/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 285 ms] (4) QTRS (5) DependencyPairsProof [EQUIVALENT, 962 ms] (6) QDP (7) QDPOrderProof [EQUIVALENT, 1362 ms] (8) QDP (9) DependencyGraphProof [EQUIVALENT, 0 ms] (10) AND (11) QDP (12) QDPOrderProof [EQUIVALENT, 229 ms] (13) QDP (14) DependencyGraphProof [EQUIVALENT, 0 ms] (15) AND (16) QDP (17) QDPOrderProof [EQUIVALENT, 42 ms] (18) QDP (19) QDPOrderProof [EQUIVALENT, 40 ms] (20) QDP (21) QDPOrderProof [EQUIVALENT, 40 ms] (22) QDP (23) MRRProof [EQUIVALENT, 416 ms] (24) QDP (25) PisEmptyProof [EQUIVALENT, 0 ms] (26) YES (27) QDP (28) QDPOrderProof [EQUIVALENT, 48 ms] (29) QDP (30) QDPOrderProof [EQUIVALENT, 50 ms] (31) QDP (32) MRRProof [EQUIVALENT, 1611 ms] (33) QDP (34) QDPOrderProof [EQUIVALENT, 24 ms] (35) QDP (36) PisEmptyProof [EQUIVALENT, 0 ms] (37) YES (38) QDP (39) QDPOrderProof [EQUIVALENT, 26 ms] (40) QDP (41) QDPOrderProof [EQUIVALENT, 62 ms] (42) QDP (43) QDPOrderProof [EQUIVALENT, 48 ms] (44) QDP (45) QDPOrderProof [EQUIVALENT, 69 ms] (46) QDP (47) DependencyGraphProof [EQUIVALENT, 0 ms] (48) AND (49) QDP (50) QDPOrderProof [EQUIVALENT, 54 ms] (51) QDP (52) QDPOrderProof [EQUIVALENT, 50 ms] (53) QDP (54) QDPOrderProof [EQUIVALENT, 40 ms] (55) QDP (56) QDPOrderProof [EQUIVALENT, 53 ms] (57) QDP (58) QDPOrderProof [EQUIVALENT, 43 ms] (59) QDP (60) QDPOrderProof [EQUIVALENT, 1662 ms] (61) QDP (62) MRRProof [EQUIVALENT, 3647 ms] (63) QDP (64) MRRProof [EQUIVALENT, 113 ms] (65) QDP (66) MRRProof [EQUIVALENT, 90 ms] (67) QDP (68) MRRProof [EQUIVALENT, 155 ms] (69) QDP (70) DependencyGraphProof [EQUIVALENT, 0 ms] (71) TRUE (72) QDP (73) QDPOrderProof [EQUIVALENT, 46 ms] (74) QDP (75) QDPOrderProof [EQUIVALENT, 0 ms] (76) QDP (77) QDPOrderProof [EQUIVALENT, 37 ms] (78) QDP (79) QDPOrderProof [EQUIVALENT, 46 ms] (80) QDP (81) QDPOrderProof [EQUIVALENT, 44 ms] (82) QDP (83) QDPOrderProof [EQUIVALENT, 43 ms] (84) QDP (85) MRRProof [EQUIVALENT, 1663 ms] (86) QDP (87) QDPOrderProof [EQUIVALENT, 15 ms] (88) QDP (89) MRRProof [EQUIVALENT, 442 ms] (90) QDP (91) DependencyGraphProof [EQUIVALENT, 0 ms] (92) AND (93) QDP (94) MRRProof [EQUIVALENT, 91 ms] (95) QDP (96) MRRProof [EQUIVALENT, 88 ms] (97) QDP (98) MRRProof [EQUIVALENT, 371 ms] (99) QDP (100) MRRProof [EQUIVALENT, 81 ms] (101) QDP (102) QDPOrderProof [EQUIVALENT, 1268 ms] (103) QDP (104) PisEmptyProof [EQUIVALENT, 0 ms] (105) YES (106) QDP (107) MRRProof [EQUIVALENT, 78 ms] (108) QDP (109) MRRProof [EQUIVALENT, 77 ms] (110) QDP (111) MRRProof [EQUIVALENT, 79 ms] (112) QDP (113) QDPOrderProof [EQUIVALENT, 400 ms] (114) QDP (115) PisEmptyProof [EQUIVALENT, 0 ms] (116) YES (117) QDP (118) QDPOrderProof [EQUIVALENT, 52 ms] (119) QDP (120) QDPOrderProof [EQUIVALENT, 53 ms] (121) QDP (122) MRRProof [EQUIVALENT, 512 ms] (123) QDP (124) QDPOrderProof [EQUIVALENT, 0 ms] (125) QDP (126) PisEmptyProof [EQUIVALENT, 0 ms] (127) YES (128) QDP (129) QDPOrderProof [EQUIVALENT, 60 ms] (130) QDP (131) QDPOrderProof [EQUIVALENT, 52 ms] (132) QDP (133) QDPOrderProof [EQUIVALENT, 51 ms] (134) QDP (135) MRRProof [EQUIVALENT, 3812 ms] (136) QDP (137) MRRProof [EQUIVALENT, 71 ms] (138) QDP (139) MRRProof [EQUIVALENT, 100 ms] (140) QDP (141) MRRProof [EQUIVALENT, 256 ms] (142) QDP (143) DependencyGraphProof [EQUIVALENT, 0 ms] (144) QDP (145) MRRProof [EQUIVALENT, 55 ms] (146) QDP (147) MRRProof [EQUIVALENT, 41 ms] (148) QDP (149) QDPOrderProof [EQUIVALENT, 407 ms] (150) QDP (151) QDPOrderProof [EQUIVALENT, 727 ms] (152) QDP (153) PisEmptyProof [EQUIVALENT, 0 ms] (154) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0(0(1(0(1(2(0(3(3(4(x1)))))))))) -> 4(2(3(4(2(1(1(4(2(0(x1)))))))))) 0(0(1(5(2(3(0(1(5(5(x1)))))))))) -> 0(0(1(3(1(2(5(0(5(5(x1)))))))))) 0(0(4(0(1(0(2(0(3(3(x1)))))))))) -> 0(0(4(0(1(0(0(2(3(3(x1)))))))))) 0(0(5(3(4(5(1(5(3(0(x1)))))))))) -> 0(0(3(1(5(4(5(5(3(0(x1)))))))))) 0(2(2(0(1(0(3(4(0(0(x1)))))))))) -> 0(2(2(0(1(3(0(0(4(0(x1)))))))))) 0(3(2(0(5(0(4(3(5(3(x1)))))))))) -> 0(3(2(0(0(5(4(3(5(3(x1)))))))))) 0(4(3(3(5(1(2(4(3(3(x1)))))))))) -> 0(4(3(3(1(5(2(4(3(3(x1)))))))))) 0(5(2(4(0(0(1(3(4(3(x1)))))))))) -> 0(5(4(2(0(0(1(3(4(3(x1)))))))))) 0(5(5(0(4(0(3(1(2(1(x1)))))))))) -> 0(5(4(5(0(0(3(1(2(1(x1)))))))))) 1(0(0(0(4(0(0(5(0(4(x1)))))))))) -> 2(4(5(5(1(0(5(5(0(2(x1)))))))))) 1(0(0(5(3(4(0(1(4(3(x1)))))))))) -> 1(0(0(5(1(3(0(4(4(3(x1)))))))))) 1(0(1(0(5(5(0(1(3(2(x1)))))))))) -> 1(0(1(5(0(5(0(1(3(2(x1)))))))))) 1(1(3(2(1(0(5(3(3(4(x1)))))))))) -> 1(1(2(3(1(0(5(3(3(4(x1)))))))))) 1(2(1(5(1(5(1(3(4(3(x1)))))))))) -> 1(2(1(3(5(4(5(1(3(1(x1)))))))))) 1(3(0(2(3(2(4(1(2(0(x1)))))))))) -> 1(3(0(1(2(3(2(4(2(0(x1)))))))))) 1(3(2(2(2(5(2(0(1(0(x1)))))))))) -> 1(3(2(2(2(2(5(0(1(0(x1)))))))))) 1(4(5(2(2(4(0(0(3(5(x1)))))))))) -> 4(2(5(0(1(3(0(4(2(5(x1)))))))))) 1(5(0(4(1(3(2(3(3(3(x1)))))))))) -> 1(5(4(0(3(1(2(3(3(3(x1)))))))))) 1(5(1(2(0(2(5(1(3(2(x1)))))))))) -> 1(5(1(0(2(2(5(1(3(2(x1)))))))))) 1(5(3(0(5(5(4(0(4(0(x1)))))))))) -> 1(5(3(0(5(4(5(0(4(0(x1)))))))))) 2(0(0(4(3(2(1(3(3(4(x1)))))))))) -> 2(0(0(3(4(2(1(3(3(4(x1)))))))))) 2(1(0(3(3(2(1(2(2(5(x1)))))))))) -> 2(1(0(3(2(3(1(2(2(5(x1)))))))))) 2(1(2(4(4(3(5(2(4(1(x1)))))))))) -> 2(1(2(4(3(1(4(4(2(5(x1)))))))))) 2(1(3(4(1(3(2(4(2(1(x1)))))))))) -> 2(1(3(4(1(2(3(4(2(1(x1)))))))))) 2(1(4(0(5(3(4(5(5(0(x1)))))))))) -> 2(4(1(5(0(3(4(5(5(0(x1)))))))))) 2(3(5(0(1(3(4(5(5(2(x1)))))))))) -> 2(3(5(0(3(1(4(5(5(2(x1)))))))))) 2(4(0(1(4(1(4(3(3(4(x1)))))))))) -> 2(4(0(4(4(3(1(1(3(4(x1)))))))))) 2(5(1(1(4(5(4(0(4(2(x1)))))))))) -> 2(5(1(1(5(4(4(0(4(2(x1)))))))))) 2(5(1(3(0(3(4(3(5(0(x1)))))))))) -> 2(5(1(3(3(4(0(3(5(0(x1)))))))))) 3(0(1(4(0(3(5(1(4(5(x1)))))))))) -> 3(0(1(1(3(4(5(0(4(5(x1)))))))))) 3(0(2(0(4(3(1(4(3(1(x1)))))))))) -> 1(0(5(0(0(5(2(5(5(2(x1)))))))))) 3(0(4(3(2(4(0(5(2(0(x1)))))))))) -> 3(0(0(2(4(5(0(4(2(3(x1)))))))))) 3(2(0(1(2(4(0(0(5(3(x1)))))))))) -> 3(2(0(1(4(2(3(0(5(0(x1)))))))))) 3(3(5(2(1(5(3(0(4(5(x1)))))))))) -> 3(3(1(4(2(5(5(3(0(5(x1)))))))))) 3(4(2(1(4(3(3(1(3(5(x1)))))))))) -> 3(1(1(3(5(5(0(3(0(5(x1)))))))))) 3(4(5(0(5(1(4(0(5(3(x1)))))))))) -> 3(4(5(0(5(4(1(0(5(3(x1)))))))))) 3(5(0(3(4(2(0(0(3(0(x1)))))))))) -> 3(5(0(3(0(4(2(0(3(0(x1)))))))))) 3(5(1(3(2(0(2(4(2(3(x1)))))))))) -> 3(2(3(1(4(5(0(2(3(2(x1)))))))))) 4(0(4(2(4(3(4(3(4(1(x1)))))))))) -> 4(4(2(3(4(0(3(4(4(1(x1)))))))))) 4(1(3(2(2(0(2(0(1(3(x1)))))))))) -> 4(2(3(2(1(0(2(0(1(3(x1)))))))))) 4(2(5(0(5(1(0(3(1(0(x1)))))))))) -> 4(2(5(0(5(0(1(3(1(0(x1)))))))))) 4(3(2(0(0(0(3(0(0(3(x1)))))))))) -> 4(3(0(2(0(0(3(0(0(3(x1)))))))))) 4(4(0(0(4(3(3(5(3(2(x1)))))))))) -> 4(4(0(4(0(3(3(5(3(2(x1)))))))))) 4(5(1(0(4(3(5(5(5(1(x1)))))))))) -> 4(5(1(0(3(4(5(5(5(1(x1)))))))))) 4(5(1(3(5(1(3(4(1(2(x1)))))))))) -> 4(5(1(3(5(1(3(1(4(2(x1)))))))))) 5(0(3(5(2(3(0(3(4(4(x1)))))))))) -> 5(0(3(5(2(3(3(0(4(4(x1)))))))))) 5(1(0(1(3(2(1(0(2(0(x1)))))))))) -> 5(1(0(1(2(3(1(0(0(2(x1)))))))))) 5(1(1(5(1(5(1(1(4(3(x1)))))))))) -> 5(1(1(5(1(3(1(5(4(1(x1)))))))))) 5(3(0(0(3(4(4(2(3(2(x1)))))))))) -> 5(3(0(0(4(3(4(2(3(2(x1)))))))))) 5(5(3(1(5(5(3(2(0(5(x1)))))))))) -> 5(5(3(1(5(5(2(3(0(5(x1)))))))))) 0(3(5(4(1(5(1(5(1(2(0(x1))))))))))) -> 3(1(3(1(0(5(1(2(4(5(x1)))))))))) 1(2(4(0(4(3(4(0(0(3(2(x1))))))))))) -> 1(1(4(1(4(4(3(2(1(1(x1)))))))))) 1(3(2(3(3(0(2(4(2(3(2(x1))))))))))) -> 2(0(2(5(4(5(5(5(0(5(x1)))))))))) 2(0(4(3(4(5(3(0(0(0(5(x1))))))))))) -> 2(4(4(4(2(0(4(2(4(2(x1)))))))))) 2(4(0(0(5(5(1(3(5(1(4(x1))))))))))) -> 2(2(0(0(0(5(3(2(0(5(x1)))))))))) 3(0(2(4(1(1(2(1(5(2(0(x1))))))))))) -> 2(1(5(3(3(1(1(2(0(2(x1)))))))))) 3(2(4(4(2(4(5(4(1(5(1(x1))))))))))) -> 2(1(4(1(5(5(4(2(0(2(x1)))))))))) 4(0(1(2(3(1(0(2(2(4(5(x1))))))))))) -> 0(5(2(4(5(0(4(5(5(0(x1)))))))))) 4(2(0(3(4(2(1(1(3(3(0(x1))))))))))) -> 1(0(5(5(3(2(2(1(5(4(x1)))))))))) 4(2(0(5(4(5(5(3(2(5(4(x1))))))))))) -> 4(1(3(2(0(0(5(2(4(1(x1)))))))))) 4(2(1(5(3(5(4(1(4(4(3(x1))))))))))) -> 4(0(5(2(3(5(4(3(2(2(x1)))))))))) 4(4(4(0(3(3(4(3(4(5(4(x1))))))))))) -> 3(2(2(4(0(0(1(0(4(5(x1)))))))))) 4(5(5(4(4(4(2(5(2(5(3(x1))))))))))) -> 0(4(0(3(2(4(1(0(0(0(x1)))))))))) 5(3(2(0(0(5(3(1(4(1(0(x1))))))))))) -> 4(4(5(5(0(1(0(2(3(0(x1)))))))))) 5(5(4(0(2(4(5(2(3(4(3(x1))))))))))) -> 1(0(2(1(4(5(1(2(4(5(x1)))))))))) 1(0(2(1(4(3(4(5(2(3(3(1(x1)))))))))))) -> 4(4(5(5(2(2(2(4(0(3(x1)))))))))) 1(2(1(1(1(1(2(5(2(2(4(5(x1)))))))))))) -> 3(2(0(1(0(1(1(0(0(0(x1)))))))))) 3(2(0(0(4(2(3(3(5(5(2(2(x1)))))))))))) -> 1(3(0(3(4(1(3(4(2(3(x1)))))))))) 3(2(0(0(5(4(5(0(5(4(2(5(x1)))))))))))) -> 0(3(0(4(0(1(4(4(4(4(x1)))))))))) 3(2(0(2(0(3(4(4(4(4(1(3(x1)))))))))))) -> 1(1(1(0(5(3(0(0(1(5(x1)))))))))) 4(0(5(0(1(1(0(4(3(5(3(5(x1)))))))))))) -> 2(5(3(4(0(4(4(5(3(1(x1)))))))))) 5(2(0(2(5(5(1(5(5(1(5(1(x1)))))))))))) -> 1(1(3(4(0(5(4(3(2(4(x1)))))))))) 5(2(1(5(5(5(2(0(4(5(2(4(x1)))))))))))) -> 1(1(4(5(4(3(4(2(2(5(x1)))))))))) 0(0(2(0(3(5(5(2(2(1(1(3(2(x1))))))))))))) -> 4(3(2(5(5(3(2(0(4(3(x1)))))))))) 1(2(1(0(2(5(4(2(2(2(5(5(4(x1))))))))))))) -> 3(3(0(0(0(5(3(2(2(3(x1)))))))))) 5(2(2(1(2(1(1(5(0(1(3(3(1(x1))))))))))))) -> 4(0(5(2(0(3(2(3(1(3(x1)))))))))) Q is empty. ---------------------------------------- (1) QTRS Reverse (EQUIVALENT) We applied the QTRS Reverse Processor [REVERSE]. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 4(3(3(0(2(1(0(1(0(0(x1)))))))))) -> 0(2(4(1(1(2(4(3(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 4(0(5(0(0(4(0(0(0(1(x1)))))))))) -> 2(0(5(5(0(1(5(5(4(2(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 1(3(4(1(3(4(0(2(0(3(x1)))))))))) -> 2(5(5(2(5(0(0(5(0(1(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(3(1(3(3(4(1(2(4(3(x1)))))))))) -> 5(0(3(0(5(5(3(1(1(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) 0(2(1(5(1(5(1(4(5(3(0(x1))))))))))) -> 5(4(2(1(5(0(1(3(1(3(x1)))))))))) 2(3(0(0(4(3(4(0(4(2(1(x1))))))))))) -> 1(1(2(3(4(4(1(4(1(1(x1)))))))))) 2(3(2(4(2(0(3(3(2(3(1(x1))))))))))) -> 5(0(5(5(5(4(5(2(0(2(x1)))))))))) 5(0(0(0(3(5(4(3(4(0(2(x1))))))))))) -> 2(4(2(4(0(2(4(4(4(2(x1)))))))))) 4(1(5(3(1(5(5(0(0(4(2(x1))))))))))) -> 5(0(2(3(5(0(0(0(2(2(x1)))))))))) 0(2(5(1(2(1(1(4(2(0(3(x1))))))))))) -> 2(0(2(1(1(3(3(5(1(2(x1)))))))))) 1(5(1(4(5(4(2(4(4(2(3(x1))))))))))) -> 2(0(2(4(5(5(1(4(1(2(x1)))))))))) 5(4(2(2(0(1(3(2(1(0(4(x1))))))))))) -> 0(5(5(4(0(5(4(2(5(0(x1)))))))))) 0(3(3(1(1(2(4(3(0(2(4(x1))))))))))) -> 4(5(1(2(2(3(5(5(0(1(x1)))))))))) 4(5(2(3(5(5(4(5(0(2(4(x1))))))))))) -> 1(4(2(5(0(0(2(3(1(4(x1)))))))))) 3(4(4(1(4(5(3(5(1(2(4(x1))))))))))) -> 2(2(3(4(5(3(2(5(0(4(x1)))))))))) 4(5(4(3(4(3(3(0(4(4(4(x1))))))))))) -> 5(4(0(1(0(0(4(2(2(3(x1)))))))))) 3(5(2(5(2(4(4(4(5(5(4(x1))))))))))) -> 0(0(0(1(4(2(3(0(4(0(x1)))))))))) 0(1(4(1(3(5(0(0(2(3(5(x1))))))))))) -> 0(3(2(0(1(0(5(5(4(4(x1)))))))))) 3(4(3(2(5(4(2(0(4(5(5(x1))))))))))) -> 5(4(2(1(5(4(1(2(0(1(x1)))))))))) 1(3(3(2(5(4(3(4(1(2(0(1(x1)))))))))))) -> 3(0(4(2(2(2(5(5(4(4(x1)))))))))) 5(4(2(2(5(2(1(1(1(1(2(1(x1)))))))))))) -> 0(0(0(1(1(0(1(0(2(3(x1)))))))))) 2(2(5(5(3(3(2(4(0(0(2(3(x1)))))))))))) -> 3(2(4(3(1(4(3(0(3(1(x1)))))))))) 5(2(4(5(0(5(4(5(0(0(2(3(x1)))))))))))) -> 4(4(4(4(1(0(4(0(3(0(x1)))))))))) 3(1(4(4(4(4(3(0(2(0(2(3(x1)))))))))))) -> 5(1(0(0(3(5(0(1(1(1(x1)))))))))) 5(3(5(3(4(0(1(1(0(5(0(4(x1)))))))))))) -> 1(3(5(4(4(0(4(3(5(2(x1)))))))))) 1(5(1(5(5(1(5(5(2(0(2(5(x1)))))))))))) -> 4(2(3(4(5(0(4(3(1(1(x1)))))))))) 4(2(5(4(0(2(5(5(5(1(2(5(x1)))))))))))) -> 5(2(2(4(3(4(5(4(1(1(x1)))))))))) 2(3(1(1(2(2(5(5(3(0(2(0(0(x1))))))))))))) -> 3(4(0(2(3(5(5(2(3(4(x1)))))))))) 4(5(5(2(2(2(4(5(2(0(1(2(1(x1))))))))))))) -> 3(2(2(3(5(0(0(0(3(3(x1)))))))))) 1(3(3(1(0(5(1(1(2(1(2(2(5(x1))))))))))))) -> 3(1(3(2(3(0(2(5(0(4(x1)))))))))) Q is empty. ---------------------------------------- (3) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 9 + x_1 POL(1(x_1)) = 5 + x_1 POL(2(x_1)) = 8 + x_1 POL(3(x_1)) = 9 + x_1 POL(4(x_1)) = 9 + x_1 POL(5(x_1)) = 8 + x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 4(3(3(0(2(1(0(1(0(0(x1)))))))))) -> 0(2(4(1(1(2(4(3(2(4(x1)))))))))) 4(0(5(0(0(4(0(0(0(1(x1)))))))))) -> 2(0(5(5(0(1(5(5(4(2(x1)))))))))) 1(3(4(1(3(4(0(2(0(3(x1)))))))))) -> 2(5(5(2(5(0(0(5(0(1(x1)))))))))) 5(3(1(3(3(4(1(2(4(3(x1)))))))))) -> 5(0(3(0(5(5(3(1(1(3(x1)))))))))) 0(2(1(5(1(5(1(4(5(3(0(x1))))))))))) -> 5(4(2(1(5(0(1(3(1(3(x1)))))))))) 2(3(0(0(4(3(4(0(4(2(1(x1))))))))))) -> 1(1(2(3(4(4(1(4(1(1(x1)))))))))) 2(3(2(4(2(0(3(3(2(3(1(x1))))))))))) -> 5(0(5(5(5(4(5(2(0(2(x1)))))))))) 5(0(0(0(3(5(4(3(4(0(2(x1))))))))))) -> 2(4(2(4(0(2(4(4(4(2(x1)))))))))) 4(1(5(3(1(5(5(0(0(4(2(x1))))))))))) -> 5(0(2(3(5(0(0(0(2(2(x1)))))))))) 0(2(5(1(2(1(1(4(2(0(3(x1))))))))))) -> 2(0(2(1(1(3(3(5(1(2(x1)))))))))) 1(5(1(4(5(4(2(4(4(2(3(x1))))))))))) -> 2(0(2(4(5(5(1(4(1(2(x1)))))))))) 5(4(2(2(0(1(3(2(1(0(4(x1))))))))))) -> 0(5(5(4(0(5(4(2(5(0(x1)))))))))) 0(3(3(1(1(2(4(3(0(2(4(x1))))))))))) -> 4(5(1(2(2(3(5(5(0(1(x1)))))))))) 4(5(2(3(5(5(4(5(0(2(4(x1))))))))))) -> 1(4(2(5(0(0(2(3(1(4(x1)))))))))) 3(4(4(1(4(5(3(5(1(2(4(x1))))))))))) -> 2(2(3(4(5(3(2(5(0(4(x1)))))))))) 4(5(4(3(4(3(3(0(4(4(4(x1))))))))))) -> 5(4(0(1(0(0(4(2(2(3(x1)))))))))) 3(5(2(5(2(4(4(4(5(5(4(x1))))))))))) -> 0(0(0(1(4(2(3(0(4(0(x1)))))))))) 0(1(4(1(3(5(0(0(2(3(5(x1))))))))))) -> 0(3(2(0(1(0(5(5(4(4(x1)))))))))) 3(4(3(2(5(4(2(0(4(5(5(x1))))))))))) -> 5(4(2(1(5(4(1(2(0(1(x1)))))))))) 1(3(3(2(5(4(3(4(1(2(0(1(x1)))))))))))) -> 3(0(4(2(2(2(5(5(4(4(x1)))))))))) 5(4(2(2(5(2(1(1(1(1(2(1(x1)))))))))))) -> 0(0(0(1(1(0(1(0(2(3(x1)))))))))) 2(2(5(5(3(3(2(4(0(0(2(3(x1)))))))))))) -> 3(2(4(3(1(4(3(0(3(1(x1)))))))))) 5(2(4(5(0(5(4(5(0(0(2(3(x1)))))))))))) -> 4(4(4(4(1(0(4(0(3(0(x1)))))))))) 3(1(4(4(4(4(3(0(2(0(2(3(x1)))))))))))) -> 5(1(0(0(3(5(0(1(1(1(x1)))))))))) 5(3(5(3(4(0(1(1(0(5(0(4(x1)))))))))))) -> 1(3(5(4(4(0(4(3(5(2(x1)))))))))) 1(5(1(5(5(1(5(5(2(0(2(5(x1)))))))))))) -> 4(2(3(4(5(0(4(3(1(1(x1)))))))))) 4(2(5(4(0(2(5(5(5(1(2(5(x1)))))))))))) -> 5(2(2(4(3(4(5(4(1(1(x1)))))))))) 2(3(1(1(2(2(5(5(3(0(2(0(0(x1))))))))))))) -> 3(4(0(2(3(5(5(2(3(4(x1)))))))))) 4(5(5(2(2(2(4(5(2(0(1(2(1(x1))))))))))))) -> 3(2(2(3(5(0(0(0(3(3(x1)))))))))) 1(3(3(1(0(5(1(1(2(1(2(2(5(x1))))))))))))) -> 3(1(3(2(3(0(2(5(0(4(x1)))))))))) ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. ---------------------------------------- (5) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: 5^1(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5^1(5(0(5(2(1(3(1(0(0(x1)))))))))) 5^1(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5^1(0(5(2(1(3(1(0(0(x1))))))))) 5^1(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 0^1(5(2(1(3(1(0(0(x1)))))))) 5^1(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5^1(2(1(3(1(0(0(x1))))))) 5^1(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 2^1(1(3(1(0(0(x1)))))) 5^1(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 1^1(3(1(0(0(x1))))) 5^1(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 3^1(1(0(0(x1)))) 3^1(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3^1(3(2(0(0(1(0(4(0(0(x1)))))))))) 3^1(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3^1(2(0(0(1(0(4(0(0(x1))))))))) 3^1(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 2^1(0(0(1(0(4(0(0(x1)))))))) 3^1(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 0^1(0(1(0(4(0(0(x1))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0^1(3(5(5(4(5(1(3(0(0(x1)))))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 3^1(5(5(4(5(1(3(0(0(x1))))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 5^1(5(4(5(1(3(0(0(x1)))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 5^1(4(5(1(3(0(0(x1))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 4^1(5(1(3(0(0(x1)))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 5^1(1(3(0(0(x1))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 1^1(3(0(0(x1)))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 3^1(0(0(x1))) 0^1(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0^1(4(0(0(3(1(0(2(2(0(x1)))))))))) 0^1(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 4^1(0(0(3(1(0(2(2(0(x1))))))))) 0^1(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0^1(0(3(1(0(2(2(0(x1)))))))) 0^1(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0^1(3(1(0(2(2(0(x1))))))) 0^1(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 3^1(1(0(2(2(0(x1)))))) 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3^1(5(3(4(5(0(0(2(3(0(x1)))))))))) 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 5^1(3(4(5(0(0(2(3(0(x1))))))))) 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3^1(4(5(0(0(2(3(0(x1)))))))) 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 4^1(5(0(0(2(3(0(x1))))))) 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 5^1(0(0(2(3(0(x1)))))) 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 0^1(0(2(3(0(x1))))) 