864.86/274.92 YES 883.72/279.68 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 883.72/279.68 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 883.72/279.68 883.72/279.68 883.72/279.68 Termination w.r.t. Q of the given QTRS could be proven: 883.72/279.68 883.72/279.68 (0) QTRS 883.72/279.68 (1) DependencyPairsProof [EQUIVALENT, 248 ms] 883.72/279.68 (2) QDP 883.72/279.68 (3) DependencyGraphProof [EQUIVALENT, 14 ms] 883.72/279.68 (4) QDP 883.72/279.68 (5) QDPOrderProof [EQUIVALENT, 418 ms] 883.72/279.68 (6) QDP 883.72/279.68 (7) QDPOrderProof [EQUIVALENT, 131 ms] 883.72/279.68 (8) QDP 883.72/279.68 (9) QDPOrderProof [EQUIVALENT, 146 ms] 883.72/279.68 (10) QDP 883.72/279.68 (11) QDPOrderProof [EQUIVALENT, 123 ms] 883.72/279.68 (12) QDP 883.72/279.68 (13) QDPOrderProof [EQUIVALENT, 84 ms] 883.72/279.68 (14) QDP 883.72/279.68 (15) QDPOrderProof [EQUIVALENT, 708 ms] 883.72/279.68 (16) QDP 883.72/279.68 (17) QDPOrderProof [EQUIVALENT, 1646 ms] 883.72/279.68 (18) QDP 883.72/279.68 (19) TransformationProof [EQUIVALENT, 0 ms] 883.72/279.68 (20) QDP 883.72/279.68 (21) DependencyGraphProof [EQUIVALENT, 0 ms] 883.72/279.68 (22) QDP 883.72/279.68 (23) TransformationProof [EQUIVALENT, 0 ms] 883.72/279.68 (24) QDP 883.72/279.68 (25) DependencyGraphProof [EQUIVALENT, 0 ms] 883.72/279.68 (26) QDP 883.72/279.68 (27) TransformationProof [EQUIVALENT, 0 ms] 883.72/279.68 (28) QDP 883.72/279.68 (29) DependencyGraphProof [EQUIVALENT, 0 ms] 883.72/279.68 (30) QDP 883.72/279.68 (31) TransformationProof [EQUIVALENT, 0 ms] 883.72/279.68 (32) QDP 883.72/279.68 (33) DependencyGraphProof [EQUIVALENT, 0 ms] 883.72/279.68 (34) QDP 883.72/279.68 (35) TransformationProof [EQUIVALENT, 0 ms] 883.72/279.68 (36) QDP 883.72/279.68 (37) DependencyGraphProof [EQUIVALENT, 0 ms] 883.72/279.68 (38) QDP 883.72/279.68 (39) TransformationProof [EQUIVALENT, 0 ms] 883.72/279.68 (40) QDP 883.72/279.68 (41) DependencyGraphProof [EQUIVALENT, 0 ms] 883.72/279.68 (42) QDP 883.72/279.68 (43) TransformationProof [EQUIVALENT, 0 ms] 883.72/279.68 (44) QDP 883.72/279.68 (45) DependencyGraphProof [EQUIVALENT, 0 ms] 883.72/279.68 (46) QDP 883.72/279.68 (47) TransformationProof [EQUIVALENT, 22 ms] 883.72/279.68 (48) QDP 883.72/279.68 (49) DependencyGraphProof [EQUIVALENT, 0 ms] 883.72/279.68 (50) QDP 883.72/279.68 (51) TransformationProof [EQUIVALENT, 0 ms] 883.72/279.68 (52) QDP 883.72/279.68 (53) DependencyGraphProof [EQUIVALENT, 0 ms] 883.72/279.68 (54) QDP 883.72/279.68 (55) TransformationProof [EQUIVALENT, 0 ms] 883.72/279.68 (56) QDP 883.72/279.68 (57) DependencyGraphProof [EQUIVALENT, 0 ms] 883.72/279.68 (58) QDP 883.72/279.68 (59) TransformationProof [EQUIVALENT, 26 ms] 883.72/279.68 (60) QDP 883.72/279.68 (61) DependencyGraphProof [EQUIVALENT, 0 ms] 883.72/279.68 (62) QDP 883.72/279.68 (63) TransformationProof [EQUIVALENT, 0 ms] 883.72/279.68 (64) QDP 883.72/279.68 (65) DependencyGraphProof [EQUIVALENT, 0 ms] 883.72/279.68 (66) QDP 883.72/279.68 (67) TransformationProof [EQUIVALENT, 19 ms] 883.72/279.68 (68) QDP 883.72/279.68 (69) DependencyGraphProof [EQUIVALENT, 0 ms] 883.72/279.68 (70) QDP 883.72/279.68 (71) TransformationProof [EQUIVALENT, 0 ms] 883.72/279.68 (72) QDP 883.72/279.68 (73) DependencyGraphProof [EQUIVALENT, 0 ms] 883.72/279.68 (74) QDP 883.72/279.68 (75) TransformationProof [EQUIVALENT, 26 ms] 883.72/279.68 (76) QDP 883.72/279.68 (77) DependencyGraphProof [EQUIVALENT, 0 ms] 883.72/279.68 (78) QDP 883.72/279.68 (79) TransformationProof [EQUIVALENT, 10 ms] 883.72/279.68 (80) QDP 883.72/279.68 (81) DependencyGraphProof [EQUIVALENT, 0 ms] 883.72/279.68 (82) QDP 883.72/279.68 (83) TransformationProof [EQUIVALENT, 0 ms] 883.72/279.68 (84) QDP 883.72/279.68 (85) DependencyGraphProof [EQUIVALENT, 0 ms] 883.72/279.68 (86) QDP 883.72/279.68 (87) TransformationProof [EQUIVALENT, 22 ms] 883.72/279.68 (88) QDP 883.72/279.68 (89) DependencyGraphProof [EQUIVALENT, 0 ms] 883.72/279.68 (90) QDP 883.72/279.68 (91) TransformationProof [EQUIVALENT, 0 ms] 883.72/279.68 (92) QDP 883.72/279.68 (93) DependencyGraphProof [EQUIVALENT, 0 ms] 883.72/279.68 (94) QDP 883.72/279.68 (95) TransformationProof [EQUIVALENT, 19 ms] 883.72/279.68 (96) QDP 883.72/279.68 (97) DependencyGraphProof [EQUIVALENT, 0 ms] 883.72/279.68 (98) QDP 883.72/279.68 (99) TransformationProof [EQUIVALENT, 31 ms] 883.72/279.68 (100) QDP 883.72/279.68 (101) DependencyGraphProof [EQUIVALENT, 0 ms] 883.72/279.68 (102) QDP 883.72/279.68 (103) QDPOrderProof [EQUIVALENT, 4807 ms] 883.72/279.68 (104) QDP 883.72/279.68 (105) DependencyGraphProof [EQUIVALENT, 0 ms] 883.72/279.68 (106) AND 883.72/279.68 (107) QDP 883.72/279.68 (108) QDPOrderProof [EQUIVALENT, 824 ms] 883.72/279.68 (109) QDP 883.72/279.68 (110) PisEmptyProof [EQUIVALENT, 0 ms] 883.72/279.68 (111) YES 883.72/279.68 (112) QDP 883.72/279.68 (113) QDPOrderProof [EQUIVALENT, 23 ms] 883.72/279.68 (114) QDP 883.72/279.68 (115) QDPOrderProof [EQUIVALENT, 20 ms] 883.72/279.68 (116) QDP 883.72/279.68 (117) DependencyGraphProof [EQUIVALENT, 0 ms] 883.72/279.68 (118) QDP 883.72/279.68 (119) QDPOrderProof [EQUIVALENT, 0 ms] 883.72/279.68 (120) QDP 883.72/279.68 (121) PisEmptyProof [EQUIVALENT, 0 ms] 883.72/279.68 (122) YES 883.72/279.68 883.72/279.68 883.72/279.68 ---------------------------------------- 883.72/279.68 883.72/279.68 (0) 883.72/279.68 Obligation: 883.72/279.68 Q restricted rewrite system: 883.72/279.68 The TRS R consists of the following rules: 883.72/279.68 883.72/279.68 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.72/279.68 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.72/279.68 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.72/279.68 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.72/279.68 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.72/279.68 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.72/279.68 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.72/279.68 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.72/279.68 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.72/279.68 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.72/279.68 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.72/279.68 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.72/279.68 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.72/279.68 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.72/279.68 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.72/279.68 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.72/279.68 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.72/279.68 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.72/279.68 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.72/279.68 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.72/279.68 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.72/279.68 883.72/279.68 Q is empty. 883.72/279.68 883.72/279.68 ---------------------------------------- 883.72/279.68 883.72/279.68 (1) DependencyPairsProof (EQUIVALENT) 883.72/279.68 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 883.72/279.68 ---------------------------------------- 883.72/279.68 883.72/279.68 (2) 883.72/279.68 Obligation: 883.72/279.68 Q DP problem: 883.72/279.68 The TRS P consists of the following rules: 883.72/279.68 883.72/279.68 5^1(5(x1)) -> 0^1(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.72/279.68 5^1(5(x1)) -> 5^1(4(0(2(5(4(5(2(1(x1))))))))) 883.72/279.68 5^1(5(x1)) -> 4^1(0(2(5(4(5(2(1(x1)))))))) 883.72/279.68 5^1(5(x1)) -> 0^1(2(5(4(5(2(1(x1))))))) 883.72/279.68 5^1(5(x1)) -> 2^1(5(4(5(2(1(x1)))))) 883.72/279.68 5^1(5(x1)) -> 5^1(4(5(2(1(x1))))) 883.72/279.68 5^1(5(x1)) -> 4^1(5(2(1(x1)))) 883.72/279.68 5^1(5(x1)) -> 5^1(2(1(x1))) 883.72/279.68 5^1(5(x1)) -> 2^1(1(x1)) 883.72/279.68 5^1(5(x1)) -> 1^1(x1) 883.72/279.68 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.72/279.68 5^1(5(x1)) -> 1^1(1(1(1(4(4(0(4(x1)))))))) 883.72/279.68 5^1(5(x1)) -> 1^1(1(1(4(4(0(4(x1))))))) 883.72/279.68 5^1(5(x1)) -> 1^1(1(4(4(0(4(x1)))))) 883.72/279.68 5^1(5(x1)) -> 1^1(4(4(0(4(x1))))) 883.72/279.68 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.72/279.68 5^1(5(x1)) -> 4^1(0(4(x1))) 883.72/279.68 5^1(5(x1)) -> 0^1(4(x1)) 883.72/279.68 5^1(5(x1)) -> 4^1(x1) 883.72/279.68 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.72/279.68 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.72/279.68 2^1(5(5(x1))) -> 5^1(4(4(0(0(1(1(2(x1)))))))) 883.72/279.68 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.72/279.68 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.72/279.68 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.72/279.68 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.72/279.68 2^1(5(5(x1))) -> 1^1(1(2(x1))) 883.72/279.68 2^1(5(5(x1))) -> 1^1(2(x1)) 883.72/279.68 2^1(5(5(x1))) -> 2^1(x1) 883.72/279.68 5^1(2(4(x1))) -> 0^1(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.72/279.68 5^1(2(4(x1))) -> 5^1(0(2(3(3(4(2(4(2(x1))))))))) 883.72/279.68 5^1(2(4(x1))) -> 0^1(2(3(3(4(2(4(2(x1)))))))) 883.72/279.68 5^1(2(4(x1))) -> 2^1(3(3(4(2(4(2(x1))))))) 883.72/279.68 5^1(2(4(x1))) -> 4^1(2(4(2(x1)))) 883.72/279.68 5^1(2(4(x1))) -> 2^1(4(2(x1))) 883.72/279.68 5^1(2(4(x1))) -> 4^1(2(x1)) 883.72/279.68 5^1(2(4(x1))) -> 2^1(x1) 883.72/279.68 5^1(5(2(x1))) -> 0^1(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.72/279.68 5^1(5(2(x1))) -> 1^1(3(2(3(0(3(2(5(3(x1))))))))) 883.72/279.68 5^1(5(2(x1))) -> 2^1(3(0(3(2(5(3(x1))))))) 883.72/279.68 5^1(5(2(x1))) -> 0^1(3(2(5(3(x1))))) 883.72/279.68 5^1(5(2(x1))) -> 2^1(5(3(x1))) 883.72/279.68 5^1(5(2(x1))) -> 5^1(3(x1)) 883.72/279.68 5^1(5(3(x1))) -> 0^1(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.72/279.68 5^1(5(3(x1))) -> 5^1(4(4(1(0(1(5(0(x1)))))))) 883.72/279.68 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.72/279.68 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.72/279.68 5^1(5(3(x1))) -> 1^1(0(1(5(0(x1))))) 883.72/279.68 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.72/279.68 5^1(5(3(x1))) -> 1^1(5(0(x1))) 883.72/279.68 5^1(5(3(x1))) -> 5^1(0(x1)) 883.72/279.68 5^1(5(3(x1))) -> 0^1(x1) 883.72/279.68 5^1(5(5(x1))) -> 5^1(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.72/279.68 5^1(5(5(x1))) -> 4^1(1(0(1(4(5(0(0(x1)))))))) 883.72/279.68 5^1(5(5(x1))) -> 1^1(0(1(4(5(0(0(x1))))))) 883.72/279.68 5^1(5(5(x1))) -> 0^1(1(4(5(0(0(x1)))))) 883.72/279.68 5^1(5(5(x1))) -> 1^1(4(5(0(0(x1))))) 883.72/279.68 5^1(5(5(x1))) -> 4^1(5(0(0(x1)))) 883.72/279.68 5^1(5(5(x1))) -> 5^1(0(0(x1))) 883.72/279.68 5^1(5(5(x1))) -> 0^1(0(x1)) 883.72/279.68 5^1(5(5(x1))) -> 0^1(x1) 883.72/279.68 2^1(5(0(4(x1)))) -> 4^1(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.72/279.68 2^1(5(0(4(x1)))) -> 4^1(3(2(4(4(5(1(0(0(x1))))))))) 883.72/279.68 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.72/279.68 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.72/279.68 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.72/279.68 2^1(5(0(4(x1)))) -> 5^1(1(0(0(x1)))) 883.72/279.68 2^1(5(0(4(x1)))) -> 1^1(0(0(x1))) 883.72/279.68 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.72/279.68 2^1(5(0(4(x1)))) -> 0^1(x1) 883.72/279.68 4^1(5(2(4(x1)))) -> 4^1(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.72/279.68 4^1(5(2(4(x1)))) -> 1^1(5(5(2(0(3(1(3(3(x1))))))))) 883.72/279.68 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.72/279.68 4^1(5(2(4(x1)))) -> 5^1(2(0(3(1(3(3(x1))))))) 883.72/279.68 4^1(5(2(4(x1)))) -> 2^1(0(3(1(3(3(x1)))))) 883.72/279.68 4^1(5(2(4(x1)))) -> 0^1(3(1(3(3(x1))))) 883.72/279.68 4^1(5(2(4(x1)))) -> 1^1(3(3(x1))) 883.72/279.68 4^1(5(5(5(x1)))) -> 1^1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.72/279.68 4^1(5(5(5(x1)))) -> 5^1(1(2(0(3(2(1(0(5(x1))))))))) 883.72/279.68 4^1(5(5(5(x1)))) -> 1^1(2(0(3(2(1(0(5(x1)))))))) 883.72/279.68 4^1(5(5(5(x1)))) -> 2^1(0(3(2(1(0(5(x1))))))) 883.72/279.68 4^1(5(5(5(x1)))) -> 0^1(3(2(1(0(5(x1)))))) 883.72/279.68 4^1(5(5(5(x1)))) -> 2^1(1(0(5(x1)))) 883.72/279.68 4^1(5(5(5(x1)))) -> 1^1(0(5(x1))) 883.72/279.68 4^1(5(5(5(x1)))) -> 0^1(5(x1)) 883.72/279.68 0^1(2(5(3(4(x1))))) -> 2^1(4(3(1(5(1(1(3(4(x1))))))))) 883.72/279.68 0^1(2(5(3(4(x1))))) -> 4^1(3(1(5(1(1(3(4(x1)))))))) 883.72/279.68 0^1(2(5(3(4(x1))))) -> 1^1(5(1(1(3(4(x1)))))) 883.72/279.68 0^1(2(5(3(4(x1))))) -> 5^1(1(1(3(4(x1))))) 883.72/279.68 0^1(2(5(3(4(x1))))) -> 1^1(1(3(4(x1)))) 883.72/279.68 0^1(2(5(3(4(x1))))) -> 1^1(3(4(x1))) 883.72/279.68 2^1(5(5(3(4(x1))))) -> 4^1(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.72/279.68 2^1(5(5(3(4(x1))))) -> 5^1(4(3(1(4(0(2(4(4(x1))))))))) 883.72/279.68 2^1(5(5(3(4(x1))))) -> 4^1(3(1(4(0(2(4(4(x1)))))))) 883.72/279.68 2^1(5(5(3(4(x1))))) -> 1^1(4(0(2(4(4(x1)))))) 883.72/279.68 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.72/279.68 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.72/279.68 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.72/279.68 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.72/279.68 5^1(5(5(1(4(x1))))) -> 0^1(5(0(4(3(4(4(0(x1)))))))) 883.72/279.68 5^1(5(5(1(4(x1))))) -> 5^1(0(4(3(4(4(0(x1))))))) 883.72/279.68 5^1(5(5(1(4(x1))))) -> 0^1(4(3(4(4(0(x1)))))) 883.72/279.68 5^1(5(5(1(4(x1))))) -> 4^1(3(4(4(0(x1))))) 883.72/279.68 5^1(5(5(1(4(x1))))) -> 4^1(4(0(x1))) 883.72/279.68 5^1(5(5(1(4(x1))))) -> 4^1(0(x1)) 883.72/279.68 5^1(5(5(1(4(x1))))) -> 0^1(x1) 883.72/279.68 0^1(4(4(5(5(5(x1)))))) -> 0^1(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.72/279.68 0^1(4(4(5(5(5(x1)))))) -> 4^1(4(4(3(3(4(1(3(1(x1))))))))) 883.72/279.68 0^1(4(4(5(5(5(x1)))))) -> 4^1(4(3(3(4(1(3(1(x1)))))))) 883.72/279.68 0^1(4(4(5(5(5(x1)))))) -> 4^1(3(3(4(1(3(1(x1))))))) 883.72/279.68 0^1(4(4(5(5(5(x1)))))) -> 4^1(1(3(1(x1)))) 883.72/279.68 0^1(4(4(5(5(5(x1)))))) -> 1^1(3(1(x1))) 883.72/279.68 0^1(4(4(5(5(5(x1)))))) -> 1^1(x1) 883.72/279.68 1^1(2(4(5(2(4(x1)))))) -> 5^1(3(0(4(0(3(1(3(x1)))))))) 883.72/279.68 1^1(2(4(5(2(4(x1)))))) -> 0^1(4(0(3(1(3(x1)))))) 883.72/279.68 1^1(2(4(5(2(4(x1)))))) -> 4^1(0(3(1(3(x1))))) 883.72/279.68 1^1(2(4(5(2(4(x1)))))) -> 0^1(3(1(3(x1)))) 883.72/279.68 1^1(2(4(5(2(4(x1)))))) -> 1^1(3(x1)) 883.72/279.68 4^1(1(5(5(0(4(x1)))))) -> 1^1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.72/279.68 4^1(1(5(5(0(4(x1)))))) -> 0^1(3(0(4(2(4(4(3(4(x1))))))))) 883.72/279.68 4^1(1(5(5(0(4(x1)))))) -> 0^1(4(2(4(4(3(4(x1))))))) 883.72/279.68 4^1(1(5(5(0(4(x1)))))) -> 4^1(2(4(4(3(4(x1)))))) 883.72/279.68 4^1(1(5(5(0(4(x1)))))) -> 2^1(4(4(3(4(x1))))) 883.72/279.68 4^1(1(5(5(0(4(x1)))))) -> 4^1(4(3(4(x1)))) 883.72/279.68 4^1(1(5(5(0(4(x1)))))) -> 4^1(3(4(x1))) 883.72/279.68 4^1(2(5(5(1(5(x1)))))) -> 2^1(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.72/279.68 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(1(1(3(4(2(5(x1)))))))) 883.72/279.68 4^1(2(5(5(1(5(x1)))))) -> 2^1(1(1(3(4(2(5(x1))))))) 883.72/279.68 4^1(2(5(5(1(5(x1)))))) -> 1^1(1(3(4(2(5(x1)))))) 883.72/279.68 4^1(2(5(5(1(5(x1)))))) -> 1^1(3(4(2(5(x1))))) 883.72/279.68 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.72/279.68 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.72/279.68 5^1(2(5(5(0(4(x1)))))) -> 0^1(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.72/279.68 5^1(2(5(5(0(4(x1)))))) -> 4^1(2(3(3(5(2(1(4(4(x1))))))))) 883.72/279.68 5^1(2(5(5(0(4(x1)))))) -> 2^1(3(3(5(2(1(4(4(x1)))))))) 883.72/279.68 5^1(2(5(5(0(4(x1)))))) -> 5^1(2(1(4(4(x1))))) 883.72/279.68 5^1(2(5(5(0(4(x1)))))) -> 2^1(1(4(4(x1)))) 883.72/279.68 5^1(2(5(5(0(4(x1)))))) -> 1^1(4(4(x1))) 883.72/279.68 5^1(2(5(5(0(4(x1)))))) -> 4^1(4(x1)) 883.72/279.68 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.72/279.68 5^1(5(2(4(5(0(x1)))))) -> 1^1(1(4(2(4(0(4(2(0(x1))))))))) 883.72/279.68 5^1(5(2(4(5(0(x1)))))) -> 1^1(4(2(4(0(4(2(0(x1)))))))) 883.72/279.68 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.72/279.68 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.72/279.68 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.72/279.68 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.72/279.68 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.72/279.68 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.72/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 5^1(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.72/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.72/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 5^1(1(0(1(2(0(5(x1))))))) 883.72/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 1^1(0(1(2(0(5(x1)))))) 883.72/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.72/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 1^1(2(0(5(x1)))) 883.72/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.72/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.72/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.72/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.72/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.72/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(4(5(1(2(2(1(x1))))))) 883.72/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(5(1(2(2(1(x1)))))) 883.72/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(1(2(2(1(x1))))) 883.72/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 1^1(2(2(1(x1)))) 883.72/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(2(1(x1))) 883.72/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) 883.72/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 1^1(x1) 883.72/279.68 883.72/279.68 The TRS R consists of the following rules: 883.72/279.68 883.72/279.68 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.72/279.68 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.72/279.68 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.72/279.68 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.72/279.68 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.72/279.68 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.72/279.68 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.72/279.68 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.72/279.68 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.72/279.68 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.72/279.68 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.72/279.68 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.72/279.68 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.72/279.68 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.72/279.68 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.72/279.68 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.72/279.68 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.72/279.68 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.72/279.68 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.72/279.68 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.72/279.68 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.72/279.68 883.72/279.68 Q is empty. 883.72/279.68 We have to consider all minimal (P,Q,R)-chains. 883.72/279.68 ---------------------------------------- 883.72/279.68 883.72/279.68 (3) DependencyGraphProof (EQUIVALENT) 883.72/279.68 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 86 less nodes. 883.72/279.68 ---------------------------------------- 883.72/279.68 883.72/279.68 (4) 883.72/279.68 Obligation: 883.72/279.68 Q DP problem: 883.72/279.68 The TRS P consists of the following rules: 883.72/279.68 883.72/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.72/279.68 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.72/279.68 4^1(5(2(4(x1)))) -> 4^1(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.72/279.68 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.72/279.68 5^1(5(x1)) -> 0^1(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.72/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 5^1(1(0(1(2(0(5(x1))))))) 883.72/279.68 5^1(5(x1)) -> 5^1(4(0(2(5(4(5(2(1(x1))))))))) 883.72/279.68 5^1(5(x1)) -> 4^1(0(2(5(4(5(2(1(x1)))))))) 883.72/279.68 4^1(5(5(5(x1)))) -> 2^1(1(0(5(x1)))) 883.72/279.68 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.72/279.68 2^1(5(5(x1))) -> 5^1(4(4(0(0(1(1(2(x1)))))))) 883.72/279.68 5^1(5(x1)) -> 0^1(2(5(4(5(2(1(x1))))))) 883.72/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.72/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.72/279.68 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.72/279.68 4^1(5(5(5(x1)))) -> 0^1(5(x1)) 883.72/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.72/279.68 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.72/279.68 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.72/279.68 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.72/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.72/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.72/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.72/279.68 5^1(5(x1)) -> 2^1(5(4(5(2(1(x1)))))) 883.72/279.68 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.72/279.68 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.72/279.68 2^1(5(5(x1))) -> 2^1(x1) 883.72/279.68 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.72/279.68 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.72/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(4(5(1(2(2(1(x1))))))) 883.72/279.68 5^1(5(x1)) -> 5^1(4(5(2(1(x1))))) 883.72/279.68 5^1(5(x1)) -> 4^1(5(2(1(x1)))) 883.72/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(5(1(2(2(1(x1)))))) 883.72/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(1(2(2(1(x1))))) 883.72/279.68 5^1(5(x1)) -> 5^1(2(1(x1))) 883.72/279.68 5^1(5(x1)) -> 2^1(1(x1)) 883.72/279.68 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.72/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(2(1(x1))) 883.72/279.68 2^1(5(0(4(x1)))) -> 5^1(1(0(0(x1)))) 883.72/279.68 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.72/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) 883.72/279.68 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.72/279.68 2^1(5(0(4(x1)))) -> 0^1(x1) 883.72/279.68 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.72/279.68 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.72/279.68 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.72/279.68 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.72/279.68 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.72/279.68 5^1(5(x1)) -> 4^1(0(4(x1))) 883.72/279.68 5^1(5(x1)) -> 0^1(4(x1)) 883.72/279.68 5^1(5(x1)) -> 4^1(x1) 883.72/279.68 5^1(2(4(x1))) -> 4^1(2(4(2(x1)))) 883.72/279.68 5^1(2(4(x1))) -> 2^1(4(2(x1))) 883.72/279.68 5^1(2(4(x1))) -> 4^1(2(x1)) 883.72/279.68 5^1(2(4(x1))) -> 2^1(x1) 883.72/279.68 5^1(5(3(x1))) -> 5^1(4(4(1(0(1(5(0(x1)))))))) 883.72/279.68 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.72/279.68 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.72/279.68 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.72/279.68 5^1(5(3(x1))) -> 5^1(0(x1)) 883.72/279.68 5^1(5(3(x1))) -> 0^1(x1) 883.72/279.68 5^1(5(5(x1))) -> 4^1(1(0(1(4(5(0(0(x1)))))))) 883.72/279.68 5^1(5(5(x1))) -> 0^1(1(4(5(0(0(x1)))))) 883.72/279.68 5^1(5(5(x1))) -> 4^1(5(0(0(x1)))) 883.72/279.68 5^1(5(5(x1))) -> 5^1(0(0(x1))) 883.72/279.68 5^1(5(5(x1))) -> 0^1(0(x1)) 883.72/279.68 5^1(5(5(x1))) -> 0^1(x1) 883.72/279.68 5^1(5(5(1(4(x1))))) -> 4^1(4(0(x1))) 883.72/279.68 5^1(5(5(1(4(x1))))) -> 4^1(0(x1)) 883.72/279.68 5^1(5(5(1(4(x1))))) -> 0^1(x1) 883.72/279.68 5^1(2(5(5(0(4(x1)))))) -> 5^1(2(1(4(4(x1))))) 883.72/279.68 5^1(2(5(5(0(4(x1)))))) -> 2^1(1(4(4(x1)))) 883.72/279.68 5^1(2(5(5(0(4(x1)))))) -> 4^1(4(x1)) 883.72/279.68 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.72/279.68 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.72/279.68 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.72/279.68 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.72/279.68 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.72/279.68 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.72/279.68 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.72/279.68 883.72/279.68 The TRS R consists of the following rules: 883.72/279.68 883.72/279.68 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.72/279.68 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.72/279.68 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.72/279.68 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.72/279.68 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.72/279.68 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.72/279.68 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.72/279.68 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.72/279.68 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.72/279.68 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.72/279.68 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.72/279.68 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.72/279.68 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.72/279.68 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.72/279.68 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.72/279.68 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.72/279.68 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.72/279.68 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.72/279.68 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.72/279.68 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.72/279.68 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.72/279.68 883.72/279.68 Q is empty. 883.72/279.68 We have to consider all minimal (P,Q,R)-chains. 883.72/279.68 ---------------------------------------- 883.72/279.68 883.72/279.68 (5) QDPOrderProof (EQUIVALENT) 883.72/279.68 We use the reduction pair processor [LPAR04,JAR06]. 883.72/279.68 883.72/279.68 883.72/279.68 The following pairs can be oriented strictly and are deleted. 883.72/279.68 883.72/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 5^1(1(0(1(2(0(5(x1))))))) 883.72/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(1(2(2(1(x1))))) 883.72/279.68 2^1(5(0(4(x1)))) -> 5^1(1(0(0(x1)))) 883.72/279.68 5^1(5(3(x1))) -> 5^1(4(4(1(0(1(5(0(x1)))))))) 883.72/279.68 5^1(2(5(5(0(4(x1)))))) -> 5^1(2(1(4(4(x1))))) 883.72/279.68 5^1(2(5(5(0(4(x1)))))) -> 2^1(1(4(4(x1)))) 883.72/279.68 5^1(2(5(5(0(4(x1)))))) -> 4^1(4(x1)) 883.72/279.68 The remaining pairs can at least be oriented weakly. 883.72/279.68 Used ordering: Polynomial interpretation [POLO]: 883.72/279.68 883.72/279.68 POL(0(x_1)) = 1 883.72/279.68 POL(0^1(x_1)) = 1 883.72/279.68 POL(1(x_1)) = 0 883.72/279.68 POL(2(x_1)) = 1 + x_1 883.72/279.68 POL(2^1(x_1)) = 1 883.72/279.68 POL(3(x_1)) = 0 883.72/279.68 POL(4(x_1)) = x_1 883.72/279.68 POL(4^1(x_1)) = 1 883.72/279.68 POL(5(x_1)) = 1 883.72/279.68 POL(5^1(x_1)) = x_1 883.72/279.68 883.72/279.68 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 883.72/279.68 883.72/279.68 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.72/279.68 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.72/279.68 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.72/279.68 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.72/279.68 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.72/279.68 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.72/279.68 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.72/279.68 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.72/279.68 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.72/279.68 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.72/279.68 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.72/279.68 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.72/279.68 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.72/279.68 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.72/279.68 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.72/279.68 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.72/279.68 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.72/279.68 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.72/279.68 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.72/279.68 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.72/279.68 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.72/279.68 883.72/279.68 883.72/279.68 ---------------------------------------- 883.72/279.68 883.72/279.68 (6) 883.72/279.68 Obligation: 883.72/279.68 Q DP problem: 883.72/279.68 The TRS P consists of the following rules: 883.72/279.68 883.72/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.72/279.68 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.72/279.68 4^1(5(2(4(x1)))) -> 4^1(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.72/279.68 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.72/279.68 5^1(5(x1)) -> 0^1(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.72/279.68 5^1(5(x1)) -> 5^1(4(0(2(5(4(5(2(1(x1))))))))) 883.72/279.68 5^1(5(x1)) -> 4^1(0(2(5(4(5(2(1(x1)))))))) 883.72/279.68 4^1(5(5(5(x1)))) -> 2^1(1(0(5(x1)))) 883.72/279.68 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.72/279.68 2^1(5(5(x1))) -> 5^1(4(4(0(0(1(1(2(x1)))))))) 883.72/279.68 5^1(5(x1)) -> 0^1(2(5(4(5(2(1(x1))))))) 883.72/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.72/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.72/279.68 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.72/279.68 4^1(5(5(5(x1)))) -> 0^1(5(x1)) 883.72/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.72/279.68 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.72/279.68 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.72/279.68 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.72/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.72/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.72/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.72/279.68 5^1(5(x1)) -> 2^1(5(4(5(2(1(x1)))))) 883.72/279.68 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.72/279.68 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.68 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.68 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.93/279.68 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(4(5(1(2(2(1(x1))))))) 883.93/279.68 5^1(5(x1)) -> 5^1(4(5(2(1(x1))))) 883.93/279.68 5^1(5(x1)) -> 4^1(5(2(1(x1)))) 883.93/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(5(1(2(2(1(x1)))))) 883.93/279.68 5^1(5(x1)) -> 5^1(2(1(x1))) 883.93/279.68 5^1(5(x1)) -> 2^1(1(x1)) 883.93/279.68 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(2(1(x1))) 883.93/279.68 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) 883.93/279.68 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.68 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.68 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.68 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.68 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.68 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.68 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.68 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.68 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.68 5^1(5(x1)) -> 4^1(x1) 883.93/279.68 5^1(2(4(x1))) -> 4^1(2(4(2(x1)))) 883.93/279.68 5^1(2(4(x1))) -> 2^1(4(2(x1))) 883.93/279.68 5^1(2(4(x1))) -> 4^1(2(x1)) 883.93/279.68 5^1(2(4(x1))) -> 2^1(x1) 883.93/279.68 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.68 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.68 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.68 5^1(5(3(x1))) -> 5^1(0(x1)) 883.93/279.68 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.68 5^1(5(5(x1))) -> 4^1(1(0(1(4(5(0(0(x1)))))))) 883.93/279.68 5^1(5(5(x1))) -> 0^1(1(4(5(0(0(x1)))))) 883.93/279.68 5^1(5(5(x1))) -> 4^1(5(0(0(x1)))) 883.93/279.68 5^1(5(5(x1))) -> 5^1(0(0(x1))) 883.93/279.68 5^1(5(5(x1))) -> 0^1(0(x1)) 883.93/279.68 5^1(5(5(x1))) -> 0^1(x1) 883.93/279.68 5^1(5(5(1(4(x1))))) -> 4^1(4(0(x1))) 883.93/279.68 5^1(5(5(1(4(x1))))) -> 4^1(0(x1)) 883.93/279.68 5^1(5(5(1(4(x1))))) -> 0^1(x1) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.68 883.93/279.68 The TRS R consists of the following rules: 883.93/279.68 883.93/279.68 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.68 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.68 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.68 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.68 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.68 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.68 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.68 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.68 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.68 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.68 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.68 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.68 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.68 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.68 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.68 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.68 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.68 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.68 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.68 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.68 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.68 883.93/279.68 Q is empty. 