YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) DependencyPairsProof [EQUIVALENT, 648 ms] (4) QDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) QDP (7) TransformationProof [EQUIVALENT, 0 ms] (8) QDP (9) DependencyGraphProof [EQUIVALENT, 0 ms] (10) QDP (11) TransformationProof [EQUIVALENT, 0 ms] (12) QDP (13) DependencyGraphProof [EQUIVALENT, 0 ms] (14) QDP (15) TransformationProof [EQUIVALENT, 0 ms] (16) QDP (17) DependencyGraphProof [EQUIVALENT, 0 ms] (18) QDP (19) TransformationProof [EQUIVALENT, 0 ms] (20) QDP (21) DependencyGraphProof [EQUIVALENT, 0 ms] (22) QDP (23) TransformationProof [EQUIVALENT, 0 ms] (24) QDP (25) DependencyGraphProof [EQUIVALENT, 0 ms] (26) QDP (27) TransformationProof [EQUIVALENT, 0 ms] (28) QDP (29) DependencyGraphProof [EQUIVALENT, 0 ms] (30) QDP (31) TransformationProof [EQUIVALENT, 0 ms] (32) QDP (33) DependencyGraphProof [EQUIVALENT, 0 ms] (34) QDP (35) TransformationProof [EQUIVALENT, 0 ms] (36) QDP (37) DependencyGraphProof [EQUIVALENT, 0 ms] (38) QDP (39) TransformationProof [EQUIVALENT, 0 ms] (40) QDP (41) DependencyGraphProof [EQUIVALENT, 0 ms] (42) QDP (43) TransformationProof [EQUIVALENT, 1 ms] (44) QDP (45) DependencyGraphProof [EQUIVALENT, 0 ms] (46) QDP (47) TransformationProof [EQUIVALENT, 0 ms] (48) QDP (49) DependencyGraphProof [EQUIVALENT, 0 ms] (50) QDP (51) TransformationProof [EQUIVALENT, 0 ms] (52) QDP (53) DependencyGraphProof [EQUIVALENT, 0 ms] (54) QDP (55) TransformationProof [EQUIVALENT, 0 ms] (56) QDP (57) DependencyGraphProof [EQUIVALENT, 0 ms] (58) QDP (59) TransformationProof [EQUIVALENT, 0 ms] (60) QDP (61) DependencyGraphProof [EQUIVALENT, 0 ms] (62) QDP (63) TransformationProof [EQUIVALENT, 0 ms] (64) QDP (65) DependencyGraphProof [EQUIVALENT, 0 ms] (66) QDP (67) TransformationProof [EQUIVALENT, 0 ms] (68) QDP (69) DependencyGraphProof [EQUIVALENT, 0 ms] (70) QDP (71) TransformationProof [EQUIVALENT, 0 ms] (72) QDP (73) DependencyGraphProof [EQUIVALENT, 0 ms] (74) QDP (75) TransformationProof [EQUIVALENT, 100 ms] (76) QDP (77) DependencyGraphProof [EQUIVALENT, 0 ms] (78) QDP (79) TransformationProof [EQUIVALENT, 57 ms] (80) QDP (81) DependencyGraphProof [EQUIVALENT, 0 ms] (82) QDP (83) TransformationProof [EQUIVALENT, 28 ms] (84) QDP (85) DependencyGraphProof [EQUIVALENT, 0 ms] (86) QDP (87) TransformationProof [EQUIVALENT, 74 ms] (88) QDP (89) DependencyGraphProof [EQUIVALENT, 0 ms] (90) QDP (91) TransformationProof [EQUIVALENT, 69 ms] (92) QDP (93) DependencyGraphProof [EQUIVALENT, 0 ms] (94) QDP (95) TransformationProof [EQUIVALENT, 64 ms] (96) QDP (97) DependencyGraphProof [EQUIVALENT, 0 ms] (98) QDP (99) TransformationProof [EQUIVALENT, 52 ms] (100) QDP (101) DependencyGraphProof [EQUIVALENT, 0 ms] (102) QDP (103) TransformationProof [EQUIVALENT, 70 ms] (104) QDP (105) DependencyGraphProof [EQUIVALENT, 0 ms] (106) QDP (107) TransformationProof [EQUIVALENT, 0 ms] (108) QDP (109) DependencyGraphProof [EQUIVALENT, 0 ms] (110) QDP (111) TransformationProof [EQUIVALENT, 47 ms] (112) QDP (113) DependencyGraphProof [EQUIVALENT, 0 ms] (114) QDP (115) TransformationProof [EQUIVALENT, 59 ms] (116) QDP (117) DependencyGraphProof [EQUIVALENT, 0 ms] (118) QDP (119) TransformationProof [EQUIVALENT, 21 ms] (120) QDP (121) DependencyGraphProof [EQUIVALENT, 0 ms] (122) QDP (123) TransformationProof [EQUIVALENT, 12 ms] (124) QDP (125) DependencyGraphProof [EQUIVALENT, 0 ms] (126) QDP (127) TransformationProof [EQUIVALENT, 38 ms] (128) QDP (129) DependencyGraphProof [EQUIVALENT, 0 ms] (130) QDP (131) TransformationProof [EQUIVALENT, 63 ms] (132) QDP (133) DependencyGraphProof [EQUIVALENT, 0 ms] (134) QDP (135) TransformationProof [EQUIVALENT, 55 ms] (136) QDP (137) DependencyGraphProof [EQUIVALENT, 0 ms] (138) QDP (139) TransformationProof [EQUIVALENT, 40 ms] (140) QDP (141) DependencyGraphProof [EQUIVALENT, 0 ms] (142) QDP (143) QDPOrderProof [EQUIVALENT, 120 ms] (144) QDP (145) DependencyGraphProof [EQUIVALENT, 0 ms] (146) QDP (147) QDPOrderProof [EQUIVALENT, 2720 ms] (148) QDP (149) QDPOrderProof [EQUIVALENT, 836 ms] (150) QDP (151) PisEmptyProof [EQUIVALENT, 0 ms] (152) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0(0(1(0(x1)))) -> 0(1(2(0(2(0(x1)))))) 0(1(0(0(x1)))) -> 1(2(0(0(0(x1))))) 0(1(0(1(x1)))) -> 1(1(0(3(0(2(x1)))))) 0(4(1(0(x1)))) -> 0(3(1(4(0(x1))))) 0(4(1(0(x1)))) -> 1(4(0(0(5(x1))))) 0(4(1(0(x1)))) -> 1(5(0(4(0(x1))))) 0(4(1(0(x1)))) -> 0(5(1(4(5(0(x1)))))) 0(4(1(0(x1)))) -> 0(5(1(5(4(0(x1)))))) 