3^1(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3^1(3(4(2(5(1(3(3(4(0(x1)))))))))) 3^1(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3^1(4(2(5(1(3(3(4(0(x1))))))))) 3^1(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 4^1(2(5(1(3(3(4(0(x1)))))))) 3^1(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 2^1(5(1(3(3(4(0(x1))))))) 3^1(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 5^1(1(3(3(4(0(x1)))))) 3^1(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 1^1(3(3(4(0(x1))))) 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3^1(4(3(1(0(0(2(4(5(0(x1)))))))))) 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 4^1(3(1(0(0(2(4(5(0(x1))))))))) 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3^1(1(0(0(2(4(5(0(x1)))))))) 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 1^1(0(0(2(4(5(0(x1))))))) 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 0^1(0(2(4(5(0(x1)))))) 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 0^1(2(4(5(0(x1))))) 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 2^1(4(5(0(x1)))) 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 4^1(5(0(x1))) 1^1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1^1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1^1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 2^1(1(3(0(0(5(4(5(0(x1))))))))) 1^1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1^1(3(0(0(5(4(5(0(x1)))))))) 1^1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 3^1(0(0(5(4(5(0(x1))))))) 1^1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 0^1(0(5(4(5(0(x1)))))) 1^1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 0^1(5(4(5(0(x1))))) 1^1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 5^1(4(5(0(x1)))) 1^1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 4^1(5(0(x1))) 3^1(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3^1(4(4(0(3(1(5(0(0(1(x1)))))))))) 3^1(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 4^1(4(0(3(1(5(0(0(1(x1))))))))) 3^1(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 4^1(0(3(1(5(0(0(1(x1)))))))) 3^1(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 0^1(3(1(5(0(0(1(x1))))))) 3^1(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3^1(1(5(0(0(1(x1)))))) 3^1(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 1^1(5(0(0(1(x1))))) 2^1(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2^1(3(1(0(5(0(5(1(0(1(x1)))))))))) 2^1(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 3^1(1(0(5(0(5(1(0(1(x1))))))))) 2^1(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 1^1(0(5(0(5(1(0(1(x1)))))))) 2^1(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 0^1(5(0(5(1(0(1(x1))))))) 2^1(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 5^1(0(5(1(0(1(x1)))))) 2^1(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 0^1(5(1(0(1(x1))))) 2^1(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 5^1(1(0(1(x1)))) 4^1(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4^1(3(3(5(0(1(3(2(1(1(x1)))))))))) 4^1(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 3^1(3(5(0(1(3(2(1(1(x1))))))))) 4^1(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 3^1(5(0(1(3(2(1(1(x1)))))))) 4^1(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 5^1(0(1(3(2(1(1(x1))))))) 4^1(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 0^1(1(3(2(1(1(x1)))))) 4^1(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 1^1(3(2(1(1(x1))))) 4^1(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 3^1(2(1(1(x1)))) 4^1(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 2^1(1(1(x1))) 3^1(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1^1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3^1(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 3^1(1(5(4(5(3(1(2(1(x1))))))))) 3^1(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1^1(5(4(5(3(1(2(1(x1)))))))) 3^1(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 5^1(4(5(3(1(2(1(x1))))))) 3^1(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 4^1(5(3(1(2(1(x1)))))) 3^1(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 5^1(3(1(2(1(x1))))) 3^1(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 3^1(1(2(1(x1)))) 0^1(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0^1(2(4(2(3(2(1(0(3(1(x1)))))))))) 0^1(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 2^1(4(2(3(2(1(0(3(1(x1))))))))) 0^1(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 4^1(2(3(2(1(0(3(1(x1)))))))) 0^1(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 2^1(3(2(1(0(3(1(x1))))))) 0^1(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 3^1(2(1(0(3(1(x1)))))) 0^1(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 2^1(1(0(3(1(x1))))) 0^1(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 1^1(0(3(1(x1)))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0^1(1(0(5(2(2(2(2(3(1(x1)))))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 1^1(0(5(2(2(2(2(3(1(x1))))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0^1(5(2(2(2(2(3(1(x1)))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 5^1(2(2(2(2(3(1(x1))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 2^1(2(2(2(3(1(x1)))))) 5^1(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5^1(2(4(0(3(1(0(5(2(4(x1)))))))))) 5^1(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 2^1(4(0(3(1(0(5(2(4(x1))))))))) 5^1(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 4^1(0(3(1(0(5(2(4(x1)))))))) 5^1(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 0^1(3(1(0(5(2(4(x1))))))) 5^1(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 3^1(1(0(5(2(4(x1)))))) 5^1(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 1^1(0(5(2(4(x1))))) 5^1(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 0^1(5(2(4(x1)))) 5^1(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5^1(2(4(x1))) 5^1(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 2^1(4(x1)) 5^1(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 4^1(x1) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3^1(3(3(2(1(3(0(4(5(1(x1)))))))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3^1(3(2(1(3(0(4(5(1(x1))))))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3^1(2(1(3(0(4(5(1(x1)))))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 2^1(1(3(0(4(5(1(x1))))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 1^1(3(0(4(5(1(x1)))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3^1(0(4(5(1(x1))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 0^1(4(5(1(x1)))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 4^1(5(1(x1))) 2^1(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2^1(3(1(5(2(2(0(1(5(1(x1)))))))))) 2^1(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 3^1(1(5(2(2(0(1(5(1(x1))))))))) 2^1(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 1^1(5(2(2(0(1(5(1(x1)))))))) 2^1(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 5^1(2(2(0(1(5(1(x1))))))) 2^1(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2^1(2(0(1(5(1(x1)))))) 2^1(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2^1(0(1(5(1(x1))))) 2^1(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 0^1(1(5(1(x1)))) 0^1(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0^1(4(0(5(4(5(0(3(5(1(x1)))))))))) 0^1(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 4^1(0(5(4(5(0(3(5(1(x1))))))))) 0^1(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0^1(5(4(5(0(3(5(1(x1)))))))) 0^1(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 5^1(4(5(0(3(5(1(x1))))))) 0^1(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 4^1(5(0(3(5(1(x1)))))) 4^1(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4^1(3(3(1(2(4(3(0(0(2(x1)))))))))) 4^1(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 3^1(3(1(2(4(3(0(0(2(x1))))))))) 4^1(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 3^1(1(2(4(3(0(0(2(x1)))))))) 4^1(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 1^1(2(4(3(0(0(2(x1))))))) 4^1(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 2^1(4(3(0(0(2(x1)))))) 4^1(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4^1(3(0(0(2(x1))))) 4^1(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 3^1(0(0(2(x1)))) 5^1(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5^1(2(2(1(3(2(3(0(1(2(x1)))))))))) 5^1(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 2^1(2(1(3(2(3(0(1(2(x1))))))))) 5^1(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 2^1(1(3(2(3(0(1(2(x1)))))))) 5^1(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 1^1(3(2(3(0(1(2(x1))))))) 5^1(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 3^1(2(3(0(1(2(x1)))))) 5^1(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 2^1(3(0(1(2(x1))))) 1^1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5^1(2(4(4(1(3(4(2(1(2(x1)))))))))) 1^1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 2^1(4(4(1(3(4(2(1(2(x1))))))))) 1^1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 4^1(4(1(3(4(2(1(2(x1)))))))) 1^1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 4^1(1(3(4(2(1(2(x1))))))) 1^1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 1^1(3(4(2(1(2(x1)))))) 1^1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 3^1(4(2(1(2(x1))))) 1^1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1^1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1^1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 2^1(4(3(2(1(4(3(1(2(x1))))))))) 1^1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 4^1(3(2(1(4(3(1(2(x1)))))))) 1^1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 3^1(2(1(4(3(1(2(x1))))))) 1^1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 2^1(1(4(3(1(2(x1)))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0^1(5(5(4(3(0(5(1(4(2(x1)))))))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 5^1(5(4(3(0(5(1(4(2(x1))))))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 5^1(4(3(0(5(1(4(2(x1)))))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 4^1(3(0(5(1(4(2(x1))))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 3^1(0(5(1(4(2(x1)))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0^1(5(1(4(2(x1))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 5^1(1(4(2(x1)))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 1^1(4(2(x1))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 4^1(2(x1)) 2^1(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2^1(5(5(4(1(3(0(5(3(2(x1)))))))))) 2^1(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 5^1(5(4(1(3(0(5(3(2(x1))))))))) 2^1(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 5^1(4(1(3(0(5(3(2(x1)))))))) 2^1(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 4^1(1(3(0(5(3(2(x1))))))) 2^1(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 1^1(3(0(5(3(2(x1)))))) 2^1(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 3^1(0(5(3(2(x1))))) 4^1(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4^1(3(1(1(3(4(4(0(4(2(x1)))))))))) 4^1(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 3^1(1(1(3(4(4(0(4(2(x1))))))))) 4^1(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 1^1(1(3(4(4(0(4(2(x1)))))))) 4^1(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 1^1(3(4(4(0(4(2(x1))))))) 4^1(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 3^1(4(4(0(4(2(x1)))))) 4^1(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4^1(4(0(4(2(x1))))) 4^1(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4^1(0(4(2(x1)))) 2^1(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2^1(4(0(4(4(5(1(1(5(2(x1)))))))))) 2^1(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 4^1(0(4(4(5(1(1(5(2(x1))))))))) 2^1(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 0^1(4(4(5(1(1(5(2(x1)))))))) 2^1(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 4^1(4(5(1(1(5(2(x1))))))) 2^1(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 4^1(5(1(1(5(2(x1)))))) 2^1(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 5^1(1(1(5(2(x1))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0^1(5(3(0(4(3(3(1(5(2(x1)))))))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 5^1(3(0(4(3(3(1(5(2(x1))))))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 3^1(0(4(3(3(1(5(2(x1)))))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0^1(4(3(3(1(5(2(x1))))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 4^1(3(3(1(5(2(x1)))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 3^1(3(1(5(2(x1))))) 5^1(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5^1(4(0(5(4(3(1(1(0(3(x1)))))))))) 5^1(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 4^1(0(5(4(3(1(1(0(3(x1))))))))) 5^1(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 0^1(5(4(3(1(1(0(3(x1)))))))) 5^1(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5^1(4(3(1(1(0(3(x1))))))) 5^1(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 4^1(3(1(1(0(3(x1)))))) 5^1(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 3^1(1(1(0(3(x1))))) 5^1(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 1^1(1(0(3(x1)))) 0^1(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3^1(2(4(0(5(4(2(0(0(3(x1)))))))))) 0^1(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 2^1(4(0(5(4(2(0(0(3(x1))))))))) 0^1(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 4^1(0(5(4(2(0(0(3(x1)))))))) 0^1(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 0^1(5(4(2(0(0(3(x1))))))) 0^1(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 5^1(4(2(0(0(3(x1)))))) 0^1(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 4^1(2(0(0(3(x1))))) 0^1(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 2^1(0(0(3(x1)))) 0^1(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 0^1(0(3(x1))) 3^1(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0^1(5(0(3(2(4(1(0(2(3(x1)))))))))) 3^1(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 5^1(0(3(2(4(1(0(2(3(x1))))))))) 3^1(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0^1(3(2(4(1(0(2(3(x1)))))))) 3^1(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 3^1(2(4(1(0(2(3(x1))))))) 3^1(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 2^1(4(1(0(2(3(x1)))))) 3^1(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 4^1(1(0(2(3(x1))))) 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5^1(0(3(5(5(2(4(1(3(3(x1)))))))))) 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 0^1(3(5(5(2(4(1(3(3(x1))))))))) 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 3^1(5(5(2(4(1(3(3(x1)))))))) 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5^1(5(2(4(1(3(3(x1))))))) 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5^1(2(4(1(3(3(x1)))))) 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 2^1(4(1(3(3(x1))))) 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 4^1(1(3(3(x1)))) 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 1^1(3(3(x1))) 3^1(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3^1(5(0(1(4(5(0(5(4(3(x1)))))))))) 3^1(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 5^1(0(1(4(5(0(5(4(3(x1))))))))) 3^1(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 0^1(1(4(5(0(5(4(3(x1)))))))) 3^1(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 1^1(4(5(0(5(4(3(x1))))))) 3^1(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 4^1(5(0(5(4(3(x1)))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0^1(3(0(2(4(0(3(0(5(3(x1)))))))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 3^1(0(2(4(0(3(0(5(3(x1))))))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0^1(2(4(0(3(0(5(3(x1)))))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 2^1(4(0(3(0(5(3(x1))))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 4^1(0(3(0(5(3(x1)))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0^1(3(0(5(3(x1))))) 3^1(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2^1(3(2(0(5(4(1(3(2(3(x1)))))))))) 3^1(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 3^1(2(0(5(4(1(3(2(3(x1))))))))) 3^1(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2^1(0(5(4(1(3(2(3(x1)))))))) 3^1(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 0^1(5(4(1(3(2(3(x1))))))) 3^1(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 5^1(4(1(3(2(3(x1)))))) 3^1(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 4^1(1(3(2(3(x1))))) 3^1(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 1^1(3(2(3(x1)))) 3^1(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 3^1(2(3(x1))) 3^1(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2^1(3(x1)) 1^1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1^1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1^1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 4^1(4(3(0(4(3(2(4(4(x1))))))))) 