883.93/279.68 We have to consider all minimal (P,Q,R)-chains. 883.93/279.68 ---------------------------------------- 883.93/279.68 883.93/279.68 (7) QDPOrderProof (EQUIVALENT) 883.93/279.68 We use the reduction pair processor [LPAR04,JAR06]. 883.93/279.68 883.93/279.68 883.93/279.68 The following pairs can be oriented strictly and are deleted. 883.93/279.68 883.93/279.68 5^1(5(x1)) -> 5^1(2(1(x1))) 883.93/279.68 The remaining pairs can at least be oriented weakly. 883.93/279.68 Used ordering: Polynomial interpretation [POLO]: 883.93/279.68 883.93/279.68 POL(0(x_1)) = 1 883.93/279.68 POL(0^1(x_1)) = 1 883.93/279.68 POL(1(x_1)) = 0 883.93/279.68 POL(2(x_1)) = x_1 883.93/279.68 POL(2^1(x_1)) = 1 883.93/279.68 POL(3(x_1)) = 0 883.93/279.68 POL(4(x_1)) = 1 883.93/279.68 POL(4^1(x_1)) = 1 883.93/279.68 POL(5(x_1)) = 1 883.93/279.68 POL(5^1(x_1)) = x_1 883.93/279.68 883.93/279.68 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 883.93/279.68 883.93/279.68 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.68 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.68 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.68 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.68 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.68 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.68 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.68 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.68 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.68 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.68 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.68 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.68 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.68 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.68 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.68 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.68 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.68 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.68 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.68 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.68 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.68 883.93/279.68 883.93/279.68 ---------------------------------------- 883.93/279.68 883.93/279.68 (8) 883.93/279.68 Obligation: 883.93/279.68 Q DP problem: 883.93/279.68 The TRS P consists of the following rules: 883.93/279.68 883.93/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.68 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.68 4^1(5(2(4(x1)))) -> 4^1(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.68 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.68 5^1(5(x1)) -> 0^1(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.68 5^1(5(x1)) -> 5^1(4(0(2(5(4(5(2(1(x1))))))))) 883.93/279.68 5^1(5(x1)) -> 4^1(0(2(5(4(5(2(1(x1)))))))) 883.93/279.68 4^1(5(5(5(x1)))) -> 2^1(1(0(5(x1)))) 883.93/279.68 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.68 2^1(5(5(x1))) -> 5^1(4(4(0(0(1(1(2(x1)))))))) 883.93/279.68 5^1(5(x1)) -> 0^1(2(5(4(5(2(1(x1))))))) 883.93/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.68 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.68 4^1(5(5(5(x1)))) -> 0^1(5(x1)) 883.93/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.68 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.68 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.68 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.68 5^1(5(x1)) -> 2^1(5(4(5(2(1(x1)))))) 883.93/279.68 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.68 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.68 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.68 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.93/279.68 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(4(5(1(2(2(1(x1))))))) 883.93/279.68 5^1(5(x1)) -> 5^1(4(5(2(1(x1))))) 883.93/279.68 5^1(5(x1)) -> 4^1(5(2(1(x1)))) 883.93/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(5(1(2(2(1(x1)))))) 883.93/279.68 5^1(5(x1)) -> 2^1(1(x1)) 883.93/279.68 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(2(1(x1))) 883.93/279.68 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) 883.93/279.68 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.68 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.68 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.68 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.68 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.68 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.68 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.68 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.68 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.68 5^1(5(x1)) -> 4^1(x1) 883.93/279.68 5^1(2(4(x1))) -> 4^1(2(4(2(x1)))) 883.93/279.68 5^1(2(4(x1))) -> 2^1(4(2(x1))) 883.93/279.68 5^1(2(4(x1))) -> 4^1(2(x1)) 883.93/279.68 5^1(2(4(x1))) -> 2^1(x1) 883.93/279.68 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.68 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.68 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.68 5^1(5(3(x1))) -> 5^1(0(x1)) 883.93/279.68 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.68 5^1(5(5(x1))) -> 4^1(1(0(1(4(5(0(0(x1)))))))) 883.93/279.68 5^1(5(5(x1))) -> 0^1(1(4(5(0(0(x1)))))) 883.93/279.68 5^1(5(5(x1))) -> 4^1(5(0(0(x1)))) 883.93/279.68 5^1(5(5(x1))) -> 5^1(0(0(x1))) 883.93/279.68 5^1(5(5(x1))) -> 0^1(0(x1)) 883.93/279.68 5^1(5(5(x1))) -> 0^1(x1) 883.93/279.68 5^1(5(5(1(4(x1))))) -> 4^1(4(0(x1))) 883.93/279.68 5^1(5(5(1(4(x1))))) -> 4^1(0(x1)) 883.93/279.68 5^1(5(5(1(4(x1))))) -> 0^1(x1) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.68 883.93/279.68 The TRS R consists of the following rules: 883.93/279.68 883.93/279.68 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.68 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.68 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.68 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.68 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.68 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.68 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.68 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.68 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.68 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.68 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.68 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.68 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.68 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.68 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.68 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.68 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.68 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.68 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.68 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.68 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.68 883.93/279.68 Q is empty. 883.93/279.68 We have to consider all minimal (P,Q,R)-chains. 883.93/279.68 ---------------------------------------- 883.93/279.68 883.93/279.68 (9) QDPOrderProof (EQUIVALENT) 883.93/279.68 We use the reduction pair processor [LPAR04,JAR06]. 883.93/279.68 883.93/279.68 883.93/279.68 The following pairs can be oriented strictly and are deleted. 883.93/279.68 883.93/279.68 5^1(2(4(x1))) -> 4^1(2(4(2(x1)))) 883.93/279.68 5^1(2(4(x1))) -> 2^1(4(2(x1))) 883.93/279.68 5^1(2(4(x1))) -> 4^1(2(x1)) 883.93/279.68 5^1(2(4(x1))) -> 2^1(x1) 883.93/279.68 The remaining pairs can at least be oriented weakly. 883.93/279.68 Used ordering: Polynomial interpretation [POLO]: 883.93/279.68 883.93/279.68 POL(0(x_1)) = 1 883.93/279.68 POL(0^1(x_1)) = 1 883.93/279.68 POL(1(x_1)) = 0 883.93/279.68 POL(2(x_1)) = 1 + x_1 883.93/279.68 POL(2^1(x_1)) = 1 883.93/279.68 POL(3(x_1)) = 0 883.93/279.68 POL(4(x_1)) = 1 883.93/279.68 POL(4^1(x_1)) = 1 883.93/279.68 POL(5(x_1)) = 1 883.93/279.68 POL(5^1(x_1)) = x_1 883.93/279.68 883.93/279.68 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 883.93/279.68 883.93/279.68 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.68 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.68 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.68 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.68 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.68 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.68 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.68 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.68 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.68 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.68 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.68 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.68 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.68 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.68 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.68 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.68 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.68 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.68 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.68 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.68 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.68 883.93/279.68 883.93/279.68 ---------------------------------------- 883.93/279.68 883.93/279.68 (10) 883.93/279.68 Obligation: 883.93/279.68 Q DP problem: 883.93/279.68 The TRS P consists of the following rules: 883.93/279.68 883.93/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.68 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.68 4^1(5(2(4(x1)))) -> 4^1(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.68 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.68 5^1(5(x1)) -> 0^1(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.68 5^1(5(x1)) -> 5^1(4(0(2(5(4(5(2(1(x1))))))))) 883.93/279.68 5^1(5(x1)) -> 4^1(0(2(5(4(5(2(1(x1)))))))) 883.93/279.68 4^1(5(5(5(x1)))) -> 2^1(1(0(5(x1)))) 883.93/279.68 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.68 2^1(5(5(x1))) -> 5^1(4(4(0(0(1(1(2(x1)))))))) 883.93/279.68 5^1(5(x1)) -> 0^1(2(5(4(5(2(1(x1))))))) 883.93/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.68 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.68 4^1(5(5(5(x1)))) -> 0^1(5(x1)) 883.93/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.68 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.68 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.68 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.68 5^1(5(x1)) -> 2^1(5(4(5(2(1(x1)))))) 883.93/279.68 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.68 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.68 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.68 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.93/279.68 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(4(5(1(2(2(1(x1))))))) 883.93/279.68 5^1(5(x1)) -> 5^1(4(5(2(1(x1))))) 883.93/279.68 5^1(5(x1)) -> 4^1(5(2(1(x1)))) 883.93/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(5(1(2(2(1(x1)))))) 883.93/279.68 5^1(5(x1)) -> 2^1(1(x1)) 883.93/279.68 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(2(1(x1))) 883.93/279.68 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) 883.93/279.68 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.68 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.68 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.68 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.68 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.68 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.68 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.68 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.68 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.68 5^1(5(x1)) -> 4^1(x1) 883.93/279.68 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.68 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.68 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.68 5^1(5(3(x1))) -> 5^1(0(x1)) 883.93/279.68 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.68 5^1(5(5(x1))) -> 4^1(1(0(1(4(5(0(0(x1)))))))) 883.93/279.68 5^1(5(5(x1))) -> 0^1(1(4(5(0(0(x1)))))) 883.93/279.68 5^1(5(5(x1))) -> 4^1(5(0(0(x1)))) 883.93/279.68 5^1(5(5(x1))) -> 5^1(0(0(x1))) 883.93/279.68 5^1(5(5(x1))) -> 0^1(0(x1)) 883.93/279.68 5^1(5(5(x1))) -> 0^1(x1) 883.93/279.68 5^1(5(5(1(4(x1))))) -> 4^1(4(0(x1))) 883.93/279.68 5^1(5(5(1(4(x1))))) -> 4^1(0(x1)) 883.93/279.68 5^1(5(5(1(4(x1))))) -> 0^1(x1) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.68 883.93/279.68 The TRS R consists of the following rules: 883.93/279.68 883.93/279.68 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.68 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.68 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.68 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.68 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.68 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.68 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.68 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.68 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.68 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.68 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.68 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.68 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.68 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.68 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.68 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.68 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.68 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.68 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.68 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.68 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.68 883.93/279.68 Q is empty. 883.93/279.68 We have to consider all minimal (P,Q,R)-chains. 883.93/279.68 ---------------------------------------- 883.93/279.68 883.93/279.68 (11) QDPOrderProof (EQUIVALENT) 883.93/279.68 We use the reduction pair processor [LPAR04,JAR06]. 883.93/279.68 883.93/279.68 883.93/279.68 The following pairs can be oriented strictly and are deleted. 883.93/279.68 883.93/279.68 5^1(5(x1)) -> 5^1(4(0(2(5(4(5(2(1(x1))))))))) 883.93/279.68 2^1(5(5(x1))) -> 5^1(4(4(0(0(1(1(2(x1)))))))) 883.93/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(4(5(1(2(2(1(x1))))))) 883.93/279.68 5^1(5(x1)) -> 5^1(4(5(2(1(x1))))) 883.93/279.68 The remaining pairs can at least be oriented weakly. 883.93/279.68 Used ordering: Polynomial interpretation [POLO]: 883.93/279.68 883.93/279.68 POL(0(x_1)) = 1 883.93/279.68 POL(0^1(x_1)) = 1 883.93/279.68 POL(1(x_1)) = 0 883.93/279.68 POL(2(x_1)) = 0 883.93/279.68 POL(2^1(x_1)) = 1 883.93/279.68 POL(3(x_1)) = 0 883.93/279.68 POL(4(x_1)) = 0 883.93/279.68 POL(4^1(x_1)) = 1 883.93/279.68 POL(5(x_1)) = 1 883.93/279.68 POL(5^1(x_1)) = x_1 883.93/279.68 883.93/279.68 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 883.93/279.68 883.93/279.68 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.68 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.68 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.68 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.68 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.68 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.68 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.68 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.68 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.68 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.68 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.68 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.68 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.68 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.68 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.68 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.68 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.68 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.68 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.68 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.68 883.93/279.68 883.93/279.68 ---------------------------------------- 883.93/279.68 883.93/279.68 (12) 883.93/279.68 Obligation: 883.93/279.68 Q DP problem: 883.93/279.68 The TRS P consists of the following rules: 883.93/279.68 883.93/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.68 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.68 4^1(5(2(4(x1)))) -> 4^1(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.68 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.68 5^1(5(x1)) -> 0^1(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.68 5^1(5(x1)) -> 4^1(0(2(5(4(5(2(1(x1)))))))) 883.93/279.68 4^1(5(5(5(x1)))) -> 2^1(1(0(5(x1)))) 883.93/279.68 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.68 5^1(5(x1)) -> 0^1(2(5(4(5(2(1(x1))))))) 883.93/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.68 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.68 4^1(5(5(5(x1)))) -> 0^1(5(x1)) 883.93/279.68 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.68 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.68 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.68 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.68 5^1(5(x1)) -> 2^1(5(4(5(2(1(x1)))))) 883.93/279.68 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.68 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.68 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.68 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.93/279.68 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.68 5^1(5(x1)) -> 4^1(5(2(1(x1)))) 883.93/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(5(1(2(2(1(x1)))))) 883.93/279.68 5^1(5(x1)) -> 2^1(1(x1)) 883.93/279.68 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(2(1(x1))) 883.93/279.68 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.68 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) 883.93/279.68 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.68 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.68 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.68 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.68 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.68 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.68 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.68 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.68 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.68 5^1(5(x1)) -> 4^1(x1) 883.93/279.68 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.68 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.68 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.68 5^1(5(3(x1))) -> 5^1(0(x1)) 883.93/279.68 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.68 5^1(5(5(x1))) -> 4^1(1(0(1(4(5(0(0(x1)))))))) 883.93/279.68 5^1(5(5(x1))) -> 0^1(1(4(5(0(0(x1)))))) 883.93/279.68 5^1(5(5(x1))) -> 4^1(5(0(0(x1)))) 883.93/279.68 5^1(5(5(x1))) -> 5^1(0(0(x1))) 883.93/279.68 5^1(5(5(x1))) -> 0^1(0(x1)) 883.93/279.68 5^1(5(5(x1))) -> 0^1(x1) 883.93/279.68 5^1(5(5(1(4(x1))))) -> 4^1(4(0(x1))) 883.93/279.68 5^1(5(5(1(4(x1))))) -> 4^1(0(x1)) 883.93/279.68 5^1(5(5(1(4(x1))))) -> 0^1(x1) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.68 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.68 883.93/279.68 The TRS R consists of the following rules: 883.93/279.68 883.93/279.68 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.68 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.68 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.68 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.68 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.68 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.68 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.68 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.68 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.68 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.68 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.68 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.68 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.68 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.68 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.68 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.68 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.68 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.68 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.68 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.68 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.68 883.93/279.68 Q is empty. 883.93/279.68 We have to consider all minimal (P,Q,R)-chains. 883.93/279.68 ---------------------------------------- 883.93/279.68 883.93/279.68 (13) QDPOrderProof (EQUIVALENT) 883.93/279.68 We use the reduction pair processor [LPAR04,JAR06]. 883.93/279.68 883.93/279.68 883.93/279.68 The following pairs can be oriented strictly and are deleted. 883.93/279.68 883.93/279.68 5^1(5(5(x1))) -> 4^1(1(0(1(4(5(0(0(x1)))))))) 883.93/279.68 5^1(5(5(x1))) -> 0^1(1(4(5(0(0(x1)))))) 883.93/279.68 5^1(5(5(x1))) -> 4^1(5(0(0(x1)))) 883.93/279.68 5^1(5(5(x1))) -> 5^1(0(0(x1))) 883.93/279.68 5^1(5(5(x1))) -> 0^1(0(x1)) 883.93/279.68 5^1(5(5(x1))) -> 0^1(x1) 883.93/279.68 5^1(5(5(1(4(x1))))) -> 4^1(4(0(x1))) 883.93/279.68 5^1(5(5(1(4(x1))))) -> 4^1(0(x1)) 883.93/279.68 5^1(5(5(1(4(x1))))) -> 0^1(x1) 883.93/279.68 The remaining pairs can at least be oriented weakly. 883.93/279.68 Used ordering: Polynomial interpretation [POLO]: 883.93/279.68 883.93/279.68 POL(0(x_1)) = 1 883.93/279.68 POL(0^1(x_1)) = 1 883.93/279.68 POL(1(x_1)) = 0 883.93/279.68 POL(2(x_1)) = 0 883.93/279.68 POL(2^1(x_1)) = 1 883.93/279.68 POL(3(x_1)) = 0 883.93/279.69 POL(4(x_1)) = 0 883.93/279.69 POL(4^1(x_1)) = 1 883.93/279.69 POL(5(x_1)) = 1 + x_1 883.93/279.69 POL(5^1(x_1)) = x_1 883.93/279.69 883.93/279.69 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 883.93/279.69 883.93/279.69 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.69 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.69 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.69 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.69 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.69 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.69 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.69 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.69 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.69 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.69 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.69 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.69 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.69 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.69 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.69 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.69 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.69 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.69 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.69 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.69 883.93/279.69 883.93/279.69 ---------------------------------------- 883.93/279.69 883.93/279.69 (14) 883.93/279.69 Obligation: 883.93/279.69 Q DP problem: 883.93/279.69 The TRS P consists of the following rules: 883.93/279.69 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.69 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.69 4^1(5(2(4(x1)))) -> 4^1(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.69 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.69 5^1(5(x1)) -> 0^1(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.69 5^1(5(x1)) -> 4^1(0(2(5(4(5(2(1(x1)))))))) 883.93/279.69 4^1(5(5(5(x1)))) -> 2^1(1(0(5(x1)))) 883.93/279.69 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.69 5^1(5(x1)) -> 0^1(2(5(4(5(2(1(x1))))))) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.69 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.69 4^1(5(5(5(x1)))) -> 0^1(5(x1)) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.69 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.69 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.69 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.69 5^1(5(x1)) -> 2^1(5(4(5(2(1(x1)))))) 883.93/279.69 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.69 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.69 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.69 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.93/279.69 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.69 5^1(5(x1)) -> 4^1(5(2(1(x1)))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(5(1(2(2(1(x1)))))) 883.93/279.69 5^1(5(x1)) -> 2^1(1(x1)) 883.93/279.69 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(2(1(x1))) 883.93/279.69 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) 883.93/279.69 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.69 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.69 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.69 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.69 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.69 5^1(5(x1)) -> 4^1(x1) 883.93/279.69 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.69 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.69 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.69 5^1(5(3(x1))) -> 5^1(0(x1)) 883.93/279.69 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.69 883.93/279.69 The TRS R consists of the following rules: 883.93/279.69 883.93/279.69 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.69 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.69 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.69 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.69 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.69 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.69 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.69 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.69 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.69 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.69 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.69 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.69 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.69 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.69 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.69 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.69 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.69 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.69 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.69 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.69 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.69 883.93/279.69 Q is empty. 883.93/279.69 We have to consider all minimal (P,Q,R)-chains. 883.93/279.69 ---------------------------------------- 883.93/279.69 883.93/279.69 (15) QDPOrderProof (EQUIVALENT) 883.93/279.69 We use the reduction pair processor [LPAR04,JAR06]. 883.93/279.69 883.93/279.69 883.93/279.69 The following pairs can be oriented strictly and are deleted. 883.93/279.69 883.93/279.69 5^1(5(3(x1))) -> 5^1(0(x1)) 883.93/279.69 The remaining pairs can at least be oriented weakly. 