U31(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[5A]] * x_2 + [[4A]] * x_3 >>> <<< POL(U32(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U33(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U34(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U35(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U36(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(U62(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(U71(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U72(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U81(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U82(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U83(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U84(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U91(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U92(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(0) = [[0A]] >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) mark(isNatKind(X)) -> active(isNatKind(X)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) mark(isNat(X)) -> active(isNat(X)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) mark(U16(X)) -> active(U16(mark(X))) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) mark(U23(X)) -> active(U23(mark(X))) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) active(U72(tt, N)) -> mark(N) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) mark(U36(X)) -> active(U36(mark(X))) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) mark(U42(X)) -> active(U42(mark(X))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) mark(U51(X)) -> active(U51(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) mark(U62(X)) -> active(U62(mark(X))) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) active(x(N, 0)) -> mark(U91(isNat(N), N)) mark(s(X)) -> active(s(mark(X))) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(tt) -> active(tt) mark(0) -> active(0) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) 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) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) active(U16(tt)) -> mark(tt) active(U23(tt)) -> mark(tt) active(U36(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U62(tt)) -> mark(tt) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNatKind(0)) -> mark(tt) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(active(X)) -> U36(X) U36(mark(X)) -> U36(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U92(active(X)) -> U92(X) U92(mark(X)) -> U92(X) U71(X1, mark(X2)) -> U71(X1, X2) U71(mark(X1), X2) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U91(X1, mark(X2)) -> U91(X1, X2) U91(mark(X1), X2) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) ---------------------------------------- (262) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U11(X1, X2, X3)) -> MARK(X1) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (263) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U11(X1, X2, X3)) -> MARK(X1) 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(isNatKind(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(ACTIVE(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(isNat(x_1)) = [[0A]] + [[1A]] * x_1 >>> <<< POL(U11(x_1, x_2, x_3)) = [[1A]] + [[1A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(tt) = [[0A]] >>> <<< POL(U12(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U13(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U14(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U15(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(mark(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U16(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U21(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(U22(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(U23(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U41(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U42(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(plus(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U51(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(s(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(x(x_1, x_2)) = [[1A]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(U61(x_1, x_2)) = [[1A]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(active(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U101(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U102(x_1, x_2, x_3)) = [[1A]] + [[0A]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U103(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U104(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U31(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[1A]] * x_2 + [[2A]] * x_3 >>> <<< POL(U32(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U33(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U34(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U35(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U36(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(U62(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U71(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U72(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U81(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U82(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U83(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U84(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U91(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U92(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(0) = [[0A]] >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) mark(isNatKind(X)) -> active(isNatKind(X)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) mark(isNat(X)) -> active(isNat(X)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) mark(U16(X)) -> active(U16(mark(X))) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) mark(U23(X)) -> active(U23(mark(X))) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) active(U72(tt, N)) -> mark(N) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) mark(U36(X)) -> active(U36(mark(X))) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) mark(U42(X)) -> active(U42(mark(X))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) mark(U51(X)) -> active(U51(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) mark(U62(X)) -> active(U62(mark(X))) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) active(x(N, 0)) -> mark(U91(isNat(N), N)) mark(s(X)) -> active(s(mark(X))) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(tt) -> active(tt) mark(0) -> active(0) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) 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) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) active(U16(tt)) -> mark(tt) active(U23(tt)) -> mark(tt) active(U36(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U62(tt)) -> mark(tt) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNatKind(0)) -> mark(tt) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(active(X)) -> U36(X) U36(mark(X)) -> U36(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U92(active(X)) -> U92(X) U92(mark(X)) -> U92(X) U71(X1, mark(X2)) -> U71(X1, X2) U71(mark(X1), X2) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U91(X1, mark(X2)) -> U91(X1, X2) U91(mark(X1), X2) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) ---------------------------------------- (264) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), V2)) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (265) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(isNatKind(x(V1, V2))) -> MARK(U61(isNatKind(V1), 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(isNatKind(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(ACTIVE(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(isNat(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U11(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(tt) = [[0A]] >>> <<< POL(U12(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U13(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U14(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U15(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(mark(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U16(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U21(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U22(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U23(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U41(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U42(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(plus(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U51(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(s(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(x(x_1, x_2)) = [[1A]] + [[5A]] * x_1 + [[1A]] * x_2 >>> <<< POL(U61(x_1, x_2)) = [[-I]] + [[3A]] * x_1 + [[0A]] * x_2 >>> <<< POL(active(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U101(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[5A]] * x_3 >>> <<< POL(U102(x_1, x_2, x_3)) = [[-I]] + [[1A]] * x_1 + [[1A]] * x_2 + [[5A]] * x_3 >>> <<< POL(U103(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[5A]] * x_3 >>> <<< POL(U104(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[5A]] * x_3 >>> <<< POL(U31(x_1, x_2, x_3)) = [[0A]] + [[1A]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U32(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U33(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U34(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U35(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U36(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(U62(x_1)) = [[3A]] + [[0A]] * x_1 >>> <<< POL(U71(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U72(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U81(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U82(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U83(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U84(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U91(x_1, x_2)) = [[0A]] + [[1A]] * x_1 + [[5A]] * x_2 >>> <<< POL(U92(x_1)) = [[1A]] + [[-I]] * x_1 >>> <<< POL(0) = [[0A]] >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) mark(isNatKind(X)) -> active(isNatKind(X)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) mark(isNat(X)) -> active(isNat(X)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) mark(U16(X)) -> active(U16(mark(X))) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) mark(U23(X)) -> active(U23(mark(X))) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) active(U72(tt, N)) -> mark(N) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) mark(U36(X)) -> active(U36(mark(X))) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) mark(U42(X)) -> active(U42(mark(X))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) mark(U51(X)) -> active(U51(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) mark(U62(X)) -> active(U62(mark(X))) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) active(x(N, 0)) -> mark(U91(isNat(N), N)) mark(s(X)) -> active(s(mark(X))) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(tt) -> active(tt) mark(0) -> active(0) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) 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) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) active(U16(tt)) -> mark(tt) active(U23(tt)) -> mark(tt) active(U36(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U62(tt)) -> mark(tt) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNatKind(0)) -> mark(tt) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(active(X)) -> U36(X) U36(mark(X)) -> U36(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U92(active(X)) -> U92(X) U92(mark(X)) -> U92(X) U71(X1, mark(X2)) -> U71(X1, X2) U71(mark(X1), X2) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U91(X1, mark(X2)) -> U91(X1, X2) U91(mark(X1), X2) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) ---------------------------------------- (266) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) MARK(U23(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (267) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U41(tt, V2)) -> MARK(U42(isNatKind(V2))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(MARK(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(isNatKind(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(ACTIVE(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(isNat(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U11(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[4A]] * x_3 >>> <<< POL(tt) = [[0A]] >>> <<< POL(U12(x_1, x_2, x_3)) = [[2A]] + [[0A]] * x_1 + [[0A]] * x_2 + [[4A]] * x_3 >>> <<< POL(U13(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U14(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U15(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(mark(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U16(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U21(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U22(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U23(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U41(x_1, x_2)) = [[2A]] + [[0A]] * x_1 + [[4A]] * x_2 >>> <<< POL(U42(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(plus(x_1, x_2)) = [[2A]] + [[0A]] * x_1 + [[4A]] * x_2 >>> <<< POL(U51(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(s(x_1)) = [[2A]] + [[0A]] * x_1 >>> <<< POL(active(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U101(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[4A]] * x_3 >>> <<< POL(U102(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[4A]] * x_3 >>> <<< POL(U103(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[4A]] * x_3 >>> <<< POL(U104(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[4A]] * x_3 >>> <<< POL(x(x_1, x_2)) = [[0A]] + [[4A]] * x_1 + [[1A]] * x_2 >>> <<< POL(U31(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[4A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U32(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[4A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U33(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[3A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U34(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[2A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U35(x_1, x_2)) = [[-I]] + [[1A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U36(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(U61(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U62(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(U71(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U72(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U81(x_1, x_2, x_3)) = [[-I]] + [[2A]] * x_1 + [[4A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U82(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[4A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U83(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[4A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U84(x_1, x_2, x_3)) = [[2A]] + [[-I]] * x_1 + [[4A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U91(x_1, x_2)) = [[0A]] + [[4A]] * x_1 + [[-I]] * x_2 >>> <<< POL(U92(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(0) = [[0A]] >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) mark(isNatKind(X)) -> active(isNatKind(X)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) mark(isNat(X)) -> active(isNat(X)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) mark(U16(X)) -> active(U16(mark(X))) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) mark(U23(X)) -> active(U23(mark(X))) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) active(U72(tt, N)) -> mark(N) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) mark(U36(X)) -> active(U36(mark(X))) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) mark(U42(X)) -> active(U42(mark(X))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) mark(U51(X)) -> active(U51(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) mark(U62(X)) -> active(U62(mark(X))) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) active(x(N, 0)) -> mark(U91(isNat(N), N)) mark(s(X)) -> active(s(mark(X))) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(tt) -> active(tt) mark(0) -> active(0) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) 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) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) active(U16(tt)) -> mark(tt) active(U23(tt)) -> mark(tt) active(U36(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U62(tt)) -> mark(tt) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNatKind(0)) -> mark(tt) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(active(X)) -> U36(X) U36(mark(X)) -> U36(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U92(active(X)) -> U92(X) U92(mark(X)) -> U92(X) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(mark(X1), X2) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U91(X1, mark(X2)) -> U91(X1, X2) U91(mark(X1), X2) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) ---------------------------------------- (268) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) MARK(U23(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (269) 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)) 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( MARK_1(x_1) ) = 1 POL( U11_3(x_1, ..., x_3) ) = 2 POL( U12_3(x_1, ..., x_3) ) = 2 POL( U13_3(x_1, ..., x_3) ) = 2 POL( U14_3(x_1, ..., x_3) ) = 2 POL( U21_2(x_1, x_2) ) = 2 POL( U22_2(x_1, x_2) ) = 2 POL( U41_2(x_1, x_2) ) = max{0, -2} POL( U51_1(x_1) ) = 0 POL( isNatKind_1(x_1) ) = 2 POL( active_1(x_1) ) = max{0, x_1 - 2} POL( mark_1(x_1) ) = 2 POL( U15_2(x_1, x_2) ) = 2 POL( U16_1(x_1) ) = max{0, -2} POL( U23_1(x_1) ) = 0 POL( isNat_1(x_1) ) = 2 POL( U101_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( tt ) = 0 POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2} POL( U103_3(x_1, ..., x_3) ) = x_2 + 2x_3 POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2} POL( plus_2(x_1, x_2) ) = 2 POL( x_2(x_1, x_2) ) = 2 POL( U31_3(x_1, ..., x_3) ) = max{0, 2x_2 + x_3 - 1} POL( U32_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 + 2x_3 - 2} POL( U33_3(x_1, ..., x_3) ) = x_2 + 2x_3 POL( U34_3(x_1, ..., x_3) ) = x_1 + x_2 + 2 POL( U35_2(x_1, x_2) ) = max{0, x_2 - 2} POL( U36_1(x_1) ) = 0 POL( U42_1(x_1) ) = 2 POL( U61_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2} POL( U62_1(x_1) ) = 0 POL( U71_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( U72_2(x_1, x_2) ) = max{0, x_2 - 2} POL( U81_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_3 - 2} POL( U82_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( U83_3(x_1, ..., x_3) ) = 2 POL( U84_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + x_3 - 2} POL( s_1(x_1) ) = max{0, x_1 - 2} POL( U91_2(x_1, x_2) ) = max{0, x_2 - 2} POL( U92_1(x_1) ) = 1 POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) 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) ---------------------------------------- (270) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) MARK(U23(X)) -> MARK(X) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (271) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U42(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( MARK_1(x_1) ) = x_1 POL( U11_3(x_1, ..., x_3) ) = max{0, -2} POL( U12_3(x_1, ..., x_3) ) = 2x_1 POL( U13_3(x_1, ..., x_3) ) = max{0, -2} POL( U14_3(x_1, ..., x_3) ) = max{0, -2} POL( U21_2(x_1, x_2) ) = max{0, -2} POL( U22_2(x_1, x_2) ) = max{0, -2} POL( U41_2(x_1, x_2) ) = x_1 POL( U51_1(x_1) ) = 2x_1 POL( isNatKind_1(x_1) ) = 0 POL( active_1(x_1) ) = x_1 + 2 POL( mark_1(x_1) ) = 2x_1 POL( U15_2(x_1, x_2) ) = 2x_1 POL( U16_1(x_1) ) = x_1 POL( U23_1(x_1) ) = x_1 POL( isNat_1(x_1) ) = 0 POL( U101_3(x_1, ..., x_3) ) = max{0, x_1 - 2} POL( tt ) = 0 POL( U102_3(x_1, ..., x_3) ) = max{0, x_2 - 2} POL( U103_3(x_1, ..., x_3) ) = 2 POL( U104_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 - 2} POL( plus_2(x_1, x_2) ) = 2 POL( x_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U31_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2} POL( U32_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2 POL( U33_3(x_1, ..., x_3) ) = max{0, 2x_2 + x_3 - 2} POL( U34_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( U35_2(x_1, x_2) ) = 2x_1 + x_2 + 2 POL( U36_1(x_1) ) = 2 POL( U42_1(x_1) ) = x_1 + 2 POL( U61_2(x_1, x_2) ) = max{0, x_2 - 2} POL( U62_1(x_1) ) = max{0, -2} POL( U71_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2} POL( U72_2(x_1, x_2) ) = 2 POL( U81_3(x_1, ..., x_3) ) = x_1 + 2 POL( U82_3(x_1, ..., x_3) ) = 2 POL( U83_3(x_1, ..., x_3) ) = 2 POL( U84_3(x_1, ..., x_3) ) = 2 POL( s_1(x_1) ) = x_1 + 2 POL( U91_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U92_1(x_1) ) = 2 POL( 0 ) = 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) 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) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) ---------------------------------------- (272) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U12(X1, X2, X3)) -> MARK(X1) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) MARK(U23(X)) -> MARK(X) MARK(U41(X1, X2)) -> MARK(X1) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (273) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U12(X1, X2, X3)) -> MARK(X1) 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(isNatKind(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(ACTIVE(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(isNat(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(U11(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(tt) = [[0A]] >>> <<< POL(U12(x_1, x_2, x_3)) = [[1A]] + [[1A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U13(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U14(x_1, x_2, x_3)) = [[1A]] + [[0A]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U15(x_1, x_2)) = [[1A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(mark(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U16(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U21(x_1, x_2)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U22(x_1, x_2)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U23(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U41(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(plus(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U51(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(s(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(active(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U101(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U102(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U103(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U104(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(x(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(U31(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U32(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U33(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U34(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U35(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U36(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(U42(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U61(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(U62(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U71(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U72(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U81(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U82(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U83(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U84(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U91(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U92(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(0) = [[0A]] >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) mark(isNatKind(X)) -> active(isNatKind(X)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) mark(isNat(X)) -> active(isNat(X)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) mark(U16(X)) -> active(U16(mark(X))) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) mark(U23(X)) -> active(U23(mark(X))) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) active(U72(tt, N)) -> mark(N) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) mark(U36(X)) -> active(U36(mark(X))) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) mark(U42(X)) -> active(U42(mark(X))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) mark(U51(X)) -> active(U51(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) mark(U62(X)) -> active(U62(mark(X))) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) active(x(N, 0)) -> mark(U91(isNat(N), N)) mark(s(X)) -> active(s(mark(X))) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(tt) -> active(tt) mark(0) -> active(0) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) 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) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) active(U16(tt)) -> mark(tt) active(U23(tt)) -> mark(tt) active(U36(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U62(tt)) -> mark(tt) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNatKind(0)) -> mark(tt) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(active(X)) -> U36(X) U36(mark(X)) -> U36(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U92(active(X)) -> U92(X) U92(mark(X)) -> U92(X) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(mark(X1), X2) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U91(X1, mark(X2)) -> U91(X1, X2) U91(mark(X1), X2) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) ---------------------------------------- (274) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) MARK(U23(X)) -> MARK(X) MARK(U41(X1, X2)) -> MARK(X1) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (275) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U16(X)) -> MARK(X) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(MARK(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(isNatKind(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(ACTIVE(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(isNat(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U11(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[1A]] * x_3 >>> <<< POL(tt) = [[0A]] >>> <<< POL(U12(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U13(x_1, x_2, x_3)) = [[1A]] + [[0A]] * x_1 + [[0A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U14(x_1, x_2, x_3)) = [[1A]] + [[0A]] * x_1 + [[0A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U15(x_1, x_2)) = [[1A]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(mark(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U16(x_1)) = [[1A]] + [[1A]] * x_1 >>> <<< POL(U21(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U22(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U23(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U41(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(plus(x_1, x_2)) = [[1A]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(U51(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(s(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(active(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U101(x_1, x_2, x_3)) = [[1A]] + [[0A]] * x_1 + [[0A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U102(x_1, x_2, x_3)) = [[1A]] + [[0A]] * x_1 + [[0A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U103(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U104(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[1A]] * x_3 >>> <<< POL(x(x_1, x_2)) = [[1A]] + [[1A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U31(x_1, x_2, x_3)) = [[0A]] + [[0A]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U32(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U33(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U34(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 >>> <<< POL(U35(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U36(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(U42(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U61(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U62(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(U71(x_1, x_2)) = [[1A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U72(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U81(x_1, x_2, x_3)) = [[1A]] + [[0A]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U82(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U83(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U84(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U91(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U92(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(0) = [[0A]] >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) mark(isNatKind(X)) -> active(isNatKind(X)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) mark(isNat(X)) -> active(isNat(X)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) mark(U16(X)) -> active(U16(mark(X))) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) mark(U23(X)) -> active(U23(mark(X))) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) active(U72(tt, N)) -> mark(N) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) mark(U36(X)) -> active(U36(mark(X))) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) mark(U42(X)) -> active(U42(mark(X))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) mark(U51(X)) -> active(U51(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) mark(U62(X)) -> active(U62(mark(X))) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) active(x(N, 0)) -> mark(U91(isNat(N), N)) mark(s(X)) -> active(s(mark(X))) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(tt) -> active(tt) mark(0) -> active(0) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) 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) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) active(U16(tt)) -> mark(tt) active(U23(tt)) -> mark(tt) active(U36(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U62(tt)) -> mark(tt) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNatKind(0)) -> mark(tt) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(active(X)) -> U36(X) U36(mark(X)) -> U36(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U92(active(X)) -> U92(X) U92(mark(X)) -> U92(X) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(mark(X1), X2) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U91(X1, mark(X2)) -> U91(X1, X2) U91(mark(X1), X2) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) ---------------------------------------- (276) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) ACTIVE(U15(tt, V2)) -> MARK(U16(isNat(V2))) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) MARK(U23(X)) -> MARK(X) MARK(U41(X1, X2)) -> MARK(X1) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (277) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U15(tt, V2)) -> MARK(U16(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(isNatKind(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(ACTIVE(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(isNat(x_1)) = [[1A]] + [[2A]] * x_1 >>> <<< POL(U11(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[2A]] * x_2 + [[-I]] * x_3 >>> <<< POL(tt) = [[0A]] >>> <<< POL(U12(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[2A]] * x_2 + [[-I]] * x_3 >>> <<< POL(U13(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[2A]] * x_2 + [[-I]] * x_3 >>> <<< POL(U14(x_1, x_2, x_3)) = [[1A]] + [[-I]] * x_1 + [[2A]] * x_2 + [[-I]] * x_3 >>> <<< POL(U15(x_1, x_2)) = [[1A]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(mark(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U16(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(U21(x_1, x_2)) = [[1A]] + [[-I]] * x_1 + [[2A]] * x_2 >>> <<< POL(U22(x_1, x_2)) = [[1A]] + [[0A]] * x_1 + [[2A]] * x_2 >>> <<< POL(U23(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U41(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(plus(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(U51(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(s(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(active(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U101(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U102(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U103(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(U104(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 >>> <<< POL(x(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U31(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U32(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U33(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U34(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 + [[1A]] * x_3 >>> <<< POL(U35(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U36(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(U42(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(U61(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(U62(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(U71(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U72(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U81(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U82(x_1, x_2, x_3)) = [[0A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U83(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U84(x_1, x_2, x_3)) = [[-I]] + [[-I]] * x_1 + [[-I]] * x_2 + [[0A]] * x_3 >>> <<< POL(U91(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(U92(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(0) = [[0A]] >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) mark(isNatKind(X)) -> active(isNatKind(X)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) mark(isNat(X)) -> active(isNat(X)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) mark(U16(X)) -> active(U16(mark(X))) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) mark(U23(X)) -> active(U23(mark(X))) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) active(U72(tt, N)) -> mark(N) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) mark(U36(X)) -> active(U36(mark(X))) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) mark(U42(X)) -> active(U42(mark(X))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) mark(U51(X)) -> active(U51(mark(X))) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) mark(U62(X)) -> active(U62(mark(X))) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) active(x(N, 0)) -> mark(U91(isNat(N), N)) mark(s(X)) -> active(s(mark(X))) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(tt) -> active(tt) mark(0) -> active(0) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) 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) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) active(U16(tt)) -> mark(tt) active(U23(tt)) -> mark(tt) active(U36(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U62(tt)) -> mark(tt) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNatKind(0)) -> mark(tt) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(X1, mark(X2)) -> U35(X1, X2) U35(mark(X1), X2) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(active(X)) -> U36(X) U36(mark(X)) -> U36(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(active(X)) -> s(X) s(mark(X)) -> s(X) U92(active(X)) -> U92(X) U92(mark(X)) -> U92(X) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(mark(X1), X2) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U91(X1, mark(X2)) -> U91(X1, X2) U91(mark(X1), X2) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) ---------------------------------------- (278) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) MARK(U15(X1, X2)) -> ACTIVE(U15(mark(X1), X2)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) MARK(U23(X)) -> MARK(X) MARK(U41(X1, X2)) -> MARK(X1) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (279) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U15(X1, X2)) -> ACTIVE(U15(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( MARK_1(x_1) ) = 2 POL( U11_3(x_1, ..., x_3) ) = 2 POL( U12_3(x_1, ..., x_3) ) = 2 POL( U13_3(x_1, ..., x_3) ) = 2 POL( U14_3(x_1, ..., x_3) ) = 2 POL( U21_2(x_1, x_2) ) = 2 