0(4(1(0(x1)))) -> 1(5(0(4(0(5(x1)))))) 4(2(1(0(x1)))) -> 1(2(5(0(4(x1))))) 4(2(1(0(x1)))) -> 1(4(2(5(0(x1))))) 4(2(1(0(x1)))) -> 1(4(5(2(0(x1))))) 4(2(1(0(x1)))) -> 2(0(3(1(4(x1))))) 4(2(1(0(x1)))) -> 2(0(5(1(4(x1))))) 4(2(1(0(x1)))) -> 2(1(2(0(4(x1))))) 4(2(1(0(x1)))) -> 2(1(4(0(5(x1))))) 4(2(1(0(x1)))) -> 3(0(2(1(4(x1))))) 4(2(1(0(x1)))) -> 4(1(2(2(0(x1))))) 4(2(1(0(x1)))) -> 4(3(2(1(0(x1))))) 4(2(1(0(x1)))) -> 4(5(1(2(0(x1))))) 4(2(1(0(x1)))) -> 5(0(2(1(4(x1))))) 4(2(1(0(x1)))) -> 5(4(1(2(0(x1))))) 4(2(1(0(x1)))) -> 2(1(5(2(0(4(x1)))))) 4(2(1(0(x1)))) -> 4(1(2(5(2(0(x1)))))) 4(2(1(0(x1)))) -> 4(3(0(2(1(4(x1)))))) 4(3(0(0(x1)))) -> 3(0(2(0(4(x1))))) 4(3(0(0(x1)))) -> 3(2(0(4(0(x1))))) 4(4(1(0(x1)))) -> 4(0(5(1(4(x1))))) 4(4(1(0(x1)))) -> 4(1(2(0(4(x1))))) 4(4(1(0(x1)))) -> 4(1(4(0(5(x1))))) 4(4(1(0(x1)))) -> 4(1(5(0(4(x1))))) 4(4(1(0(x1)))) -> 4(1(5(4(0(x1))))) 0(0(2(1(0(x1))))) -> 0(1(2(0(2(0(x1)))))) 0(4(2(1(0(x1))))) -> 0(4(1(2(1(0(x1)))))) 0(4(2(1(0(x1))))) -> 1(2(0(3(4(0(x1)))))) 0(4(2(1(0(x1))))) -> 4(0(3(1(0(2(x1)))))) 0(4(4(1(0(x1))))) -> 1(4(4(0(5(0(x1)))))) 0(4(4(1(0(x1))))) -> 4(0(1(2(0(4(x1)))))) 0(4(4(1(0(x1))))) -> 5(0(4(1(4(0(x1)))))) 1(0(1(1(4(x1))))) -> 1(1(1(4(0(5(x1)))))) 1(0(4(1(0(x1))))) -> 4(0(5(1(1(0(x1)))))) 1(1(3(0(0(x1))))) -> 1(3(1(0(2(0(x1)))))) 1(3(0(0(1(x1))))) -> 0(1(5(1(0(3(x1)))))) 1(4(1(0(0(x1))))) -> 5(1(1(4(0(0(x1)))))) 1(4(2(1(0(x1))))) -> 4(1(2(1(2(0(x1)))))) 1(4(2(1(0(x1))))) -> 4(1(2(1(5(0(x1)))))) 1(4(3(0(0(x1))))) -> 3(2(1(0(4(0(x1)))))) 4(0(2(1(0(x1))))) -> 2(0(2(1(4(0(x1)))))) 4(1(0(1(0(x1))))) -> 1(0(4(3(1(0(x1)))))) 4(1(3(0(0(x1))))) -> 3(1(0(4(5(0(x1)))))) 4(1(3(0(0(x1))))) -> 4(3(1(0(2(0(x1)))))) 4(2(0(1(0(x1))))) -> 1(4(3(0(2(0(x1)))))) 4(2(0(1(0(x1))))) -> 1(5(0(2(0(4(x1)))))) 4(2(0(1(0(x1))))) -> 3(0(2(1(4(0(x1)))))) 4(2(1(0(0(x1))))) -> 2(1(4(2(0(0(x1)))))) 4(2(1(0(1(x1))))) -> 1(2(0(4(1(5(x1)))))) 4(2(1(0(4(x1))))) -> 4(1(2(2(0(4(x1)))))) 4(2(1(0(4(x1))))) -> 4(1(5(2(4(0(x1)))))) 4(2(1(1(0(x1))))) -> 4(1(2(1(5(0(x1)))))) 4(2(2(1(0(x1))))) -> 1(2(5(0(2(4(x1)))))) 4(2(2(1(0(x1))))) -> 1(4(2(2(0(5(x1)))))) 4(2(2(1(0(x1))))) -> 2(1(3(2(4(0(x1)))))) 4(2(2(1(0(x1))))) -> 2(1(4(1(2(0(x1)))))) 4(2(3(0(0(x1))))) -> 3(2(5(4(0(0(x1)))))) 4(2(4(1(0(x1))))) -> 1(4(4(2(0(5(x1)))))) 4(2(4(1(0(x1))))) -> 4(0(1(2(4(4(x1)))))) 4(2(4(1(0(x1))))) -> 4(1(2(0(4(5(x1)))))) 4(2(4(1(0(x1))))) -> 4(1(3(4(0(2(x1)))))) 4(2(4(1(0(x1))))) -> 4(1(4(2(0(5(x1)))))) 4(2(4(1(0(x1))))) -> 4(1(4(5(2(0(x1)))))) 4(3(1(2(1(x1))))) -> 3(2(2(1(4(1(x1)))))) 4(3(5(0(0(x1))))) -> 3(0(5(4(0(0(x1)))))) 4(3(5(0(0(x1))))) -> 5(0(3(1(4(0(x1)))))) 4(4(0(1(0(x1))))) -> 4(0(3(1(4(0(x1)))))) 4(4(1(0(1(x1))))) -> 4(0(3(1(4(1(x1)))))) 4(4(2(1(0(x1))))) -> 1(0(3(4(2(4(x1)))))) 4(4(2(1(0(x1))))) -> 4(0(3(1(4(2(x1)))))) 4(4(3(0(0(x1))))) -> 2(0(3(0(4(4(x1)))))) 4(4(3(0(0(x1))))) -> 4(3(0(4(5(0(x1)))))) 4(4(4(1(0(x1))))) -> 4(4(3(1(0(4(x1)))))) 4(5(2(1(0(x1))))) -> 3(2(1(4(5(0(x1)))))) 4(5(4(1(0(x1))))) -> 4(1(4(0(5(0(x1)))))) Q is empty. ---------------------------------------- (1) QTRS Reverse (EQUIVALENT) We applied the QTRS Reverse Processor [REVERSE]. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (4) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 0^1(2(0(2(1(0(x1)))))) 0^1(1(0(0(x1)))) -> 0^1(2(1(0(x1)))) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 0^1(0(1(0(x1)))) -> 0^1(0(0(2(1(x1))))) 0^1(0(1(0(x1)))) -> 0^1(0(2(1(x1)))) 0^1(0(1(0(x1)))) -> 0^1(2(1(x1))) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(0(x1)))) -> 0^1(3(0(1(1(x1))))) 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 1^1(0(1(0(x1)))) -> 1^1(1(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 0^1(4(1(3(0(x1))))) 0^1(1(4(0(x1)))) -> 4^1(1(3(0(x1)))) 0^1(1(4(0(x1)))) -> 1^1(3(0(x1))) 0^1(1(4(0(x1)))) -> 0^1(0(4(1(x1)))) 0^1(1(4(0(x1)))) -> 0^1(4(1(x1))) 0^1(1(4(0(x1)))) -> 4^1(1(x1)) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 0^1(4(0(5(1(x1))))) 0^1(1(4(0(x1)))) -> 4^1(0(5(1(x1)))) 0^1(1(4(0(x1)))) -> 0^1(5(1(x1))) 0^1(1(4(0(x1)))) -> 0^1(5(4(1(5(0(x1)))))) 0^1(1(4(0(x1)))) -> 4^1(1(5(0(x1)))) 0^1(1(4(0(x1)))) -> 1^1(5(0(x1))) 0^1(1(4(0(x1)))) -> 0^1(4(5(1(5(0(x1)))))) 0^1(1(4(0(x1)))) -> 4^1(5(1(5(0(x1))))) 0^1(1(2(4(x1)))) -> 4^1(0(5(2(1(x1))))) 0^1(1(2(4(x1)))) -> 0^1(5(2(1(x1)))) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 0^1(5(2(4(1(x1))))) 0^1(1(2(4(x1)))) -> 4^1(1(x1)) 0^1(1(2(4(x1)))) -> 0^1(2(5(4(1(x1))))) 0^1(1(2(4(x1)))) -> 4^1(1(3(0(2(x1))))) 0^1(1(2(4(x1)))) -> 