883.93/279.69 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 883.93/279.69 883.93/279.69 POL( 0^1_1(x_1) ) = 1 883.93/279.69 POL( 2^1_1(x_1) ) = 1 883.93/279.69 POL( 4^1_1(x_1) ) = 1 883.93/279.69 POL( 5^1_1(x_1) ) = max{0, x_1 - 1} 883.93/279.69 POL( 5_1(x_1) ) = x_1 + 2 883.93/279.69 POL( 0_1(x_1) ) = x_1 883.93/279.69 POL( 4_1(x_1) ) = max{0, -2} 883.93/279.69 POL( 2_1(x_1) ) = max{0, -2} 883.93/279.69 POL( 1_1(x_1) ) = max{0, x_1 - 2} 883.93/279.69 POL( 3_1(x_1) ) = x_1 883.93/279.69 883.93/279.69 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 883.93/279.69 883.93/279.69 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.69 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.69 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.69 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.69 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.69 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.69 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.69 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.69 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.69 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.69 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.69 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.69 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.69 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.69 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.69 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.69 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.69 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.69 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.69 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.69 883.93/279.69 883.93/279.69 ---------------------------------------- 883.93/279.69 883.93/279.69 (16) 883.93/279.69 Obligation: 883.93/279.69 Q DP problem: 883.93/279.69 The TRS P consists of the following rules: 883.93/279.69 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.69 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.69 4^1(5(2(4(x1)))) -> 4^1(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.69 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.69 5^1(5(x1)) -> 0^1(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.69 5^1(5(x1)) -> 4^1(0(2(5(4(5(2(1(x1)))))))) 883.93/279.69 4^1(5(5(5(x1)))) -> 2^1(1(0(5(x1)))) 883.93/279.69 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.69 5^1(5(x1)) -> 0^1(2(5(4(5(2(1(x1))))))) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.69 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.69 4^1(5(5(5(x1)))) -> 0^1(5(x1)) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.69 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.69 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.69 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.69 5^1(5(x1)) -> 2^1(5(4(5(2(1(x1)))))) 883.93/279.69 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.69 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.69 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.69 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.93/279.69 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.69 5^1(5(x1)) -> 4^1(5(2(1(x1)))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(5(1(2(2(1(x1)))))) 883.93/279.69 5^1(5(x1)) -> 2^1(1(x1)) 883.93/279.69 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(2(1(x1))) 883.93/279.69 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) 883.93/279.69 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.69 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.69 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.69 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.69 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.69 5^1(5(x1)) -> 4^1(x1) 883.93/279.69 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.69 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.69 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.69 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.69 883.93/279.69 The TRS R consists of the following rules: 883.93/279.69 883.93/279.69 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.69 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.69 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.69 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.69 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.69 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.69 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.69 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.69 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.69 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.69 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.69 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.69 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.69 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.69 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.69 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.69 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.69 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.69 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.69 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.69 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.69 883.93/279.69 Q is empty. 883.93/279.69 We have to consider all minimal (P,Q,R)-chains. 883.93/279.69 ---------------------------------------- 883.93/279.69 883.93/279.69 (17) QDPOrderProof (EQUIVALENT) 883.93/279.69 We use the reduction pair processor [LPAR04,JAR06]. 883.93/279.69 883.93/279.69 883.93/279.69 The following pairs can be oriented strictly and are deleted. 883.93/279.69 883.93/279.69 4^1(5(5(5(x1)))) -> 2^1(1(0(5(x1)))) 883.93/279.69 4^1(5(5(5(x1)))) -> 0^1(5(x1)) 883.93/279.69 The remaining pairs can at least be oriented weakly. 883.93/279.69 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 883.93/279.69 883.93/279.69 POL( 0^1_1(x_1) ) = max{0, -2} 883.93/279.69 POL( 2^1_1(x_1) ) = max{0, -2} 883.93/279.69 POL( 4^1_1(x_1) ) = max{0, x_1 - 2} 883.93/279.69 POL( 5^1_1(x_1) ) = max{0, x_1 - 1} 883.93/279.69 POL( 5_1(x_1) ) = 2x_1 + 1 883.93/279.69 POL( 0_1(x_1) ) = 1 883.93/279.69 POL( 4_1(x_1) ) = max{0, -2} 883.93/279.69 POL( 2_1(x_1) ) = max{0, -2} 883.93/279.69 POL( 1_1(x_1) ) = max{0, -2} 883.93/279.69 POL( 3_1(x_1) ) = max{0, -2} 883.93/279.69 883.93/279.69 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 883.93/279.69 883.93/279.69 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.69 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.69 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.69 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.69 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.69 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.69 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.69 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.69 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.69 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.69 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.69 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.69 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.69 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.69 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.69 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.69 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.69 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.69 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.69 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.69 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.69 883.93/279.69 883.93/279.69 ---------------------------------------- 883.93/279.69 883.93/279.69 (18) 883.93/279.69 Obligation: 883.93/279.69 Q DP problem: 883.93/279.69 The TRS P consists of the following rules: 883.93/279.69 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.69 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.69 4^1(5(2(4(x1)))) -> 4^1(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.69 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.69 5^1(5(x1)) -> 0^1(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.69 5^1(5(x1)) -> 4^1(0(2(5(4(5(2(1(x1)))))))) 883.93/279.69 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.69 5^1(5(x1)) -> 0^1(2(5(4(5(2(1(x1))))))) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.69 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.69 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.69 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.69 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.69 5^1(5(x1)) -> 2^1(5(4(5(2(1(x1)))))) 883.93/279.69 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.69 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.69 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.69 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.93/279.69 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.69 5^1(5(x1)) -> 4^1(5(2(1(x1)))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(5(1(2(2(1(x1)))))) 883.93/279.69 5^1(5(x1)) -> 2^1(1(x1)) 883.93/279.69 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(2(1(x1))) 883.93/279.69 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) 883.93/279.69 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.69 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.69 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.69 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.69 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.69 5^1(5(x1)) -> 4^1(x1) 883.93/279.69 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.69 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.69 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.69 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.69 883.93/279.69 The TRS R consists of the following rules: 883.93/279.69 883.93/279.69 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.69 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.69 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.69 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.69 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.69 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.69 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.69 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.69 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.69 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.69 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.69 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.69 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.69 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.69 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.69 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.69 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.69 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.69 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.69 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.69 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.69 883.93/279.69 Q is empty. 883.93/279.69 We have to consider all minimal (P,Q,R)-chains. 883.93/279.69 ---------------------------------------- 883.93/279.69 883.93/279.69 (19) TransformationProof (EQUIVALENT) 883.93/279.69 By narrowing [LPAR04] the rule 4^1(5(2(4(x1)))) -> 4^1(1(5(5(2(0(3(1(3(3(x1)))))))))) at position [0] we obtained the following new rules [LPAR04]: 883.93/279.69 883.93/279.69 (4^1(5(2(4(y0)))) -> 4^1(1(0(5(4(0(2(5(4(5(2(1(2(0(3(1(3(3(y0)))))))))))))))))),4^1(5(2(4(y0)))) -> 4^1(1(0(5(4(0(2(5(4(5(2(1(2(0(3(1(3(3(y0))))))))))))))))))) 883.93/279.69 (4^1(5(2(4(y0)))) -> 4^1(1(3(4(1(1(1(1(4(4(0(4(2(0(3(1(3(3(y0)))))))))))))))))),4^1(5(2(4(y0)))) -> 4^1(1(3(4(1(1(1(1(4(4(0(4(2(0(3(1(3(3(y0))))))))))))))))))) 883.93/279.69 (4^1(5(2(4(y0)))) -> 4^1(1(0(1(3(2(3(0(3(2(5(3(0(3(1(3(3(y0))))))))))))))))),4^1(5(2(4(y0)))) -> 4^1(1(0(1(3(2(3(0(3(2(5(3(0(3(1(3(3(y0)))))))))))))))))) 883.93/279.69 883.93/279.69 883.93/279.69 ---------------------------------------- 883.93/279.69 883.93/279.69 (20) 883.93/279.69 Obligation: 883.93/279.69 Q DP problem: 883.93/279.69 The TRS P consists of the following rules: 883.93/279.69 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.69 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.69 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.69 5^1(5(x1)) -> 0^1(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.69 5^1(5(x1)) -> 4^1(0(2(5(4(5(2(1(x1)))))))) 883.93/279.69 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.69 5^1(5(x1)) -> 0^1(2(5(4(5(2(1(x1))))))) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.69 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.69 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.69 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.69 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.69 5^1(5(x1)) -> 2^1(5(4(5(2(1(x1)))))) 883.93/279.69 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.69 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.69 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.69 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.93/279.69 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.69 5^1(5(x1)) -> 4^1(5(2(1(x1)))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(5(1(2(2(1(x1)))))) 883.93/279.69 5^1(5(x1)) -> 2^1(1(x1)) 883.93/279.69 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(2(1(x1))) 883.93/279.69 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) 883.93/279.69 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.69 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.69 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.69 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.69 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.69 5^1(5(x1)) -> 4^1(x1) 883.93/279.69 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.69 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.69 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.69 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.69 4^1(5(2(4(y0)))) -> 4^1(1(0(5(4(0(2(5(4(5(2(1(2(0(3(1(3(3(y0)))))))))))))))))) 883.93/279.69 4^1(5(2(4(y0)))) -> 4^1(1(3(4(1(1(1(1(4(4(0(4(2(0(3(1(3(3(y0)))))))))))))))))) 883.93/279.69 4^1(5(2(4(y0)))) -> 4^1(1(0(1(3(2(3(0(3(2(5(3(0(3(1(3(3(y0))))))))))))))))) 883.93/279.69 883.93/279.69 The TRS R consists of the following rules: 883.93/279.69 883.93/279.69 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.69 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.69 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.69 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.69 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.69 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.69 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.69 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.69 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.69 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.69 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.69 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.69 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.69 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.69 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.69 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.69 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.69 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.69 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.69 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.69 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.69 883.93/279.69 Q is empty. 883.93/279.69 We have to consider all minimal (P,Q,R)-chains. 883.93/279.69 ---------------------------------------- 883.93/279.69 883.93/279.69 (21) DependencyGraphProof (EQUIVALENT) 883.93/279.69 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 883.93/279.69 ---------------------------------------- 883.93/279.69 883.93/279.69 (22) 883.93/279.69 Obligation: 883.93/279.69 Q DP problem: 883.93/279.69 The TRS P consists of the following rules: 883.93/279.69 883.93/279.69 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.69 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.69 5^1(5(x1)) -> 0^1(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.69 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.69 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.69 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.69 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.69 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.69 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.69 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.69 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.69 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.93/279.69 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.69 5^1(5(x1)) -> 4^1(0(2(5(4(5(2(1(x1)))))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(5(1(2(2(1(x1)))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(2(1(x1))) 883.93/279.69 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) 883.93/279.69 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.69 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.69 5^1(5(x1)) -> 0^1(2(5(4(5(2(1(x1))))))) 883.93/279.69 5^1(5(x1)) -> 2^1(5(4(5(2(1(x1)))))) 883.93/279.69 5^1(5(x1)) -> 4^1(5(2(1(x1)))) 883.93/279.69 5^1(5(x1)) -> 2^1(1(x1)) 883.93/279.69 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.69 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.69 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.69 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.69 5^1(5(x1)) -> 4^1(x1) 883.93/279.69 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.69 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.69 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.69 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.69 883.93/279.69 The TRS R consists of the following rules: 883.93/279.69 883.93/279.69 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.69 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.69 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.69 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.69 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.69 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.69 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.69 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.69 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.69 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.69 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.69 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.69 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.69 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.69 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.69 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.69 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.69 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.69 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.69 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.69 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.69 883.93/279.69 Q is empty. 883.93/279.69 We have to consider all minimal (P,Q,R)-chains. 883.93/279.69 ---------------------------------------- 883.93/279.69 883.93/279.69 (23) TransformationProof (EQUIVALENT) 883.93/279.69 By narrowing [LPAR04] the rule 5^1(5(x1)) -> 0^1(5(4(0(2(5(4(5(2(1(x1)))))))))) at position [0] we obtained the following new rules [LPAR04]: 883.93/279.69 883.93/279.69 (5^1(5(2(4(5(2(4(x0))))))) -> 0^1(5(4(0(2(5(4(5(2(3(3(5(3(0(4(0(3(1(3(x0))))))))))))))))))),5^1(5(2(4(5(2(4(x0))))))) -> 0^1(5(4(0(2(5(4(5(2(3(3(5(3(0(4(0(3(1(3(x0)))))))))))))))))))) 883.93/279.69 883.93/279.69 883.93/279.69 ---------------------------------------- 883.93/279.69 883.93/279.69 (24) 883.93/279.69 Obligation: 883.93/279.69 Q DP problem: 883.93/279.69 The TRS P consists of the following rules: 883.93/279.69 883.93/279.69 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.69 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.69 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.69 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.69 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.69 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.69 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.69 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.69 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.69 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.69 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.93/279.69 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.69 5^1(5(x1)) -> 4^1(0(2(5(4(5(2(1(x1)))))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(5(1(2(2(1(x1)))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(2(1(x1))) 883.93/279.69 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) 883.93/279.69 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.69 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.69 5^1(5(x1)) -> 0^1(2(5(4(5(2(1(x1))))))) 883.93/279.69 5^1(5(x1)) -> 2^1(5(4(5(2(1(x1)))))) 883.93/279.69 5^1(5(x1)) -> 4^1(5(2(1(x1)))) 883.93/279.69 5^1(5(x1)) -> 2^1(1(x1)) 883.93/279.69 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.69 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.69 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.69 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.69 5^1(5(x1)) -> 4^1(x1) 883.93/279.69 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.69 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.69 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.69 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.69 5^1(5(2(4(5(2(4(x0))))))) -> 0^1(5(4(0(2(5(4(5(2(3(3(5(3(0(4(0(3(1(3(x0))))))))))))))))))) 883.93/279.69 883.93/279.69 The TRS R consists of the following rules: 883.93/279.69 883.93/279.69 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.69 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.69 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.69 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.69 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.69 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.69 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.69 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.69 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.69 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.69 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.69 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.69 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.69 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.69 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.69 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.69 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.69 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.69 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.69 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.69 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.69 883.93/279.69 Q is empty. 883.93/279.69 We have to consider all minimal (P,Q,R)-chains. 883.93/279.69 ---------------------------------------- 883.93/279.69 883.93/279.69 (25) DependencyGraphProof (EQUIVALENT) 883.93/279.69 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 883.93/279.69 ---------------------------------------- 883.93/279.69 883.93/279.69 (26) 883.93/279.69 Obligation: 883.93/279.69 Q DP problem: 883.93/279.69 The TRS P consists of the following rules: 883.93/279.69 883.93/279.69 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.69 5^1(5(x1)) -> 4^1(0(2(5(4(5(2(1(x1)))))))) 883.93/279.69 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.69 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.69 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.69 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.69 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.69 5^1(5(x1)) -> 0^1(2(5(4(5(2(1(x1))))))) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.69 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(5(1(2(2(1(x1)))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(2(1(x1))) 883.93/279.69 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.69 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.69 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.69 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.93/279.69 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) 883.93/279.69 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.69 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.69 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.69 5^1(5(x1)) -> 2^1(5(4(5(2(1(x1)))))) 883.93/279.69 5^1(5(x1)) -> 4^1(5(2(1(x1)))) 883.93/279.69 5^1(5(x1)) -> 2^1(1(x1)) 883.93/279.69 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.69 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.69 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.69 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.69 5^1(5(x1)) -> 4^1(x1) 883.93/279.69 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.69 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.69 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.69 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.69 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.69 883.93/279.69 The TRS R consists of the following rules: 883.93/279.69 883.93/279.69 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.69 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.69 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.69 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.69 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.69 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.69 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.69 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.69 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.69 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.69 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.69 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.69 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.69 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.69 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.69 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.69 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.69 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.69 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.69 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.69 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.69 883.93/279.69 Q is empty. 883.93/279.69 We have to consider all minimal (P,Q,R)-chains. 883.93/279.69 ---------------------------------------- 883.93/279.69 883.93/279.69 (27) TransformationProof (EQUIVALENT) 883.93/279.69 By narrowing [LPAR04] the rule 5^1(5(x1)) -> 4^1(0(2(5(4(5(2(1(x1)))))))) at position [0] we obtained the following new rules [LPAR04]: 883.93/279.69 883.93/279.69 (5^1(5(2(4(5(2(4(x0))))))) -> 4^1(0(2(5(4(5(2(3(3(5(3(0(4(0(3(1(3(x0))))))))))))))))),5^1(5(2(4(5(2(4(x0))))))) -> 4^1(0(2(5(4(5(2(3(3(5(3(0(4(0(3(1(3(x0)))))))))))))))))) 883.93/279.69 883.93/279.69 883.93/279.69 ---------------------------------------- 883.93/279.69 883.93/279.69 (28) 883.93/279.69 Obligation: 883.93/279.69 Q DP problem: 883.93/279.69 The TRS P consists of the following rules: 883.93/279.69 883.93/279.69 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.69 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.69 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.69 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.69 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.69 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.69 5^1(5(x1)) -> 0^1(2(5(4(5(2(1(x1))))))) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.69 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(5(1(2(2(1(x1)))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(2(1(x1))) 883.93/279.69 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.69 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.69 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.69 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.69 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.93/279.69 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.69 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) 883.93/279.69 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.69 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.69 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.69 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.69 5^1(5(x1)) -> 2^1(5(4(5(2(1(x1)))))) 883.93/279.69 5^1(5(x1)) -> 4^1(5(2(1(x1)))) 883.93/279.69 5^1(5(x1)) -> 2^1(1(x1)) 883.93/279.69 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.70 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.70 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.70 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.70 5^1(5(x1)) -> 4^1(x1) 883.93/279.70 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.70 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.70 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.70 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.70 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.70 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.70 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.70 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.70 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.70 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.70 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.70 5^1(5(2(4(5(2(4(x0))))))) -> 4^1(0(2(5(4(5(2(3(3(5(3(0(4(0(3(1(3(x0))))))))))))))))) 883.93/279.70 883.93/279.70 The TRS R consists of the following rules: 883.93/279.70 883.93/279.70 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.70 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.70 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.70 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.70 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.70 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.70 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.70 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.70 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.70 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.70 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.70 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.70 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.70 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.70 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.70 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.70 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.70 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.70 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.70 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.70 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.70 883.93/279.70 Q is empty. 883.93/279.70 We have to consider all minimal (P,Q,R)-chains. 