1^1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 4^1(3(0(4(3(2(4(4(x1)))))))) 1^1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 3^1(0(4(3(2(4(4(x1))))))) 1^1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 0^1(4(3(2(4(4(x1)))))) 1^1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 4^1(3(2(4(4(x1))))) 1^1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 3^1(2(4(4(x1)))) 1^1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 2^1(4(4(x1))) 1^1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 4^1(4(x1)) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3^1(1(0(2(0(1(2(3(2(4(x1)))))))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 1^1(0(2(0(1(2(3(2(4(x1))))))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 0^1(2(0(1(2(3(2(4(x1)))))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 2^1(0(1(2(3(2(4(x1))))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 0^1(1(2(3(2(4(x1)))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 1^1(2(3(2(4(x1))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 2^1(3(2(4(x1)))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3^1(2(4(x1))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 2^1(4(x1)) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(1(3(1(0(5(0(5(2(4(x1)))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 1^1(3(1(0(5(0(5(2(4(x1))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 3^1(1(0(5(0(5(2(4(x1)))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 1^1(0(5(0(5(2(4(x1))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(5(0(5(2(4(x1)))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3^1(0(0(3(0(0(2(0(3(4(x1)))))))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 0^1(0(3(0(0(2(0(3(4(x1))))))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 0^1(3(0(0(2(0(3(4(x1)))))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3^1(0(0(2(0(3(4(x1))))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 0^1(0(2(0(3(4(x1)))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 0^1(2(0(3(4(x1))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 2^1(0(3(4(x1)))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 0^1(3(4(x1))) 2^1(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2^1(3(5(3(3(0(4(0(4(4(x1)))))))))) 2^1(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 3^1(5(3(3(0(4(0(4(4(x1))))))))) 2^1(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 5^1(3(3(0(4(0(4(4(x1)))))))) 2^1(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 3^1(3(0(4(0(4(4(x1))))))) 2^1(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 3^1(0(4(0(4(4(x1)))))) 2^1(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 0^1(4(0(4(4(x1))))) 2^1(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 4^1(0(4(4(x1)))) 1^1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1^1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1^1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 5^1(5(5(4(3(0(1(5(4(x1))))))))) 1^1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 5^1(5(4(3(0(1(5(4(x1)))))))) 1^1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 5^1(4(3(0(1(5(4(x1))))))) 1^1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 4^1(3(0(1(5(4(x1)))))) 1^1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 3^1(0(1(5(4(x1))))) 2^1(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2^1(4(1(3(1(5(3(1(5(4(x1)))))))))) 2^1(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 4^1(1(3(1(5(3(1(5(4(x1))))))))) 2^1(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 1^1(3(1(5(3(1(5(4(x1)))))))) 4^1(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4^1(4(0(3(3(2(5(3(0(5(x1)))))))))) 4^1(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4^1(0(3(3(2(5(3(0(5(x1))))))))) 4^1(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 0^1(3(3(2(5(3(0(5(x1)))))))) 4^1(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 3^1(3(2(5(3(0(5(x1))))))) 0^1(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2^1(0(0(1(3(2(1(0(1(5(x1)))))))))) 0^1(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 0^1(0(1(3(2(1(0(1(5(x1))))))))) 0^1(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 0^1(1(3(2(1(0(1(5(x1)))))))) 0^1(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 1^1(3(2(1(0(1(5(x1))))))) 0^1(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 3^1(2(1(0(1(5(x1)))))) 0^1(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2^1(1(0(1(5(x1))))) 3^1(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1^1(4(5(1(3(1(5(1(1(5(x1)))))))))) 3^1(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 4^1(5(1(3(1(5(1(1(5(x1))))))))) 3^1(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 5^1(1(3(1(5(1(1(5(x1)))))))) 3^1(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1^1(3(1(5(1(1(5(x1))))))) 3^1(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 3^1(1(5(1(1(5(x1)))))) 2^1(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2^1(3(2(4(3(4(0(0(3(5(x1)))))))))) 2^1(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 3^1(2(4(3(4(0(0(3(5(x1))))))))) 2^1(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2^1(4(3(4(0(0(3(5(x1)))))))) 2^1(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 4^1(3(4(0(0(3(5(x1))))))) 2^1(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 3^1(4(0(0(3(5(x1)))))) 2^1(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 4^1(0(0(3(5(x1))))) 5^1(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5^1(0(3(2(5(5(1(3(5(5(x1)))))))))) 5^1(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 0^1(3(2(5(5(1(3(5(5(x1))))))))) 5^1(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 3^1(2(5(5(1(3(5(5(x1)))))))) 5^1(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 2^1(5(5(1(3(5(5(x1))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (7) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 5^1(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 2^1(1(3(1(0(0(x1)))))) 5^1(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 1^1(3(1(0(0(x1))))) 5^1(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 3^1(1(0(0(x1)))) 3^1(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 2^1(0(0(1(0(4(0(0(x1)))))))) 3^1(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 0^1(0(1(0(4(0(0(x1))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 4^1(5(1(3(0(0(x1)))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 5^1(1(3(0(0(x1))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 1^1(3(0(0(x1)))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 3^1(0(0(x1))) 0^1(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 4^1(0(0(3(1(0(2(2(0(x1))))))))) 0^1(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0^1(0(3(1(0(2(2(0(x1)))))))) 0^1(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0^1(3(1(0(2(2(0(x1))))))) 0^1(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 3^1(1(0(2(2(0(x1)))))) 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 4^1(5(0(0(2(3(0(x1))))))) 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 5^1(0(0(2(3(0(x1)))))) 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 0^1(0(2(3(0(x1))))) 3^1(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 4^1(2(5(1(3(3(4(0(x1)))))))) 3^1(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 2^1(5(1(3(3(4(0(x1))))))) 3^1(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 5^1(1(3(3(4(0(x1)))))) 3^1(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 1^1(3(3(4(0(x1))))) 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 4^1(3(1(0(0(2(4(5(0(x1))))))))) 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3^1(1(0(0(2(4(5(0(x1)))))))) 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 1^1(0(0(2(4(5(0(x1))))))) 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 0^1(0(2(4(5(0(x1)))))) 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 0^1(2(4(5(0(x1))))) 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 2^1(4(5(0(x1)))) 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 4^1(5(0(x1))) 1^1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 2^1(1(3(0(0(5(4(5(0(x1))))))))) 1^1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1^1(3(0(0(5(4(5(0(x1)))))))) 1^1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 3^1(0(0(5(4(5(0(x1))))))) 1^1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 0^1(0(5(4(5(0(x1)))))) 1^1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 0^1(5(4(5(0(x1))))) 1^1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 5^1(4(5(0(x1)))) 1^1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 4^1(5(0(x1))) 3^1(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 4^1(4(0(3(1(5(0(0(1(x1))))))))) 3^1(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 4^1(0(3(1(5(0(0(1(x1)))))))) 3^1(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 0^1(3(1(5(0(0(1(x1))))))) 3^1(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3^1(1(5(0(0(1(x1)))))) 3^1(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 1^1(5(0(0(1(x1))))) 4^1(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 2^1(1(1(x1))) 3^1(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 4^1(5(3(1(2(1(x1)))))) 3^1(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 5^1(3(1(2(1(x1))))) 3^1(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 3^1(1(2(1(x1)))) 0^1(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 2^1(4(2(3(2(1(0(3(1(x1))))))))) 0^1(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 4^1(2(3(2(1(0(3(1(x1)))))))) 0^1(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 2^1(3(2(1(0(3(1(x1))))))) 0^1(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 3^1(2(1(0(3(1(x1)))))) 0^1(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 2^1(1(0(3(1(x1))))) 0^1(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 1^1(0(3(1(x1)))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 2^1(2(2(2(3(1(x1)))))) 5^1(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 2^1(4(0(3(1(0(5(2(4(x1))))))))) 5^1(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 4^1(0(3(1(0(5(2(4(x1)))))))) 5^1(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 0^1(3(1(0(5(2(4(x1))))))) 5^1(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 3^1(1(0(5(2(4(x1)))))) 5^1(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 1^1(0(5(2(4(x1))))) 5^1(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 0^1(5(2(4(x1)))) 5^1(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5^1(2(4(x1))) 5^1(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 2^1(4(x1)) 5^1(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 4^1(x1) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 2^1(1(3(0(4(5(1(x1))))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 1^1(3(0(4(5(1(x1)))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3^1(0(4(5(1(x1))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 0^1(4(5(1(x1)))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 4^1(5(1(x1))) 2^1(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2^1(2(0(1(5(1(x1)))))) 2^1(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2^1(0(1(5(1(x1))))) 2^1(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 0^1(1(5(1(x1)))) 0^1(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 4^1(0(5(4(5(0(3(5(1(x1))))))))) 0^1(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0^1(5(4(5(0(3(5(1(x1)))))))) 0^1(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 5^1(4(5(0(3(5(1(x1))))))) 0^1(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 4^1(5(0(3(5(1(x1)))))) 4^1(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 2^1(4(3(0(0(2(x1)))))) 4^1(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4^1(3(0(0(2(x1))))) 4^1(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 3^1(0(0(2(x1)))) 5^1(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 2^1(2(1(3(2(3(0(1(2(x1))))))))) 5^1(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 2^1(1(3(2(3(0(1(2(x1)))))))) 5^1(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 1^1(3(2(3(0(1(2(x1))))))) 5^1(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 3^1(2(3(0(1(2(x1)))))) 5^1(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 2^1(3(0(1(2(x1))))) 1^1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 2^1(4(4(1(3(4(2(1(2(x1))))))))) 1^1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 4^1(4(1(3(4(2(1(2(x1)))))))) 1^1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 4^1(1(3(4(2(1(2(x1))))))) 1^1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 1^1(3(4(2(1(2(x1)))))) 1^1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 3^1(4(2(1(2(x1))))) 1^1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 2^1(4(3(2(1(4(3(1(2(x1))))))))) 1^1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 4^1(3(2(1(4(3(1(2(x1)))))))) 1^1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 3^1(2(1(4(3(1(2(x1))))))) 1^1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 2^1(1(4(3(1(2(x1)))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 4^1(3(0(5(1(4(2(x1))))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 3^1(0(5(1(4(2(x1)))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0^1(5(1(4(2(x1))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 5^1(1(4(2(x1)))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 1^1(4(2(x1))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 4^1(2(x1)) 2^1(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 4^1(1(3(0(5(3(2(x1))))))) 2^1(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 1^1(3(0(5(3(2(x1)))))) 2^1(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 3^1(0(5(3(2(x1))))) 4^1(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4^1(4(0(4(2(x1))))) 4^1(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4^1(0(4(2(x1)))) 2^1(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 4^1(0(4(4(5(1(1(5(2(x1))))))))) 2^1(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 0^1(4(4(5(1(1(5(2(x1)))))))) 2^1(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 4^1(4(5(1(1(5(2(x1))))))) 2^1(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 4^1(5(1(1(5(2(x1)))))) 2^1(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 5^1(1(1(5(2(x1))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 4^1(3(3(1(5(2(x1)))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 3^1(3(1(5(2(x1))))) 5^1(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 4^1(0(5(4(3(1(1(0(3(x1))))))))) 5^1(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 0^1(5(4(3(1(1(0(3(x1)))))))) 5^1(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5^1(4(3(1(1(0(3(x1))))))) 5^1(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 4^1(3(1(1(0(3(x1)))))) 5^1(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 3^1(1(1(0(3(x1))))) 5^1(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 1^1(1(0(3(x1)))) 0^1(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 2^1(4(0(5(4(2(0(0(3(x1))))))))) 0^1(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 4^1(0(5(4(2(0(0(3(x1)))))))) 0^1(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 0^1(5(4(2(0(0(3(x1))))))) 0^1(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 5^1(4(2(0(0(3(x1)))))) 0^1(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 4^1(2(0(0(3(x1))))) 0^1(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 2^1(0(0(3(x1)))) 0^1(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 0^1(0(3(x1))) 3^1(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 2^1(4(1(0(2(3(x1)))))) 3^1(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 4^1(1(0(2(3(x1))))) 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 2^1(4(1(3(3(x1))))) 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 4^1(1(3(3(x1)))) 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 1^1(3(3(x1))) 3^1(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 4^1(5(0(5(4(3(x1)))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 2^1(4(0(3(0(5(3(x1))))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 4^1(0(3(0(5(3(x1)))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0^1(3(0(5(3(x1))))) 3^1(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2^1(3(2(0(5(4(1(3(2(3(x1)))))))))) 3^1(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 3^1(2(0(5(4(1(3(2(3(x1))))))))) 3^1(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2^1(0(5(4(1(3(2(3(x1)))))))) 3^1(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 0^1(5(4(1(3(2(3(x1))))))) 3^1(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 5^1(4(1(3(2(3(x1)))))) 3^1(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 4^1(1(3(2(3(x1))))) 3^1(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 1^1(3(2(3(x1)))) 3^1(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 