1^1(3(0(2(x1)))) 0^1(1(2(4(x1)))) -> 0^1(2(x1)) 0^1(1(2(4(x1)))) -> 4^1(1(5(0(2(x1))))) 0^1(1(2(4(x1)))) -> 1^1(5(0(2(x1)))) 0^1(1(2(4(x1)))) -> 4^1(0(2(1(2(x1))))) 0^1(1(2(4(x1)))) -> 0^1(2(1(2(x1)))) 0^1(1(2(4(x1)))) -> 1^1(2(x1)) 0^1(1(2(4(x1)))) -> 0^1(4(1(2(x1)))) 0^1(1(2(4(x1)))) -> 4^1(1(2(x1))) 0^1(1(2(4(x1)))) -> 4^1(1(2(0(3(x1))))) 0^1(1(2(4(x1)))) -> 1^1(2(0(3(x1)))) 0^1(1(2(4(x1)))) -> 0^1(3(x1)) 0^1(1(2(4(x1)))) -> 0^1(2(2(1(4(x1))))) 0^1(1(2(4(x1)))) -> 1^1(4(x1)) 0^1(1(2(4(x1)))) -> 0^1(1(2(3(4(x1))))) 0^1(1(2(4(x1)))) -> 1^1(2(3(4(x1)))) 0^1(1(2(4(x1)))) -> 0^1(2(1(5(4(x1))))) 0^1(1(2(4(x1)))) -> 1^1(5(4(x1))) 0^1(1(2(4(x1)))) -> 4^1(1(2(0(5(x1))))) 0^1(1(2(4(x1)))) -> 1^1(2(0(5(x1)))) 0^1(1(2(4(x1)))) -> 0^1(5(x1)) 0^1(1(2(4(x1)))) -> 0^1(2(1(4(5(x1))))) 0^1(1(2(4(x1)))) -> 1^1(4(5(x1))) 0^1(1(2(4(x1)))) -> 4^1(5(x1)) 0^1(1(2(4(x1)))) -> 4^1(0(2(5(1(2(x1)))))) 0^1(1(2(4(x1)))) -> 0^1(2(5(1(2(x1))))) 0^1(1(2(4(x1)))) -> 0^1(2(5(2(1(4(x1)))))) 0^1(1(2(4(x1)))) -> 4^1(1(2(0(3(4(x1)))))) 0^1(1(2(4(x1)))) -> 1^1(2(0(3(4(x1))))) 0^1(1(2(4(x1)))) -> 0^1(3(4(x1))) 0^1(0(3(4(x1)))) -> 4^1(0(2(0(3(x1))))) 0^1(0(3(4(x1)))) -> 0^1(2(0(3(x1)))) 0^1(0(3(4(x1)))) -> 0^1(3(x1)) 0^1(0(3(4(x1)))) -> 0^1(4(0(2(3(x1))))) 0^1(0(3(4(x1)))) -> 4^1(0(2(3(x1)))) 0^1(0(3(4(x1)))) -> 0^1(2(3(x1))) 0^1(1(4(4(x1)))) -> 4^1(1(5(0(4(x1))))) 0^1(1(4(4(x1)))) -> 1^1(5(0(4(x1)))) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 4^1(0(2(1(4(x1))))) 0^1(1(4(4(x1)))) -> 0^1(2(1(4(x1)))) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 0^1(1(4(4(x1)))) -> 4^1(0(5(1(4(x1))))) 0^1(1(4(4(x1)))) -> 0^1(5(1(4(x1)))) 0^1(1(4(4(x1)))) -> 0^1(4(5(1(4(x1))))) 0^1(1(4(4(x1)))) -> 4^1(5(1(4(x1)))) 0^1(1(2(0(0(x1))))) -> 0^1(2(0(2(1(0(x1)))))) 0^1(1(2(0(0(x1))))) -> 0^1(2(1(0(x1)))) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(2(1(4(0(x1))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 0^1(4(3(0(2(1(x1)))))) 0^1(1(2(4(0(x1))))) -> 4^1(3(0(2(1(x1))))) 0^1(1(2(4(0(x1))))) -> 0^1(2(1(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(1(3(0(4(x1))))) 0^1(1(2(4(0(x1))))) -> 1^1(3(0(4(x1)))) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(5(0(4(4(1(x1)))))) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 4^1(0(2(1(0(4(x1)))))) 0^1(1(4(4(0(x1))))) -> 0^1(2(1(0(4(x1))))) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(1(4(0(5(x1)))))) 0^1(1(4(4(0(x1))))) -> 4^1(1(4(0(5(x1))))) 0^1(1(4(4(0(x1))))) -> 1^1(4(0(5(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(0(5(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(5(x1)) 4^1(1(1(0(1(x1))))) -> 0^1(4(1(1(1(x1))))) 4^1(1(1(0(1(x1))))) -> 4^1(1(1(1(x1)))) 4^1(1(1(0(1(x1))))) -> 1^1(1(1(x1))) 4^1(1(1(0(1(x1))))) -> 1^1(1(x1)) 0^1(1(4(0(1(x1))))) -> 0^1(1(1(5(0(4(x1)))))) 0^1(1(4(0(1(x1))))) -> 1^1(1(5(0(4(x1))))) 0^1(1(4(0(1(x1))))) -> 1^1(5(0(4(x1)))) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(0(3(1(1(x1))))) -> 0^1(2(0(1(3(1(x1)))))) 0^1(0(3(1(1(x1))))) -> 0^1(1(3(1(x1)))) 0^1(0(3(1(1(x1))))) -> 1^1(3(1(x1))) 1^1(0(0(3(1(x1))))) -> 0^1(1(5(1(0(x1))))) 1^1(0(0(3(1(x1))))) -> 1^1(5(1(0(x1)))) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(0(1(4(1(x1))))) -> 0^1(0(4(1(1(5(x1)))))) 0^1(0(1(4(1(x1))))) -> 0^1(4(1(1(5(x1))))) 0^1(0(1(4(1(x1))))) -> 4^1(1(1(5(x1)))) 0^1(0(1(4(1(x1))))) -> 1^1(1(5(x1))) 0^1(0(1(4(1(x1))))) -> 1^1(5(x1)) 0^1(1(2(4(1(x1))))) -> 0^1(2(1(2(1(4(x1)))))) 0^1(1(2(4(1(x1))))) -> 1^1(2(1(4(x1)))) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 0^1(5(1(2(1(4(x1)))))) 0^1(0(3(4(1(x1))))) -> 0^1(4(0(1(2(3(x1)))))) 0^1(0(3(4(1(x1))))) -> 4^1(0(1(2(3(x1))))) 0^1(0(3(4(1(x1))))) -> 0^1(1(2(3(x1)))) 0^1(0(3(4(1(x1))))) -> 1^1(2(3(x1))) 0^1(1(2(0(4(x1))))) -> 0^1(4(1(2(0(2(x1)))))) 0^1(1(2(0(4(x1))))) -> 4^1(1(2(0(2(x1))))) 0^1(1(2(0(4(x1))))) -> 1^1(2(0(2(x1)))) 0^1(1(2(0(4(x1))))) -> 0^1(2(x1)) 0^1(1(0(1(4(x1))))) -> 0^1(1(3(4(0(1(x1)))))) 0^1(1(0(1(4(x1))))) -> 1^1(3(4(0(1(x1))))) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(0(3(1(4(x1))))) -> 0^1(5(4(0(1(3(x1)))))) 0^1(0(3(1(4(x1))))) -> 4^1(0(1(3(x1)))) 0^1(0(3(1(4(x1))))) -> 0^1(1(3(x1))) 0^1(0(3(1(4(x1))))) -> 1^1(3(x1)) 0^1(0(3(1(4(x1))))) -> 0^1(2(0(1(3(4(x1)))))) 0^1(0(3(1(4(x1))))) -> 0^1(1(3(4(x1)))) 0^1(0(3(1(4(x1))))) -> 1^1(3(4(x1))) 0^1(1(0(2(4(x1))))) -> 0^1(2(0(3(4(1(x1)))))) 0^1(1(0(2(4(x1))))) -> 0^1(3(4(1(x1)))) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(0(2(0(5(1(x1)))))) 0^1(1(0(2(4(x1))))) -> 0^1(2(0(5(1(x1))))) 