883.93/279.70 ---------------------------------------- 883.93/279.70 883.93/279.70 (29) DependencyGraphProof (EQUIVALENT) 883.93/279.73 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 883.93/279.73 ---------------------------------------- 883.93/279.73 883.93/279.73 (30) 883.93/279.73 Obligation: 883.93/279.73 Q DP problem: 883.93/279.73 The TRS P consists of the following rules: 883.93/279.73 883.93/279.73 5^1(5(x1)) -> 0^1(2(5(4(5(2(1(x1))))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.73 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.73 5^1(5(x1)) -> 2^1(5(4(5(2(1(x1)))))) 883.93/279.73 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.73 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.73 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.73 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.73 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.73 5^1(5(x1)) -> 4^1(5(2(1(x1)))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(5(1(2(2(1(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(2(1(x1))) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 2^1(1(x1)) 883.93/279.73 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.73 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.73 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.73 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 4^1(x1) 883.93/279.73 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.73 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.73 883.93/279.73 The TRS R consists of the following rules: 883.93/279.73 883.93/279.73 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.73 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.73 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.73 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.73 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.73 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.73 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.73 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.73 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.73 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.73 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.73 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.73 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.73 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.73 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.73 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.73 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.73 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.73 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 883.93/279.73 Q is empty. 883.93/279.73 We have to consider all minimal (P,Q,R)-chains. 883.93/279.73 ---------------------------------------- 883.93/279.73 883.93/279.73 (31) TransformationProof (EQUIVALENT) 883.93/279.73 By narrowing [LPAR04] the rule 5^1(5(x1)) -> 0^1(2(5(4(5(2(1(x1))))))) at position [0] we obtained the following new rules [LPAR04]: 883.93/279.73 883.93/279.73 (5^1(5(2(4(5(2(4(x0))))))) -> 0^1(2(5(4(5(2(3(3(5(3(0(4(0(3(1(3(x0)))))))))))))))),5^1(5(2(4(5(2(4(x0))))))) -> 0^1(2(5(4(5(2(3(3(5(3(0(4(0(3(1(3(x0))))))))))))))))) 883.93/279.73 883.93/279.73 883.93/279.73 ---------------------------------------- 883.93/279.73 883.93/279.73 (32) 883.93/279.73 Obligation: 883.93/279.73 Q DP problem: 883.93/279.73 The TRS P consists of the following rules: 883.93/279.73 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.73 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.73 5^1(5(x1)) -> 2^1(5(4(5(2(1(x1)))))) 883.93/279.73 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.73 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.73 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.73 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.73 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.73 5^1(5(x1)) -> 4^1(5(2(1(x1)))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(5(1(2(2(1(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(2(1(x1))) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 2^1(1(x1)) 883.93/279.73 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.73 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.73 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.73 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 4^1(x1) 883.93/279.73 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.73 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.73 5^1(5(2(4(5(2(4(x0))))))) -> 0^1(2(5(4(5(2(3(3(5(3(0(4(0(3(1(3(x0)))))))))))))))) 883.93/279.73 883.93/279.73 The TRS R consists of the following rules: 883.93/279.73 883.93/279.73 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.73 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.73 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.73 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.73 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.73 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.73 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.73 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.73 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.73 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.73 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.73 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.73 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.73 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.73 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.73 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.73 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.73 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.73 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 883.93/279.73 Q is empty. 883.93/279.73 We have to consider all minimal (P,Q,R)-chains. 883.93/279.73 ---------------------------------------- 883.93/279.73 883.93/279.73 (33) DependencyGraphProof (EQUIVALENT) 883.93/279.73 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 883.93/279.73 ---------------------------------------- 883.93/279.73 883.93/279.73 (34) 883.93/279.73 Obligation: 883.93/279.73 Q DP problem: 883.93/279.73 The TRS P consists of the following rules: 883.93/279.73 883.93/279.73 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.73 5^1(5(x1)) -> 2^1(5(4(5(2(1(x1)))))) 883.93/279.73 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.73 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.73 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.73 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.73 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.73 5^1(5(x1)) -> 4^1(5(2(1(x1)))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(5(1(2(2(1(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(2(1(x1))) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 2^1(1(x1)) 883.93/279.73 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.73 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.73 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.73 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 4^1(x1) 883.93/279.73 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.73 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.73 883.93/279.73 The TRS R consists of the following rules: 883.93/279.73 883.93/279.73 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.73 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.73 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.73 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.73 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.73 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.73 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.73 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.73 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.73 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.73 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.73 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.73 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.73 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.73 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.73 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.73 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.73 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.73 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 883.93/279.73 Q is empty. 883.93/279.73 We have to consider all minimal (P,Q,R)-chains. 883.93/279.73 ---------------------------------------- 883.93/279.73 883.93/279.73 (35) TransformationProof (EQUIVALENT) 883.93/279.73 By narrowing [LPAR04] the rule 5^1(5(x1)) -> 2^1(5(4(5(2(1(x1)))))) at position [0] we obtained the following new rules [LPAR04]: 883.93/279.73 883.93/279.73 (5^1(5(2(4(5(2(4(x0))))))) -> 2^1(5(4(5(2(3(3(5(3(0(4(0(3(1(3(x0))))))))))))))),5^1(5(2(4(5(2(4(x0))))))) -> 2^1(5(4(5(2(3(3(5(3(0(4(0(3(1(3(x0)))))))))))))))) 883.93/279.73 883.93/279.73 883.93/279.73 ---------------------------------------- 883.93/279.73 883.93/279.73 (36) 883.93/279.73 Obligation: 883.93/279.73 Q DP problem: 883.93/279.73 The TRS P consists of the following rules: 883.93/279.73 883.93/279.73 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.73 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.73 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.73 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.73 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.73 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.73 5^1(5(x1)) -> 4^1(5(2(1(x1)))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(5(1(2(2(1(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(2(1(x1))) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 2^1(1(x1)) 883.93/279.73 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.73 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.73 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.73 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 4^1(x1) 883.93/279.73 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.73 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.73 5^1(5(2(4(5(2(4(x0))))))) -> 2^1(5(4(5(2(3(3(5(3(0(4(0(3(1(3(x0))))))))))))))) 883.93/279.73 883.93/279.73 The TRS R consists of the following rules: 883.93/279.73 883.93/279.73 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.73 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.73 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.73 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.73 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.73 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.73 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.73 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.73 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.73 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.73 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.73 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.73 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.73 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.73 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.73 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.73 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.73 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.73 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 883.93/279.73 Q is empty. 883.93/279.73 We have to consider all minimal (P,Q,R)-chains. 883.93/279.73 ---------------------------------------- 883.93/279.73 883.93/279.73 (37) DependencyGraphProof (EQUIVALENT) 883.93/279.73 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 883.93/279.73 ---------------------------------------- 883.93/279.73 883.93/279.73 (38) 883.93/279.73 Obligation: 883.93/279.73 Q DP problem: 883.93/279.73 The TRS P consists of the following rules: 883.93/279.73 883.93/279.73 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.73 5^1(5(x1)) -> 4^1(5(2(1(x1)))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.73 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.73 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.73 5^1(5(x1)) -> 2^1(1(x1)) 883.93/279.73 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(5(1(2(2(1(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(2(1(x1))) 883.93/279.73 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.73 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.73 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.73 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.73 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.73 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 4^1(x1) 883.93/279.73 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.73 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.73 883.93/279.73 The TRS R consists of the following rules: 883.93/279.73 883.93/279.73 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.73 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.73 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.73 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.73 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.73 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.73 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.73 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.73 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.73 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.73 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.73 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.73 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.73 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.73 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.73 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.73 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.73 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.73 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 883.93/279.73 Q is empty. 883.93/279.73 We have to consider all minimal (P,Q,R)-chains. 883.93/279.73 ---------------------------------------- 883.93/279.73 883.93/279.73 (39) TransformationProof (EQUIVALENT) 883.93/279.73 By narrowing [LPAR04] the rule 5^1(5(x1)) -> 4^1(5(2(1(x1)))) at position [0] we obtained the following new rules [LPAR04]: 883.93/279.73 883.93/279.73 (5^1(5(2(4(5(2(4(x0))))))) -> 4^1(5(2(3(3(5(3(0(4(0(3(1(3(x0))))))))))))),5^1(5(2(4(5(2(4(x0))))))) -> 4^1(5(2(3(3(5(3(0(4(0(3(1(3(x0)))))))))))))) 883.93/279.73 883.93/279.73 883.93/279.73 ---------------------------------------- 883.93/279.73 883.93/279.73 (40) 883.93/279.73 Obligation: 883.93/279.73 Q DP problem: 883.93/279.73 The TRS P consists of the following rules: 883.93/279.73 883.93/279.73 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.73 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.73 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.73 5^1(5(x1)) -> 2^1(1(x1)) 883.93/279.73 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(5(1(2(2(1(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(2(1(x1))) 883.93/279.73 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.73 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.73 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.73 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.73 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.73 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 4^1(x1) 883.93/279.73 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.73 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.73 5^1(5(2(4(5(2(4(x0))))))) -> 4^1(5(2(3(3(5(3(0(4(0(3(1(3(x0))))))))))))) 883.93/279.73 883.93/279.73 The TRS R consists of the following rules: 883.93/279.73 883.93/279.73 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.73 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.73 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.73 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.73 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.73 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.73 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.73 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.73 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.73 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.73 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.73 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.73 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.73 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.73 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.73 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.73 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.73 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.73 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 883.93/279.73 Q is empty. 883.93/279.73 We have to consider all minimal (P,Q,R)-chains. 883.93/279.73 ---------------------------------------- 883.93/279.73 883.93/279.73 (41) DependencyGraphProof (EQUIVALENT) 883.93/279.73 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 883.93/279.73 ---------------------------------------- 883.93/279.73 883.93/279.73 (42) 883.93/279.73 Obligation: 883.93/279.73 Q DP problem: 883.93/279.73 The TRS P consists of the following rules: 883.93/279.73 883.93/279.73 5^1(5(x1)) -> 2^1(1(x1)) 883.93/279.73 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.73 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.73 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.73 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.73 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(5(1(2(2(1(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(2(1(x1))) 883.93/279.73 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.73 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.73 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.73 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 4^1(x1) 883.93/279.73 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.73 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.73 883.93/279.73 The TRS R consists of the following rules: 883.93/279.73 883.93/279.73 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.73 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.73 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.73 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.73 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.73 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.73 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.73 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.73 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.73 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.73 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.73 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.73 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.73 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.73 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.73 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.73 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.73 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.73 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 883.93/279.73 Q is empty. 883.93/279.73 We have to consider all minimal (P,Q,R)-chains. 883.93/279.73 ---------------------------------------- 883.93/279.73 883.93/279.73 (43) TransformationProof (EQUIVALENT) 883.93/279.73 By narrowing [LPAR04] the rule 5^1(5(x1)) -> 2^1(1(x1)) at position [0] we obtained the following new rules [LPAR04]: 883.93/279.73 883.93/279.73 (5^1(5(2(4(5(2(4(x0))))))) -> 2^1(3(3(5(3(0(4(0(3(1(3(x0))))))))))),5^1(5(2(4(5(2(4(x0))))))) -> 2^1(3(3(5(3(0(4(0(3(1(3(x0)))))))))))) 883.93/279.73 883.93/279.73 883.93/279.73 ---------------------------------------- 883.93/279.73 883.93/279.73 (44) 883.93/279.73 Obligation: 883.93/279.73 Q DP problem: 883.93/279.73 The TRS P consists of the following rules: 883.93/279.73 883.93/279.73 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.73 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.73 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.73 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.73 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(5(1(2(2(1(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(2(1(x1))) 883.93/279.73 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.73 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.73 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.73 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 4^1(x1) 883.93/279.73 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.73 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.73 5^1(5(2(4(5(2(4(x0))))))) -> 2^1(3(3(5(3(0(4(0(3(1(3(x0))))))))))) 883.93/279.73 883.93/279.73 The TRS R consists of the following rules: 883.93/279.73 883.93/279.73 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.73 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.73 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.73 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.73 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.73 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.73 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.73 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.73 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.73 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.73 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.73 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.73 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.73 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.73 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.73 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.73 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.73 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.73 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 883.93/279.73 Q is empty. 883.93/279.73 We have to consider all minimal (P,Q,R)-chains. 883.93/279.73 ---------------------------------------- 883.93/279.73 883.93/279.73 (45) DependencyGraphProof (EQUIVALENT) 883.93/279.73 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 883.93/279.73 ---------------------------------------- 883.93/279.73 883.93/279.73 (46) 883.93/279.73 Obligation: 883.93/279.73 Q DP problem: 883.93/279.73 The TRS P consists of the following rules: 883.93/279.73 883.93/279.73 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.73 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.73 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.73 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.73 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(5(1(2(2(1(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(2(1(x1))) 883.93/279.73 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) 883.93/279.73 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.73 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.73 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.73 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 4^1(x1) 883.93/279.73 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.73 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.73 883.93/279.73 The TRS R consists of the following rules: 883.93/279.73 883.93/279.73 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.73 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.73 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.73 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.73 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.73 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.73 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.73 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.73 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.73 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.73 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.73 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.73 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.73 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.73 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.73 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.73 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.73 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.73 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 883.93/279.73 Q is empty. 883.93/279.73 We have to consider all minimal (P,Q,R)-chains. 883.93/279.73 ---------------------------------------- 883.93/279.73 883.93/279.73 (47) TransformationProof (EQUIVALENT) 883.93/279.73 By narrowing [LPAR04] the rule 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(5(1(2(2(1(x1)))))) at position [0] we obtained the following new rules [LPAR04]: 883.93/279.73 883.93/279.73 (4^1(4(5(2(4(2(2(2(4(5(2(4(x0)))))))))))) -> 4^1(5(1(2(2(3(3(5(3(0(4(0(3(1(3(x0))))))))))))))),4^1(4(5(2(4(2(2(2(4(5(2(4(x0)))))))))))) -> 4^1(5(1(2(2(3(3(5(3(0(4(0(3(1(3(x0)))))))))))))))) 883.93/279.73 883.93/279.73 883.93/279.73 ---------------------------------------- 883.93/279.73 883.93/279.73 (48) 883.93/279.73 Obligation: 883.93/279.73 Q DP problem: 883.93/279.73 The TRS P consists of the following rules: 883.93/279.73 883.93/279.73 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.73 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.73 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.73 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.73 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(2(1(x1))) 883.93/279.73 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) 883.93/279.73 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.73 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.73 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.73 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 4^1(x1) 883.93/279.73 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.73 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.73 4^1(4(5(2(4(2(2(2(4(5(2(4(x0)))))))))))) -> 4^1(5(1(2(2(3(3(5(3(0(4(0(3(1(3(x0))))))))))))))) 883.93/279.73 883.93/279.73 The TRS R consists of the following rules: 883.93/279.73 883.93/279.73 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.73 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.73 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.73 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.73 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.73 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.73 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.73 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.73 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.73 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.73 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.73 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.73 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.73 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.73 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.73 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.73 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.73 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.73 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 883.93/279.73 Q is empty. 883.93/279.73 We have to consider all minimal (P,Q,R)-chains. 883.93/279.73 ---------------------------------------- 883.93/279.73 883.93/279.73 (49) DependencyGraphProof (EQUIVALENT) 883.93/279.73 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 883.93/279.73 ---------------------------------------- 883.93/279.73 883.93/279.73 (50) 883.93/279.73 Obligation: 883.93/279.73 Q DP problem: 883.93/279.73 The TRS P consists of the following rules: 883.93/279.73 883.93/279.73 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.73 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.73 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.73 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.73 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.73 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(2(1(x1))) 883.93/279.73 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) 883.93/279.73 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.73 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.73 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 4^1(x1) 883.93/279.73 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.73 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.73 883.93/279.73 The TRS R consists of the following rules: 883.93/279.73 883.93/279.73 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.73 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.73 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.73 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.73 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.73 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.73 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.73 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.73 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.73 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.73 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.73 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.73 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.73 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.73 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.73 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.73 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.73 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.73 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 883.93/279.73 Q is empty. 883.93/279.73 We have to consider all minimal (P,Q,R)-chains. 883.93/279.73 ---------------------------------------- 883.93/279.73 883.93/279.73 (51) TransformationProof (EQUIVALENT) 883.93/279.73 By narrowing [LPAR04] the rule 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(2(1(x1))) at position [0] we obtained the following new rules [LPAR04]: 883.93/279.73 883.93/279.73 (4^1(4(5(2(4(2(2(2(4(5(2(4(x0)))))))))))) -> 2^1(2(3(3(5(3(0(4(0(3(1(3(x0)))))))))))),4^1(4(5(2(4(2(2(2(4(5(2(4(x0)))))))))))) -> 2^1(2(3(3(5(3(0(4(0(3(1(3(x0))))))))))))) 883.93/279.73 883.93/279.73 883.93/279.73 ---------------------------------------- 883.93/279.73 883.93/279.73 (52) 883.93/279.73 Obligation: 883.93/279.73 Q DP problem: 883.93/279.73 The TRS P consists of the following rules: 883.93/279.73 883.93/279.73 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.73 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.73 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.73 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.73 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.73 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.73 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) 883.93/279.73 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.73 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.73 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 4^1(x1) 883.93/279.73 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.73 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.73 4^1(4(5(2(4(2(2(2(4(5(2(4(x0)))))))))))) -> 2^1(2(3(3(5(3(0(4(0(3(1(3(x0)))))))))))) 883.93/279.73 883.93/279.73 The TRS R consists of the following rules: 883.93/279.73 883.93/279.73 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.73 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.73 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.73 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.73 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.73 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.73 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.73 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.73 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.73 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.73 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.73 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.73 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.73 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.73 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.73 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.73 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.73 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.73 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 883.93/279.73 Q is empty. 