3^1(2(3(x1))) 3^1(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2^1(3(x1)) 1^1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 4^1(4(3(0(4(3(2(4(4(x1))))))))) 1^1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 4^1(3(0(4(3(2(4(4(x1)))))))) 1^1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 3^1(0(4(3(2(4(4(x1))))))) 1^1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 0^1(4(3(2(4(4(x1)))))) 1^1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 4^1(3(2(4(4(x1))))) 1^1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 3^1(2(4(4(x1)))) 1^1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 2^1(4(4(x1))) 1^1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 4^1(4(x1)) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 2^1(0(1(2(3(2(4(x1))))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 0^1(1(2(3(2(4(x1)))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 1^1(2(3(2(4(x1))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 2^1(3(2(4(x1)))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3^1(2(4(x1))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 2^1(4(x1)) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 2^1(0(3(4(x1)))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 0^1(3(4(x1))) 2^1(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 4^1(0(4(4(x1)))) 1^1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 4^1(3(0(1(5(4(x1)))))) 1^1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 3^1(0(1(5(4(x1))))) 2^1(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 4^1(1(3(1(5(3(1(5(4(x1))))))))) 2^1(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 1^1(3(1(5(3(1(5(4(x1)))))))) 4^1(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4^1(0(3(3(2(5(3(0(5(x1))))))))) 4^1(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 0^1(3(3(2(5(3(0(5(x1)))))))) 4^1(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 3^1(3(2(5(3(0(5(x1))))))) 0^1(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2^1(0(0(1(3(2(1(0(1(5(x1)))))))))) 0^1(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 0^1(0(1(3(2(1(0(1(5(x1))))))))) 0^1(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 0^1(1(3(2(1(0(1(5(x1)))))))) 0^1(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 1^1(3(2(1(0(1(5(x1))))))) 0^1(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 3^1(2(1(0(1(5(x1)))))) 0^1(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2^1(1(0(1(5(x1))))) 3^1(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 4^1(5(1(3(1(5(1(1(5(x1))))))))) 3^1(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 5^1(1(3(1(5(1(1(5(x1)))))))) 3^1(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1^1(3(1(5(1(1(5(x1))))))) 3^1(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 3^1(1(5(1(1(5(x1)))))) 2^1(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2^1(4(3(4(0(0(3(5(x1)))))))) 2^1(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 4^1(3(4(0(0(3(5(x1))))))) 2^1(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 3^1(4(0(0(3(5(x1)))))) 2^1(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 4^1(0(0(3(5(x1))))) 5^1(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 2^1(5(5(1(3(5(5(x1))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(0^1(x_1)) = x_1 POL(1(x_1)) = x_1 POL(1^1(x_1)) = x_1 POL(2(x_1)) = 1 + x_1 POL(2^1(x_1)) = x_1 POL(3(x_1)) = x_1 POL(3^1(x_1)) = x_1 POL(4(x_1)) = 1 + x_1 POL(4^1(x_1)) = x_1 POL(5(x_1)) = x_1 POL(5^1(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: 5^1(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5^1(5(0(5(2(1(3(1(0(0(x1)))))))))) 5^1(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5^1(0(5(2(1(3(1(0(0(x1))))))))) 5^1(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 0^1(5(2(1(3(1(0(0(x1)))))))) 5^1(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5^1(2(1(3(1(0(0(x1))))))) 3^1(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3^1(3(2(0(0(1(0(4(0(0(x1)))))))))) 3^1(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3^1(2(0(0(1(0(4(0(0(x1))))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0^1(3(5(5(4(5(1(3(0(0(x1)))))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 3^1(5(5(4(5(1(3(0(0(x1))))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 5^1(5(4(5(1(3(0(0(x1)))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 5^1(4(5(1(3(0(0(x1))))))) 0^1(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0^1(4(0(0(3(1(0(2(2(0(x1)))))))))) 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3^1(5(3(4(5(0(0(2(3(0(x1)))))))))) 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 5^1(3(4(5(0(0(2(3(0(x1))))))))) 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3^1(4(5(0(0(2(3(0(x1)))))))) 3^1(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3^1(3(4(2(5(1(3(3(4(0(x1)))))))))) 3^1(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3^1(4(2(5(1(3(3(4(0(x1))))))))) 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3^1(4(3(1(0(0(2(4(5(0(x1)))))))))) 1^1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1^1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3^1(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3^1(4(4(0(3(1(5(0(0(1(x1)))))))))) 2^1(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2^1(3(1(0(5(0(5(1(0(1(x1)))))))))) 2^1(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 3^1(1(0(5(0(5(1(0(1(x1))))))))) 2^1(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 1^1(0(5(0(5(1(0(1(x1)))))))) 2^1(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 0^1(5(0(5(1(0(1(x1))))))) 2^1(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 5^1(0(5(1(0(1(x1)))))) 2^1(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 0^1(5(1(0(1(x1))))) 2^1(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 5^1(1(0(1(x1)))) 4^1(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4^1(3(3(5(0(1(3(2(1(1(x1)))))))))) 4^1(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 3^1(3(5(0(1(3(2(1(1(x1))))))))) 4^1(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 3^1(5(0(1(3(2(1(1(x1)))))))) 4^1(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 5^1(0(1(3(2(1(1(x1))))))) 4^1(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 0^1(1(3(2(1(1(x1)))))) 4^1(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 1^1(3(2(1(1(x1))))) 4^1(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 3^1(2(1(1(x1)))) 3^1(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1^1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3^1(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 3^1(1(5(4(5(3(1(2(1(x1))))))))) 3^1(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1^1(5(4(5(3(1(2(1(x1)))))))) 3^1(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 5^1(4(5(3(1(2(1(x1))))))) 0^1(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0^1(2(4(2(3(2(1(0(3(1(x1)))))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0^1(1(0(5(2(2(2(2(3(1(x1)))))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 1^1(0(5(2(2(2(2(3(1(x1))))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0^1(5(2(2(2(2(3(1(x1)))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 5^1(2(2(2(2(3(1(x1))))))) 5^1(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5^1(2(4(0(3(1(0(5(2(4(x1)))))))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3^1(3(3(2(1(3(0(4(5(1(x1)))))))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3^1(3(2(1(3(0(4(5(1(x1))))))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3^1(2(1(3(0(4(5(1(x1)))))))) 2^1(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2^1(3(1(5(2(2(0(1(5(1(x1)))))))))) 2^1(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 3^1(1(5(2(2(0(1(5(1(x1))))))))) 2^1(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 1^1(5(2(2(0(1(5(1(x1)))))))) 2^1(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 5^1(2(2(0(1(5(1(x1))))))) 0^1(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0^1(4(0(5(4(5(0(3(5(1(x1)))))))))) 4^1(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4^1(3(3(1(2(4(3(0(0(2(x1)))))))))) 4^1(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 3^1(3(1(2(4(3(0(0(2(x1))))))))) 4^1(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 3^1(1(2(4(3(0(0(2(x1)))))))) 4^1(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 1^1(2(4(3(0(0(2(x1))))))) 5^1(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5^1(2(2(1(3(2(3(0(1(2(x1)))))))))) 1^1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5^1(2(4(4(1(3(4(2(1(2(x1)))))))))) 1^1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1^1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0^1(5(5(4(3(0(5(1(4(2(x1)))))))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 5^1(5(4(3(0(5(1(4(2(x1))))))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 5^1(4(3(0(5(1(4(2(x1)))))))) 2^1(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2^1(5(5(4(1(3(0(5(3(2(x1)))))))))) 2^1(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 5^1(5(4(1(3(0(5(3(2(x1))))))))) 2^1(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 5^1(4(1(3(0(5(3(2(x1)))))))) 4^1(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4^1(3(1(1(3(4(4(0(4(2(x1)))))))))) 4^1(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 3^1(1(1(3(4(4(0(4(2(x1))))))))) 4^1(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 1^1(1(3(4(4(0(4(2(x1)))))))) 4^1(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 1^1(3(4(4(0(4(2(x1))))))) 4^1(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 3^1(4(4(0(4(2(x1)))))) 2^1(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2^1(4(0(4(4(5(1(1(5(2(x1)))))))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0^1(5(3(0(4(3(3(1(5(2(x1)))))))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 5^1(3(0(4(3(3(1(5(2(x1))))))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 3^1(0(4(3(3(1(5(2(x1)))))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0^1(4(3(3(1(5(2(x1))))))) 5^1(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5^1(4(0(5(4(3(1(1(0(3(x1)))))))))) 0^1(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3^1(2(4(0(5(4(2(0(0(3(x1)))))))))) 3^1(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0^1(5(0(3(2(4(1(0(2(3(x1)))))))))) 3^1(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 5^1(0(3(2(4(1(0(2(3(x1))))))))) 3^1(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0^1(3(2(4(1(0(2(3(x1)))))))) 3^1(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 3^1(2(4(1(0(2(3(x1))))))) 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5^1(0(3(5(5(2(4(1(3(3(x1)))))))))) 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 0^1(3(5(5(2(4(1(3(3(x1))))))))) 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 3^1(5(5(2(4(1(3(3(x1)))))))) 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5^1(5(2(4(1(3(3(x1))))))) 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5^1(2(4(1(3(3(x1)))))) 3^1(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3^1(5(0(1(4(5(0(5(4(3(x1)))))))))) 3^1(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 5^1(0(1(4(5(0(5(4(3(x1))))))))) 3^1(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 0^1(1(4(5(0(5(4(3(x1)))))))) 3^1(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 1^1(4(5(0(5(4(3(x1))))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0^1(3(0(2(4(0(3(0(5(3(x1)))))))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 3^1(0(2(4(0(3(0(5(3(x1))))))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0^1(2(4(0(3(0(5(3(x1)))))))) 1^1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1^1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3^1(1(0(2(0(1(2(3(2(4(x1)))))))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 1^1(0(2(0(1(2(3(2(4(x1))))))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 0^1(2(0(1(2(3(2(4(x1)))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(1(3(1(0(5(0(5(2(4(x1)))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 1^1(3(1(0(5(0(5(2(4(x1))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 3^1(1(0(5(0(5(2(4(x1)))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 1^1(0(5(0(5(2(4(x1))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(5(0(5(2(4(x1)))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3^1(0(0(3(0(0(2(0(3(4(x1)))))))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 0^1(0(3(0(0(2(0(3(4(x1))))))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 0^1(3(0(0(2(0(3(4(x1)))))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3^1(0(0(2(0(3(4(x1))))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 0^1(0(2(0(3(4(x1)))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 0^1(2(0(3(4(x1))))) 2^1(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2^1(3(5(3(3(0(4(0(4(4(x1)))))))))) 2^1(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 3^1(5(3(3(0(4(0(4(4(x1))))))))) 2^1(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 5^1(3(3(0(4(0(4(4(x1)))))))) 2^1(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 3^1(3(0(4(0(4(4(x1))))))) 2^1(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 3^1(0(4(0(4(4(x1)))))) 2^1(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 0^1(4(0(4(4(x1))))) 1^1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1^1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1^1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 5^1(5(5(4(3(0(1(5(4(x1))))))))) 1^1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 5^1(5(4(3(0(1(5(4(x1)))))))) 1^1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 5^1(4(3(0(1(5(4(x1))))))) 2^1(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2^1(4(1(3(1(5(3(1(5(4(x1)))))))))) 4^1(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4^1(4(0(3(3(2(5(3(0(5(x1)))))))))) 3^1(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1^1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2^1(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2^1(3(2(4(3(4(0(0(3(5(x1)))))))))) 2^1(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 3^1(2(4(3(4(0(0(3(5(x1))))))))) 5^1(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5^1(0(3(2(5(5(1(3(5(5(x1)))))))))) 5^1(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 0^1(3(2(5(5(1(3(5(5(x1))))))))) 5^1(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 3^1(2(5(5(1(3(5(5(x1)))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 30 less nodes. ---------------------------------------- (10) Complex Obligation (AND) ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: 5^1(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5^1(0(5(2(1(3(1(0(0(x1))))))))) 5^1(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5^1(5(0(5(2(1(3(1(0(0(x1)))))))))) 5^1(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 0^1(5(2(1(3(1(0(0(x1)))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0^1(3(5(5(4(5(1(3(0(0(x1)))))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 3^1(5(5(4(5(1(3(0(0(x1))))))))) 3^1(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3^1(3(2(0(0(1(0(4(0(0(x1)))))))))) 3^1(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3^1(2(0(0(1(0(4(0(0(x1))))))))) 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3^1(5(3(4(5(0(0(2(3(0(x1)))))))))) 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 5^1(3(4(5(0(0(2(3(0(x1))))))))) 5^1(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5^1(2(1(3(1(0(0(x1))))))) 5^1(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5^1(2(4(0(3(1(0(5(2(4(x1)))))))))) 5^1(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5^1(2(2(1(3(2(3(0(1(2(x1)))))))))) 5^1(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5^1(4(0(5(4(3(1(1(0(3(x1)))))))))) 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5^1(0(3(5(5(2(4(1(3(3(x1)))))))))) 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 0^1(3(5(5(2(4(1(3(3(x1))))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 5^1(5(4(5(1(3(0(0(x1)))))))) 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 3^1(5(5(2(4(1(3(3(x1)))))))) 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3^1(4(5(0(0(2(3(0(x1)))))))) 3^1(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3^1(3(4(2(5(1(3(3(4(0(x1)))))))))) 3^1(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3^1(4(2(5(1(3(3(4(0(x1))))))))) 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3^1(4(3(1(0(0(2(4(5(0(x1)))))))))) 3^1(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3^1(4(4(0(3(1(5(0(0(1(x1)))))))))) 3^1(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1^1(3(1(5(4(5(3(1(2(1(x1)))))))))) 1^1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1^1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1^1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5^1(2(4(4(1(3(4(2(1(2(x1)))))))))) 