POL( U22_2(x_1, x_2) ) = 2 POL( U41_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( U51_1(x_1) ) = max{0, -2} POL( isNatKind_1(x_1) ) = 2 POL( active_1(x_1) ) = 2 POL( mark_1(x_1) ) = 2 POL( U15_2(x_1, x_2) ) = max{0, -2} POL( U23_1(x_1) ) = 0 POL( isNat_1(x_1) ) = 2 POL( U101_3(x_1, ..., x_3) ) = max{0, x_1 - 2} POL( tt ) = 0 POL( U102_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 - 2} POL( U103_3(x_1, ..., x_3) ) = 2 POL( U104_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2 POL( plus_2(x_1, x_2) ) = max{0, x_1 - 2} POL( x_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 POL( U16_1(x_1) ) = 2 POL( U31_3(x_1, ..., x_3) ) = max{0, x_2 + 2x_3 - 2} POL( U32_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 - 2} POL( U33_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2} POL( U34_3(x_1, ..., x_3) ) = 2 POL( U35_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 POL( U36_1(x_1) ) = 2x_1 + 2 POL( U42_1(x_1) ) = x_1 + 1 POL( U61_2(x_1, x_2) ) = 2x_1 + 2 POL( U62_1(x_1) ) = 2 POL( U71_2(x_1, x_2) ) = x_2 POL( U72_2(x_1, x_2) ) = x_1 + 1 POL( U81_3(x_1, ..., x_3) ) = x_3 POL( U82_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( U83_3(x_1, ..., x_3) ) = 2x_2 + 2 POL( U84_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2} POL( s_1(x_1) ) = max{0, x_1 - 2} POL( U91_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2} POL( U92_1(x_1) ) = 0 POL( 0 ) = 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) ---------------------------------------- (280) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) MARK(U23(X)) -> MARK(X) MARK(U41(X1, X2)) -> MARK(X1) ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) MARK(U51(X)) -> MARK(X) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (281) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(isNat(plus(V1, V2))) -> MARK(U11(isNatKind(V1), V1, V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1)) 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( MARK_1(x_1) ) = max{0, x_1 - 1} POL( U11_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2 POL( U12_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2 POL( U13_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2 POL( U14_3(x_1, ..., x_3) ) = 2x_2 + x_3 + 2 POL( U21_2(x_1, x_2) ) = 2x_2 + 2 POL( U22_2(x_1, x_2) ) = 2x_2 + 2 POL( U41_2(x_1, x_2) ) = 2x_1 + 1 POL( U51_1(x_1) ) = x_1 + 1 POL( isNatKind_1(x_1) ) = 0 POL( active_1(x_1) ) = 2x_1 + 2 POL( mark_1(x_1) ) = x_1 POL( U15_2(x_1, x_2) ) = x_1 + 1 POL( U23_1(x_1) ) = x_1 + 1 POL( isNat_1(x_1) ) = 2x_1 + 1 POL( U101_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2 POL( tt ) = 0 POL( U102_3(x_1, ..., x_3) ) = x_1 + x_3 + 2 POL( U103_3(x_1, ..., x_3) ) = 2x_2 + 1 POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2} POL( plus_2(x_1, x_2) ) = x_1 + 2x_2 + 2 POL( x_2(x_1, x_2) ) = max{0, -2} POL( U16_1(x_1) ) = 2 POL( U31_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2} POL( U32_3(x_1, ..., x_3) ) = max{0, x_1 - 2} POL( U33_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + 2x_3 - 2} POL( U34_3(x_1, ..., x_3) ) = max{0, x_3 - 2} POL( U35_2(x_1, x_2) ) = 2 POL( U36_1(x_1) ) = 2x_1 POL( U42_1(x_1) ) = max{0, -2} POL( U61_2(x_1, x_2) ) = x_2 + 2 POL( U62_1(x_1) ) = 2 POL( U71_2(x_1, x_2) ) = max{0, x_2 - 2} POL( U72_2(x_1, x_2) ) = max{0, x_2 - 2} POL( U81_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_3 - 1} POL( U82_3(x_1, ..., x_3) ) = 2x_2 + 2 POL( U83_3(x_1, ..., x_3) ) = x_2 + x_3 + 2 POL( U84_3(x_1, ..., x_3) ) = x_2 + 2 POL( s_1(x_1) ) = 2x_1 + 2 POL( U91_2(x_1, x_2) ) = 2 POL( U92_1(x_1) ) = x_1 + 2 POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) 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) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) ---------------------------------------- (282) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) MARK(U23(X)) -> MARK(X) MARK(U41(X1, X2)) -> MARK(X1) MARK(U51(X)) -> MARK(X) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (283) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(isNat(X)) -> ACTIVE(isNat(X)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2} POL( MARK_1(x_1) ) = 2 POL( U12_3(x_1, ..., x_3) ) = 2 POL( U13_3(x_1, ..., x_3) ) = 2 POL( U14_3(x_1, ..., x_3) ) = 2 POL( U22_2(x_1, x_2) ) = 2 POL( U41_2(x_1, x_2) ) = x_1 + 2 POL( U51_1(x_1) ) = max{0, -2} POL( isNatKind_1(x_1) ) = 2 POL( active_1(x_1) ) = max{0, 2x_1 - 2} POL( mark_1(x_1) ) = 2 POL( U15_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( U23_1(x_1) ) = max{0, x_1 - 2} POL( isNat_1(x_1) ) = 0 POL( U11_3(x_1, ..., x_3) ) = 2 POL( U21_2(x_1, x_2) ) = 2 POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( tt ) = 2 POL( U102_3(x_1, ..., x_3) ) = x_2 + x_3 POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + 2x_3 - 2} POL( U104_3(x_1, ..., x_3) ) = max{0, x_3 - 1} POL( plus_2(x_1, x_2) ) = x_1 + x_2 POL( x_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U16_1(x_1) ) = 2x_1 POL( U31_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + x_3 - 2} POL( U32_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2} POL( U33_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2} POL( U34_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2} POL( U35_2(x_1, x_2) ) = 2 POL( U36_1(x_1) ) = 2 POL( U42_1(x_1) ) = max{0, -2} POL( U61_2(x_1, x_2) ) = x_1 + 2 POL( U62_1(x_1) ) = 2 POL( U71_2(x_1, x_2) ) = max{0, x_2 - 2} POL( U72_2(x_1, x_2) ) = 2x_2 + 2 POL( U81_3(x_1, ..., x_3) ) = 2x_2 + 2 POL( U82_3(x_1, ..., x_3) ) = 2 POL( U83_3(x_1, ..., x_3) ) = 2 POL( U84_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2} POL( s_1(x_1) ) = 2 POL( U91_2(x_1, x_2) ) = x_1 + x_2 POL( U92_1(x_1) ) = max{0, 2x_1 - 1} POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) ---------------------------------------- (284) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U11(X1, X2, X3)) -> ACTIVE(U11(mark(X1), X2, X3)) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) MARK(U23(X)) -> MARK(X) MARK(U41(X1, X2)) -> MARK(X1) MARK(U51(X)) -> MARK(X) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (285) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U11(X1, X2, X3)) -> ACTIVE(U11(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) ) = 0 POL( MARK_1(x_1) ) = max{0, 2x_1 - 2} POL( U12_3(x_1, ..., x_3) ) = 1 POL( U13_3(x_1, ..., x_3) ) = 0 POL( U14_3(x_1, ..., x_3) ) = 1 POL( U22_2(x_1, x_2) ) = 0 POL( U41_2(x_1, x_2) ) = 2x_1 + 1 POL( U51_1(x_1) ) = 2x_1 + 1 POL( isNatKind_1(x_1) ) = 0 POL( active_1(x_1) ) = x_1 + 2 POL( mark_1(x_1) ) = 2x_1 + 2 POL( U15_2(x_1, x_2) ) = 2x_1 + 1 POL( U23_1(x_1) ) = 2x_1 + 1 POL( U11_3(x_1, ..., x_3) ) = x_2 + 2 POL( U21_2(x_1, x_2) ) = max{0, -2} POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2} POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2} POL( U16_1(x_1) ) = max{0, 2x_1 - 2} POL( U35_2(x_1, x_2) ) = 2 POL( U36_1(x_1) ) = 2 POL( U71_2(x_1, x_2) ) = 2x_1 + 2 POL( U81_3(x_1, ..., x_3) ) = max{0, x_3 - 2} POL( U83_3(x_1, ..., x_3) ) = max{0, x_1 - 2} POL( U91_2(x_1, x_2) ) = 2x_1 + 2 POL( isNat_1(x_1) ) = 0 POL( tt ) = 1 POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_3 - 2} POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 - 2} POL( plus_2(x_1, x_2) ) = 2 POL( x_2(x_1, x_2) ) = 2 POL( U31_3(x_1, ..., x_3) ) = max{0, x_2 + 2x_3 - 2} POL( U32_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2} POL( U33_3(x_1, ..., x_3) ) = max{0, x_2 - 2} POL( U34_3(x_1, ..., x_3) ) = max{0, x_2 + 2x_3 - 2} POL( U42_1(x_1) ) = 2 POL( U61_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2} POL( U62_1(x_1) ) = max{0, 2x_1 - 2} POL( U72_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U82_3(x_1, ..., x_3) ) = max{0, 2x_2 + x_3 - 2} POL( U84_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2} POL( s_1(x_1) ) = 2 POL( U92_1(x_1) ) = max{0, -2} POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) 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) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) ---------------------------------------- (286) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) MARK(U23(X)) -> MARK(X) MARK(U41(X1, X2)) -> MARK(X1) MARK(U51(X)) -> MARK(X) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (287) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U11(tt, V1, V2)) -> MARK(U12(isNatKind(V1), V1, V2)) 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( MARK_1(x_1) ) = max{0, -2} POL( U12_3(x_1, ..., x_3) ) = max{0, -2} POL( U13_3(x_1, ..., x_3) ) = 0 POL( U14_3(x_1, ..., x_3) ) = 1 POL( U22_2(x_1, x_2) ) = max{0, -2} POL( U41_2(x_1, x_2) ) = max{0, x_2 - 2} POL( U51_1(x_1) ) = max{0, -2} POL( isNatKind_1(x_1) ) = 0 POL( active_1(x_1) ) = 2 POL( mark_1(x_1) ) = max{0, -2} POL( U15_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U23_1(x_1) ) = max{0, 2x_1 - 2} POL( U21_2(x_1, x_2) ) = max{0, -2} POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2} POL( U103_3(x_1, ..., x_3) ) = 2 POL( U16_1(x_1) ) = 2 POL( U35_2(x_1, x_2) ) = max{0, x_2 - 2} POL( U36_1(x_1) ) = 2 POL( U71_2(x_1, x_2) ) = x_2 + 2 POL( U81_3(x_1, ..., x_3) ) = max{0, x_3 - 2} POL( U83_3(x_1, ..., x_3) ) = 2x_1 + 2x_3 + 2 POL( U91_2(x_1, x_2) ) = max{0, x_2 - 2} POL( isNat_1(x_1) ) = 0 POL( tt ) = 2 POL( U102_3(x_1, ..., x_3) ) = 2 POL( U104_3(x_1, ..., x_3) ) = max{0, x_2 - 2} POL( plus_2(x_1, x_2) ) = 2 POL( x_2(x_1, x_2) ) = 2 POL( U11_3(x_1, ..., x_3) ) = 2x_1 + 2 POL( U31_3(x_1, ..., x_3) ) = 2x_2 + 2 POL( U32_3(x_1, ..., x_3) ) = 2 POL( U33_3(x_1, ..., x_3) ) = 2 POL( U34_3(x_1, ..., x_3) ) = 2 POL( U42_1(x_1) ) = max{0, -2} POL( U61_2(x_1, x_2) ) = max{0, x_2 - 2} POL( U62_1(x_1) ) = 2 POL( U72_2(x_1, x_2) ) = 2x_1 + 2 POL( U82_3(x_1, ..., x_3) ) = max{0, x_2 + 2x_3 - 2} POL( U84_3(x_1, ..., x_3) ) = max{0, x_3 - 2} POL( s_1(x_1) ) = 0 POL( U92_1(x_1) ) = 2 POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(mark(X1), X2) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) ---------------------------------------- (288) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) MARK(U23(X)) -> MARK(X) MARK(U41(X1, X2)) -> MARK(X1) MARK(U51(X)) -> MARK(X) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (289) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U12(X1, X2, X3)) -> ACTIVE(U12(mark(X1), X2, X3)) MARK(U21(X1, X2)) -> ACTIVE(U21(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, -2} POL( MARK_1(x_1) ) = max{0, 2x_1 - 2} POL( U13_3(x_1, ..., x_3) ) = max{0, -2} POL( U14_3(x_1, ..., x_3) ) = 2x_1 + 1 POL( U22_2(x_1, x_2) ) = max{0, -2} POL( U41_2(x_1, x_2) ) = x_1 + 1 POL( U51_1(x_1) ) = x_1 + 1 POL( isNatKind_1(x_1) ) = 0 POL( active_1(x_1) ) = 2x_1 + 2 POL( mark_1(x_1) ) = 2x_1 + 2 POL( U15_2(x_1, x_2) ) = 2x_1 + 1 POL( U23_1(x_1) ) = 2x_1 + 1 POL( U12_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( U21_2(x_1, x_2) ) = 2 POL( U101_3(x_1, ..., x_3) ) = 2x_1 + 2 POL( U103_3(x_1, ..., x_3) ) = 2 POL( U16_1(x_1) ) = 2 POL( U35_2(x_1, x_2) ) = 2x_2 + 2 POL( U36_1(x_1) ) = 2 POL( U71_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U81_3(x_1, ..., x_3) ) = max{0, x_2 - 2} POL( U83_3(x_1, ..., x_3) ) = 2 POL( U91_2(x_1, x_2) ) = 2x_2 + 2 POL( isNat_1(x_1) ) = 0 POL( tt ) = 0 POL( U102_3(x_1, ..., x_3) ) = 2 POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_1 - 1} POL( plus_2(x_1, x_2) ) = 2 POL( x_2(x_1, x_2) ) = 2x_2 + 2 POL( U11_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( U31_3(x_1, ..., x_3) ) = x_1 + 2 POL( U32_3(x_1, ..., x_3) ) = max{0, x_1 + x_3 - 2} POL( U33_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2} POL( U34_3(x_1, ..., x_3) ) = 2 POL( U42_1(x_1) ) = 2 POL( U61_2(x_1, x_2) ) = 2x_1 + 2 POL( U62_1(x_1) ) = max{0, -2} POL( U72_2(x_1, x_2) ) = x_1 + x_2 + 2 POL( U82_3(x_1, ..., x_3) ) = 2x_3 + 2 POL( U84_3(x_1, ..., x_3) ) = 2x_1 + 2 POL( s_1(x_1) ) = 2 POL( U92_1(x_1) ) = 2x_1 POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) 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) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) ---------------------------------------- (290) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) MARK(U23(X)) -> MARK(X) MARK(U41(X1, X2)) -> MARK(X1) MARK(U51(X)) -> MARK(X) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (291) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U12(tt, V1, V2)) -> MARK(U13(isNatKind(V2), V1, V2)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = 2x_1 POL( MARK_1(x_1) ) = max{0, -2} POL( U13_3(x_1, ..., x_3) ) = max{0, -2} POL( U14_3(x_1, ..., x_3) ) = max{0, -2} POL( U22_2(x_1, x_2) ) = max{0, -2} POL( U41_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( U51_1(x_1) ) = max{0, 2x_1 - 2} POL( isNatKind_1(x_1) ) = 0 POL( active_1(x_1) ) = max{0, x_1 - 2} POL( mark_1(x_1) ) = 2 POL( U15_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U23_1(x_1) ) = max{0, -2} POL( U101_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2} POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( U16_1(x_1) ) = 0 POL( U35_2(x_1, x_2) ) = 2x_1 + x_2 + 2 POL( U36_1(x_1) ) = max{0, 2x_1 - 2} POL( U71_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2} POL( U81_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2} POL( U83_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2 POL( U91_2(x_1, x_2) ) = 2x_2 + 2 POL( isNat_1(x_1) ) = 0 POL( tt ) = 0 POL( U102_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_3 - 2} POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + 2x_3 - 2} POL( plus_2(x_1, x_2) ) = 2 POL( x_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U11_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2} POL( U12_3(x_1, ..., x_3) ) = 1 POL( U21_2(x_1, x_2) ) = max{0, -2} POL( U31_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_3 - 2} POL( U32_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 - 2} POL( U33_3(x_1, ..., x_3) ) = x_1 + x_3 + 2 POL( U34_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 2 POL( U42_1(x_1) ) = 2 POL( U61_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( U62_1(x_1) ) = 0 POL( U72_2(x_1, x_2) ) = 2x_2 + 2 POL( U82_3(x_1, ..., x_3) ) = 2x_2 + 2 POL( U84_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_3 - 2} POL( s_1(x_1) ) = 0 POL( U92_1(x_1) ) = 0 POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) ---------------------------------------- (292) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) MARK(U23(X)) -> MARK(X) MARK(U41(X1, X2)) -> MARK(X1) MARK(U51(X)) -> MARK(X) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (293) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1)) 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( MARK_1(x_1) ) = max{0, -2} POL( U14_3(x_1, ..., x_3) ) = max{0, -2} POL( U22_2(x_1, x_2) ) = max{0, -2} POL( U41_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2} POL( U51_1(x_1) ) = max{0, -2} POL( isNatKind_1(x_1) ) = 0 POL( active_1(x_1) ) = max{0, 2x_1 - 2} POL( mark_1(x_1) ) = max{0, 2x_1 - 2} POL( U15_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U23_1(x_1) ) = max{0, -2} POL( U13_3(x_1, ..., x_3) ) = 1 POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + x_3 - 2} POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( U16_1(x_1) ) = max{0, 2x_1 - 2} POL( U35_2(x_1, x_2) ) = 2 POL( U36_1(x_1) ) = 2 POL( U71_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2} POL( U81_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_3 - 2} POL( U83_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2} POL( U91_2(x_1, x_2) ) = max{0, x_2 - 2} POL( isNat_1(x_1) ) = 2x_1 POL( tt ) = 0 POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2} POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_3 - 2} POL( plus_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( x_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U11_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 + x_3 - 2} POL( U12_3(x_1, ..., x_3) ) = x_1 + x_3 + 2 POL( U21_2(x_1, x_2) ) = x_1 + 2 POL( U31_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + x_3 - 2} POL( U32_3(x_1, ..., x_3) ) = max{0, x_2 - 2} POL( U33_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_3 - 2} POL( U34_3(x_1, ..., x_3) ) = 2 POL( U42_1(x_1) ) = max{0, x_1 - 2} POL( U61_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U62_1(x_1) ) = 2 POL( U72_2(x_1, x_2) ) = max{0, x_2 - 2} POL( U82_3(x_1, ..., x_3) ) = 2x_1 + x_2 + x_3 + 2 POL( U84_3(x_1, ..., x_3) ) = max{0, x_3 - 2} POL( s_1(x_1) ) = 2 POL( U92_1(x_1) ) = 2 POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) ---------------------------------------- (294) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) MARK(U23(X)) -> MARK(X) MARK(U41(X1, X2)) -> MARK(X1) MARK(U51(X)) -> MARK(X) ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (295) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(isNatKind(plus(V1, V2))) -> MARK(U41(isNatKind(V1), V2)) 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 - 2} POL( MARK_1(x_1) ) = max{0, x_1 - 2} POL( U14_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2 POL( U41_2(x_1, x_2) ) = x_1 + 2 POL( U51_1(x_1) ) = 2x_1 + 2 POL( isNatKind_1(x_1) ) = 2x_1 + 2 POL( active_1(x_1) ) = 2x_1 POL( mark_1(x_1) ) = x_1 + 1 POL( U15_2(x_1, x_2) ) = 2x_1 + 2 POL( U23_1(x_1) ) = x_1 + 2 POL( U13_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2 POL( U22_2(x_1, x_2) ) = x_2 + 2 POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_2 - 1} POL( U103_3(x_1, ..., x_3) ) = x_1 + 2 POL( U16_1(x_1) ) = max{0, 2x_1 - 2} POL( U35_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( U36_1(x_1) ) = max{0, -2} POL( U71_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U81_3(x_1, ..., x_3) ) = 2x_2 + 2 POL( U83_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 1 POL( U91_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( isNat_1(x_1) ) = x_1 POL( tt ) = 2 POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( plus_2(x_1, x_2) ) = x_1 + 2 POL( x_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( U11_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 1} POL( U12_3(x_1, ..., x_3) ) = max{0, x_2 + 2x_3 - 2} POL( U21_2(x_1, x_2) ) = max{0, -2} POL( U31_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2} POL( U32_3(x_1, ..., x_3) ) = max{0, x_2 + 