883.93/279.73 We have to consider all minimal (P,Q,R)-chains. 883.93/279.73 ---------------------------------------- 883.93/279.73 883.93/279.73 (53) DependencyGraphProof (EQUIVALENT) 883.93/279.73 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 883.93/279.73 ---------------------------------------- 883.93/279.73 883.93/279.73 (54) 883.93/279.73 Obligation: 883.93/279.73 Q DP problem: 883.93/279.73 The TRS P consists of the following rules: 883.93/279.73 883.93/279.73 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.73 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.73 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.73 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.73 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) 883.93/279.73 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.73 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.73 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.73 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.73 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 4^1(x1) 883.93/279.73 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.73 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.73 883.93/279.73 The TRS R consists of the following rules: 883.93/279.73 883.93/279.73 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.73 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.73 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.73 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.73 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.73 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.73 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.73 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.73 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.73 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.73 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.73 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.73 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.73 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.73 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.73 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.73 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.73 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.73 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 883.93/279.73 Q is empty. 883.93/279.73 We have to consider all minimal (P,Q,R)-chains. 883.93/279.73 ---------------------------------------- 883.93/279.73 883.93/279.73 (55) TransformationProof (EQUIVALENT) 883.93/279.73 By narrowing [LPAR04] the rule 4^1(4(5(2(4(2(2(x1))))))) -> 2^1(1(x1)) at position [0] we obtained the following new rules [LPAR04]: 883.93/279.73 883.93/279.73 (4^1(4(5(2(4(2(2(2(4(5(2(4(x0)))))))))))) -> 2^1(3(3(5(3(0(4(0(3(1(3(x0))))))))))),4^1(4(5(2(4(2(2(2(4(5(2(4(x0)))))))))))) -> 2^1(3(3(5(3(0(4(0(3(1(3(x0)))))))))))) 883.93/279.73 883.93/279.73 883.93/279.73 ---------------------------------------- 883.93/279.73 883.93/279.73 (56) 883.93/279.73 Obligation: 883.93/279.73 Q DP problem: 883.93/279.73 The TRS P consists of the following rules: 883.93/279.73 883.93/279.73 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.73 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.73 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.73 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.73 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.73 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.73 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.73 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.73 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.73 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.73 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.73 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.73 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.73 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.73 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.73 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.73 5^1(5(x1)) -> 4^1(x1) 883.93/279.73 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.73 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.73 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.73 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.73 4^1(4(5(2(4(2(2(2(4(5(2(4(x0)))))))))))) -> 2^1(3(3(5(3(0(4(0(3(1(3(x0))))))))))) 883.93/279.73 883.93/279.73 The TRS R consists of the following rules: 883.93/279.73 883.93/279.73 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.73 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.73 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.73 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.73 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.73 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.73 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.73 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.73 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.73 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.73 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.73 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.73 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.73 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.73 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.73 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.73 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.73 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.73 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.73 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.73 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.73 883.93/279.73 Q is empty. 883.93/279.73 We have to consider all minimal (P,Q,R)-chains. 883.93/279.73 ---------------------------------------- 883.93/279.74 883.93/279.74 (57) DependencyGraphProof (EQUIVALENT) 883.93/279.74 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 883.93/279.74 ---------------------------------------- 883.93/279.74 883.93/279.74 (58) 883.93/279.74 Obligation: 883.93/279.74 Q DP problem: 883.93/279.74 The TRS P consists of the following rules: 883.93/279.74 883.93/279.74 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.74 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.74 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.74 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.74 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.74 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.74 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.74 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.74 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.74 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.74 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.74 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.74 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.74 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.74 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.74 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.74 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.74 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.74 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.74 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.74 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.74 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) 883.93/279.74 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.74 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.74 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.74 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.74 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.74 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.74 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.74 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.74 5^1(5(x1)) -> 4^1(x1) 883.93/279.74 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.74 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.74 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.74 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.74 883.93/279.74 The TRS R consists of the following rules: 883.93/279.74 883.93/279.74 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.74 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.74 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.74 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.74 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.74 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.74 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.74 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.74 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.74 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.74 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.74 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.74 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.74 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.74 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.74 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.74 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.74 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.74 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.74 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.74 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.74 883.93/279.74 Q is empty. 883.93/279.74 We have to consider all minimal (P,Q,R)-chains. 883.93/279.74 ---------------------------------------- 883.93/279.74 883.93/279.74 (59) TransformationProof (EQUIVALENT) 883.93/279.74 By narrowing [LPAR04] the rule 2^1(5(0(4(x1)))) -> 2^1(4(4(5(1(0(0(x1))))))) at position [0] we obtained the following new rules [LPAR04]: 883.93/279.74 883.93/279.74 (2^1(5(0(4(2(5(3(4(x0)))))))) -> 2^1(4(4(5(1(0(3(2(4(3(1(5(1(1(3(4(x0)))))))))))))))),2^1(5(0(4(2(5(3(4(x0)))))))) -> 2^1(4(4(5(1(0(3(2(4(3(1(5(1(1(3(4(x0))))))))))))))))) 883.93/279.74 (2^1(5(0(4(4(4(5(5(5(x0))))))))) -> 2^1(4(4(5(1(0(0(4(4(4(3(3(4(1(3(1(x0)))))))))))))))),2^1(5(0(4(4(4(5(5(5(x0))))))))) -> 2^1(4(4(5(1(0(0(4(4(4(3(3(4(1(3(1(x0))))))))))))))))) 883.93/279.74 (2^1(5(0(4(1(5(5(5(3(5(x0)))))))))) -> 2^1(4(4(5(1(0(5(3(2(5(1(0(1(2(0(5(x0)))))))))))))))),2^1(5(0(4(1(5(5(5(3(5(x0)))))))))) -> 2^1(4(4(5(1(0(5(3(2(5(1(0(1(2(0(5(x0))))))))))))))))) 883.93/279.74 883.93/279.74 883.93/279.74 ---------------------------------------- 883.93/279.74 883.93/279.74 (60) 883.93/279.74 Obligation: 883.93/279.74 Q DP problem: 883.93/279.74 The TRS P consists of the following rules: 883.93/279.74 883.93/279.74 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.74 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.74 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.74 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.74 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.74 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.74 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.74 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.74 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.74 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.74 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.74 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.74 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.74 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.74 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.74 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.74 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.74 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.74 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.74 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.74 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.74 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.74 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.74 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.74 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.74 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.74 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.74 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.74 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.74 5^1(5(x1)) -> 4^1(x1) 883.93/279.74 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.74 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.74 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.74 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.74 2^1(5(0(4(2(5(3(4(x0)))))))) -> 2^1(4(4(5(1(0(3(2(4(3(1(5(1(1(3(4(x0)))))))))))))))) 883.93/279.74 2^1(5(0(4(4(4(5(5(5(x0))))))))) -> 2^1(4(4(5(1(0(0(4(4(4(3(3(4(1(3(1(x0)))))))))))))))) 883.93/279.74 2^1(5(0(4(1(5(5(5(3(5(x0)))))))))) -> 2^1(4(4(5(1(0(5(3(2(5(1(0(1(2(0(5(x0)))))))))))))))) 883.93/279.74 883.93/279.74 The TRS R consists of the following rules: 883.93/279.74 883.93/279.74 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.74 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.74 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.74 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.74 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.74 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.74 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.74 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.74 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.74 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.74 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.74 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.74 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.74 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.74 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.74 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.74 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.74 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.74 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.74 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.74 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.74 883.93/279.74 Q is empty. 883.93/279.74 We have to consider all minimal (P,Q,R)-chains. 883.93/279.74 ---------------------------------------- 883.93/279.74 883.93/279.74 (61) DependencyGraphProof (EQUIVALENT) 883.93/279.74 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 883.93/279.74 ---------------------------------------- 883.93/279.74 883.93/279.74 (62) 883.93/279.74 Obligation: 883.93/279.74 Q DP problem: 883.93/279.74 The TRS P consists of the following rules: 883.93/279.74 883.93/279.74 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.74 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.74 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.74 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.74 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.74 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.74 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.74 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.74 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.74 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.74 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.74 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.74 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.74 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.74 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.74 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.74 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.74 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.74 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.74 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.74 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.74 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) 883.93/279.74 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.74 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.74 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.74 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.74 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.74 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.74 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.74 5^1(5(x1)) -> 4^1(x1) 883.93/279.74 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.74 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.74 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.74 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.74 883.93/279.74 The TRS R consists of the following rules: 883.93/279.74 883.93/279.74 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.74 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.74 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.74 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.74 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.74 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.74 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.74 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.74 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.74 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.74 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.74 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.74 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.74 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.74 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.74 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.74 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.74 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.74 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.74 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.74 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.74 883.93/279.74 Q is empty. 883.93/279.74 We have to consider all minimal (P,Q,R)-chains. 883.93/279.74 ---------------------------------------- 883.93/279.74 883.93/279.74 (63) TransformationProof (EQUIVALENT) 883.93/279.74 By narrowing [LPAR04] the rule 2^1(5(0(4(x1)))) -> 4^1(4(5(1(0(0(x1)))))) at position [0] we obtained the following new rules [LPAR04]: 883.93/279.74 883.93/279.74 (2^1(5(0(4(2(5(3(4(x0)))))))) -> 4^1(4(5(1(0(3(2(4(3(1(5(1(1(3(4(x0))))))))))))))),2^1(5(0(4(2(5(3(4(x0)))))))) -> 4^1(4(5(1(0(3(2(4(3(1(5(1(1(3(4(x0)))))))))))))))) 883.93/279.74 (2^1(5(0(4(4(4(5(5(5(x0))))))))) -> 4^1(4(5(1(0(0(4(4(4(3(3(4(1(3(1(x0))))))))))))))),2^1(5(0(4(4(4(5(5(5(x0))))))))) -> 4^1(4(5(1(0(0(4(4(4(3(3(4(1(3(1(x0)))))))))))))))) 883.93/279.74 (2^1(5(0(4(1(5(5(5(3(5(x0)))))))))) -> 4^1(4(5(1(0(5(3(2(5(1(0(1(2(0(5(x0))))))))))))))),2^1(5(0(4(1(5(5(5(3(5(x0)))))))))) -> 4^1(4(5(1(0(5(3(2(5(1(0(1(2(0(5(x0)))))))))))))))) 883.93/279.74 883.93/279.74 883.93/279.74 ---------------------------------------- 883.93/279.74 883.93/279.74 (64) 883.93/279.74 Obligation: 883.93/279.74 Q DP problem: 883.93/279.74 The TRS P consists of the following rules: 883.93/279.74 883.93/279.74 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.74 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.74 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.74 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.74 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.74 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.74 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.74 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.74 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.74 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.74 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.74 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.74 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.74 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.74 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.74 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.74 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.74 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.74 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.74 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.74 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.74 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.74 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.74 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.74 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.74 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.74 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.74 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.74 5^1(5(x1)) -> 4^1(x1) 883.93/279.74 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.74 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.74 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.74 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.74 2^1(5(0(4(2(5(3(4(x0)))))))) -> 4^1(4(5(1(0(3(2(4(3(1(5(1(1(3(4(x0))))))))))))))) 883.93/279.74 2^1(5(0(4(4(4(5(5(5(x0))))))))) -> 4^1(4(5(1(0(0(4(4(4(3(3(4(1(3(1(x0))))))))))))))) 883.93/279.74 2^1(5(0(4(1(5(5(5(3(5(x0)))))))))) -> 4^1(4(5(1(0(5(3(2(5(1(0(1(2(0(5(x0))))))))))))))) 883.93/279.74 883.93/279.74 The TRS R consists of the following rules: 883.93/279.74 883.93/279.74 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.74 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.74 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.74 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.74 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.74 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.74 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.74 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.74 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.74 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.74 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.74 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.74 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.74 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.74 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.74 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.74 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.74 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.74 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.74 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.74 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.74 883.93/279.74 Q is empty. 883.93/279.74 We have to consider all minimal (P,Q,R)-chains. 883.93/279.74 ---------------------------------------- 883.93/279.74 883.93/279.74 (65) DependencyGraphProof (EQUIVALENT) 883.93/279.74 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 883.93/279.74 ---------------------------------------- 883.93/279.74 883.93/279.74 (66) 883.93/279.74 Obligation: 883.93/279.74 Q DP problem: 883.93/279.74 The TRS P consists of the following rules: 883.93/279.74 883.93/279.74 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.74 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.74 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.74 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.74 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.74 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.74 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.74 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.74 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.74 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.74 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.74 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.74 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.74 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.74 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.74 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.74 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.74 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.74 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.74 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.74 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.74 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) 883.93/279.74 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.74 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.74 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.74 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.74 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.74 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.74 5^1(5(x1)) -> 4^1(x1) 883.93/279.74 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.74 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.74 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.74 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.74 883.93/279.74 The TRS R consists of the following rules: 883.93/279.74 883.93/279.74 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.74 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.74 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.74 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.74 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.74 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.74 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.74 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.74 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.74 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.74 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.74 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.74 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.74 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.74 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.74 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.74 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.74 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.74 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.74 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.74 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.74 883.93/279.74 Q is empty. 883.93/279.74 We have to consider all minimal (P,Q,R)-chains. 883.93/279.74 ---------------------------------------- 883.93/279.74 883.93/279.74 (67) TransformationProof (EQUIVALENT) 883.93/279.74 By narrowing [LPAR04] the rule 2^1(5(0(4(x1)))) -> 4^1(5(1(0(0(x1))))) at position [0] we obtained the following new rules [LPAR04]: 883.93/279.74 883.93/279.74 (2^1(5(0(4(2(5(3(4(x0)))))))) -> 4^1(5(1(0(3(2(4(3(1(5(1(1(3(4(x0)))))))))))))),2^1(5(0(4(2(5(3(4(x0)))))))) -> 4^1(5(1(0(3(2(4(3(1(5(1(1(3(4(x0))))))))))))))) 883.93/279.74 (2^1(5(0(4(4(4(5(5(5(x0))))))))) -> 4^1(5(1(0(0(4(4(4(3(3(4(1(3(1(x0)))))))))))))),2^1(5(0(4(4(4(5(5(5(x0))))))))) -> 4^1(5(1(0(0(4(4(4(3(3(4(1(3(1(x0))))))))))))))) 883.93/279.74 (2^1(5(0(4(1(5(5(5(3(5(x0)))))))))) -> 4^1(5(1(0(5(3(2(5(1(0(1(2(0(5(x0)))))))))))))),2^1(5(0(4(1(5(5(5(3(5(x0)))))))))) -> 4^1(5(1(0(5(3(2(5(1(0(1(2(0(5(x0))))))))))))))) 883.93/279.74 883.93/279.74 883.93/279.74 ---------------------------------------- 883.93/279.74 883.93/279.74 (68) 883.93/279.74 Obligation: 883.93/279.74 Q DP problem: 883.93/279.74 The TRS P consists of the following rules: 883.93/279.74 883.93/279.74 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.74 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.74 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.74 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.74 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.74 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.74 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.74 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.74 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.74 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.74 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.74 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.74 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.74 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.74 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.74 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.74 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.74 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.74 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.74 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.74 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.74 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.74 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.74 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.74 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.74 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.74 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.74 5^1(5(x1)) -> 4^1(x1) 883.93/279.74 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.74 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.74 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.74 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.74 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.74 2^1(5(0(4(2(5(3(4(x0)))))))) -> 4^1(5(1(0(3(2(4(3(1(5(1(1(3(4(x0)))))))))))))) 883.93/279.74 2^1(5(0(4(4(4(5(5(5(x0))))))))) -> 4^1(5(1(0(0(4(4(4(3(3(4(1(3(1(x0)))))))))))))) 883.93/279.74 2^1(5(0(4(1(5(5(5(3(5(x0)))))))))) -> 4^1(5(1(0(5(3(2(5(1(0(1(2(0(5(x0)))))))))))))) 883.93/279.74 883.93/279.74 The TRS R consists of the following rules: 883.93/279.74 883.93/279.74 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.75 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.75 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.75 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.75 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.75 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.75 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.75 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.75 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.75 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.75 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.75 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.75 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.75 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.75 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.75 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.75 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.75 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.75 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.75 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.75 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.75 883.93/279.75 Q is empty. 