0^1(1(0(2(4(x1))))) -> 0^1(5(1(x1))) 0^1(1(0(2(4(x1))))) -> 0^1(4(1(2(0(3(x1)))))) 0^1(1(0(2(4(x1))))) -> 4^1(1(2(0(3(x1))))) 0^1(1(0(2(4(x1))))) -> 1^1(2(0(3(x1)))) 0^1(1(0(2(4(x1))))) -> 0^1(3(x1)) 0^1(0(1(2(4(x1))))) -> 0^1(0(2(4(1(2(x1)))))) 0^1(0(1(2(4(x1))))) -> 0^1(2(4(1(2(x1))))) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(0(1(2(4(x1))))) -> 1^1(2(x1)) 1^1(0(1(2(4(x1))))) -> 1^1(4(0(2(1(x1))))) 1^1(0(1(2(4(x1))))) -> 4^1(0(2(1(x1)))) 1^1(0(1(2(4(x1))))) -> 0^1(2(1(x1))) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 4^1(0(1(2(4(x1))))) -> 4^1(0(2(2(1(4(x1)))))) 4^1(0(1(2(4(x1))))) -> 0^1(2(2(1(4(x1))))) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 4^1(0(1(2(4(x1))))) -> 0^1(4(2(5(1(4(x1)))))) 4^1(0(1(2(4(x1))))) -> 4^1(2(5(1(4(x1))))) 0^1(1(1(2(4(x1))))) -> 0^1(5(1(2(1(4(x1)))))) 0^1(1(1(2(4(x1))))) -> 1^1(2(1(4(x1)))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 4^1(2(0(5(2(1(x1)))))) 0^1(1(2(2(4(x1))))) -> 0^1(5(2(1(x1)))) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 0^1(2(2(4(1(x1))))) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 0^1(4(2(3(1(2(x1)))))) 0^1(1(2(2(4(x1))))) -> 4^1(2(3(1(2(x1))))) 0^1(1(2(2(4(x1))))) -> 1^1(2(x1)) 0^1(1(2(2(4(x1))))) -> 0^1(2(1(4(1(2(x1)))))) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(0(3(2(4(x1))))) -> 0^1(0(4(5(2(3(x1)))))) 0^1(0(3(2(4(x1))))) -> 0^1(4(5(2(3(x1))))) 0^1(0(3(2(4(x1))))) -> 4^1(5(2(3(x1)))) 0^1(1(4(2(4(x1))))) -> 0^1(2(4(4(1(x1))))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 4^1(4(2(1(0(4(x1)))))) 0^1(1(4(2(4(x1))))) -> 4^1(2(1(0(4(x1))))) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(0(2(1(4(x1))))) 0^1(1(4(2(4(x1))))) -> 0^1(2(1(4(x1)))) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 0^1(4(3(1(4(x1))))) 0^1(1(4(2(4(x1))))) -> 4^1(3(1(4(x1)))) 0^1(1(4(2(4(x1))))) -> 0^1(2(4(1(4(x1))))) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(2(5(4(1(4(x1)))))) 1^1(2(1(3(4(x1))))) -> 1^1(4(1(2(2(3(x1)))))) 1^1(2(1(3(4(x1))))) -> 4^1(1(2(2(3(x1))))) 1^1(2(1(3(4(x1))))) -> 1^1(2(2(3(x1)))) 0^1(0(5(3(4(x1))))) -> 0^1(0(4(5(0(3(x1)))))) 0^1(0(5(3(4(x1))))) -> 0^1(4(5(0(3(x1))))) 0^1(0(5(3(4(x1))))) -> 4^1(5(0(3(x1)))) 0^1(0(5(3(4(x1))))) -> 0^1(3(x1)) 0^1(0(5(3(4(x1))))) -> 0^1(4(1(3(0(5(x1)))))) 0^1(0(5(3(4(x1))))) -> 4^1(1(3(0(5(x1))))) 0^1(0(5(3(4(x1))))) -> 1^1(3(0(5(x1)))) 0^1(0(5(3(4(x1))))) -> 0^1(5(x1)) 0^1(1(0(4(4(x1))))) -> 0^1(4(1(3(0(4(x1)))))) 0^1(1(0(4(4(x1))))) -> 4^1(1(3(0(4(x1))))) 0^1(1(0(4(4(x1))))) -> 1^1(3(0(4(x1)))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 1^1(0(1(4(4(x1))))) -> 1^1(4(1(3(0(4(x1)))))) 1^1(0(1(4(4(x1))))) -> 4^1(1(3(0(4(x1))))) 1^1(0(1(4(4(x1))))) -> 1^1(3(0(4(x1)))) 1^1(0(1(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 4^1(2(4(3(0(1(x1)))))) 0^1(1(2(4(4(x1))))) -> 4^1(3(0(1(x1)))) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 4^1(1(3(0(4(x1))))) 0^1(1(2(4(4(x1))))) -> 1^1(3(0(4(x1)))) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(0(3(4(4(x1))))) -> 4^1(4(0(3(0(2(x1)))))) 0^1(0(3(4(4(x1))))) -> 4^1(0(3(0(2(x1))))) 0^1(0(3(4(4(x1))))) -> 0^1(3(0(2(x1)))) 0^1(0(3(4(4(x1))))) -> 0^1(2(x1)) 0^1(0(3(4(4(x1))))) -> 0^1(5(4(0(3(4(x1)))))) 0^1(0(3(4(4(x1))))) -> 4^1(0(3(4(x1)))) 0^1(0(3(4(4(x1))))) -> 0^1(3(4(x1))) 0^1(1(4(4(4(x1))))) -> 4^1(0(1(3(4(4(x1)))))) 0^1(1(4(4(4(x1))))) -> 0^1(1(3(4(4(x1))))) 0^1(1(4(4(4(x1))))) -> 1^1(3(4(4(x1)))) 0^1(1(2(5(4(x1))))) -> 0^1(5(4(1(2(3(x1)))))) 0^1(1(2(5(4(x1))))) -> 4^1(1(2(3(x1)))) 0^1(1(2(5(4(x1))))) -> 1^1(2(3(x1))) 0^1(1(4(5(4(x1))))) -> 0^1(5(0(4(1(4(x1)))))) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 187 less nodes. ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(0(x1)))) -> 1^1(1(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(4(0(x1)))) -> 0^1(0(4(1(x1)))) 0^1(1(4(0(x1)))) -> 0^1(4(1(x1))) 0^1(1(4(0(x1)))) -> 4^1(1(x1)) 4^1(1(1(0(1(x1))))) -> 0^1(4(1(1(1(x1))))) 0^1(1(4(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 1^1(0(1(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 4^1(1(x1)) 4^1(1(1(0(1(x1))))) -> 4^1(1(1(1(x1)))) 4^1(1(1(0(1(x1))))) -> 1^1(1(1(x1))) 4^1(1(1(0(1(x1))))) -> 1^1(1(x1)) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(x1)))) -> 0^1(4(1(2(x1)))) 0^1(1(2(4(x1)))) -> 4^1(1(2(x1))) 0^1(1(2(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (7) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 1^1(0(1(0(x1)))) -> 1^1(1(x1)) at position [0] we obtained the following new rules [LPAR04]: (1^1(0(1(0(0(1(0(x0))))))) -> 1^1(2(0(3(0(1(1(x0))))))),1^1(0(1(0(0(1(0(x0))))))) -> 1^1(2(0(3(0(1(1(x0)))))))) (1^1(0(1(0(0(0(3(1(x0)))))))) -> 1^1(3(0(1(5(1(0(x0))))))),1^1(0(1(0(0(0(3(1(x0)))))))) -> 1^1(3(0(1(5(1(0(x0)))))))) (1^1(0(1(0(0(1(2(4(x0)))))))) -> 1^1(5(1(4(0(2(1(x0))))))),1^1(0(1(0(0(1(2(4(x0)))))))) -> 1^1(5(1(4(0(2(1(x0)))))))) (1^1(0(1(0(2(1(3(4(x0)))))))) -> 1^1(1(4(1(2(2(3(x0))))))),1^1(0(1(0(2(1(3(4(x0)))))))) -> 1^1(1(4(1(2(2(3(x0)))))))) (1^1(0(1(0(0(1(4(4(x0)))))))) -> 1^1(1(4(1(3(0(4(x0))))))),1^1(0(1(0(0(1(4(4(x0)))))))) -> 1^1(1(4(1(3(0(4(x0)))))))) ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(4(0(x1)))) -> 0^1(0(4(1(x1)))) 0^1(1(4(0(x1)))) -> 0^1(4(1(x1))) 0^1(1(4(0(x1)))) -> 4^1(1(x1)) 4^1(1(1(0(1(x1))))) -> 0^1(4(1(1(1(x1))))) 0^1(1(4(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 1^1(0(1(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 4^1(1(x1)) 4^1(1(1(0(1(x1))))) -> 4^1(1(1(1(x1)))) 4^1(1(1(0(1(x1))))) -> 1^1(1(1(x1))) 4^1(1(1(0(1(x1))))) -> 1^1(1(x1)) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(x1)))) -> 0^1(4(1(2(x1)))) 0^1(1(2(4(x1)))) -> 4^1(1(2(x1))) 0^1(1(2(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 1^1(0(1(0(0(1(0(x0))))))) -> 1^1(2(0(3(0(1(1(x0))))))) 1^1(0(1(0(0(0(3(1(x0)))))))) -> 1^1(3(0(1(5(1(0(x0))))))) 1^1(0(1(0(0(1(2(4(x0)))))))) -> 1^1(5(1(4(0(2(1(x0))))))) 1^1(0(1(0(2(1(3(4(x0)))))))) -> 1^1(1(4(1(2(2(3(x0))))))) 1^1(0(1(0(0(1(4(4(x0)))))))) -> 1^1(1(4(1(3(0(4(x0))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes. ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 1^1(0(1(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(x1)))) -> 0^1(0(4(1(x1)))) 0^1(1(4(0(x1)))) -> 0^1(4(1(x1))) 0^1(1(4(0(x1)))) -> 4^1(1(x1)) 4^1(1(1(0(1(x1))))) -> 0^1(4(1(1(1(x1))))) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 4^1(1(x1)) 4^1(1(1(0(1(x1))))) -> 4^1(1(1(1(x1)))) 4^1(1(1(0(1(x1))))) -> 1^1(1(1(x1))) 4^1(1(1(0(1(x1))))) -> 1^1(1(x1)) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(x1)))) -> 0^1(4(1(2(x1)))) 0^1(1(2(4(x1)))) -> 4^1(1(2(x1))) 0^1(1(2(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 1^1(0(1(4(4(x1))))) -> 0^1(4(x1)) at position [0] we obtained the following new rules [LPAR04]: (1^1(0(1(4(4(1(1(0(1(x0))))))))) -> 0^1(5(0(4(1(1(1(x0))))))),1^1(0(1(4(4(1(1(0(1(x0))))))))) -> 0^1(5(0(4(1(1(1(x0)))))))) (1^1(0(1(4(4(0(1(2(4(x0))))))))) -> 0^1(4(0(2(2(1(4(x0))))))),1^1(0(1(4(4(0(1(2(4(x0))))))))) -> 0^1(4(0(2(2(1(4(x0)))))))) (1^1(0(1(4(4(0(1(2(4(x0))))))))) -> 0^1(0(4(2(5(1(4(x0))))))),1^1(0(1(4(4(0(1(2(4(x0))))))))) -> 0^1(0(4(2(5(1(4(x0)))))))) ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 0^1(0(4(1(x1)))) 0^1(1(4(0(x1)))) -> 0^1(4(1(x1))) 0^1(1(4(0(x1)))) -> 4^1(1(x1)) 4^1(1(1(0(1(x1))))) -> 0^1(4(1(1(1(x1))))) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 4^1(1(x1)) 4^1(1(1(0(1(x1))))) -> 4^1(1(1(1(x1)))) 4^1(1(1(0(1(x1))))) -> 1^1(1(1(x1))) 4^1(1(1(0(1(x1))))) -> 1^1(1(x1)) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(x1)))) -> 0^1(4(1(2(x1)))) 0^1(1(2(4(x1)))) -> 4^1(1(2(x1))) 0^1(1(2(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 1^1(0(1(4(4(1(1(0(1(x0))))))))) -> 0^1(5(0(4(1(1(1(x0))))))) 1^1(0(1(4(4(0(1(2(4(x0))))))))) -> 0^1(4(0(2(2(1(4(x0))))))) 1^1(0(1(4(4(0(1(2(4(x0))))))))) -> 0^1(0(4(2(5(1(4(x0))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(4(0(x1)))) -> 0^1(0(4(1(x1)))) 0^1(1(4(0(x1)))) -> 0^1(4(1(x1))) 0^1(1(4(0(x1)))) -> 4^1(1(x1)) 4^1(1(1(0(1(x1))))) -> 0^1(4(1(1(1(x1))))) 0^1(1(4(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 4^1(1(x1)) 4^1(1(1(0(1(x1))))) -> 4^1(1(1(1(x1)))) 4^1(1(1(0(1(x1))))) -> 1^1(1(1(x1))) 4^1(1(1(0(1(x1))))) -> 1^1(1(x1)) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(x1)))) -> 0^1(4(1(2(x1)))) 0^1(1(2(4(x1)))) -> 4^1(1(2(x1))) 0^1(1(2(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (15) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 0^1(1(4(0(x1)))) -> 0^1(0(4(1(x1)))) at position [0] we obtained the following new rules [LPAR04]: (0^1(1(4(0(1(0(1(x0))))))) -> 0^1(0(5(0(4(1(1(1(x0)))))))),0^1(1(4(0(1(0(1(x0))))))) -> 0^1(0(5(0(4(1(1(1(x0))))))))) (0^1(1(4(0(0(1(0(x0))))))) -> 0^1(0(4(2(0(3(0(1(1(x0))))))))),0^1(1(4(0(0(1(0(x0))))))) -> 0^1(0(4(2(0(3(0(1(1(x0)))))))))) (0^1(1(4(0(0(0(3(1(x0)))))))) -> 0^1(0(4(3(0(1(5(1(0(x0))))))))),0^1(1(4(0(0(0(3(1(x0)))))))) -> 0^1(0(4(3(0(1(5(1(0(x0)))))))))) (0^1(1(4(0(0(1(2(4(x0)))))))) -> 0^1(0(4(5(1(4(0(2(1(x0))))))))),0^1(1(4(0(0(1(2(4(x0)))))))) -> 