2x_3 - 1} POL( U33_3(x_1, ..., x_3) ) = 2x_1 + x_2 + x_3 + 2 POL( U34_3(x_1, ..., x_3) ) = max{0, x_2 - 2} POL( U42_1(x_1) ) = max{0, -2} POL( U61_2(x_1, x_2) ) = max{0, -2} POL( U62_1(x_1) ) = max{0, -2} POL( U72_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U82_3(x_1, ..., x_3) ) = max{0, x_3 - 2} POL( U84_3(x_1, ..., x_3) ) = x_1 + 2 POL( s_1(x_1) ) = 2x_1 + 2 POL( U92_1(x_1) ) = max{0, -2} POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U22(X1, mark(X2)) -> U22(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) 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) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) ---------------------------------------- (296) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) MARK(U23(X)) -> MARK(X) MARK(U41(X1, X2)) -> MARK(X1) MARK(U51(X)) -> MARK(X) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (297) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U13(X1, X2, X3)) -> ACTIVE(U13(mark(X1), X2, X3)) MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) MARK(U41(X1, X2)) -> MARK(X1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, -2} POL( MARK_1(x_1) ) = max{0, 2x_1 - 2} POL( U14_3(x_1, ..., x_3) ) = max{0, -2} POL( U51_1(x_1) ) = 2x_1 + 1 POL( isNatKind_1(x_1) ) = 0 POL( active_1(x_1) ) = 2x_1 + 2 POL( mark_1(x_1) ) = x_1 POL( U15_2(x_1, x_2) ) = x_1 + 1 POL( U23_1(x_1) ) = x_1 + 1 POL( U13_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 2 POL( U22_2(x_1, x_2) ) = x_2 + 2 POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + x_3 - 2} POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2} POL( U16_1(x_1) ) = 2x_1 + 2 POL( U35_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2} POL( U36_1(x_1) ) = 0 POL( U71_2(x_1, x_2) ) = 2x_1 + 2 POL( U81_3(x_1, ..., x_3) ) = max{0, x_3 - 2} POL( U83_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + 2x_3 - 2} POL( U91_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2} POL( isNat_1(x_1) ) = 0 POL( tt ) = 1 POL( U102_3(x_1, ..., x_3) ) = 2 POL( U104_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2x_3 + 2 POL( plus_2(x_1, x_2) ) = x_1 + 2 POL( x_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 1} POL( U11_3(x_1, ..., x_3) ) = 2 POL( U12_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2} POL( U21_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U31_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 + x_3 - 2} POL( U32_3(x_1, ..., x_3) ) = x_1 + x_2 + 2x_3 + 1 POL( U33_3(x_1, ..., x_3) ) = max{0, -2} POL( U34_3(x_1, ..., x_3) ) = x_2 + 2 POL( U41_2(x_1, x_2) ) = 2x_1 + 2 POL( U42_1(x_1) ) = 2 POL( U61_2(x_1, x_2) ) = 2 POL( U62_1(x_1) ) = 2 POL( U72_2(x_1, x_2) ) = max{0, x_2 - 2} POL( U82_3(x_1, ..., x_3) ) = 2 POL( U84_3(x_1, ..., x_3) ) = 2x_1 + x_2 + x_3 + 2 POL( s_1(x_1) ) = max{0, 2x_1 - 2} POL( U92_1(x_1) ) = 2 POL( 0 ) = 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) ---------------------------------------- (298) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U51(X)) -> MARK(X) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (299) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U13(tt, V1, V2)) -> MARK(U14(isNatKind(V2), V1, V2)) ACTIVE(U14(tt, V1, V2)) -> MARK(U15(isNat(V1), V2)) ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1))) The remaining pairs can at least be oriented weakly. Used ordering: Combined order from the following AFS and order. MARK(x1) = x1 isNatKind(x1) = isNatKind ACTIVE(x1) = x1 U13(x1, x2, x3) = U13 U14(x1, x2, x3) = U14 U15(x1, x2) = x1 isNat(x1) = isNat U22(x1, x2) = U22 U23(x1) = x1 U51(x1) = x1 mark(x1) = x1 active(x1) = x1 Knuth-Bendix order [KBO] with precedence:trivial and weight map: isNat=1 isNatKind=2 U14=2 U13=3 U22=2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U15(X1, mark(X2)) -> U15(X1, X2) U15(mark(X1), X2) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) ---------------------------------------- (300) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U51(X)) -> MARK(X) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (301) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U23(X)) -> MARK(X) The remaining pairs can at least be oriented weakly. Used ordering: Combined order from the following AFS and order. MARK(x1) = x1 isNatKind(x1) = isNatKind ACTIVE(x1) = x1 U14(x1, x2, x3) = U14 U15(x1, x2) = x1 U23(x1) = U23(x1) U51(x1) = x1 active(x1) = x1 mark(x1) = x1 Knuth-Bendix order [KBO] with precedence:trivial and weight map: U23_1=1 isNatKind=1 U14=2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) ---------------------------------------- (302) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) MARK(U15(X1, X2)) -> MARK(X1) MARK(U51(X)) -> MARK(X) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (303) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U15(X1, X2)) -> MARK(X1) The remaining pairs can at least be oriented weakly. Used ordering: Combined order from the following AFS and order. MARK(x1) = x1 isNatKind(x1) = isNatKind ACTIVE(x1) = x1 U14(x1, x2, x3) = U14 U15(x1, x2) = U15(x1) U51(x1) = x1 active(x1) = x1 mark(x1) = x1 Knuth-Bendix order [KBO] with precedence:trivial and weight map: U15_1=1 isNatKind=1 U14=2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) ---------------------------------------- (304) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(U14(X1, X2, X3)) -> ACTIVE(U14(mark(X1), X2, X3)) MARK(U51(X)) -> MARK(X) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (305) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U14(X1, X2, X3)) -> ACTIVE(U14(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) ) = max{0, x_1 - 1} POL( MARK_1(x_1) ) = 1 POL( U51_1(x_1) ) = max{0, x_1 - 2} POL( isNatKind_1(x_1) ) = 2 POL( active_1(x_1) ) = max{0, x_1 - 2} POL( mark_1(x_1) ) = max{0, -2} POL( U14_3(x_1, ..., x_3) ) = 0 POL( U101_3(x_1, ..., x_3) ) = 2x_1 + x_3 + 2 POL( tt ) = 2 POL( U102_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2 POL( U103_3(x_1, ..., x_3) ) = 2x_1 + 2 POL( isNat_1(x_1) ) = 1 POL( U104_3(x_1, ..., x_3) ) = max{0, x_2 + 2x_3 - 2} POL( plus_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( x_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( U11_3(x_1, ..., x_3) ) = x_3 + 2 POL( U12_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2 POL( U13_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2} POL( U15_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U16_1(x_1) ) = 2 POL( U21_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U22_2(x_1, x_2) ) = x_1 + 2 POL( U23_1(x_1) ) = 2 POL( U31_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2} POL( U32_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2} POL( U33_3(x_1, ..., x_3) ) = 2 POL( U34_3(x_1, ..., x_3) ) = x_1 + x_2 + 2 POL( U35_2(x_1, x_2) ) = 2 POL( U36_1(x_1) ) = max{0, -2} POL( U41_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U42_1(x_1) ) = 0 POL( U61_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2} POL( U62_1(x_1) ) = 2 POL( U71_2(x_1, x_2) ) = 2 POL( U72_2(x_1, x_2) ) = max{0, x_2 - 2} POL( U81_3(x_1, ..., x_3) ) = x_2 + x_3 + 2 POL( U82_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2} POL( U83_3(x_1, ..., x_3) ) = 2 POL( U84_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_3 - 2} POL( s_1(x_1) ) = 1 POL( U91_2(x_1, x_2) ) = 2 POL( U92_1(x_1) ) = max{0, -2} POL( 0 ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) ---------------------------------------- (306) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(U51(X)) -> MARK(X) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) The TRS R consists of the following rules: active(U101(tt, M, N)) -> mark(U102(isNatKind(M), M, N)) active(U102(tt, M, N)) -> mark(U103(isNat(N), M, N)) active(U103(tt, M, N)) -> mark(U104(isNatKind(N), M, N)) active(U104(tt, M, N)) -> mark(plus(x(N, M), N)) active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2)) active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2)) active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2)) active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2)) active(U15(tt, V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) active(U22(tt, V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt, V1, V2)) -> mark(U32(isNatKind(V1), V1, V2)) active(U32(tt, V1, V2)) -> mark(U33(isNatKind(V2), V1, V2)) active(U33(tt, V1, V2)) -> mark(U34(isNatKind(V2), V1, V2)) active(U34(tt, V1, V2)) -> mark(U35(isNat(V1), V2)) active(U35(tt, V2)) -> mark(U36(isNat(V2))) active(U36(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatKind(V2))) active(U42(tt)) -> mark(tt) active(U51(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatKind(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, N)) -> mark(U72(isNatKind(N), N)) active(U72(tt, N)) -> mark(N) active(U81(tt, M, N)) -> mark(U82(isNatKind(M), M, N)) active(U82(tt, M, N)) -> mark(U83(isNat(N), M, N)) active(U83(tt, M, N)) -> mark(U84(isNatKind(N), M, N)) active(U84(tt, M, N)) -> mark(s(plus(N, M))) active(U91(tt, N)) -> mark(U92(isNatKind(N))) active(U92(tt)) -> mark(0) active(isNat(0)) -> mark(tt) active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) active(isNat(x(V1, V2))) -> mark(U31(isNatKind(V1), V1, V2)) active(isNatKind(0)) -> mark(tt) active(isNatKind(plus(V1, V2))) -> mark(U41(isNatKind(V1), V2)) active(isNatKind(s(V1))) -> mark(U51(isNatKind(V1))) active(isNatKind(x(V1, V2))) -> mark(U61(isNatKind(V1), V2)) active(plus(N, 