883.93/279.75 We have to consider all minimal (P,Q,R)-chains. 883.93/279.75 ---------------------------------------- 883.93/279.75 883.93/279.75 (69) DependencyGraphProof (EQUIVALENT) 883.93/279.75 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 883.93/279.75 ---------------------------------------- 883.93/279.75 883.93/279.75 (70) 883.93/279.75 Obligation: 883.93/279.75 Q DP problem: 883.93/279.75 The TRS P consists of the following rules: 883.93/279.75 883.93/279.75 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.75 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.75 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.75 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.75 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.75 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.75 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.75 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.75 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.75 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.75 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.75 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.75 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.75 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.75 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.75 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.75 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.75 2^1(5(0(4(x1)))) -> 0^1(0(x1)) 883.93/279.75 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.75 5^1(5(x1)) -> 4^1(x1) 883.93/279.75 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.75 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.75 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.75 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.75 883.93/279.75 The TRS R consists of the following rules: 883.93/279.75 883.93/279.75 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.75 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.75 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.75 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.75 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.75 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.75 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.75 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.75 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.75 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.75 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.75 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.75 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.75 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.75 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.75 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.75 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.75 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.75 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.75 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.75 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.75 883.93/279.75 Q is empty. 883.93/279.75 We have to consider all minimal (P,Q,R)-chains. 883.93/279.75 ---------------------------------------- 883.93/279.75 883.93/279.75 (71) TransformationProof (EQUIVALENT) 883.93/279.75 By narrowing [LPAR04] the rule 2^1(5(0(4(x1)))) -> 0^1(0(x1)) at position [0] we obtained the following new rules [LPAR04]: 883.93/279.75 883.93/279.75 (2^1(5(0(4(2(5(3(4(x0)))))))) -> 0^1(3(2(4(3(1(5(1(1(3(4(x0))))))))))),2^1(5(0(4(2(5(3(4(x0)))))))) -> 0^1(3(2(4(3(1(5(1(1(3(4(x0)))))))))))) 883.93/279.75 (2^1(5(0(4(4(4(5(5(5(x0))))))))) -> 0^1(0(4(4(4(3(3(4(1(3(1(x0))))))))))),2^1(5(0(4(4(4(5(5(5(x0))))))))) -> 0^1(0(4(4(4(3(3(4(1(3(1(x0)))))))))))) 883.93/279.75 (2^1(5(0(4(1(5(5(5(3(5(x0)))))))))) -> 0^1(5(3(2(5(1(0(1(2(0(5(x0))))))))))),2^1(5(0(4(1(5(5(5(3(5(x0)))))))))) -> 0^1(5(3(2(5(1(0(1(2(0(5(x0)))))))))))) 883.93/279.75 883.93/279.75 883.93/279.75 ---------------------------------------- 883.93/279.75 883.93/279.75 (72) 883.93/279.75 Obligation: 883.93/279.75 Q DP problem: 883.93/279.75 The TRS P consists of the following rules: 883.93/279.75 883.93/279.75 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.75 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.75 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.75 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.75 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.75 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.75 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.75 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.75 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.75 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.75 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.75 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.75 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.75 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.75 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.75 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.75 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.75 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.75 5^1(5(x1)) -> 4^1(x1) 883.93/279.75 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.75 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.75 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.75 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.75 2^1(5(0(4(2(5(3(4(x0)))))))) -> 0^1(3(2(4(3(1(5(1(1(3(4(x0))))))))))) 883.93/279.75 2^1(5(0(4(4(4(5(5(5(x0))))))))) -> 0^1(0(4(4(4(3(3(4(1(3(1(x0))))))))))) 883.93/279.75 2^1(5(0(4(1(5(5(5(3(5(x0)))))))))) -> 0^1(5(3(2(5(1(0(1(2(0(5(x0))))))))))) 883.93/279.75 883.93/279.75 The TRS R consists of the following rules: 883.93/279.75 883.93/279.75 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.75 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.75 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.75 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.75 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.75 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.75 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.75 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.75 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.75 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.75 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.75 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.75 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.75 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.75 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.75 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.75 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.75 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.75 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.75 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.75 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.75 883.93/279.75 Q is empty. 883.93/279.75 We have to consider all minimal (P,Q,R)-chains. 883.93/279.75 ---------------------------------------- 883.93/279.75 883.93/279.75 (73) DependencyGraphProof (EQUIVALENT) 883.93/279.75 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 883.93/279.75 ---------------------------------------- 883.93/279.75 883.93/279.75 (74) 883.93/279.75 Obligation: 883.93/279.75 Q DP problem: 883.93/279.75 The TRS P consists of the following rules: 883.93/279.75 883.93/279.75 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.75 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.75 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.75 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.75 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.75 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.75 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.75 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.75 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.75 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.75 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.75 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.75 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.75 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.75 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.75 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.75 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.75 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.75 5^1(5(x1)) -> 4^1(x1) 883.93/279.75 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.75 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.75 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.75 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.75 883.93/279.75 The TRS R consists of the following rules: 883.93/279.75 883.93/279.75 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.75 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.75 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.75 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.75 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.75 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.75 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.75 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.75 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.75 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.75 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.75 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.75 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.75 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.75 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.75 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.75 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.75 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.75 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.75 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.75 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.75 883.93/279.75 Q is empty. 883.93/279.75 We have to consider all minimal (P,Q,R)-chains. 883.93/279.75 ---------------------------------------- 883.93/279.75 883.93/279.75 (75) TransformationProof (EQUIVALENT) 883.93/279.75 By narrowing [LPAR04] the rule 5^1(5(2(4(5(0(x1)))))) -> 2^1(1(1(4(2(4(0(4(2(0(x1)))))))))) at position [0] we obtained the following new rules [LPAR04]: 883.93/279.75 883.93/279.75 (5^1(5(2(4(5(0(2(5(3(4(x0)))))))))) -> 2^1(1(1(4(2(4(0(4(2(3(2(4(3(1(5(1(1(3(4(x0))))))))))))))))))),5^1(5(2(4(5(0(2(5(3(4(x0)))))))))) -> 2^1(1(1(4(2(4(0(4(2(3(2(4(3(1(5(1(1(3(4(x0)))))))))))))))))))) 883.93/279.75 (5^1(5(2(4(5(0(4(4(5(5(5(x0))))))))))) -> 2^1(1(1(4(2(4(0(4(2(0(4(4(4(3(3(4(1(3(1(x0))))))))))))))))))),5^1(5(2(4(5(0(4(4(5(5(5(x0))))))))))) -> 2^1(1(1(4(2(4(0(4(2(0(4(4(4(3(3(4(1(3(1(x0)))))))))))))))))))) 883.93/279.75 (5^1(5(2(4(5(0(1(5(5(5(3(5(x0)))))))))))) -> 2^1(1(1(4(2(4(0(4(2(5(3(2(5(1(0(1(2(0(5(x0))))))))))))))))))),5^1(5(2(4(5(0(1(5(5(5(3(5(x0)))))))))))) -> 2^1(1(1(4(2(4(0(4(2(5(3(2(5(1(0(1(2(0(5(x0)))))))))))))))))))) 883.93/279.75 883.93/279.75 883.93/279.75 ---------------------------------------- 883.93/279.75 883.93/279.75 (76) 883.93/279.75 Obligation: 883.93/279.75 Q DP problem: 883.93/279.75 The TRS P consists of the following rules: 883.93/279.75 883.93/279.75 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.75 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.75 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.75 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.75 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.75 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.75 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.75 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.75 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.75 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.75 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.75 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.75 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.75 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.75 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.75 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.75 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.75 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.75 5^1(5(x1)) -> 4^1(x1) 883.93/279.75 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.75 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.75 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.75 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.75 5^1(5(2(4(5(0(2(5(3(4(x0)))))))))) -> 2^1(1(1(4(2(4(0(4(2(3(2(4(3(1(5(1(1(3(4(x0))))))))))))))))))) 883.93/279.75 5^1(5(2(4(5(0(4(4(5(5(5(x0))))))))))) -> 2^1(1(1(4(2(4(0(4(2(0(4(4(4(3(3(4(1(3(1(x0))))))))))))))))))) 883.93/279.75 5^1(5(2(4(5(0(1(5(5(5(3(5(x0)))))))))))) -> 2^1(1(1(4(2(4(0(4(2(5(3(2(5(1(0(1(2(0(5(x0))))))))))))))))))) 883.93/279.75 883.93/279.75 The TRS R consists of the following rules: 883.93/279.75 883.93/279.75 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.75 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.75 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.75 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.75 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.75 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.75 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.75 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.75 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.75 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.75 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.75 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.75 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.75 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.75 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.75 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.75 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.75 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.75 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.75 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.75 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.75 883.93/279.75 Q is empty. 883.93/279.75 We have to consider all minimal (P,Q,R)-chains. 883.93/279.75 ---------------------------------------- 883.93/279.75 883.93/279.75 (77) DependencyGraphProof (EQUIVALENT) 883.93/279.75 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 883.93/279.75 ---------------------------------------- 883.93/279.75 883.93/279.75 (78) 883.93/279.75 Obligation: 883.93/279.75 Q DP problem: 883.93/279.75 The TRS P consists of the following rules: 883.93/279.75 883.93/279.75 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.75 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.75 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.75 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.75 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.75 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.75 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.75 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.75 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.75 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.75 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.75 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.75 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.75 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.75 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.75 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.75 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.75 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.75 5^1(5(x1)) -> 4^1(x1) 883.93/279.75 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.75 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.75 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.75 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.75 883.93/279.75 The TRS R consists of the following rules: 883.93/279.75 883.93/279.75 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.75 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.75 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.75 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.75 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.75 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.75 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.75 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.75 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.75 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.75 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.75 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.75 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.75 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.75 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.75 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.75 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.75 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.75 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.75 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.75 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.75 883.93/279.75 Q is empty. 883.93/279.75 We have to consider all minimal (P,Q,R)-chains. 883.93/279.75 ---------------------------------------- 883.93/279.75 883.93/279.75 (79) TransformationProof (EQUIVALENT) 883.93/279.75 By narrowing [LPAR04] the rule 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(4(0(4(2(0(x1))))))) at position [0] we obtained the following new rules [LPAR04]: 883.93/279.75 883.93/279.75 (5^1(5(2(4(5(0(2(5(3(4(x0)))))))))) -> 4^1(2(4(0(4(2(3(2(4(3(1(5(1(1(3(4(x0)))))))))))))))),5^1(5(2(4(5(0(2(5(3(4(x0)))))))))) -> 4^1(2(4(0(4(2(3(2(4(3(1(5(1(1(3(4(x0))))))))))))))))) 883.93/279.75 (5^1(5(2(4(5(0(4(4(5(5(5(x0))))))))))) -> 4^1(2(4(0(4(2(0(4(4(4(3(3(4(1(3(1(x0)))))))))))))))),5^1(5(2(4(5(0(4(4(5(5(5(x0))))))))))) -> 4^1(2(4(0(4(2(0(4(4(4(3(3(4(1(3(1(x0))))))))))))))))) 883.93/279.75 (5^1(5(2(4(5(0(1(5(5(5(3(5(x0)))))))))))) -> 4^1(2(4(0(4(2(5(3(2(5(1(0(1(2(0(5(x0)))))))))))))))),5^1(5(2(4(5(0(1(5(5(5(3(5(x0)))))))))))) -> 4^1(2(4(0(4(2(5(3(2(5(1(0(1(2(0(5(x0))))))))))))))))) 883.93/279.75 883.93/279.75 883.93/279.75 ---------------------------------------- 883.93/279.75 883.93/279.75 (80) 883.93/279.75 Obligation: 883.93/279.75 Q DP problem: 883.93/279.75 The TRS P consists of the following rules: 883.93/279.75 883.93/279.75 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.75 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.75 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.75 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.75 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.75 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.75 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.75 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.75 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.75 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.75 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.75 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.75 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.75 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.75 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.75 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.75 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.75 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.75 5^1(5(x1)) -> 4^1(x1) 883.93/279.75 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.75 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.75 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.75 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.75 5^1(5(2(4(5(0(2(5(3(4(x0)))))))))) -> 4^1(2(4(0(4(2(3(2(4(3(1(5(1(1(3(4(x0)))))))))))))))) 883.93/279.75 5^1(5(2(4(5(0(4(4(5(5(5(x0))))))))))) -> 4^1(2(4(0(4(2(0(4(4(4(3(3(4(1(3(1(x0)))))))))))))))) 883.93/279.75 5^1(5(2(4(5(0(1(5(5(5(3(5(x0)))))))))))) -> 4^1(2(4(0(4(2(5(3(2(5(1(0(1(2(0(5(x0)))))))))))))))) 883.93/279.75 883.93/279.75 The TRS R consists of the following rules: 883.93/279.75 883.93/279.75 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.75 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.75 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.75 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.75 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.75 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.75 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.75 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.75 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.75 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.75 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.75 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.75 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.75 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.75 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.75 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.75 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.75 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.75 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.75 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.75 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.75 883.93/279.75 Q is empty. 883.93/279.75 We have to consider all minimal (P,Q,R)-chains. 883.93/279.75 ---------------------------------------- 883.93/279.75 883.93/279.75 (81) DependencyGraphProof (EQUIVALENT) 883.93/279.75 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 883.93/279.75 ---------------------------------------- 883.93/279.75 883.93/279.75 (82) 883.93/279.75 Obligation: 883.93/279.75 Q DP problem: 883.93/279.75 The TRS P consists of the following rules: 883.93/279.75 883.93/279.75 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.75 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.75 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.75 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.75 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.75 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.75 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.75 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.75 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.75 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.75 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.75 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.75 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.75 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.75 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.75 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.75 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.75 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.75 5^1(5(x1)) -> 4^1(x1) 883.93/279.75 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.75 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.75 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.75 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.75 883.93/279.75 The TRS R consists of the following rules: 883.93/279.75 883.93/279.75 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.75 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.75 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.75 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.75 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.75 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.75 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.75 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.75 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.75 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.75 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.75 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.75 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.75 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.75 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.75 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.75 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.75 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.75 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.75 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.75 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.75 883.93/279.75 Q is empty. 883.93/279.75 We have to consider all minimal (P,Q,R)-chains. 883.93/279.75 ---------------------------------------- 883.93/279.75 883.93/279.75 (83) TransformationProof (EQUIVALENT) 883.93/279.75 By narrowing [LPAR04] the rule 5^1(5(2(4(5(0(x1)))))) -> 2^1(4(0(4(2(0(x1)))))) at position [0] we obtained the following new rules [LPAR04]: 883.93/279.75 883.93/279.75 (5^1(5(2(4(5(0(2(5(3(4(x0)))))))))) -> 2^1(4(0(4(2(3(2(4(3(1(5(1(1(3(4(x0))))))))))))))),5^1(5(2(4(5(0(2(5(3(4(x0)))))))))) -> 2^1(4(0(4(2(3(2(4(3(1(5(1(1(3(4(x0)))))))))))))))) 883.93/279.75 (5^1(5(2(4(5(0(4(4(5(5(5(x0))))))))))) -> 2^1(4(0(4(2(0(4(4(4(3(3(4(1(3(1(x0))))))))))))))),5^1(5(2(4(5(0(4(4(5(5(5(x0))))))))))) -> 2^1(4(0(4(2(0(4(4(4(3(3(4(1(3(1(x0)))))))))))))))) 883.93/279.75 (5^1(5(2(4(5(0(1(5(5(5(3(5(x0)))))))))))) -> 2^1(4(0(4(2(5(3(2(5(1(0(1(2(0(5(x0))))))))))))))),5^1(5(2(4(5(0(1(5(5(5(3(5(x0)))))))))))) -> 2^1(4(0(4(2(5(3(2(5(1(0(1(2(0(5(x0)))))))))))))))) 883.93/279.75 883.93/279.75 883.93/279.75 ---------------------------------------- 883.93/279.75 883.93/279.75 (84) 883.93/279.75 Obligation: 883.93/279.75 Q DP problem: 883.93/279.75 The TRS P consists of the following rules: 883.93/279.75 883.93/279.75 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.75 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.75 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.75 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.75 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.75 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.75 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.75 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.75 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.75 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.75 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.75 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.75 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.75 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.75 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.75 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.75 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.75 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.75 5^1(5(x1)) -> 4^1(x1) 883.93/279.75 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.75 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.75 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.75 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.75 5^1(5(2(4(5(0(2(5(3(4(x0)))))))))) -> 2^1(4(0(4(2(3(2(4(3(1(5(1(1(3(4(x0))))))))))))))) 883.93/279.75 5^1(5(2(4(5(0(4(4(5(5(5(x0))))))))))) -> 2^1(4(0(4(2(0(4(4(4(3(3(4(1(3(1(x0))))))))))))))) 883.93/279.75 5^1(5(2(4(5(0(1(5(5(5(3(5(x0)))))))))))) -> 2^1(4(0(4(2(5(3(2(5(1(0(1(2(0(5(x0))))))))))))))) 883.93/279.75 883.93/279.75 The TRS R consists of the following rules: 883.93/279.75 883.93/279.75 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.75 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.75 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.75 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.75 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.75 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.75 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.75 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.75 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.75 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.75 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.75 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.75 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.75 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.75 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.75 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.75 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.75 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.75 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.75 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.75 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.75 883.93/279.75 Q is empty. 883.93/279.75 We have to consider all minimal (P,Q,R)-chains. 883.93/279.75 ---------------------------------------- 883.93/279.75 883.93/279.75 (85) DependencyGraphProof (EQUIVALENT) 883.93/279.75 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 883.93/279.75 ---------------------------------------- 883.93/279.75 883.93/279.75 (86) 883.93/279.75 Obligation: 883.93/279.75 Q DP problem: 883.93/279.75 The TRS P consists of the following rules: 883.93/279.75 883.93/279.75 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.75 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.75 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.75 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.75 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.75 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.75 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.75 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.75 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.75 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.75 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.75 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.75 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.75 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.75 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.75 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.75 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.75 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.75 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.75 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.75 5^1(5(x1)) -> 4^1(x1) 883.93/279.75 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.75 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.75 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.75 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.75 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.75 883.93/279.75 The TRS R consists of the following rules: 883.93/279.75 883.93/279.75 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.75 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.75 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.75 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.75 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.75 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.75 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.75 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.75 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.75 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.76 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.76 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.76 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.76 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.76 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.76 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.76 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.76 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.76 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.76 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.76 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.76 883.93/279.76 Q is empty. 