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5^1(5(2(4(1(3(3(x1))))))) 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5^1(2(4(1(3(3(x1)))))) 5^1(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5^1(0(3(2(5(5(1(3(5(5(x1)))))))))) 5^1(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 0^1(3(2(5(5(1(3(5(5(x1))))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 5^1(4(5(1(3(0(0(x1))))))) 5^1(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 3^1(2(5(5(1(3(5(5(x1)))))))) 3^1(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 3^1(1(5(4(5(3(1(2(1(x1))))))))) 3^1(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1^1(5(4(5(3(1(2(1(x1)))))))) 1^1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1^1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1^1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1^1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1^1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1^1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1^1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 5^1(5(5(4(3(0(1(5(4(x1))))))))) 1^1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 5^1(5(4(3(0(1(5(4(x1)))))))) 1^1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 5^1(4(3(0(1(5(4(x1))))))) 3^1(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 5^1(4(5(3(1(2(1(x1))))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3^1(3(3(2(1(3(0(4(5(1(x1)))))))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3^1(3(2(1(3(0(4(5(1(x1))))))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3^1(2(1(3(0(4(5(1(x1)))))))) 3^1(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0^1(5(0(3(2(4(1(0(2(3(x1)))))))))) 0^1(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0^1(4(0(0(3(1(0(2(2(0(x1)))))))))) 0^1(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0^1(2(4(2(3(2(1(0(3(1(x1)))))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0^1(1(0(5(2(2(2(2(3(1(x1)))))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 1^1(0(5(2(2(2(2(3(1(x1))))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0^1(5(2(2(2(2(3(1(x1)))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 5^1(2(2(2(2(3(1(x1))))))) 0^1(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0^1(4(0(5(4(5(0(3(5(1(x1)))))))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0^1(5(5(4(3(0(5(1(4(2(x1)))))))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 5^1(5(4(3(0(5(1(4(2(x1))))))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 5^1(4(3(0(5(1(4(2(x1)))))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0^1(5(3(0(4(3(3(1(5(2(x1)))))))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 5^1(3(0(4(3(3(1(5(2(x1))))))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 3^1(0(4(3(3(1(5(2(x1)))))))) 3^1(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 5^1(0(3(2(4(1(0(2(3(x1))))))))) 3^1(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0^1(3(2(4(1(0(2(3(x1)))))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0^1(4(3(3(1(5(2(x1))))))) 0^1(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3^1(2(4(0(5(4(2(0(0(3(x1)))))))))) 3^1(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 3^1(2(4(1(0(2(3(x1))))))) 3^1(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3^1(5(0(1(4(5(0(5(4(3(x1)))))))))) 3^1(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 5^1(0(1(4(5(0(5(4(3(x1))))))))) 3^1(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 0^1(1(4(5(0(5(4(3(x1)))))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0^1(3(0(2(4(0(3(0(5(3(x1)))))))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 3^1(0(2(4(0(3(0(5(3(x1))))))))) 3^1(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 1^1(4(5(0(5(4(3(x1))))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3^1(1(0(2(0(1(2(3(2(4(x1)))))))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 1^1(0(2(0(1(2(3(2(4(x1))))))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 0^1(2(0(1(2(3(2(4(x1)))))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0^1(2(4(0(3(0(5(3(x1)))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(1(3(1(0(5(0(5(2(4(x1)))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 1^1(3(1(0(5(0(5(2(4(x1))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 3^1(1(0(5(0(5(2(4(x1)))))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3^1(0(0(3(0(0(2(0(3(4(x1)))))))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 0^1(0(3(0(0(2(0(3(4(x1))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 1^1(0(5(0(5(2(4(x1))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(5(0(5(2(4(x1)))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 0^1(3(0(0(2(0(3(4(x1)))))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3^1(0(0(2(0(3(4(x1))))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 0^1(0(2(0(3(4(x1)))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 0^1(2(0(3(4(x1))))) 3^1(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1^1(4(5(1(3(1(5(1(1(5(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (12) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 5^1(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5^1(0(5(2(1(3(1(0(0(x1))))))))) 5^1(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 0^1(5(2(1(3(1(0(0(x1)))))))) 3^1(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3^1(2(0(0(1(0(4(0(0(x1))))))))) 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 5^1(3(4(5(0(0(2(3(0(x1))))))))) 5^1(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5^1(2(1(3(1(0(0(x1))))))) 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 0^1(3(5(5(2(4(1(3(3(x1))))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 5^1(5(4(5(1(3(0(0(x1)))))))) 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 3^1(5(5(2(4(1(3(3(x1)))))))) 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3^1(4(5(0(0(2(3(0(x1)))))))) 3^1(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3^1(4(2(5(1(3(3(4(0(x1))))))))) 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5^1(5(2(4(1(3(3(x1))))))) 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5^1(2(4(1(3(3(x1)))))) 5^1(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 0^1(3(2(5(5(1(3(5(5(x1))))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 5^1(4(5(1(3(0(0(x1))))))) 5^1(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 3^1(2(5(5(1(3(5(5(x1)))))))) 3^1(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1^1(5(4(5(3(1(2(1(x1)))))))) 1^1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 5^1(5(4(3(0(1(5(4(x1)))))))) 1^1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 5^1(4(3(0(1(5(4(x1))))))) 3^1(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 5^1(4(5(3(1(2(1(x1))))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3^1(3(2(1(3(0(4(5(1(x1))))))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3^1(2(1(3(0(4(5(1(x1)))))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 5^1(4(3(0(5(1(4(2(x1)))))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 3^1(0(4(3(3(1(5(2(x1)))))))) 3^1(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0^1(3(2(4(1(0(2(3(x1)))))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0^1(4(3(3(1(5(2(x1))))))) 3^1(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 3^1(2(4(1(0(2(3(x1))))))) 3^1(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 5^1(0(1(4(5(0(5(4(3(x1))))))))) 3^1(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 0^1(1(4(5(0(5(4(3(x1)))))))) 3^1(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 1^1(4(5(0(5(4(3(x1))))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 1^1(0(2(0(1(2(3(2(4(x1))))))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 0^1(2(0(1(2(3(2(4(x1)))))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0^1(2(4(0(3(0(5(3(x1)))))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 0^1(0(3(0(0(2(0(3(4(x1))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 1^1(0(5(0(5(2(4(x1))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(5(0(5(2(4(x1)))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 0^1(3(0(0(2(0(3(4(x1)))))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3^1(0(0(2(0(3(4(x1))))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 0^1(0(2(0(3(4(x1)))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 0^1(2(0(3(4(x1))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(0^1(x_1)) = x_1 POL(1(x_1)) = x_1 POL(1^1(x_1)) = x_1 POL(2(x_1)) = x_1 POL(3(x_1)) = 1 + x_1 POL(3^1(x_1)) = 1 + x_1 POL(4(x_1)) = 1 + x_1 POL(5(x_1)) = 1 + x_1 POL(5^1(x_1)) = 1 + x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: 5^1(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5^1(5(0(5(2(1(3(1(0(0(x1)))))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0^1(3(5(5(4(5(1(3(0(0(x1)))))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 3^1(5(5(4(5(1(3(0(0(x1))))))))) 3^1(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3^1(3(2(0(0(1(0(4(0(0(x1)))))))))) 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3^1(5(3(4(5(0(0(2(3(0(x1)))))))))) 5^1(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5^1(2(4(0(3(1(0(5(2(4(x1)))))))))) 5^1(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5^1(2(2(1(3(2(3(0(1(2(x1)))))))))) 5^1(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5^1(4(0(5(4(3(1(1(0(3(x1)))))))))) 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5^1(0(3(5(5(2(4(1(3(3(x1)))))))))) 3^1(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3^1(3(4(2(5(1(3(3(4(0(x1)))))))))) 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3^1(4(3(1(0(0(2(4(5(0(x1)))))))))) 3^1(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3^1(4(4(0(3(1(5(0(0(1(x1)))))))))) 3^1(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1^1(3(1(5(4(5(3(1(2(1(x1)))))))))) 1^1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1^1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1^1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5^1(2(4(4(1(3(4(2(1(2(x1)))))))))) 5^1(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5^1(0(3(2(5(5(1(3(5(5(x1)))))))))) 3^1(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 3^1(1(5(4(5(3(1(2(1(x1))))))))) 1^1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1^1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1^1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1^1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1^1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1^1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1^1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 5^1(5(5(4(3(0(1(5(4(x1))))))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3^1(3(3(2(1(3(0(4(5(1(x1)))))))))) 3^1(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0^1(5(0(3(2(4(1(0(2(3(x1)))))))))) 0^1(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0^1(4(0(0(3(1(0(2(2(0(x1)))))))))) 0^1(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0^1(2(4(2(3(2(1(0(3(1(x1)))))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0^1(1(0(5(2(2(2(2(3(1(x1)))))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 1^1(0(5(2(2(2(2(3(1(x1))))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0^1(5(2(2(2(2(3(1(x1)))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 5^1(2(2(2(2(3(1(x1))))))) 0^1(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0^1(4(0(5(4(5(0(3(5(1(x1)))))))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0^1(5(5(4(3(0(5(1(4(2(x1)))))))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 5^1(5(4(3(0(5(1(4(2(x1))))))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0^1(5(3(0(4(3(3(1(5(2(x1)))))))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 5^1(3(0(4(3(3(1(5(2(x1))))))))) 3^1(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 5^1(0(3(2(4(1(0(2(3(x1))))))))) 0^1(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3^1(2(4(0(5(4(2(0(0(3(x1)))))))))) 3^1(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3^1(5(0(1(4(5(0(5(4(3(x1)))))))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0^1(3(0(2(4(0(3(0(5(3(x1)))))))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 3^1(0(2(4(0(3(0(5(3(x1))))))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3^1(1(0(2(0(1(2(3(2(4(x1)))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(1(3(1(0(5(0(5(2(4(x1)))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 1^1(3(1(0(5(0(5(2(4(x1))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 3^1(1(0(5(0(5(2(4(x1)))))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3^1(0(0(3(0(0(2(0(3(4(x1)))))))))) 3^1(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1^1(4(5(1(3(1(5(1(1(5(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 10 less nodes. ---------------------------------------- (15) Complex Obligation (AND) ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: 5^1(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5^1(2(4(0(3(1(0(5(2(4(x1)))))))))) 5^1(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5^1(5(0(5(2(1(3(1(0(0(x1)))))))))) 5^1(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5^1(2(2(1(3(2(3(0(1(2(x1)))))))))) 5^1(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5^1(4(0(5(4(3(1(1(0(3(x1)))))))))) 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5^1(0(3(5(5(2(4(1(3(3(x1)))))))))) 5^1(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5^1(0(3(2(5(5(1(3(5(5(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (17) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 5^1(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5^1(0(3(5(5(2(4(1(3(3(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 0 POL(1(x_1)) = 0 POL(2(x_1)) = 0 POL(3(x_1)) = 0 POL(4(x_1)) = 1 POL(5(x_1)) = 0 POL(5^1(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: 5^1(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5^1(2(4(0(3(1(0(5(2(4(x1)))))))))) 5^1(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5^1(5(0(5(2(1(3(1(0(0(x1)))))))))) 5^1(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5^1(2(2(1(3(2(3(0(1(2(x1)))))))))) 5^1(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5^1(4(0(5(4(3(1(1(0(3(x1)))))))))) 5^1(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5^1(0(3(2(5(5(1(3(5(5(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 5^1(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5^1(2(4(0(3(1(0(5(2(4(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 POL(1(x_1)) = 0 POL(2(x_1)) = 0 POL(3(x_1)) = 1 POL(4(x_1)) = 0 POL(5(x_1)) = 0 POL(5^1(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: 5^1(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5^1(5(0(5(2(1(3(1(0(0(x1)))))))))) 5^1(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5^1(2(2(1(3(2(3(0(1(2(x1)))))))))) 5^1(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5^1(4(0(5(4(3(1(1(0(3(x1)))))))))) 5^1(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5^1(0(3(2(5(5(1(3(5(5(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 5^1(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5^1(5(0(5(2(1(3(1(0(0(x1)))))))))) 5^1(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5^1(4(0(5(4(3(1(1(0(3(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 0 POL(1(x_1)) = 1 POL(2(x_1)) = 0 POL(3(x_1)) = x_1 POL(4(x_1)) = 1 + x_1 POL(5(x_1)) = x_1 POL(5^1(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: 5^1(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5^1(2(2(1(3(2(3(0(1(2(x1)))))))))) 5^1(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5^1(0(3(2(5(5(1(3(5(5(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented dependency pairs: 5^1(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5^1(2(2(1(3(2(3(0(1(2(x1)))))))))) 5^1(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5^1(0(3(2(5(5(1(3(5(5(x1)))))))))) Strictly oriented rules of the TRS R: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(1(x_1)) = x_1 POL(2(x_1)) = 3 + 3*x_1 POL(3(x_1)) = 3 + 2*x_1 POL(4(x_1)) = x_1 POL(5(x_1)) = 2 + x_1 POL(5^1(x_1)) = 3*x_1 ---------------------------------------- (24) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (26) YES ---------------------------------------- (27) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1^1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1^1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1^1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1^1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1^1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1^1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1^1(5(5(5(4(3(0(1(5(4(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (28) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 