0^1(0(4(5(1(4(0(2(1(x0)))))))))) (0^1(1(4(0(2(1(3(4(x0)))))))) -> 0^1(0(4(1(4(1(2(2(3(x0))))))))),0^1(1(4(0(2(1(3(4(x0)))))))) -> 0^1(0(4(1(4(1(2(2(3(x0)))))))))) (0^1(1(4(0(0(1(4(4(x0)))))))) -> 0^1(0(4(1(4(1(3(0(4(x0))))))))),0^1(1(4(0(0(1(4(4(x0)))))))) -> 0^1(0(4(1(4(1(3(0(4(x0)))))))))) ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(4(0(x1)))) -> 0^1(4(1(x1))) 0^1(1(4(0(x1)))) -> 4^1(1(x1)) 4^1(1(1(0(1(x1))))) -> 0^1(4(1(1(1(x1))))) 0^1(1(4(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 4^1(1(x1)) 4^1(1(1(0(1(x1))))) -> 4^1(1(1(1(x1)))) 4^1(1(1(0(1(x1))))) -> 1^1(1(1(x1))) 4^1(1(1(0(1(x1))))) -> 1^1(1(x1)) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(x1)))) -> 0^1(4(1(2(x1)))) 0^1(1(2(4(x1)))) -> 4^1(1(2(x1))) 0^1(1(2(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(0(1(0(1(x0))))))) -> 0^1(0(5(0(4(1(1(1(x0)))))))) 0^1(1(4(0(0(1(0(x0))))))) -> 0^1(0(4(2(0(3(0(1(1(x0))))))))) 0^1(1(4(0(0(0(3(1(x0)))))))) -> 0^1(0(4(3(0(1(5(1(0(x0))))))))) 0^1(1(4(0(0(1(2(4(x0)))))))) -> 0^1(0(4(5(1(4(0(2(1(x0))))))))) 0^1(1(4(0(2(1(3(4(x0)))))))) -> 0^1(0(4(1(4(1(2(2(3(x0))))))))) 0^1(1(4(0(0(1(4(4(x0)))))))) -> 0^1(0(4(1(4(1(3(0(4(x0))))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (17) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 0^1(4(1(x1))) 0^1(1(4(0(x1)))) -> 4^1(1(x1)) 4^1(1(1(0(1(x1))))) -> 0^1(4(1(1(1(x1))))) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 4^1(1(x1)) 4^1(1(1(0(1(x1))))) -> 4^1(1(1(1(x1)))) 4^1(1(1(0(1(x1))))) -> 1^1(1(1(x1))) 4^1(1(1(0(1(x1))))) -> 1^1(1(x1)) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(x1)))) -> 0^1(4(1(2(x1)))) 0^1(1(2(4(x1)))) -> 4^1(1(2(x1))) 0^1(1(2(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 0^1(1(4(0(x1)))) -> 0^1(4(1(x1))) at position [0] we obtained the following new rules [LPAR04]: (0^1(1(4(0(1(0(1(x0))))))) -> 0^1(5(0(4(1(1(1(x0))))))),0^1(1(4(0(1(0(1(x0))))))) -> 0^1(5(0(4(1(1(1(x0)))))))) (0^1(1(4(0(0(1(0(x0))))))) -> 0^1(4(2(0(3(0(1(1(x0)))))))),0^1(1(4(0(0(1(0(x0))))))) -> 0^1(4(2(0(3(0(1(1(x0))))))))) (0^1(1(4(0(0(0(3(1(x0)))))))) -> 0^1(4(3(0(1(5(1(0(x0)))))))),0^1(1(4(0(0(0(3(1(x0)))))))) -> 0^1(4(3(0(1(5(1(0(x0))))))))) (0^1(1(4(0(0(1(2(4(x0)))))))) -> 0^1(4(5(1(4(0(2(1(x0)))))))),0^1(1(4(0(0(1(2(4(x0)))))))) -> 0^1(4(5(1(4(0(2(1(x0))))))))) (0^1(1(4(0(2(1(3(4(x0)))))))) -> 0^1(4(1(4(1(2(2(3(x0)))))))),0^1(1(4(0(2(1(3(4(x0)))))))) -> 0^1(4(1(4(1(2(2(3(x0))))))))) (0^1(1(4(0(0(1(4(4(x0)))))))) -> 0^1(4(1(4(1(3(0(4(x0)))))))),0^1(1(4(0(0(1(4(4(x0)))))))) -> 0^1(4(1(4(1(3(0(4(x0))))))))) ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 4^1(1(x1)) 4^1(1(1(0(1(x1))))) -> 0^1(4(1(1(1(x1))))) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 4^1(1(x1)) 4^1(1(1(0(1(x1))))) -> 4^1(1(1(1(x1)))) 4^1(1(1(0(1(x1))))) -> 1^1(1(1(x1))) 4^1(1(1(0(1(x1))))) -> 1^1(1(x1)) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(x1)))) -> 0^1(4(1(2(x1)))) 0^1(1(2(4(x1)))) -> 4^1(1(2(x1))) 0^1(1(2(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(0(1(0(1(x0))))))) -> 0^1(5(0(4(1(1(1(x0))))))) 0^1(1(4(0(0(1(0(x0))))))) -> 0^1(4(2(0(3(0(1(1(x0)))))))) 0^1(1(4(0(0(0(3(1(x0)))))))) -> 0^1(4(3(0(1(5(1(0(x0)))))))) 0^1(1(4(0(0(1(2(4(x0)))))))) -> 0^1(4(5(1(4(0(2(1(x0)))))))) 0^1(1(4(0(2(1(3(4(x0)))))))) -> 0^1(4(1(4(1(2(2(3(x0)))))))) 0^1(1(4(0(0(1(4(4(x0)))))))) -> 0^1(4(1(4(1(3(0(4(x0)))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes. ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(4(0(x1)))) -> 4^1(1(x1)) 4^1(1(1(0(1(x1))))) -> 0^1(4(1(1(1(x1))))) 0^1(1(4(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 4^1(1(x1)) 4^1(1(1(0(1(x1))))) -> 4^1(1(1(1(x1)))) 4^1(1(1(0(1(x1))))) -> 1^1(1(1(x1))) 4^1(1(1(0(1(x1))))) -> 1^1(1(x1)) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(x1)))) -> 0^1(4(1(2(x1)))) 0^1(1(2(4(x1)))) -> 4^1(1(2(x1))) 0^1(1(2(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 4^1(1(1(0(1(x1))))) -> 0^1(4(1(1(1(x1))))) at position [0] we obtained the following new rules [LPAR04]: (4^1(1(1(0(1(0(1(0(x0)))))))) -> 0^1(4(1(1(2(0(3(0(1(1(x0)))))))))),4^1(1(1(0(1(0(1(0(x0)))))))) -> 0^1(4(1(1(2(0(3(0(1(1(x0))))))))))) (4^1(1(1(0(1(0(0(3(1(x0))))))))) -> 0^1(4(1(1(3(0(1(5(1(0(x0)))))))))),4^1(1(1(0(1(0(0(3(1(x0))))))))) -> 0^1(4(1(1(3(0(1(5(1(0(x0))))))))))) (4^1(1(1(0(1(0(1(2(4(x0))))))))) -> 0^1(4(1(1(5(1(4(0(2(1(x0)))))))))),4^1(1(1(0(1(0(1(2(4(x0))))))))) -> 0^1(4(1(1(5(1(4(0(2(1(x0))))))))))) (4^1(1(1(0(1(2(1(3(4(x0))))))))) -> 