883.93/279.76 We have to consider all minimal (P,Q,R)-chains. 883.93/279.76 ---------------------------------------- 883.93/279.76 883.93/279.76 (87) TransformationProof (EQUIVALENT) 883.93/279.76 By narrowing [LPAR04] the rule 5^1(5(2(4(5(0(x1)))))) -> 4^1(0(4(2(0(x1))))) at position [0] we obtained the following new rules [LPAR04]: 883.93/279.76 883.93/279.76 (5^1(5(2(4(5(0(2(5(3(4(x0)))))))))) -> 4^1(0(4(2(3(2(4(3(1(5(1(1(3(4(x0)))))))))))))),5^1(5(2(4(5(0(2(5(3(4(x0)))))))))) -> 4^1(0(4(2(3(2(4(3(1(5(1(1(3(4(x0))))))))))))))) 883.93/279.76 (5^1(5(2(4(5(0(4(4(5(5(5(x0))))))))))) -> 4^1(0(4(2(0(4(4(4(3(3(4(1(3(1(x0)))))))))))))),5^1(5(2(4(5(0(4(4(5(5(5(x0))))))))))) -> 4^1(0(4(2(0(4(4(4(3(3(4(1(3(1(x0))))))))))))))) 883.93/279.76 (5^1(5(2(4(5(0(1(5(5(5(3(5(x0)))))))))))) -> 4^1(0(4(2(5(3(2(5(1(0(1(2(0(5(x0)))))))))))))),5^1(5(2(4(5(0(1(5(5(5(3(5(x0)))))))))))) -> 4^1(0(4(2(5(3(2(5(1(0(1(2(0(5(x0))))))))))))))) 883.93/279.76 883.93/279.76 883.93/279.76 ---------------------------------------- 883.93/279.76 883.93/279.76 (88) 883.93/279.76 Obligation: 883.93/279.76 Q DP problem: 883.93/279.76 The TRS P consists of the following rules: 883.93/279.76 883.93/279.76 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.76 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.76 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.76 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.76 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.76 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.76 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.76 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.76 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.76 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.76 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.76 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.76 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.76 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.76 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.76 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.76 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.76 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.76 5^1(5(x1)) -> 4^1(x1) 883.93/279.76 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.76 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.76 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.76 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.76 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.76 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.76 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.76 5^1(5(2(4(5(0(2(5(3(4(x0)))))))))) -> 4^1(0(4(2(3(2(4(3(1(5(1(1(3(4(x0)))))))))))))) 883.93/279.76 5^1(5(2(4(5(0(4(4(5(5(5(x0))))))))))) -> 4^1(0(4(2(0(4(4(4(3(3(4(1(3(1(x0)))))))))))))) 883.93/279.76 5^1(5(2(4(5(0(1(5(5(5(3(5(x0)))))))))))) -> 4^1(0(4(2(5(3(2(5(1(0(1(2(0(5(x0)))))))))))))) 883.93/279.76 883.93/279.76 The TRS R consists of the following rules: 883.93/279.76 883.93/279.76 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.76 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.76 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.76 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.76 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.76 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.76 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.76 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.76 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.76 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.76 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.76 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.76 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.76 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.76 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.76 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.76 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.76 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.76 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.76 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.76 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.76 883.93/279.76 Q is empty. 883.93/279.76 We have to consider all minimal (P,Q,R)-chains. 883.93/279.76 ---------------------------------------- 883.93/279.76 883.93/279.76 (89) DependencyGraphProof (EQUIVALENT) 883.93/279.76 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 883.93/279.76 ---------------------------------------- 883.93/279.76 883.93/279.76 (90) 883.93/279.76 Obligation: 883.93/279.76 Q DP problem: 883.93/279.76 The TRS P consists of the following rules: 883.93/279.76 883.93/279.76 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.76 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.76 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.76 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.76 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.76 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.76 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.76 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.76 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.76 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.76 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.76 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.76 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.76 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.76 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.76 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.76 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.76 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.76 5^1(5(x1)) -> 4^1(x1) 883.93/279.76 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.76 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.76 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.76 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.76 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) 883.93/279.76 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.76 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.76 883.93/279.76 The TRS R consists of the following rules: 883.93/279.76 883.93/279.76 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.76 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.76 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.76 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.76 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.76 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.76 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.76 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.76 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.76 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.76 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.76 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.76 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.76 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.76 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.76 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.76 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.76 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.76 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.76 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.76 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.76 883.93/279.76 Q is empty. 883.93/279.76 We have to consider all minimal (P,Q,R)-chains. 883.93/279.76 ---------------------------------------- 883.93/279.76 883.93/279.76 (91) TransformationProof (EQUIVALENT) 883.93/279.76 By narrowing [LPAR04] the rule 5^1(5(2(4(5(0(x1)))))) -> 0^1(4(2(0(x1)))) at position [0] we obtained the following new rules [LPAR04]: 883.93/279.76 883.93/279.76 (5^1(5(2(4(5(0(2(5(3(4(x0)))))))))) -> 0^1(4(2(3(2(4(3(1(5(1(1(3(4(x0))))))))))))),5^1(5(2(4(5(0(2(5(3(4(x0)))))))))) -> 0^1(4(2(3(2(4(3(1(5(1(1(3(4(x0)))))))))))))) 883.93/279.76 (5^1(5(2(4(5(0(4(4(5(5(5(x0))))))))))) -> 0^1(4(2(0(4(4(4(3(3(4(1(3(1(x0))))))))))))),5^1(5(2(4(5(0(4(4(5(5(5(x0))))))))))) -> 0^1(4(2(0(4(4(4(3(3(4(1(3(1(x0)))))))))))))) 883.93/279.76 (5^1(5(2(4(5(0(1(5(5(5(3(5(x0)))))))))))) -> 0^1(4(2(5(3(2(5(1(0(1(2(0(5(x0))))))))))))),5^1(5(2(4(5(0(1(5(5(5(3(5(x0)))))))))))) -> 0^1(4(2(5(3(2(5(1(0(1(2(0(5(x0)))))))))))))) 883.93/279.76 883.93/279.76 883.93/279.76 ---------------------------------------- 883.93/279.76 883.93/279.76 (92) 883.93/279.76 Obligation: 883.93/279.76 Q DP problem: 883.93/279.76 The TRS P consists of the following rules: 883.93/279.76 883.93/279.76 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.76 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.76 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.76 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.76 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.76 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.76 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.76 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.76 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.76 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.76 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.76 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.76 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.76 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.76 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.76 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.76 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.76 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.76 5^1(5(x1)) -> 4^1(x1) 883.93/279.76 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.76 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.76 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.76 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.76 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.76 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.76 5^1(5(2(4(5(0(2(5(3(4(x0)))))))))) -> 0^1(4(2(3(2(4(3(1(5(1(1(3(4(x0))))))))))))) 883.93/279.76 5^1(5(2(4(5(0(4(4(5(5(5(x0))))))))))) -> 0^1(4(2(0(4(4(4(3(3(4(1(3(1(x0))))))))))))) 883.93/279.76 5^1(5(2(4(5(0(1(5(5(5(3(5(x0)))))))))))) -> 0^1(4(2(5(3(2(5(1(0(1(2(0(5(x0))))))))))))) 883.93/279.76 883.93/279.76 The TRS R consists of the following rules: 883.93/279.76 883.93/279.76 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.76 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.76 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.76 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.76 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.76 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.76 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.76 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.76 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.76 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.76 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.76 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.76 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.76 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.76 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.76 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.76 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.76 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.76 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.76 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.76 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.76 883.93/279.76 Q is empty. 883.93/279.76 We have to consider all minimal (P,Q,R)-chains. 883.93/279.76 ---------------------------------------- 883.93/279.76 883.93/279.76 (93) DependencyGraphProof (EQUIVALENT) 883.93/279.76 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 883.93/279.76 ---------------------------------------- 883.93/279.76 883.93/279.76 (94) 883.93/279.76 Obligation: 883.93/279.76 Q DP problem: 883.93/279.76 The TRS P consists of the following rules: 883.93/279.76 883.93/279.76 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.76 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.76 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.76 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.76 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.76 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.76 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.76 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.76 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.76 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.76 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.76 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.76 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.76 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.76 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.76 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.76 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.76 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.76 5^1(5(x1)) -> 4^1(x1) 883.93/279.76 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.76 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.76 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.76 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.76 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) 883.93/279.76 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.76 883.93/279.76 The TRS R consists of the following rules: 883.93/279.76 883.93/279.76 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.76 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.76 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.76 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.76 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.76 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.76 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.76 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.76 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.76 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.76 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.76 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.76 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.76 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.76 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.76 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.76 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.76 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.76 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.76 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.76 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.76 883.93/279.76 Q is empty. 883.93/279.76 We have to consider all minimal (P,Q,R)-chains. 883.93/279.76 ---------------------------------------- 883.93/279.76 883.93/279.76 (95) TransformationProof (EQUIVALENT) 883.93/279.76 By narrowing [LPAR04] the rule 5^1(5(2(4(5(0(x1)))))) -> 4^1(2(0(x1))) at position [0] we obtained the following new rules [LPAR04]: 883.93/279.76 883.93/279.76 (5^1(5(2(4(5(0(2(5(3(4(x0)))))))))) -> 4^1(2(3(2(4(3(1(5(1(1(3(4(x0)))))))))))),5^1(5(2(4(5(0(2(5(3(4(x0)))))))))) -> 4^1(2(3(2(4(3(1(5(1(1(3(4(x0))))))))))))) 883.93/279.76 (5^1(5(2(4(5(0(4(4(5(5(5(x0))))))))))) -> 4^1(2(0(4(4(4(3(3(4(1(3(1(x0)))))))))))),5^1(5(2(4(5(0(4(4(5(5(5(x0))))))))))) -> 4^1(2(0(4(4(4(3(3(4(1(3(1(x0))))))))))))) 883.93/279.76 (5^1(5(2(4(5(0(1(5(5(5(3(5(x0)))))))))))) -> 4^1(2(5(3(2(5(1(0(1(2(0(5(x0)))))))))))),5^1(5(2(4(5(0(1(5(5(5(3(5(x0)))))))))))) -> 4^1(2(5(3(2(5(1(0(1(2(0(5(x0))))))))))))) 883.93/279.76 883.93/279.76 883.93/279.76 ---------------------------------------- 883.93/279.76 883.93/279.76 (96) 883.93/279.76 Obligation: 883.93/279.76 Q DP problem: 883.93/279.76 The TRS P consists of the following rules: 883.93/279.76 883.93/279.76 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.76 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.76 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.76 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.76 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.76 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.76 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.76 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.76 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.76 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.76 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.76 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.76 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.76 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.76 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.76 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.76 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.76 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.76 5^1(5(x1)) -> 4^1(x1) 883.93/279.76 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.76 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.76 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.76 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.76 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.76 5^1(5(2(4(5(0(2(5(3(4(x0)))))))))) -> 4^1(2(3(2(4(3(1(5(1(1(3(4(x0)))))))))))) 883.93/279.76 5^1(5(2(4(5(0(4(4(5(5(5(x0))))))))))) -> 4^1(2(0(4(4(4(3(3(4(1(3(1(x0)))))))))))) 883.93/279.76 5^1(5(2(4(5(0(1(5(5(5(3(5(x0)))))))))))) -> 4^1(2(5(3(2(5(1(0(1(2(0(5(x0)))))))))))) 883.93/279.76 883.93/279.76 The TRS R consists of the following rules: 883.93/279.76 883.93/279.76 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.76 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.76 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.76 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.76 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.76 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.76 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.76 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.76 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.76 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.76 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.76 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.76 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.76 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.76 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.76 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.76 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.76 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.76 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.76 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.76 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.76 883.93/279.76 Q is empty. 883.93/279.76 We have to consider all minimal (P,Q,R)-chains. 883.93/279.76 ---------------------------------------- 883.93/279.76 883.93/279.76 (97) DependencyGraphProof (EQUIVALENT) 883.93/279.76 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 883.93/279.76 ---------------------------------------- 883.93/279.76 883.93/279.76 (98) 883.93/279.76 Obligation: 883.93/279.76 Q DP problem: 883.93/279.76 The TRS P consists of the following rules: 883.93/279.76 883.93/279.76 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.76 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.76 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.76 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.76 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.76 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.76 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.76 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.76 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.76 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.76 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.76 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.76 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.76 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.76 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.76 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.76 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.76 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.76 5^1(5(x1)) -> 4^1(x1) 883.93/279.76 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.76 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.76 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.76 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.76 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) 883.93/279.76 883.93/279.76 The TRS R consists of the following rules: 883.93/279.76 883.93/279.76 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.76 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.76 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.76 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.76 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.76 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.76 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.76 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.76 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.76 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.76 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.76 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.76 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.76 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.76 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.76 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.76 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 883.93/279.76 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 883.93/279.76 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 883.93/279.76 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 883.93/279.76 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.76 883.93/279.76 Q is empty. 883.93/279.76 We have to consider all minimal (P,Q,R)-chains. 883.93/279.76 ---------------------------------------- 883.93/279.76 883.93/279.76 (99) TransformationProof (EQUIVALENT) 883.93/279.76 By narrowing [LPAR04] the rule 5^1(5(2(4(5(0(x1)))))) -> 2^1(0(x1)) at position [0] we obtained the following new rules [LPAR04]: 883.93/279.76 883.93/279.76 (5^1(5(2(4(5(0(2(5(3(4(x0)))))))))) -> 2^1(3(2(4(3(1(5(1(1(3(4(x0))))))))))),5^1(5(2(4(5(0(2(5(3(4(x0)))))))))) -> 2^1(3(2(4(3(1(5(1(1(3(4(x0)))))))))))) 883.93/279.76 (5^1(5(2(4(5(0(4(4(5(5(5(x0))))))))))) -> 2^1(0(4(4(4(3(3(4(1(3(1(x0))))))))))),5^1(5(2(4(5(0(4(4(5(5(5(x0))))))))))) -> 2^1(0(4(4(4(3(3(4(1(3(1(x0)))))))))))) 883.93/279.76 (5^1(5(2(4(5(0(1(5(5(5(3(5(x0)))))))))))) -> 2^1(5(3(2(5(1(0(1(2(0(5(x0))))))))))),5^1(5(2(4(5(0(1(5(5(5(3(5(x0)))))))))))) -> 2^1(5(3(2(5(1(0(1(2(0(5(x0)))))))))))) 883.93/279.76 883.93/279.76 883.93/279.76 ---------------------------------------- 883.93/279.76 883.93/279.76 (100) 883.93/279.76 Obligation: 883.93/279.76 Q DP problem: 883.93/279.76 The TRS P consists of the following rules: 883.93/279.76 883.93/279.76 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 883.93/279.76 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 883.93/279.76 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 883.93/279.76 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 883.93/279.76 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 883.93/279.76 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.76 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 883.93/279.76 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 883.93/279.76 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 883.93/279.76 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 883.93/279.76 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 883.93/279.76 5^1(5(x1)) -> 4^1(0(4(x1))) 883.93/279.76 5^1(5(x1)) -> 0^1(4(x1)) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 883.93/279.76 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 883.93/279.76 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 883.93/279.76 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 883.93/279.76 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 883.93/279.76 2^1(5(5(x1))) -> 2^1(x1) 883.93/279.76 2^1(5(0(4(x1)))) -> 0^1(x1) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 883.93/279.76 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 883.93/279.76 5^1(5(x1)) -> 4^1(x1) 883.93/279.76 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 883.93/279.76 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 883.93/279.76 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 883.93/279.76 5^1(5(3(x1))) -> 0^1(x1) 883.93/279.76 5^1(5(2(4(5(0(2(5(3(4(x0)))))))))) -> 2^1(3(2(4(3(1(5(1(1(3(4(x0))))))))))) 883.93/279.76 5^1(5(2(4(5(0(4(4(5(5(5(x0))))))))))) -> 2^1(0(4(4(4(3(3(4(1(3(1(x0))))))))))) 883.93/279.76 5^1(5(2(4(5(0(1(5(5(5(3(5(x0)))))))))))) -> 2^1(5(3(2(5(1(0(1(2(0(5(x0))))))))))) 883.93/279.76 883.93/279.76 The TRS R consists of the following rules: 883.93/279.76 883.93/279.76 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 883.93/279.76 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 883.93/279.76 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 883.93/279.76 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 883.93/279.76 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 883.93/279.76 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 883.93/279.76 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 883.93/279.76 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 883.93/279.76 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 883.93/279.76 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 883.93/279.76 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 883.93/279.76 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 883.93/279.76 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 883.93/279.76 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 883.93/279.76 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 883.93/279.76 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 883.93/279.76 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 884.27/279.78 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 884.27/279.78 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 884.27/279.78 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 884.27/279.78 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 884.27/279.78 884.27/279.78 Q is empty. 884.27/279.78 We have to consider all minimal (P,Q,R)-chains. 884.27/279.78 ---------------------------------------- 884.27/279.78 884.27/279.78 (101) DependencyGraphProof (EQUIVALENT) 884.27/279.78 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 884.27/279.78 ---------------------------------------- 884.27/279.78 884.27/279.78 (102) 884.27/279.78 Obligation: 884.27/279.78 Q DP problem: 884.27/279.78 The TRS P consists of the following rules: 884.27/279.78 884.27/279.78 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 884.27/279.78 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 884.27/279.78 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 884.27/279.78 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 884.27/279.78 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 884.27/279.78 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 884.27/279.78 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 884.27/279.78 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 884.27/279.78 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 884.27/279.78 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 884.27/279.78 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 884.27/279.78 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 884.27/279.78 5^1(5(x1)) -> 4^1(0(4(x1))) 884.27/279.78 5^1(5(x1)) -> 0^1(4(x1)) 884.27/279.78 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 884.27/279.78 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 884.27/279.78 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 884.27/279.78 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 884.27/279.78 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 884.27/279.78 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 884.27/279.78 2^1(5(5(x1))) -> 2^1(x1) 884.27/279.78 2^1(5(0(4(x1)))) -> 0^1(x1) 884.27/279.78 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 884.27/279.78 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 884.27/279.78 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 884.27/279.78 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 884.27/279.78 5^1(5(x1)) -> 4^1(x1) 884.27/279.78 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 884.27/279.78 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 884.27/279.78 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 884.27/279.78 5^1(5(3(x1))) -> 0^1(x1) 884.27/279.78 884.27/279.78 The TRS R consists of the following rules: 884.27/279.78 884.27/279.78 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 884.27/279.78 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 884.27/279.78 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 884.27/279.78 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 884.27/279.78 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 884.27/279.78 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 884.27/279.78 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 884.27/279.78 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 884.27/279.78 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 884.27/279.78 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 884.27/279.78 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 884.27/279.78 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 884.27/279.78 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 884.27/279.78 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 884.27/279.78 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 884.27/279.78 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 884.27/279.78 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 884.27/279.78 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 884.27/279.78 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 884.27/279.78 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 884.27/279.78 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 884.27/279.78 884.27/279.78 Q is empty. 