1^1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1^1(4(4(3(0(4(3(2(4(4(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 POL(1(x_1)) = 1 + x_1 POL(1^1(x_1)) = x_1 POL(2(x_1)) = 0 POL(3(x_1)) = 1 + x_1 POL(4(x_1)) = 1 + x_1 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) ---------------------------------------- (29) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1^1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1^1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1^1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1^1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1^1(5(5(5(4(3(0(1(5(4(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (30) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 1^1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1^1(5(5(5(4(3(0(1(5(4(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 POL(1(x_1)) = 1 POL(1^1(x_1)) = x_1 POL(2(x_1)) = 0 POL(3(x_1)) = 1 + x_1 POL(4(x_1)) = 1 POL(5(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) ---------------------------------------- (31) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1^1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1^1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1^1(2(1(3(0(0(5(4(5(0(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (32) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented dependency pairs: 1^1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1^1(2(4(3(2(1(4(3(1(2(x1)))))))))) Strictly oriented rules of the TRS R: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(1(x_1)) = x_1 POL(1^1(x_1)) = 2*x_1 POL(2(x_1)) = 2*x_1 POL(3(x_1)) = 2 + x_1 POL(4(x_1)) = x_1 POL(5(x_1)) = 3 + x_1 ---------------------------------------- (33) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1^1(2(1(3(0(0(5(4(5(0(x1)))))))))) The TRS R consists of the following rules: 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (34) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 1^1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1^1(2(1(3(0(0(5(4(5(0(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(1(x_1)) = x_1 POL(1^1(x_1)) = x_1 POL(2(x_1)) = x_1 POL(3(x_1)) = 1 + x_1 POL(4(x_1)) = 1 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) ---------------------------------------- (35) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (36) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (37) YES ---------------------------------------- (38) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3^1(3(2(0(0(1(0(4(0(0(x1)))))))))) 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3^1(5(3(4(5(0(0(2(3(0(x1)))))))))) 3^1(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3^1(3(4(2(5(1(3(3(4(0(x1)))))))))) 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3^1(4(3(1(0(0(2(4(5(0(x1)))))))))) 3^1(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3^1(4(4(0(3(1(5(0(0(1(x1)))))))))) 3^1(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 3^1(1(5(4(5(3(1(2(1(x1))))))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3^1(3(3(2(1(3(0(4(5(1(x1)))))))))) 3^1(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0^1(5(0(3(2(4(1(0(2(3(x1)))))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0^1(3(5(5(4(5(1(3(0(0(x1)))))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 3^1(5(5(4(5(1(3(0(0(x1))))))))) 3^1(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3^1(5(0(1(4(5(0(5(4(3(x1)))))))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3^1(1(0(2(0(1(2(3(2(4(x1)))))))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3^1(0(0(3(0(0(2(0(3(4(x1)))))))))) 0^1(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0^1(4(0(0(3(1(0(2(2(0(x1)))))))))) 0^1(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0^1(2(4(2(3(2(1(0(3(1(x1)))))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0^1(1(0(5(2(2(2(2(3(1(x1)))))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0^1(5(2(2(2(2(3(1(x1)))))))) 0^1(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0^1(4(0(5(4(5(0(3(5(1(x1)))))))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0^1(5(5(4(3(0(5(1(4(2(x1)))))))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0^1(5(3(0(4(3(3(1(5(2(x1)))))))))) 0^1(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3^1(2(4(0(5(4(2(0(0(3(x1)))))))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0^1(3(0(2(4(0(3(0(5(3(x1)))))))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 3^1(0(2(4(0(3(0(5(3(x1))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(1(3(1(0(5(0(5(2(4(x1)))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 3^1(1(0(5(0(5(2(4(x1)))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (39) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 3^1(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 3^1(1(5(4(5(3(1(2(1(x1))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 0 POL(0^1(x_1)) = 0 POL(1(x_1)) = 0 POL(2(x_1)) = 0 POL(3(x_1)) = 0 POL(3^1(x_1)) = x_1 POL(4(x_1)) = 1 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3^1(3(2(0(0(1(0(4(0(0(x1)))))))))) 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3^1(5(3(4(5(0(0(2(3(0(x1)))))))))) 3^1(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3^1(3(4(2(5(1(3(3(4(0(x1)))))))))) 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3^1(4(3(1(0(0(2(4(5(0(x1)))))))))) 3^1(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3^1(4(4(0(3(1(5(0(0(1(x1)))))))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3^1(3(3(2(1(3(0(4(5(1(x1)))))))))) 3^1(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0^1(5(0(3(2(4(1(0(2(3(x1)))))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0^1(3(5(5(4(5(1(3(0(0(x1)))))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 3^1(5(5(4(5(1(3(0(0(x1))))))))) 3^1(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3^1(5(0(1(4(5(0(5(4(3(x1)))))))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3^1(1(0(2(0(1(2(3(2(4(x1)))))))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3^1(0(0(3(0(0(2(0(3(4(x1)))))))))) 0^1(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0^1(4(0(0(3(1(0(2(2(0(x1)))))))))) 0^1(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0^1(2(4(2(3(2(1(0(3(1(x1)))))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0^1(1(0(5(2(2(2(2(3(1(x1)))))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0^1(5(2(2(2(2(3(1(x1)))))))) 0^1(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0^1(4(0(5(4(5(0(3(5(1(x1)))))))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0^1(5(5(4(3(0(5(1(4(2(x1)))))))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0^1(5(3(0(4(3(3(1(5(2(x1)))))))))) 0^1(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3^1(2(4(0(5(4(2(0(0(3(x1)))))))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0^1(3(0(2(4(0(3(0(5(3(x1)))))))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 3^1(0(2(4(0(3(0(5(3(x1))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(1(3(1(0(5(0(5(2(4(x1)))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 3^1(1(0(5(0(5(2(4(x1)))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (41) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3^1(2(4(0(5(4(2(0(0(3(x1)))))))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 3^1(0(2(4(0(3(0(5(3(x1))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 3^1(1(0(5(0(5(2(4(x1)))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 0 POL(0^1(x_1)) = 1 POL(1(x_1)) = x_1 POL(2(x_1)) = x_1 POL(3(x_1)) = x_1 POL(3^1(x_1)) = x_1 POL(4(x_1)) = x_1 POL(5(x_1)) = 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) ---------------------------------------- (42) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3^1(3(2(0(0(1(0(4(0(0(x1)))))))))) 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3^1(5(3(4(5(0(0(2(3(0(x1)))))))))) 3^1(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3^1(3(4(2(5(1(3(3(4(0(x1)))))))))) 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3^1(4(3(1(0(0(2(4(5(0(x1)))))))))) 3^1(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3^1(4(4(0(3(1(5(0(0(1(x1)))))))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3^1(3(3(2(1(3(0(4(5(1(x1)))))))))) 3^1(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0^1(5(0(3(2(4(1(0(2(3(x1)))))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0^1(3(5(5(4(5(1(3(0(0(x1)))))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 3^1(5(5(4(5(1(3(0(0(x1))))))))) 3^1(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3^1(5(0(1(4(5(0(5(4(3(x1)))))))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3^1(1(0(2(0(1(2(3(2(4(x1)))))))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3^1(0(0(3(0(0(2(0(3(4(x1)))))))))) 0^1(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0^1(4(0(0(3(1(0(2(2(0(x1)))))))))) 0^1(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0^1(2(4(2(3(2(1(0(3(1(x1)))))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0^1(1(0(5(2(2(2(2(3(1(x1)))))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0^1(5(2(2(2(2(3(1(x1)))))))) 0^1(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0^1(4(0(5(4(5(0(3(5(1(x1)))))))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0^1(5(5(4(3(0(5(1(4(2(x1)))))))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0^1(5(3(0(4(3(3(1(5(2(x1)))))))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0^1(3(0(2(4(0(3(0(5(3(x1)))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(1(3(1(0(5(0(5(2(4(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (43) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 3^1(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3^1(3(2(0(0(1(0(4(0(0(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 POL(0^1(x_1)) = 0 POL(1(x_1)) = 0 POL(2(x_1)) = 0 POL(3(x_1)) = 1 + x_1 POL(3^1(x_1)) = x_1 POL(4(x_1)) = 0 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) ---------------------------------------- (44) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3^1(5(3(4(5(0(0(2(3(0(x1)))))))))) 3^1(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3^1(3(4(2(5(1(3(3(4(0(x1)))))))))) 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3^1(4(3(1(0(0(2(4(5(0(x1)))))))))) 3^1(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3^1(4(4(0(3(1(5(0(0(1(x1)))))))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3^1(3(3(2(1(3(0(4(5(1(x1)))))))))) 3^1(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0^1(5(0(3(2(4(1(0(2(3(x1)))))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0^1(3(5(5(4(5(1(3(0(0(x1)))))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 3^1(5(5(4(5(1(3(0(0(x1))))))))) 3^1(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3^1(5(0(1(4(5(0(5(4(3(x1)))))))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3^1(1(0(2(0(1(2(3(2(4(x1)))))))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3^1(0(0(3(0(0(2(0(3(4(x1)))))))))) 0^1(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0^1(4(0(0(3(1(0(2(2(0(x1)))))))))) 0^1(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0^1(2(4(2(3(2(1(0(3(1(x1)))))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0^1(1(0(5(2(2(2(2(3(1(x1)))))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0^1(5(2(2(2(2(3(1(x1)))))))) 0^1(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0^1(4(0(5(4(5(0(3(5(1(x1)))))))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0^1(5(5(4(3(0(5(1(4(2(x1)))))))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0^1(5(3(0(4(3(3(1(5(2(x1)))))))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0^1(3(0(2(4(0(3(0(5(3(x1)))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(1(3(1(0(5(0(5(2(4(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (45) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 3^1(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0^1(5(0(3(2(4(1(0(2(3(x1)))))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0^1(5(2(2(2(2(3(1(x1)))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 + x_1 POL(0^1(x_1)) = 1 + x_1 POL(1(x_1)) = x_1 POL(2(x_1)) = x_1 POL(3(x_1)) = x_1 POL(3^1(x_1)) = 1 + x_1 POL(4(x_1)) = x_1 POL(5(x_1)) = 1 + x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) ---------------------------------------- (46) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3^1(5(3(4(5(0(0(2(3(0(x1)))))))))) 3^1(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3^1(3(4(2(5(1(3(3(4(0(x1)))))))))) 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3^1(4(3(1(0(0(2(4(5(0(x1)))))))))) 3^1(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3^1(4(4(0(3(1(5(0(0(1(x1)))))))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3^1(3(3(2(1(3(0(4(5(1(x1)))))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0^1(3(5(5(4(5(1(3(0(0(x1)))))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 3^1(5(5(4(5(1(3(0(0(x1))))))))) 3^1(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3^1(5(0(1(4(5(0(5(4(3(x1)))))))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3^1(1(0(2(0(1(2(3(2(4(x1)))))))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3^1(0(0(3(0(0(2(0(3(4(x1)))))))))) 0^1(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0^1(4(0(0(3(1(0(2(2(0(x1)))))))))) 0^1(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0^1(2(4(2(3(2(1(0(3(1(x1)))))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0^1(1(0(5(2(2(2(2(3(1(x1)))))))))) 0^1(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0^1(4(0(5(4(5(0(3(5(1(x1)))))))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0^1(5(5(4(3(0(5(1(4(2(x1)))))))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0^1(5(3(0(4(3(3(1(5(2(x1)))))))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0^1(3(0(2(4(0(3(0(5(3(x1)))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(1(3(1(0(5(0(5(2(4(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (47) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. ---------------------------------------- (48) Complex Obligation (AND) ---------------------------------------- (49) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3^1(3(4(2(5(1(3(3(4(0(x1)))))))))) 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3^1(5(3(4(5(0(0(2(3(0(x1)))))))))) 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3^1(4(3(1(0(0(2(4(5(0(x1)))))))))) 3^1(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3^1(4(4(0(3(1(5(0(0(1(x1)))))))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3^1(3(3(2(1(3(0(4(5(1(x1)))))))))) 3^1(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3^1(5(0(1(4(5(0(5(4(3(x1)))))))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3^1(1(0(2(0(1(2(3(2(4(x1)))))))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3^1(0(0(3(0(0(2(0(3(4(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (50) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 3^1(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3^1(4(4(0(3(1(5(0(0(1(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 0 POL(1(x_1)) = 1 POL(2(x_1)) = 0 POL(3(x_1)) = x_1 POL(3^1(x_1)) = x_1 POL(4(x_1)) = x_1 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) ---------------------------------------- (51) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3^1(3(4(2(5(1(3(3(4(0(x1)))))))))) 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3^1(5(3(4(5(0(0(2(3(0(x1)))))))))) 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3^1(4(3(1(0(0(2(4(5(0(x1)))))))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3^1(3(3(2(1(3(0(4(5(1(x1)))))))))) 3^1(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3^1(5(0(1(4(5(0(5(4(3(x1)))))))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3^1(1(0(2(0(1(2(3(2(4(x1)))))))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3^1(0(0(3(0(0(2(0(3(4(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (52) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 3^1(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3^1(5(0(1(4(5(0(5(4(3(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(1(x_1)) = 0 POL(2(x_1)) = 0 POL(3(x_1)) = 0 POL(3^1(x_1)) = x_1 POL(4(x_1)) = 1 POL(5(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) ---------------------------------------- (53) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3^1(3(4(2(5(1(3(3(4(0(x1)))))))))) 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3^1(5(3(4(5(0(0(2(3(0(x1)))))))))) 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3^1(4(3(1(0(0(2(4(5(0(x1)))))))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3^1(3(3(2(1(3(0(4(5(1(x1)))))))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3^1(1(0(2(0(1(2(3(2(4(x1)))))))))) 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3^1(0(0(3(0(0(2(0(3(4(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (54) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 