0)) -> mark(U71(isNat(N), N)) active(plus(N, s(M))) -> mark(U81(isNat(M), M, N)) active(x(N, 0)) -> mark(U91(isNat(N), N)) active(x(N, s(M))) -> mark(U101(isNat(M), M, N)) mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3)) mark(tt) -> active(tt) mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3)) mark(isNat(X)) -> active(isNat(X)) mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3)) mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2))) mark(x(X1, X2)) -> active(x(mark(X1), mark(X2))) mark(U11(X1, X2, X3)) -> active(U11(mark(X1), X2, X3)) mark(U12(X1, X2, X3)) -> active(U12(mark(X1), X2, X3)) mark(U13(X1, X2, X3)) -> active(U13(mark(X1), X2, X3)) mark(U14(X1, X2, X3)) -> active(U14(mark(X1), X2, X3)) mark(U15(X1, X2)) -> active(U15(mark(X1), X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1, X2)) -> active(U21(mark(X1), X2)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2, X3)) -> active(U31(mark(X1), X2, X3)) mark(U32(X1, X2, X3)) -> active(U32(mark(X1), X2, X3)) mark(U33(X1, X2, X3)) -> active(U33(mark(X1), X2, X3)) mark(U34(X1, X2, X3)) -> active(U34(mark(X1), X2, X3)) mark(U35(X1, X2)) -> active(U35(mark(X1), X2)) mark(U36(X)) -> active(U36(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(U51(X)) -> active(U51(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(U81(X1, X2, X3)) -> active(U81(mark(X1), X2, X3)) mark(U82(X1, X2, X3)) -> active(U82(mark(X1), X2, X3)) mark(U83(X1, X2, X3)) -> active(U83(mark(X1), X2, X3)) mark(U84(X1, X2, X3)) -> active(U84(mark(X1), X2, X3)) mark(s(X)) -> active(s(mark(X))) mark(U91(X1, X2)) -> active(U91(mark(X1), X2)) mark(U92(X)) -> active(U92(mark(X))) mark(0) -> active(0) U101(mark(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, mark(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, mark(X3)) -> U101(X1, X2, X3) U101(active(X1), X2, X3) -> U101(X1, X2, X3) U101(X1, active(X2), X3) -> U101(X1, X2, X3) U101(X1, X2, active(X3)) -> U101(X1, X2, X3) U102(mark(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, mark(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, mark(X3)) -> U102(X1, X2, X3) U102(active(X1), X2, X3) -> U102(X1, X2, X3) U102(X1, active(X2), X3) -> U102(X1, X2, X3) U102(X1, X2, active(X3)) -> U102(X1, X2, X3) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) U103(mark(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, mark(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, mark(X3)) -> U103(X1, X2, X3) U103(active(X1), X2, X3) -> U103(X1, X2, X3) U103(X1, active(X2), X3) -> U103(X1, X2, X3) U103(X1, X2, active(X3)) -> U103(X1, X2, X3) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U104(mark(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, mark(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, mark(X3)) -> U104(X1, X2, X3) U104(active(X1), X2, X3) -> U104(X1, X2, X3) U104(X1, active(X2), X3) -> U104(X1, X2, X3) U104(X1, X2, active(X3)) -> U104(X1, X2, X3) 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) 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) U11(mark(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, mark(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, mark(X3)) -> U11(X1, X2, X3) U11(active(X1), X2, X3) -> U11(X1, X2, X3) U11(X1, active(X2), X3) -> U11(X1, X2, X3) U11(X1, X2, active(X3)) -> U11(X1, X2, X3) U12(mark(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, mark(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, mark(X3)) -> U12(X1, X2, X3) U12(active(X1), X2, X3) -> U12(X1, X2, X3) U12(X1, active(X2), X3) -> U12(X1, X2, X3) U12(X1, X2, active(X3)) -> U12(X1, X2, X3) U13(mark(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, mark(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, mark(X3)) -> U13(X1, X2, X3) U13(active(X1), X2, X3) -> U13(X1, X2, X3) U13(X1, active(X2), X3) -> U13(X1, X2, X3) U13(X1, X2, active(X3)) -> U13(X1, X2, X3) U14(mark(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, mark(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, mark(X3)) -> U14(X1, X2, X3) U14(active(X1), X2, X3) -> U14(X1, X2, X3) U14(X1, active(X2), X3) -> U14(X1, X2, X3) U14(X1, X2, active(X3)) -> U14(X1, X2, X3) U15(mark(X1), X2) -> U15(X1, X2) U15(X1, mark(X2)) -> U15(X1, X2) U15(active(X1), X2) -> U15(X1, X2) U15(X1, active(X2)) -> U15(X1, X2) U16(mark(X)) -> U16(X) U16(active(X)) -> U16(X) U21(mark(X1), X2) -> U21(X1, X2) U21(X1, mark(X2)) -> U21(X1, X2) U21(active(X1), X2) -> U21(X1, X2) U21(X1, active(X2)) -> U21(X1, X2) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, mark(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, mark(X3)) -> U31(X1, X2, X3) U31(active(X1), X2, X3) -> U31(X1, X2, X3) U31(X1, active(X2), X3) -> U31(X1, X2, X3) U31(X1, X2, active(X3)) -> U31(X1, X2, X3) U32(mark(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, mark(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, mark(X3)) -> U32(X1, X2, X3) U32(active(X1), X2, X3) -> U32(X1, X2, X3) U32(X1, active(X2), X3) -> U32(X1, X2, X3) U32(X1, X2, active(X3)) -> U32(X1, X2, X3) U33(mark(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, mark(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, mark(X3)) -> U33(X1, X2, X3) U33(active(X1), X2, X3) -> U33(X1, X2, X3) U33(X1, active(X2), X3) -> U33(X1, X2, X3) U33(X1, X2, active(X3)) -> U33(X1, X2, X3) U34(mark(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, mark(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, mark(X3)) -> U34(X1, X2, X3) U34(active(X1), X2, X3) -> U34(X1, X2, X3) U34(X1, active(X2), X3) -> U34(X1, X2, X3) U34(X1, X2, active(X3)) -> U34(X1, X2, X3) U35(mark(X1), X2) -> U35(X1, X2) U35(X1, mark(X2)) -> U35(X1, X2) U35(active(X1), X2) -> U35(X1, X2) U35(X1, active(X2)) -> U35(X1, X2) U36(mark(X)) -> U36(X) U36(active(X)) -> U36(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) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) U51(mark(X)) -> U51(X) U51(active(X)) -> U51(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U81(mark(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, mark(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, mark(X3)) -> U81(X1, X2, X3) U81(active(X1), X2, X3) -> U81(X1, X2, X3) U81(X1, active(X2), X3) -> U81(X1, X2, X3) U81(X1, X2, active(X3)) -> U81(X1, X2, X3) U82(mark(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, mark(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, mark(X3)) -> U82(X1, X2, X3) U82(active(X1), X2, X3) -> U82(X1, X2, X3) U82(X1, active(X2), X3) -> U82(X1, X2, X3) U82(X1, X2, active(X3)) -> U82(X1, X2, X3) U83(mark(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, mark(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, mark(X3)) -> U83(X1, X2, X3) U83(active(X1), X2, X3) -> U83(X1, X2, X3) U83(X1, active(X2), X3) -> U83(X1, X2, X3) U83(X1, X2, active(X3)) -> U83(X1, X2, X3) U84(mark(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, mark(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, mark(X3)) -> U84(X1, X2, X3) U84(active(X1), X2, X3) -> U84(X1, X2, X3) U84(X1, active(X2), X3) -> U84(X1, X2, X3) U84(X1, X2, active(X3)) -> U84(X1, X2, X3) s(mark(X)) -> s(X) s(active(X)) -> s(X) U91(mark(X1), X2) -> U91(X1, X2) U91(X1, mark(X2)) -> U91(X1, X2) U91(active(X1), X2) -> U91(X1, X2) U91(X1, active(X2)) -> U91(X1, X2) U92(mark(X)) -> U92(X) U92(active(X)) -> U92(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (307) 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. ---------------------------------------- (308) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(U51(X)) -> MARK(X) ACTIVE(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) The TRS R consists of the following rules: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (309) 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(isNatKind(s(V1))) -> MARK(U51(isNatKind(V1))) The following rules are removed from R: isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) U51(active(X)) -> U51(X) U51(mark(X)) -> U51(X) Used ordering: POLO with Polynomial interpretation [POLO]: POL(ACTIVE(x_1)) = x_1 POL(MARK(x_1)) = x_1 POL(U51(x_1)) = x_1 POL(active(x_1)) = 2*x_1 POL(isNatKind(x_1)) = 2*x_1 POL(mark(x_1)) = 2*x_1 POL(s(x_1)) = 2 + 2*x_1 ---------------------------------------- (310) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(U51(X)) -> MARK(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (311) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (312) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U51(X)) -> MARK(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (313) 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(U51(X)) -> MARK(X) The graph contains the following edges 1 > 1 ---------------------------------------- (314) YES