884.27/279.78 We have to consider all minimal (P,Q,R)-chains. 884.27/279.78 ---------------------------------------- 884.27/279.78 884.27/279.78 (103) QDPOrderProof (EQUIVALENT) 884.27/279.78 We use the reduction pair processor [LPAR04,JAR06]. 884.27/279.78 884.27/279.78 884.27/279.78 The following pairs can be oriented strictly and are deleted. 884.27/279.78 884.27/279.78 2^1(5(5(x1))) -> 4^1(2(5(4(4(0(0(1(1(2(x1)))))))))) 884.27/279.78 2^1(5(5(x1))) -> 2^1(5(4(4(0(0(1(1(2(x1))))))))) 884.27/279.78 2^1(5(5(x1))) -> 4^1(4(0(0(1(1(2(x1))))))) 884.27/279.78 2^1(5(5(x1))) -> 4^1(0(0(1(1(2(x1)))))) 884.27/279.78 2^1(5(5(x1))) -> 0^1(0(1(1(2(x1))))) 884.27/279.78 2^1(5(5(x1))) -> 0^1(1(1(2(x1)))) 884.27/279.78 2^1(5(5(x1))) -> 2^1(x1) 884.27/279.78 2^1(5(0(4(x1)))) -> 0^1(x1) 884.27/279.78 2^1(5(5(3(4(x1))))) -> 4^1(0(2(4(4(x1))))) 884.27/279.78 2^1(5(5(3(4(x1))))) -> 0^1(2(4(4(x1)))) 884.27/279.78 2^1(5(5(3(4(x1))))) -> 2^1(4(4(x1))) 884.27/279.78 2^1(5(5(3(4(x1))))) -> 4^1(4(x1)) 884.27/279.78 The remaining pairs can at least be oriented weakly. 884.27/279.78 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 884.27/279.78 884.27/279.78 POL( 0^1_1(x_1) ) = max{0, -2} 884.27/279.78 POL( 2^1_1(x_1) ) = max{0, 2x_1 - 2} 884.27/279.78 POL( 4^1_1(x_1) ) = max{0, 2x_1 - 2} 884.27/279.78 POL( 5^1_1(x_1) ) = max{0, 2x_1 - 2} 884.27/279.78 POL( 4_1(x_1) ) = max{0, -2} 884.27/279.78 POL( 5_1(x_1) ) = x_1 + 1 884.27/279.78 POL( 2_1(x_1) ) = max{0, x_1 - 1} 884.27/279.78 POL( 0_1(x_1) ) = 1 884.27/279.78 POL( 1_1(x_1) ) = max{0, 2x_1 - 2} 884.27/279.78 POL( 3_1(x_1) ) = max{0, -2} 884.27/279.78 884.27/279.78 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 884.27/279.78 884.27/279.78 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 884.27/279.78 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 884.27/279.78 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 884.27/279.78 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 884.27/279.78 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 884.27/279.78 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 884.27/279.78 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 884.27/279.78 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 884.27/279.78 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 884.27/279.78 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 884.27/279.78 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 884.27/279.78 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 884.27/279.78 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 884.27/279.78 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 884.27/279.78 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 884.27/279.78 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 884.27/279.78 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 884.27/279.78 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 884.27/279.78 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 884.27/279.78 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 884.27/279.78 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 884.27/279.78 884.27/279.78 884.27/279.78 ---------------------------------------- 884.27/279.78 884.27/279.78 (104) 884.27/279.78 Obligation: 884.27/279.78 Q DP problem: 884.27/279.78 The TRS P consists of the following rules: 884.27/279.78 884.27/279.78 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 884.27/279.78 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 884.27/279.78 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 884.27/279.78 4^1(2(5(5(1(5(x1)))))) -> 2^1(5(x1)) 884.27/279.78 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 884.27/279.78 4^1(4(5(2(4(2(2(x1))))))) -> 0^1(5(5(4(5(1(2(2(1(x1))))))))) 884.27/279.78 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(5(1(0(1(2(0(5(x1)))))))) 884.27/279.78 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 884.27/279.78 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 884.27/279.78 5^1(5(x1)) -> 4^1(0(4(x1))) 884.27/279.78 5^1(5(x1)) -> 0^1(4(x1)) 884.27/279.78 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 884.27/279.78 0^1(1(5(5(5(3(5(x1))))))) -> 2^1(0(5(x1))) 884.27/279.78 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 884.27/279.78 5^1(5(x1)) -> 4^1(x1) 884.27/279.78 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 884.27/279.78 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 884.27/279.78 5^1(5(3(x1))) -> 0^1(1(5(0(x1)))) 884.27/279.78 5^1(5(3(x1))) -> 0^1(x1) 884.27/279.78 884.27/279.78 The TRS R consists of the following rules: 884.27/279.78 884.27/279.78 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 884.27/279.78 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 884.27/279.78 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 884.27/279.78 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 884.27/279.78 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 884.27/279.78 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 884.27/279.78 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 884.27/279.78 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 884.27/279.78 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 884.27/279.78 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 884.27/279.78 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 884.27/279.78 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 884.27/279.78 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 884.27/279.78 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 884.27/279.78 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 884.27/279.78 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 884.27/279.78 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 884.27/279.78 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 884.27/279.78 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 884.27/279.78 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 884.27/279.78 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 884.27/279.78 884.27/279.78 Q is empty. 884.27/279.78 We have to consider all minimal (P,Q,R)-chains. 884.27/279.78 ---------------------------------------- 884.27/279.78 884.27/279.78 (105) DependencyGraphProof (EQUIVALENT) 884.27/279.78 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 7 less nodes. 884.27/279.78 ---------------------------------------- 884.27/279.78 884.27/279.78 (106) 884.27/279.78 Complex Obligation (AND) 884.27/279.78 884.27/279.78 ---------------------------------------- 884.27/279.78 884.27/279.78 (107) 884.27/279.78 Obligation: 884.27/279.78 Q DP problem: 884.27/279.78 The TRS P consists of the following rules: 884.27/279.78 884.27/279.78 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 884.27/279.78 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 884.27/279.78 884.27/279.78 The TRS R consists of the following rules: 884.27/279.78 884.27/279.78 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 884.27/279.78 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 884.27/279.78 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 884.27/279.78 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 884.27/279.78 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 884.27/279.78 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 884.27/279.78 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 884.27/279.78 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 884.27/279.78 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 884.27/279.78 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 884.27/279.78 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 884.27/279.78 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 884.27/279.78 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 884.27/279.78 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 884.27/279.78 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 884.27/279.78 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 884.27/279.78 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 884.27/279.78 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 884.27/279.78 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 884.27/279.78 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 884.27/279.78 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 884.27/279.78 884.27/279.78 Q is empty. 884.27/279.78 We have to consider all minimal (P,Q,R)-chains. 884.27/279.78 ---------------------------------------- 884.27/279.78 884.27/279.78 (108) QDPOrderProof (EQUIVALENT) 884.27/279.78 We use the reduction pair processor [LPAR04,JAR06]. 884.27/279.78 884.27/279.78 884.27/279.78 The following pairs can be oriented strictly and are deleted. 884.27/279.78 884.27/279.78 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(5(x1)) 884.27/279.78 0^1(1(5(5(5(3(5(x1))))))) -> 0^1(1(2(0(5(x1))))) 884.27/279.78 The remaining pairs can at least be oriented weakly. 884.27/279.78 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 884.27/279.78 884.27/279.78 POL( 0^1_1(x_1) ) = 2x_1 + 2 884.27/279.78 POL( 5_1(x_1) ) = x_1 + 1 884.27/279.78 POL( 0_1(x_1) ) = max{0, 2x_1 - 2} 884.27/279.78 POL( 4_1(x_1) ) = max{0, -2} 884.27/279.78 POL( 2_1(x_1) ) = max{0, -2} 884.27/279.78 POL( 1_1(x_1) ) = max{0, 2x_1 - 2} 884.27/279.78 POL( 3_1(x_1) ) = max{0, 2x_1 - 2} 884.27/279.78 884.27/279.78 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 884.27/279.78 884.27/279.78 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 884.27/279.78 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 884.27/279.78 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 884.27/279.78 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 884.27/279.78 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 884.27/279.78 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 884.27/279.78 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 884.27/279.78 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 884.27/279.78 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 884.27/279.78 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 884.27/279.78 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 884.27/279.78 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 884.27/279.78 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 884.27/279.78 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 884.27/279.78 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 884.27/279.78 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 884.27/279.78 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 884.27/279.78 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 884.27/279.78 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 884.27/279.78 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 884.27/279.78 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 884.27/279.78 884.27/279.78 884.27/279.78 ---------------------------------------- 884.27/279.78 884.27/279.78 (109) 884.27/279.78 Obligation: 884.27/279.78 Q DP problem: 884.27/279.78 P is empty. 884.27/279.78 The TRS R consists of the following rules: 884.27/279.78 884.27/279.78 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 884.27/279.78 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 884.27/279.78 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 884.27/279.78 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 884.27/279.78 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 884.27/279.78 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 884.27/279.78 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 884.27/279.78 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 884.27/279.78 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 884.27/279.78 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 884.27/279.78 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 884.27/279.78 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 884.27/279.78 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 884.27/279.78 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 884.27/279.78 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 884.27/279.78 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 884.27/279.78 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 884.27/279.78 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 884.27/279.78 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 884.27/279.78 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 884.27/279.78 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 884.27/279.78 884.27/279.78 Q is empty. 884.27/279.78 We have to consider all minimal (P,Q,R)-chains. 884.27/279.78 ---------------------------------------- 884.27/279.78 884.27/279.78 (110) PisEmptyProof (EQUIVALENT) 884.27/279.78 The TRS P is empty. Hence, there is no (P,Q,R) chain. 884.27/279.78 ---------------------------------------- 884.27/279.78 884.27/279.78 (111) 884.27/279.78 YES 884.27/279.78 884.27/279.78 ---------------------------------------- 884.27/279.78 884.27/279.78 (112) 884.27/279.78 Obligation: 884.27/279.78 Q DP problem: 884.27/279.78 The TRS P consists of the following rules: 884.27/279.78 884.27/279.78 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 884.27/279.78 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 884.27/279.78 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 884.27/279.78 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 884.27/279.78 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 884.27/279.78 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 884.27/279.78 5^1(5(x1)) -> 4^1(0(4(x1))) 884.27/279.78 5^1(5(x1)) -> 4^1(x1) 884.27/279.78 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 884.27/279.78 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 884.27/279.78 884.27/279.78 The TRS R consists of the following rules: 884.27/279.78 884.27/279.78 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 884.27/279.78 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 884.27/279.78 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 884.27/279.78 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 884.27/279.78 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 884.27/279.78 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 884.27/279.78 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 884.27/279.78 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 884.27/279.78 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 884.27/279.78 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 884.27/279.78 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 884.27/279.78 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 884.27/279.78 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 884.27/279.78 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 884.27/279.78 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 884.27/279.78 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 884.27/279.78 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 884.27/279.78 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 884.27/279.78 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 884.27/279.78 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 884.27/279.78 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 884.27/279.78 884.27/279.78 Q is empty. 884.27/279.78 We have to consider all minimal (P,Q,R)-chains. 884.27/279.78 ---------------------------------------- 884.27/279.78 884.27/279.78 (113) QDPOrderProof (EQUIVALENT) 884.27/279.78 We use the reduction pair processor [LPAR04,JAR06]. 884.27/279.78 884.27/279.78 884.27/279.78 The following pairs can be oriented strictly and are deleted. 884.27/279.78 884.27/279.78 5^1(5(x1)) -> 4^1(1(1(1(1(4(4(0(4(x1))))))))) 884.27/279.78 4^1(4(5(2(4(2(2(x1))))))) -> 4^1(0(5(5(4(5(1(2(2(1(x1)))))))))) 884.27/279.78 5^1(5(x1)) -> 4^1(x1) 884.27/279.78 5^1(5(3(x1))) -> 4^1(4(1(0(1(5(0(x1))))))) 884.27/279.78 5^1(5(3(x1))) -> 4^1(1(0(1(5(0(x1)))))) 884.27/279.78 The remaining pairs can at least be oriented weakly. 884.27/279.78 Used ordering: Polynomial interpretation [POLO]: 884.27/279.78 884.27/279.78 POL(0(x_1)) = 1 884.27/279.78 POL(1(x_1)) = 0 884.27/279.78 POL(2(x_1)) = 1 884.27/279.78 POL(3(x_1)) = 0 884.27/279.78 POL(4(x_1)) = x_1 884.27/279.78 POL(4^1(x_1)) = x_1 884.27/279.78 POL(5(x_1)) = 1 + x_1 884.27/279.78 POL(5^1(x_1)) = x_1 884.27/279.78 884.27/279.78 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 884.27/279.78 884.27/279.78 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 884.27/279.78 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 884.27/279.78 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 884.27/279.78 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 884.27/279.78 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 884.27/279.78 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 884.27/279.78 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 884.27/279.78 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 884.27/279.78 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 884.27/279.78 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 884.27/279.78 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 884.27/279.78 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 884.27/279.78 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 884.27/279.78 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 884.27/279.78 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 884.27/279.78 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 884.27/279.78 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 884.27/279.78 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 884.27/279.78 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 884.27/279.78 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 884.27/279.78 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 884.27/279.78 884.27/279.78 884.27/279.78 ---------------------------------------- 884.27/279.78 884.27/279.78 (114) 884.27/279.78 Obligation: 884.27/279.78 Q DP problem: 884.27/279.78 The TRS P consists of the following rules: 884.27/279.78 884.27/279.78 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 884.27/279.78 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 884.27/279.78 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 884.27/279.78 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 884.27/279.78 5^1(5(x1)) -> 4^1(0(4(x1))) 884.27/279.78 884.27/279.78 The TRS R consists of the following rules: 884.27/279.78 884.27/279.78 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 884.27/279.78 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 884.27/279.78 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 884.27/279.78 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 884.27/279.78 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 884.27/279.78 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 884.27/279.78 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 884.27/279.78 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 884.27/279.78 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 884.27/279.78 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 884.27/279.78 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 884.27/279.78 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 884.27/279.78 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 884.27/279.78 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 884.27/279.78 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 884.27/279.78 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 884.27/279.78 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 884.27/279.78 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 884.27/279.78 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 884.27/279.78 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 884.27/279.78 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 884.27/279.78 884.27/279.78 Q is empty. 884.27/279.78 We have to consider all minimal (P,Q,R)-chains. 884.27/279.78 ---------------------------------------- 884.27/279.78 884.27/279.78 (115) QDPOrderProof (EQUIVALENT) 884.27/279.78 We use the reduction pair processor [LPAR04,JAR06]. 884.27/279.78 884.27/279.78 884.27/279.78 The following pairs can be oriented strictly and are deleted. 884.27/279.78 884.27/279.78 4^1(5(2(4(x1)))) -> 5^1(5(2(0(3(1(3(3(x1)))))))) 884.27/279.78 4^1(4(5(2(4(2(2(x1))))))) -> 5^1(5(4(5(1(2(2(1(x1)))))))) 884.27/279.78 The remaining pairs can at least be oriented weakly. 884.27/279.78 Used ordering: Polynomial interpretation [POLO]: 884.27/279.78 884.27/279.78 POL(0(x_1)) = 0 884.27/279.78 POL(1(x_1)) = 0 884.27/279.78 POL(2(x_1)) = 1 884.27/279.78 POL(3(x_1)) = 0 884.27/279.78 POL(4(x_1)) = x_1 884.27/279.78 POL(4^1(x_1)) = x_1 884.27/279.78 POL(5(x_1)) = x_1 884.27/279.78 POL(5^1(x_1)) = 0 884.27/279.78 884.27/279.78 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 884.27/279.78 884.27/279.78 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 884.27/279.78 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 884.27/279.78 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 884.27/279.78 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 884.27/279.78 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 884.27/279.78 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 884.27/279.78 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 884.27/279.78 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 884.27/279.78 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 884.27/279.78 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 884.27/279.78 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 884.27/279.78 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 884.27/279.78 884.27/279.78 884.27/279.78 ---------------------------------------- 884.27/279.78 884.27/279.78 (116) 884.27/279.78 Obligation: 884.27/279.78 Q DP problem: 884.27/279.78 The TRS P consists of the following rules: 884.27/279.78 884.27/279.78 5^1(5(x1)) -> 4^1(4(0(4(x1)))) 884.27/279.78 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 884.27/279.78 5^1(5(x1)) -> 4^1(0(4(x1))) 884.27/279.78 884.27/279.78 The TRS R consists of the following rules: 884.27/279.78 884.27/279.78 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 884.27/279.78 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 884.27/279.78 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 884.27/279.78 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 884.27/279.78 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 884.27/279.78 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 884.27/279.78 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 884.27/279.78 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 884.27/279.78 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 884.27/279.78 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 884.27/279.78 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 884.27/279.78 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 884.27/279.78 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 884.27/279.78 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 884.27/279.78 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 884.27/279.78 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 884.27/279.78 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 884.27/279.78 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 884.27/279.78 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 884.27/279.78 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 884.27/279.78 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 884.27/279.78 884.27/279.78 Q is empty. 884.27/279.78 We have to consider all minimal (P,Q,R)-chains. 884.27/279.78 ---------------------------------------- 884.27/279.78 884.27/279.78 (117) DependencyGraphProof (EQUIVALENT) 884.27/279.78 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 884.27/279.78 ---------------------------------------- 884.27/279.78 884.27/279.78 (118) 884.27/279.78 Obligation: 884.27/279.78 Q DP problem: 884.27/279.78 The TRS P consists of the following rules: 884.27/279.78 884.27/279.78 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 884.27/279.78 884.27/279.78 The TRS R consists of the following rules: 884.27/279.78 884.27/279.78 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 884.27/279.78 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 884.27/279.78 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 884.27/279.78 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 884.27/279.78 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 884.27/279.78 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 884.27/279.78 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 884.27/279.78 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 884.27/279.78 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 884.27/279.78 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 884.27/279.78 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 884.27/279.78 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 884.27/279.78 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 884.27/279.78 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 884.27/279.78 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 884.27/279.78 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 884.27/279.78 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 884.27/279.78 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 884.27/279.78 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 884.27/279.78 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 884.27/279.78 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 884.27/279.78 884.27/279.78 Q is empty. 884.27/279.78 We have to consider all minimal (P,Q,R)-chains. 884.27/279.78 ---------------------------------------- 884.27/279.78 884.27/279.78 (119) QDPOrderProof (EQUIVALENT) 884.27/279.78 We use the reduction pair processor [LPAR04,JAR06]. 884.27/279.78 884.27/279.78 884.27/279.78 The following pairs can be oriented strictly and are deleted. 884.27/279.78 884.27/279.78 4^1(2(5(5(1(5(x1)))))) -> 4^1(2(5(x1))) 884.27/279.78 The remaining pairs can at least be oriented weakly. 884.27/279.78 Used ordering: Polynomial interpretation [POLO]: 884.27/279.78 884.27/279.78 POL(0(x_1)) = x_1 884.27/279.78 POL(1(x_1)) = x_1 884.27/279.78 POL(2(x_1)) = x_1 884.27/279.78 POL(3(x_1)) = 0 884.27/279.78 POL(4(x_1)) = 1 884.27/279.78 POL(4^1(x_1)) = x_1 884.27/279.78 POL(5(x_1)) = 1 + x_1 884.27/279.78 884.27/279.78 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 884.27/279.78 884.27/279.78 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 884.27/279.78 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 884.27/279.78 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 884.27/279.78 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 884.27/279.78 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 884.27/279.78 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 884.27/279.78 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 884.27/279.78 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 884.27/279.78 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 884.27/279.78 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 884.27/279.78 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 884.27/279.78 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 884.27/279.78 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 884.27/279.78 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 884.27/279.78 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 884.27/279.78 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 884.27/279.78 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 884.27/279.78 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 884.27/279.78 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 884.27/279.78 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 884.27/279.78 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 884.27/279.78 884.27/279.78 884.27/279.78 ---------------------------------------- 884.27/279.78 884.27/279.78 (120) 884.27/279.78 Obligation: 884.27/279.78 Q DP problem: 884.27/279.78 P is empty. 884.27/279.78 The TRS R consists of the following rules: 884.27/279.78 884.27/279.78 5(5(x1)) -> 0(5(4(0(2(5(4(5(2(1(x1)))))))))) 884.27/279.78 5(5(x1)) -> 3(4(1(1(1(1(4(4(0(4(x1)))))))))) 884.27/279.78 2(5(5(x1))) -> 4(2(5(4(4(0(0(1(1(2(x1)))))))))) 884.27/279.78 5(2(4(x1))) -> 0(5(0(2(3(3(4(2(4(2(x1)))))))))) 884.27/279.78 5(5(2(x1))) -> 0(1(3(2(3(0(3(2(5(3(x1)))))))))) 884.27/279.78 5(5(3(x1))) -> 0(3(5(4(4(1(0(1(5(0(x1)))))))))) 884.27/279.78 5(5(5(x1))) -> 5(3(4(1(0(1(4(5(0(0(x1)))))))))) 884.27/279.78 2(5(0(4(x1)))) -> 4(4(3(2(4(4(5(1(0(0(x1)))))))))) 884.27/279.78 4(5(2(4(x1)))) -> 4(1(5(5(2(0(3(1(3(3(x1)))))))))) 884.27/279.78 4(5(5(5(x1)))) -> 1(5(1(2(0(3(2(1(0(5(x1)))))))))) 884.27/279.78 0(2(5(3(4(x1))))) -> 3(2(4(3(1(5(1(1(3(4(x1)))))))))) 884.27/279.78 2(5(5(3(4(x1))))) -> 4(5(4(3(1(4(0(2(4(4(x1)))))))))) 884.27/279.78 5(5(5(1(4(x1))))) -> 3(3(0(5(0(4(3(4(4(0(x1)))))))))) 884.27/279.78 0(4(4(5(5(5(x1)))))) -> 0(4(4(4(3(3(4(1(3(1(x1)))))))))) 884.27/279.78 1(2(4(5(2(4(x1)))))) -> 3(3(5(3(0(4(0(3(1(3(x1)))))))))) 884.27/279.78 4(1(5(5(0(4(x1)))))) -> 1(0(3(0(4(2(4(4(3(4(x1)))))))))) 884.27/279.78 4(2(5(5(1(5(x1)))))) -> 2(3(4(2(1(1(3(4(2(5(x1)))))))))) 884.27/279.78 5(2(5(5(0(4(x1)))))) -> 0(4(2(3(3(5(2(1(4(4(x1)))))))))) 884.27/279.78 5(5(2(4(5(0(x1)))))) -> 2(1(1(4(2(4(0(4(2(0(x1)))))))))) 884.27/279.78 0(1(5(5(5(3(5(x1))))))) -> 5(3(2(5(1(0(1(2(0(5(x1)))))))))) 884.27/279.78 4(4(5(2(4(2(2(x1))))))) -> 4(0(5(5(4(5(1(2(2(1(x1)))))))))) 884.27/279.78 884.27/279.78 Q is empty. 884.27/279.78 We have to consider all minimal (P,Q,R)-chains. 884.27/279.78 ---------------------------------------- 884.27/279.78 884.27/279.78 (121) PisEmptyProof (EQUIVALENT) 884.27/279.78 The TRS P is empty. Hence, there is no (P,Q,R) chain. 884.27/279.78 ---------------------------------------- 884.27/279.78 884.27/279.78 (122) 884.27/279.78 YES 884.46/279.90 EOF