3^1(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3^1(0(0(3(0(0(2(0(3(4(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 + x_1 POL(1(x_1)) = 0 POL(2(x_1)) = 0 POL(3(x_1)) = 1 + x_1 POL(3^1(x_1)) = x_1 POL(4(x_1)) = 0 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) ---------------------------------------- (55) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3^1(3(4(2(5(1(3(3(4(0(x1)))))))))) 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3^1(5(3(4(5(0(0(2(3(0(x1)))))))))) 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3^1(4(3(1(0(0(2(4(5(0(x1)))))))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3^1(3(3(2(1(3(0(4(5(1(x1)))))))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3^1(1(0(2(0(1(2(3(2(4(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (56) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 3^1(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3^1(4(3(1(0(0(2(4(5(0(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(1(x_1)) = x_1 POL(2(x_1)) = 0 POL(3(x_1)) = x_1 POL(3^1(x_1)) = x_1 POL(4(x_1)) = 1 + x_1 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) ---------------------------------------- (57) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3^1(3(4(2(5(1(3(3(4(0(x1)))))))))) 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3^1(5(3(4(5(0(0(2(3(0(x1)))))))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3^1(3(3(2(1(3(0(4(5(1(x1)))))))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3^1(1(0(2(0(1(2(3(2(4(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (58) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 3^1(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3^1(3(4(2(5(1(3(3(4(0(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 0 POL(1(x_1)) = 1 POL(2(x_1)) = x_1 POL(3(x_1)) = x_1 POL(3^1(x_1)) = x_1 POL(4(x_1)) = x_1 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) ---------------------------------------- (59) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3^1(5(3(4(5(0(0(2(3(0(x1)))))))))) 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3^1(3(3(2(1(3(0(4(5(1(x1)))))))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3^1(1(0(2(0(1(2(3(2(4(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (60) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 3^1(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3^1(3(3(2(1(3(0(4(5(1(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 0 POL(1(x_1)) = 2 POL(2(x_1)) = 4*x_1 POL(3(x_1)) = 4*x_1 POL(3^1(x_1)) = 2*x_1 POL(4(x_1)) = x_1 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) ---------------------------------------- (61) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3^1(5(3(4(5(0(0(2(3(0(x1)))))))))) 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3^1(1(0(2(0(1(2(3(2(4(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (62) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented dependency pairs: 3^1(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3^1(1(0(2(0(1(2(3(2(4(x1)))))))))) Strictly oriented rules of the TRS R: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(1(x_1)) = x_1 POL(2(x_1)) = 2*x_1 POL(3(x_1)) = 2 + 2*x_1 POL(3^1(x_1)) = x_1 POL(4(x_1)) = x_1 POL(5(x_1)) = 1 + x_1 ---------------------------------------- (63) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3^1(5(3(4(5(0(0(2(3(0(x1)))))))))) The TRS R consists of the following rules: 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (64) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented rules of the TRS R: 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(1(x_1)) = 2*x_1 POL(2(x_1)) = 2 + 2*x_1 POL(3(x_1)) = 2*x_1 POL(3^1(x_1)) = x_1 POL(4(x_1)) = x_1 POL(5(x_1)) = 2*x_1 ---------------------------------------- (65) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3^1(5(3(4(5(0(0(2(3(0(x1)))))))))) The TRS R consists of the following rules: 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (66) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented rules of the TRS R: 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(1(x_1)) = 2*x_1 POL(2(x_1)) = 1 + 2*x_1 POL(3(x_1)) = 2*x_1 POL(3^1(x_1)) = x_1 POL(4(x_1)) = 2*x_1 POL(5(x_1)) = 2*x_1 ---------------------------------------- (67) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3^1(5(3(4(5(0(0(2(3(0(x1)))))))))) The TRS R consists of the following rules: 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (68) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented rules of the TRS R: 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 + 2*x_1 POL(1(x_1)) = x_1 POL(2(x_1)) = x_1 POL(3(x_1)) = 2*x_1 POL(3^1(x_1)) = x_1 POL(4(x_1)) = x_1 POL(5(x_1)) = x_1 ---------------------------------------- (69) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3^1(5(3(4(5(0(0(2(3(0(x1)))))))))) The TRS R consists of the following rules: 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (70) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (71) TRUE ---------------------------------------- (72) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0^1(4(0(0(3(1(0(2(2(0(x1)))))))))) 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0^1(3(5(5(4(5(1(3(0(0(x1)))))))))) 0^1(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0^1(2(4(2(3(2(1(0(3(1(x1)))))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0^1(1(0(5(2(2(2(2(3(1(x1)))))))))) 0^1(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0^1(4(0(5(4(5(0(3(5(1(x1)))))))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0^1(5(5(4(3(0(5(1(4(2(x1)))))))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0^1(5(3(0(4(3(3(1(5(2(x1)))))))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0^1(3(0(2(4(0(3(0(5(3(x1)))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(1(3(1(0(5(0(5(2(4(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (73) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0^1(4(0(0(3(1(0(2(2(0(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 POL(0^1(x_1)) = x_1 POL(1(x_1)) = 0 POL(2(x_1)) = 0 POL(3(x_1)) = 1 POL(4(x_1)) = 0 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) ---------------------------------------- (74) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0^1(3(5(5(4(5(1(3(0(0(x1)))))))))) 0^1(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0^1(2(4(2(3(2(1(0(3(1(x1)))))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0^1(1(0(5(2(2(2(2(3(1(x1)))))))))) 0^1(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0^1(4(0(5(4(5(0(3(5(1(x1)))))))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0^1(5(5(4(3(0(5(1(4(2(x1)))))))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0^1(5(3(0(4(3(3(1(5(2(x1)))))))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0^1(3(0(2(4(0(3(0(5(3(x1)))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(1(3(1(0(5(0(5(2(4(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (75) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0^1(4(0(5(4(5(0(3(5(1(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(0^1(x_1)) = x_1 POL(1(x_1)) = 0 POL(2(x_1)) = 0 POL(3(x_1)) = 0 POL(4(x_1)) = 1 + x_1 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) ---------------------------------------- (76) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0^1(3(5(5(4(5(1(3(0(0(x1)))))))))) 0^1(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0^1(2(4(2(3(2(1(0(3(1(x1)))))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0^1(1(0(5(2(2(2(2(3(1(x1)))))))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0^1(5(5(4(3(0(5(1(4(2(x1)))))))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0^1(5(3(0(4(3(3(1(5(2(x1)))))))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0^1(3(0(2(4(0(3(0(5(3(x1)))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(1(3(1(0(5(0(5(2(4(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (77) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0^1(2(4(2(3(2(1(0(3(1(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 POL(0^1(x_1)) = x_1 POL(1(x_1)) = 1 POL(2(x_1)) = x_1 POL(3(x_1)) = 1 POL(4(x_1)) = 0 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) ---------------------------------------- (78) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0^1(3(5(5(4(5(1(3(0(0(x1)))))))))) 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0^1(1(0(5(2(2(2(2(3(1(x1)))))))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0^1(5(5(4(3(0(5(1(4(2(x1)))))))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0^1(5(3(0(4(3(3(1(5(2(x1)))))))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0^1(3(0(2(4(0(3(0(5(3(x1)))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(1(3(1(0(5(0(5(2(4(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (79) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0^1(1(0(5(2(2(2(2(3(1(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(0^1(x_1)) = x_1 POL(1(x_1)) = x_1 POL(2(x_1)) = 1 POL(3(x_1)) = 1 POL(4(x_1)) = 0 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) ---------------------------------------- (80) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0^1(3(5(5(4(5(1(3(0(0(x1)))))))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0^1(5(5(4(3(0(5(1(4(2(x1)))))))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0^1(5(3(0(4(3(3(1(5(2(x1)))))))))) 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0^1(3(0(2(4(0(3(0(5(3(x1)))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(1(3(1(0(5(0(5(2(4(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (81) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0^1(3(0(2(4(0(3(0(5(3(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 + x_1 POL(0^1(x_1)) = x_1 POL(1(x_1)) = 0 POL(2(x_1)) = 0 POL(3(x_1)) = 1 + x_1 POL(4(x_1)) = 0 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) ---------------------------------------- (82) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0^1(3(5(5(4(5(1(3(0(0(x1)))))))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0^1(5(5(4(3(0(5(1(4(2(x1)))))))))) 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0^1(5(3(0(4(3(3(1(5(2(x1)))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(1(3(1(0(5(0(5(2(4(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (83) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0^1(5(3(0(4(3(3(1(5(2(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 POL(0^1(x_1)) = x_1 POL(1(x_1)) = x_1 POL(2(x_1)) = 0 POL(3(x_1)) = 1 + x_1 POL(4(x_1)) = x_1 POL(5(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) ---------------------------------------- (84) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0^1(3(5(5(4(5(1(3(0(0(x1)))))))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0^1(5(5(4(3(0(5(1(4(2(x1)))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(1(3(1(0(5(0(5(2(4(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (85) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented rules of the TRS R: 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(0^1(x_1)) = 2*x_1 POL(1(x_1)) = x_1 POL(2(x_1)) = 2*x_1 POL(3(x_1)) = 2 + 2*x_1 POL(4(x_1)) = x_1 POL(5(x_1)) = 2 + 2*x_1 ---------------------------------------- (86) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0^1(3(5(5(4(5(1(3(0(0(x1)))))))))) 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0^1(5(5(4(3(0(5(1(4(2(x1)))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(1(3(1(0(5(0(5(2(4(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (87) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0^1(3(5(5(4(5(1(3(0(0(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 0 POL(0^1(x_1)) = x_1 POL(1(x_1)) = 1 POL(2(x_1)) = x_1 POL(3(x_1)) = 1 + x_1 POL(4(x_1)) = 0 POL(5(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) ---------------------------------------- (88) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0^1(5(5(4(3(0(5(1(4(2(x1)))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(1(3(1(0(5(0(5(2(4(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (89) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented rules of the TRS R: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 2 + 2*x_1 POL(0^1(x_1)) = 2*x_1 POL(1(x_1)) = x_1 POL(2(x_1)) = 2 + 2*x_1 POL(3(x_1)) = 2*x_1 POL(4(x_1)) = x_1 POL(5(x_1)) = x_1 ---------------------------------------- (90) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0^1(5(5(4(3(0(5(1(4(2(x1)))))))))) 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(1(3(1(0(5(0(5(2(4(x1)))))))))) The TRS R consists of the following rules: 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (91) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. ---------------------------------------- (92) Complex Obligation (AND) ---------------------------------------- (93) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0^1(5(5(4(3(0(5(1(4(2(x1)))))))))) The TRS R consists of the following rules: 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (94) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented rules of the TRS R: 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(0^1(x_1)) = x_1 POL(1(x_1)) = 2*x_1 POL(2(x_1)) = 2 + 2*x_1 POL(3(x_1)) = x_1 POL(4(x_1)) = x_1 POL(5(x_1)) = 2*x_1 ---------------------------------------- (95) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0^1(5(5(4(3(0(5(1(4(2(x1)))))))))) The TRS R consists of the following rules: 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (96) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented rules of the TRS R: 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 2*x_1 POL(0^1(x_1)) = x_1 POL(1(x_1)) = 2*x_1 POL(2(x_1)) = 2 + 2*x_1 POL(3(x_1)) = 2*x_1 POL(4(x_1)) = 2*x_1 POL(5(x_1)) = 2*x_1 ---------------------------------------- (97) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0^1(5(5(4(3(0(5(1(4(2(x1)))))))))) The TRS R consists of the following rules: 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (98) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented rules of the TRS R: 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 + 2*x_1 POL(0^1(x_1)) = x_1 POL(1(x_1)) = x_1 POL(2(x_1)) = 2*x_1 POL(3(x_1)) = 2*x_1 POL(4(x_1)) = x_1 POL(5(x_1)) = 2 + 3*x_1 ---------------------------------------- (99) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0^1(5(5(4(3(0(5(1(4(2(x1)))))))))) The TRS R consists of the following rules: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (100) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented rules of the TRS R: 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(0^1(x_1)) = x_1 POL(1(x_1)) = 2 + 2*x_1 POL(2(x_1)) = 2*x_1 POL(3(x_1)) = x_1 POL(4(x_1)) = x_1 POL(5(x_1)) = 1 + x_1 ---------------------------------------- (101) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0^1(5(5(4(3(0(5(1(4(2(x1)))))))))) The TRS R consists of the following rules: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (102) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0^1(5(5(4(3(0(5(1(4(2(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( 0_1(x_1) ) = max{0, 2x_1 - 1} POL( 0^1_1(x_1) ) = 2x_1 + 2 POL( 3_1(x_1) ) = max{0, x_1 - 2} POL( 4_1(x_1) ) = max{0, 2x_1 - 2} POL( 5_1(x_1) ) = max{0, 2x_1 - 2} POL( 1_1(x_1) ) = 2 POL( 2_1(x_1) ) = max{0, -2} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) ---------------------------------------- (103) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (104) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (105) YES ---------------------------------------- (106) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(1(3(1(0(5(0(5(2(4(x1)))))))))) The TRS R consists of the following rules: 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (107) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented rules of the TRS R: 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 2*x_1 POL(0^1(x_1)) = 2*x_1 POL(1(x_1)) = x_1 POL(2(x_1)) = 2 + 2*x_1 POL(3(x_1)) = x_1 POL(4(x_1)) = x_1 POL(5(x_1)) = 2*x_1 ---------------------------------------- (108) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(1(3(1(0(5(0(5(2(4(x1)))))))))) The TRS R consists of the following rules: 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (109) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented rules of the TRS R: 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(0^1(x_1)) = 2*x_1 POL(1(x_1)) = 2*x_1 POL(2(x_1)) = 1 + 2*x_1 POL(3(x_1)) = 2*x_1 POL(4(x_1)) = 2*x_1 POL(5(x_1)) = x_1 ---------------------------------------- (110) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(1(3(1(0(5(0(5(2(4(x1)))))))))) The