0^1(4(1(1(1(4(1(2(2(3(x0)))))))))),4^1(1(1(0(1(2(1(3(4(x0))))))))) -> 0^1(4(1(1(1(4(1(2(2(3(x0))))))))))) (4^1(1(1(0(1(0(1(4(4(x0))))))))) -> 0^1(4(1(1(1(4(1(3(0(4(x0)))))))))),4^1(1(1(0(1(0(1(4(4(x0))))))))) -> 0^1(4(1(1(1(4(1(3(0(4(x0))))))))))) ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(4(0(x1)))) -> 4^1(1(x1)) 0^1(1(4(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 4^1(1(x1)) 4^1(1(1(0(1(x1))))) -> 4^1(1(1(1(x1)))) 4^1(1(1(0(1(x1))))) -> 1^1(1(1(x1))) 4^1(1(1(0(1(x1))))) -> 1^1(1(x1)) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(x1)))) -> 0^1(4(1(2(x1)))) 0^1(1(2(4(x1)))) -> 4^1(1(2(x1))) 0^1(1(2(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 4^1(1(1(0(1(0(1(0(x0)))))))) -> 0^1(4(1(1(2(0(3(0(1(1(x0)))))))))) 4^1(1(1(0(1(0(0(3(1(x0))))))))) -> 0^1(4(1(1(3(0(1(5(1(0(x0)))))))))) 4^1(1(1(0(1(0(1(2(4(x0))))))))) -> 0^1(4(1(1(5(1(4(0(2(1(x0)))))))))) 4^1(1(1(0(1(2(1(3(4(x0))))))))) -> 0^1(4(1(1(1(4(1(2(2(3(x0)))))))))) 4^1(1(1(0(1(0(1(4(4(x0))))))))) -> 0^1(4(1(1(1(4(1(3(0(4(x0)))))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes. ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 4^1(1(x1)) 4^1(1(1(0(1(x1))))) -> 4^1(1(1(1(x1)))) 4^1(1(1(0(1(x1))))) -> 1^1(1(1(x1))) 4^1(1(1(0(1(x1))))) -> 1^1(1(x1)) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 4^1(1(x1)) 0^1(1(2(4(x1)))) -> 0^1(4(1(2(x1)))) 0^1(1(2(4(x1)))) -> 4^1(1(2(x1))) 0^1(1(2(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 4^1(1(1(0(1(x1))))) -> 4^1(1(1(1(x1)))) at position [0] we obtained the following new rules [LPAR04]: (4^1(1(1(0(1(0(1(0(x0)))))))) -> 4^1(1(1(2(0(3(0(1(1(x0))))))))),4^1(1(1(0(1(0(1(0(x0)))))))) -> 4^1(1(1(2(0(3(0(1(1(x0)))))))))) (4^1(1(1(0(1(0(0(3(1(x0))))))))) -> 4^1(1(1(3(0(1(5(1(0(x0))))))))),4^1(1(1(0(1(0(0(3(1(x0))))))))) -> 4^1(1(1(3(0(1(5(1(0(x0)))))))))) (4^1(1(1(0(1(0(1(2(4(x0))))))))) -> 4^1(1(1(5(1(4(0(2(1(x0))))))))),4^1(1(1(0(1(0(1(2(4(x0))))))))) -> 4^1(1(1(5(1(4(0(2(1(x0)))))))))) (4^1(1(1(0(1(2(1(3(4(x0))))))))) -> 4^1(1(1(1(4(1(2(2(3(x0))))))))),4^1(1(1(0(1(2(1(3(4(x0))))))))) -> 4^1(1(1(1(4(1(2(2(3(x0)))))))))) (4^1(1(1(0(1(0(1(4(4(x0))))))))) -> 4^1(1(1(1(4(1(3(0(4(x0))))))))),4^1(1(1(0(1(0(1(4(4(x0))))))))) -> 4^1(1(1(1(4(1(3(0(4(x0)))))))))) ---------------------------------------- (28) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 4^1(1(x1)) 4^1(1(1(0(1(x1))))) -> 1^1(1(1(x1))) 4^1(1(1(0(1(x1))))) -> 1^1(1(x1)) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 4^1(1(x1)) 0^1(1(2(4(x1)))) -> 0^1(4(1(2(x1)))) 0^1(1(2(4(x1)))) -> 4^1(1(2(x1))) 0^1(1(2(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 4^1(1(1(0(1(0(1(0(x0)))))))) -> 4^1(1(1(2(0(3(0(1(1(x0))))))))) 4^1(1(1(0(1(0(0(3(1(x0))))))))) -> 4^1(1(1(3(0(1(5(1(0(x0))))))))) 4^1(1(1(0(1(0(1(2(4(x0))))))))) -> 4^1(1(1(5(1(4(0(2(1(x0))))))))) 4^1(1(1(0(1(2(1(3(4(x0))))))))) -> 4^1(1(1(1(4(1(2(2(3(x0))))))))) 4^1(1(1(0(1(0(1(4(4(x0))))))))) -> 4^1(1(1(1(4(1(3(0(4(x0))))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes. ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(4(0(x1)))) -> 4^1(1(x1)) 4^1(1(1(0(1(x1))))) -> 1^1(1(1(x1))) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 4^1(1(1(0(1(x1))))) -> 1^1(1(x1)) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 4^1(1(x1)) 0^1(1(2(4(x1)))) -> 0^1(4(1(2(x1)))) 0^1(1(2(4(x1)))) -> 4^1(1(2(x1))) 0^1(1(2(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 4^1(1(1(0(1(x1))))) -> 1^1(1(1(x1))) at position [0] we obtained the following new rules [LPAR04]: (4^1(1(1(0(1(0(1(0(x0)))))))) -> 1^1(1(2(0(3(0(1(1(x0)))))))),4^1(1(1(0(1(0(1(0(x0)))))))) -> 1^1(1(2(0(3(0(1(1(x0))))))))) (4^1(1(1(0(1(0(0(3(1(x0))))))))) -> 1^1(1(3(0(1(5(1(0(x0)))))))),4^1(1(1(0(1(0(0(3(1(x0))))))))) -> 1^1(1(3(0(1(5(1(0(x0))))))))) (4^1(1(1(0(1(0(1(2(4(x0))))))))) -> 1^1(1(5(1(4(0(2(1(x0)))))))),4^1(1(1(0(1(0(1(2(4(x0))))))))) -> 1^1(1(5(1(4(0(2(1(x0))))))))) (4^1(1(1(0(1(2(1(3(4(x0))))))))) -> 1^1(1(1(4(1(2(2(3(x0)))))))),4^1(1(1(0(1(2(1(3(4(x0))))))))) -> 1^1(1(1(4(1(2(2(3(x0))))))))) (4^1(1(1(0(1(0(1(4(4(x0))))))))) -> 1^1(1(1(4(1(3(0(4(x0)))))))),4^1(1(1(0(1(0(1(4(4(x0))))))))) -> 1^1(1(1(4(1(3(0(4(x0))))))))) ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(4(0(x1)))) -> 4^1(1(x1)) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 4^1(1(1(0(1(x1))))) -> 1^1(1(x1)) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 4^1(1(x1)) 0^1(1(2(4(x1)))) -> 0^1(4(1(2(x1)))) 0^1(1(2(4(x1)))) -> 4^1(1(2(x1))) 0^1(1(2(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 