TRS R consists of the following rules: 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (111) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented rules of the TRS R: 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(0^1(x_1)) = 2*x_1 POL(1(x_1)) = 2*x_1 POL(2(x_1)) = 2*x_1 POL(3(x_1)) = 2*x_1 POL(4(x_1)) = x_1 POL(5(x_1)) = 1 + x_1 ---------------------------------------- (112) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(1(3(1(0(5(0(5(2(4(x1)))))))))) The TRS R consists of the following rules: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (113) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0^1(1(3(1(0(5(0(5(2(4(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 2*x_1 POL(0^1(x_1)) = 4*x_1 POL(1(x_1)) = 2 + 4*x_1 POL(2(x_1)) = x_1 POL(3(x_1)) = 1 + 4*x_1 POL(4(x_1)) = 0 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) ---------------------------------------- (114) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (115) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (116) YES ---------------------------------------- (117) Obligation: Q DP problem: The TRS P consists of the following rules: 4^1(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4^1(3(3(1(2(4(3(0(0(2(x1)))))))))) 4^1(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4^1(3(3(5(0(1(3(2(1(1(x1)))))))))) 4^1(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4^1(3(1(1(3(4(4(0(4(2(x1)))))))))) 4^1(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4^1(4(0(3(3(2(5(3(0(5(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (118) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 4^1(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4^1(3(1(1(3(4(4(0(4(2(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 0 POL(1(x_1)) = 1 POL(2(x_1)) = 0 POL(3(x_1)) = x_1 POL(4(x_1)) = 1 + x_1 POL(4^1(x_1)) = x_1 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) ---------------------------------------- (119) Obligation: Q DP problem: The TRS P consists of the following rules: 4^1(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4^1(3(3(1(2(4(3(0(0(2(x1)))))))))) 4^1(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4^1(3(3(5(0(1(3(2(1(1(x1)))))))))) 4^1(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4^1(4(0(3(3(2(5(3(0(5(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (120) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 4^1(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4^1(4(0(3(3(2(5(3(0(5(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 POL(1(x_1)) = 1 POL(2(x_1)) = 0 POL(3(x_1)) = 1 + x_1 POL(4(x_1)) = x_1 POL(4^1(x_1)) = x_1 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) ---------------------------------------- (121) Obligation: Q DP problem: The TRS P consists of the following rules: 4^1(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4^1(3(3(1(2(4(3(0(0(2(x1)))))))))) 4^1(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4^1(3(3(5(0(1(3(2(1(1(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (122) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented dependency pairs: 4^1(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4^1(3(3(5(0(1(3(2(1(1(x1)))))))))) Strictly oriented rules of the TRS R: 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(1(x_1)) = x_1 POL(2(x_1)) = 1 + 2*x_1 POL(3(x_1)) = 3 + x_1 POL(4(x_1)) = x_1 POL(4^1(x_1)) = x_1 POL(5(x_1)) = 3 + x_1 ---------------------------------------- (123) Obligation: Q DP problem: The TRS P consists of the following rules: 4^1(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4^1(3(3(1(2(4(3(0(0(2(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (124) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 4^1(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4^1(3(3(1(2(4(3(0(0(2(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 0 POL(1(x_1)) = x_1 POL(2(x_1)) = x_1 POL(3(x_1)) = 1 + x_1 POL(4(x_1)) = 1 POL(4^1(x_1)) = x_1 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) ---------------------------------------- (125) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (126) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (127) YES ---------------------------------------- (128) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2^1(3(1(5(2(2(0(1(5(1(x1)))))))))) 2^1(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2^1(3(1(0(5(0(5(1(0(1(x1)))))))))) 2^1(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2^1(5(5(4(1(3(0(5(3(2(x1)))))))))) 2^1(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2^1(4(0(4(4(5(1(1(5(2(x1)))))))))) 2^1(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2^1(3(5(3(3(0(4(0(4(4(x1)))))))))) 2^1(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2^1(4(1(3(1(5(3(1(5(4(x1)))))))))) 2^1(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2^1(3(2(4(3(4(0(0(3(5(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (129) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 2^1(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2^1(4(1(3(1(5(3(1(5(4(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 POL(1(x_1)) = 1 POL(2(x_1)) = 0 POL(2^1(x_1)) = x_1 POL(3(x_1)) = 1 POL(4(x_1)) = 0 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) ---------------------------------------- (130) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2^1(3(1(5(2(2(0(1(5(1(x1)))))))))) 2^1(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2^1(3(1(0(5(0(5(1(0(1(x1)))))))))) 2^1(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2^1(5(5(4(1(3(0(5(3(2(x1)))))))))) 2^1(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2^1(4(0(4(4(5(1(1(5(2(x1)))))))))) 2^1(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2^1(3(5(3(3(0(4(0(4(4(x1)))))))))) 2^1(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2^1(3(2(4(3(4(0(0(3(5(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (131) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 2^1(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2^1(3(5(3(3(0(4(0(4(4(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 0 POL(1(x_1)) = 0 POL(2(x_1)) = 0 POL(2^1(x_1)) = x_1 POL(3(x_1)) = x_1 POL(4(x_1)) = 1 POL(5(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) ---------------------------------------- (132) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2^1(3(1(5(2(2(0(1(5(1(x1)))))))))) 2^1(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2^1(3(1(0(5(0(5(1(0(1(x1)))))))))) 2^1(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2^1(5(5(4(1(3(0(5(3(2(x1)))))))))) 2^1(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2^1(4(0(4(4(5(1(1(5(2(x1)))))))))) 2^1(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2^1(3(2(4(3(4(0(0(3(5(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (133) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 2^1(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2^1(5(5(4(1(3(0(5(3(2(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 POL(1(x_1)) = 1 POL(2(x_1)) = 0 POL(2^1(x_1)) = x_1 POL(3(x_1)) = 1 + x_1 POL(4(x_1)) = x_1 POL(5(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) ---------------------------------------- (134) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2^1(3(1(5(2(2(0(1(5(1(x1)))))))))) 2^1(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2^1(3(1(0(5(0(5(1(0(1(x1)))))))))) 2^1(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2^1(4(0(4(4(5(1(1(5(2(x1)))))))))) 2^1(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2^1(3(2(4(3(4(0(0(3(5(x1)))))))))) The TRS R consists of the following rules: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (135) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented rules of the TRS R: 5(5(1(0(3(2(5(1(0(0(x1)))))))))) -> 5(5(0(5(2(1(3(1(0(0(x1)))))))))) 0(3(5(1(5(4(3(5(0(0(x1)))))))))) -> 0(3(5(5(4(5(1(3(0(0(x1)))))))))) 4(3(3(5(0(1(2(3(1(1(x1)))))))))) -> 4(3(3(5(0(1(3(2(1(1(x1)))))))))) 3(4(3(1(5(1(5(1(2(1(x1)))))))))) -> 1(3(1(5(4(5(3(1(2(1(x1)))))))))) 0(1(0(2(5(2(2(2(3(1(x1)))))))))) -> 0(1(0(5(2(2(2(2(3(1(x1)))))))))) 5(3(0(0(4(2(2(5(4(1(x1)))))))))) -> 5(2(4(0(3(1(0(5(2(4(x1)))))))))) 5(2(2(1(2(3(3(0(1(2(x1)))))))))) -> 5(2(2(1(3(2(3(0(1(2(x1)))))))))) 1(4(2(5(3(4(4(2(1(2(x1)))))))))) -> 5(2(4(4(1(3(4(2(1(2(x1)))))))))) 1(2(4(2(3(1(4(3(1(2(x1)))))))))) -> 1(2(4(3(2(1(4(3(1(2(x1)))))))))) 0(2(5(0(4(2(3(4(0(3(x1)))))))))) -> 3(2(4(0(5(4(2(0(0(3(x1)))))))))) 3(5(0(0(4(2(1(0(2(3(x1)))))))))) -> 0(5(0(3(2(4(1(0(2(3(x1)))))))))) 5(4(0(3(5(1(2(5(3(3(x1)))))))))) -> 5(0(3(5(5(2(4(1(3(3(x1)))))))))) 3(2(4(2(0(2(3(1(5(3(x1)))))))))) -> 2(3(2(0(5(4(1(3(2(3(x1)))))))))) 3(1(0(2(0(2(2(3(1(4(x1)))))))))) -> 3(1(0(2(0(1(2(3(2(4(x1)))))))))) 0(2(0(1(2(3(1(0(1(5(x1)))))))))) -> 2(0(0(1(3(2(1(0(1(5(x1)))))))))) 3(4(1(1(5(1(5(1(1(5(x1)))))))))) -> 1(4(5(1(3(1(5(1(1(5(x1)))))))))) 5(0(2(3(5(5(1(3(5(5(x1)))))))))) -> 5(0(3(2(5(5(1(3(5(5(x1)))))))))) Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(1(x_1)) = x_1 POL(2(x_1)) = 2*x_1 POL(2^1(x_1)) = x_1 POL(3(x_1)) = 1 + 2*x_1 POL(4(x_1)) = x_1 POL(5(x_1)) = 2 + 2*x_1 ---------------------------------------- (136) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2^1(3(1(5(2(2(0(1(5(1(x1)))))))))) 2^1(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2^1(3(1(0(5(0(5(1(0(1(x1)))))))))) 2^1(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2^1(4(0(4(4(5(1(1(5(2(x1)))))))))) 2^1(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2^1(3(2(4(3(4(0(0(3(5(x1)))))))))) The TRS R consists of the following rules: 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (137) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented rules of the TRS R: 0(2(1(4(2(3(2(0(3(1(x1)))))))))) -> 0(2(4(2(3(2(1(0(3(1(x1)))))))))) Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(1(x_1)) = 2*x_1 POL(2(x_1)) = 2 + x_1 POL(2^1(x_1)) = 2*x_1 POL(3(x_1)) = 2*x_1 POL(4(x_1)) = x_1 POL(5(x_1)) = 2*x_1 ---------------------------------------- (138) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2^1(3(1(5(2(2(0(1(5(1(x1)))))))))) 2^1(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2^1(3(1(0(5(0(5(1(0(1(x1)))))))))) 2^1(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2^1(4(0(4(4(5(1(1(5(2(x1)))))))))) 2^1(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2^1(3(2(4(3(4(0(0(3(5(x1)))))))))) The TRS R consists of the following rules: 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (139) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented rules of the TRS R: 3(4(3(1(0(0(4(2(5(0(x1)))))))))) -> 3(4(3(1(0(0(2(4(5(0(x1)))))))))) Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(1(x_1)) = x_1 POL(2(x_1)) = 1 + 2*x_1 POL(2^1(x_1)) = 2*x_1 POL(3(x_1)) = x_1 POL(4(x_1)) = 2*x_1 POL(5(x_1)) = x_1 ---------------------------------------- (140) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2^1(3(1(5(2(2(0(1(5(1(x1)))))))))) 2^1(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2^1(3(1(0(5(0(5(1(0(1(x1)))))))))) 2^1(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2^1(4(0(4(4(5(1(1(5(2(x1)))))))))) 2^1(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2^1(3(2(4(3(4(0(0(3(5(x1)))))))))) The TRS R consists of the following rules: 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (141) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented dependency pairs: 2^1(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2^1(3(1(5(2(2(0(1(5(1(x1)))))))))) Strictly oriented rules of the TRS R: 3(3(0(2(0(1(0(4(0(0(x1)))))))))) -> 3(3(2(0(0(1(0(4(0(0(x1)))))))))) 0(0(4(3(0(1(0(2(2(0(x1)))))))))) -> 0(4(0(0(3(1(0(2(2(0(x1)))))))))) 2(3(1(5(2(0(2(1(5(1(x1)))))))))) -> 2(3(1(5(2(2(0(1(5(1(x1)))))))))) 0(5(3(4(3(0(3(1(5(2(x1)))))))))) -> 0(5(3(0(4(3(3(1(5(2(x1)))))))))) 5(4(1(5(3(0(4(1(0(3(x1)))))))))) -> 5(4(0(5(4(3(1(1(0(3(x1)))))))))) 0(3(0(0(2(4(3(0(5(3(x1)))))))))) -> 0(3(0(2(4(0(3(0(5(3(x1)))))))))) 3(0(0(3(0(0(0(2(3(4(x1)))))))))) -> 3(0(0(3(0(0(2(0(3(4(x1)))))))))) 4(4(3(0(3(2(5(3(0(5(x1)))))))))) -> 4(4(0(3(3(2(5(3(0(5(x1)))))))))) Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 2 + 2*x_1 POL(1(x_1)) = x_1 POL(2(x_1)) = 3 + 2*x_1 POL(2^1(x_1)) = x_1 POL(3(x_1)) = 2*x_1 POL(4(x_1)) = x_1 POL(5(x_1)) = x_1 ---------------------------------------- (142) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2^1(3(1(0(5(0(5(1(0(1(x1)))))))))) 2^1(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2^1(4(0(4(4(5(1(1(5(2(x1)))))))))) 2^1(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2^1(3(2(4(3(4(0(0(3(5(x1)))))))))) The TRS R consists of the following rules: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (143) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (144) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2^1(3(2(4(3(4(0(0(3(5(x1)))))))))) 2^1(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2^1(3(1(0(5(0(5(1(0(1(x1)))))))))) The TRS R consists of the following rules: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (145) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented rules of the TRS R: 3(3(4(2(1(5(3(3(4(0(x1)))))))))) -> 3(3(4(2(5(1(3(3(4(0(x1)))))))))) Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(1(x_1)) = 2*x_1 POL(2(x_1)) = 2*x_1 POL(2^1(x_1)) = x_1 POL(3(x_1)) = 2*x_1 POL(4(x_1)) = x_1 POL(5(x_1)) = 1 + 2*x_1 ---------------------------------------- (146) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2^1(3(2(4(3(4(0(0(3(5(x1)))))))))) 2^1(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2^1(3(1(0(5(0(5(1(0(1(x1)))))))))) The TRS R consists of the following rules: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (147) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented rules of the TRS R: 1(4(3(4(3(4(2(4(0(4(x1)))))))))) -> 1(4(4(3(0(4(3(2(4(4(x1)))))))))) Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 + 2*x_1 POL(1(x_1)) = x_1 POL(2(x_1)) = 2*x_1 POL(2^1(x_1)) = x_1 POL(3(x_1)) = x_1 POL(4(x_1)) = 1 + 2*x_1 POL(5(x_1)) = x_1 ---------------------------------------- (148) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2^1(3(2(4(3(4(0(0(3(5(x1)))))))))) 2^1(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2^1(3(1(0(5(0(5(1(0(1(x1)))))))))) The TRS R consists of the following rules: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (149) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 2^1(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2^1(3(2(4(3(4(0(0(3(5(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 0 POL(1(x_1)) = 0 POL(2(x_1)) = 2*x_1 POL(2^1(x_1)) = x_1 POL(3(x_1)) = 1 + 4*x_1 POL(4(x_1)) = 3*x_1 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) ---------------------------------------- (150) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2^1(3(1(0(5(0(5(1(0(1(x1)))))))))) The TRS R consists of the following rules: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (151) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 2^1(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2^1(3(1(0(5(0(5(1(0(1(x1)))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 2 + 2*x_1 POL(1(x_1)) = x_1 POL(2(x_1)) = 0 POL(2^1(x_1)) = x_1 POL(3(x_1)) = 1 + 2*x_1 POL(4(x_1)) = 0 POL(5(x_1)) = 2*x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) ---------------------------------------- (152) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: 3(5(3(4(0(5(0(2(3(0(x1)))))))))) -> 3(5(3(4(5(0(0(2(3(0(x1)))))))))) 1(2(1(3(0(4(0(5(5(0(x1)))))))))) -> 1(2(1(3(0(0(5(4(5(0(x1)))))))))) 3(4(1(0(4(3(5(0(0(1(x1)))))))))) -> 3(4(4(0(3(1(5(0(0(1(x1)))))))))) 2(3(1(0(5(5(0(1(0(1(x1)))))))))) -> 2(3(1(0(5(0(5(1(0(1(x1)))))))))) 3(3(3(2(3(1(4(0(5(1(x1)))))))))) -> 3(3(3(2(1(3(0(4(5(1(x1)))))))))) 0(4(0(4(5(5(0(3(5(1(x1)))))))))) -> 0(4(0(5(4(5(0(3(5(1(x1)))))))))) 4(3(3(1(2(3(4(0(0(2(x1)))))))))) -> 4(3(3(1(2(4(3(0(0(2(x1)))))))))) 0(5(5(4(3(5(0(4(1(2(x1)))))))))) -> 0(5(5(4(3(0(5(1(4(2(x1)))))))))) 2(5(5(4(3(1(0(5(3(2(x1)))))))))) -> 2(5(5(4(1(3(0(5(3(2(x1)))))))))) 4(3(3(4(1(4(1(0(4(2(x1)))))))))) -> 4(3(1(1(3(4(4(0(4(2(x1)))))))))) 2(4(0(4(5(4(1(1(5(2(x1)))))))))) -> 2(4(0(4(4(5(1(1(5(2(x1)))))))))) 3(5(0(4(1(5(0(5(4(3(x1)))))))))) -> 3(5(0(1(4(5(0(5(4(3(x1)))))))))) 0(1(3(0(1(5(0(5(2(4(x1)))))))))) -> 0(1(3(1(0(5(0(5(2(4(x1)))))))))) 2(3(5(3(3(4(0(0(4(4(x1)))))))))) -> 2(3(5(3(3(0(4(0(4(4(x1)))))))))) 1(5(5(5(3(4(0(1(5(4(x1)))))))))) -> 1(5(5(5(4(3(0(1(5(4(x1)))))))))) 2(1(4(3(1(5(3(1(5(4(x1)))))))))) -> 2(4(1(3(1(5(3(1(5(4(x1)))))))))) 2(3(2(4(4(3(0(0(3(5(x1)))))))))) -> 2(3(2(4(3(4(0(0(3(5(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (153) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (154) YES