4^1(1(1(0(1(0(1(0(x0)))))))) -> 1^1(1(2(0(3(0(1(1(x0)))))))) 4^1(1(1(0(1(0(0(3(1(x0))))))))) -> 1^1(1(3(0(1(5(1(0(x0)))))))) 4^1(1(1(0(1(0(1(2(4(x0))))))))) -> 1^1(1(5(1(4(0(2(1(x0)))))))) 4^1(1(1(0(1(2(1(3(4(x0))))))))) -> 1^1(1(1(4(1(2(2(3(x0)))))))) 4^1(1(1(0(1(0(1(4(4(x0))))))))) -> 1^1(1(1(4(1(3(0(4(x0)))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes. ---------------------------------------- (34) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 4^1(1(x1)) 4^1(1(1(0(1(x1))))) -> 1^1(1(x1)) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 4^1(1(x1)) 0^1(1(2(4(x1)))) -> 0^1(4(1(2(x1)))) 0^1(1(2(4(x1)))) -> 4^1(1(2(x1))) 0^1(1(2(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (35) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 4^1(1(1(0(1(x1))))) -> 1^1(1(x1)) at position [0] we obtained the following new rules [LPAR04]: (4^1(1(1(0(1(0(1(0(x0)))))))) -> 1^1(2(0(3(0(1(1(x0))))))),4^1(1(1(0(1(0(1(0(x0)))))))) -> 1^1(2(0(3(0(1(1(x0)))))))) (4^1(1(1(0(1(0(0(3(1(x0))))))))) -> 1^1(3(0(1(5(1(0(x0))))))),4^1(1(1(0(1(0(0(3(1(x0))))))))) -> 1^1(3(0(1(5(1(0(x0)))))))) (4^1(1(1(0(1(0(1(2(4(x0))))))))) -> 1^1(5(1(4(0(2(1(x0))))))),4^1(1(1(0(1(0(1(2(4(x0))))))))) -> 1^1(5(1(4(0(2(1(x0)))))))) (4^1(1(1(0(1(2(1(3(4(x0))))))))) -> 1^1(1(4(1(2(2(3(x0))))))),4^1(1(1(0(1(2(1(3(4(x0))))))))) -> 1^1(1(4(1(2(2(3(x0)))))))) (4^1(1(1(0(1(0(1(4(4(x0))))))))) -> 1^1(1(4(1(3(0(4(x0))))))),4^1(1(1(0(1(0(1(4(4(x0))))))))) -> 1^1(1(4(1(3(0(4(x0)))))))) ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 4^1(1(x1)) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 4^1(1(x1)) 0^1(1(2(4(x1)))) -> 0^1(4(1(2(x1)))) 0^1(1(2(4(x1)))) -> 4^1(1(2(x1))) 0^1(1(2(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 4^1(1(1(0(1(0(1(0(x0)))))))) -> 1^1(2(0(3(0(1(1(x0))))))) 4^1(1(1(0(1(0(0(3(1(x0))))))))) -> 1^1(3(0(1(5(1(0(x0))))))) 4^1(1(1(0(1(0(1(2(4(x0))))))))) -> 1^1(5(1(4(0(2(1(x0))))))) 4^1(1(1(0(1(2(1(3(4(x0))))))))) -> 1^1(1(4(1(2(2(3(x0))))))) 4^1(1(1(0(1(0(1(4(4(x0))))))))) -> 1^1(1(4(1(3(0(4(x0))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (37) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes. ---------------------------------------- (38) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(4(0(x1)))) -> 4^1(1(x1)) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 4^1(1(x1)) 0^1(1(2(4(x1)))) -> 0^1(4(1(2(x1)))) 0^1(1(2(4(x1)))) -> 4^1(1(2(x1))) 0^1(1(2(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (39) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 0^1(1(4(0(x1)))) -> 4^1(1(x1)) at position [0] we obtained the following new rules [LPAR04]: (0^1(1(4(0(0(1(0(x0))))))) -> 4^1(2(0(3(0(1(1(x0))))))),0^1(1(4(0(0(1(0(x0))))))) -> 4^1(2(0(3(0(1(1(x0)))))))) (0^1(1(4(0(0(0(3(1(x0)))))))) -> 4^1(3(0(1(5(1(0(x0))))))),0^1(1(4(0(0(0(3(1(x0)))))))) -> 4^1(3(0(1(5(1(0(x0)))))))) (0^1(1(4(0(0(1(2(4(x0)))))))) -> 4^1(5(1(4(0(2(1(x0))))))),0^1(1(4(0(0(1(2(4(x0)))))))) -> 4^1(5(1(4(0(2(1(x0)))))))) (0^1(1(4(0(2(1(3(4(x0)))))))) -> 4^1(1(4(1(2(2(3(x0))))))),0^1(1(4(0(2(1(3(4(x0)))))))) -> 4^1(1(4(1(2(2(3(x0)))))))) (0^1(1(4(0(0(1(4(4(x0)))))))) -> 4^1(1(4(1(3(0(4(x0))))))),0^1(1(4(0(0(1(4(4(x0)))))))) -> 4^1(1(4(1(3(0(4(x0)))))))) ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 4^1(1(x1)) 0^1(1(2(4(x1)))) -> 0^1(4(1(2(x1)))) 0^1(1(2(4(x1)))) -> 4^1(1(2(x1))) 0^1(1(2(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(0(0(1(0(x0))))))) -> 4^1(2(0(3(0(1(1(x0))))))) 0^1(1(4(0(0(0(3(1(x0)))))))) -> 4^1(3(0(1(5(1(0(x0))))))) 0^1(1(4(0(0(1(2(4(x0)))))))) -> 4^1(5(1(4(0(2(1(x0))))))) 0^1(1(4(0(2(1(3(4(x0)))))))) -> 4^1(1(4(1(2(2(3(x0))))))) 0^1(1(4(0(0(1(4(4(x0)))))))) -> 4^1(1(4(1(3(0(4(x0))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (41) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes. ---------------------------------------- (42) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 4^1(1(x1)) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(x1)))) -> 0^1(4(1(2(x1)))) 0^1(1(2(4(x1)))) -> 4^1(1(2(x1))) 0^1(1(2(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (43) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 0^1(1(2(4(x1)))) -> 4^1(1(x1)) at position [0] we obtained the following new rules [LPAR04]: (0^1(1(2(4(0(1(0(x0))))))) -> 4^1(2(0(3(0(1(1(x0))))))),0^1(1(2(4(0(1(0(x0))))))) -> 4^1(2(0(3(0(1(1(x0)))))))) (0^1(1(2(4(0(0(3(1(x0)))))))) -> 4^1(3(0(1(5(1(0(x0))))))),0^1(1(2(4(0(0(3(1(x0)))))))) -> 4^1(3(0(1(5(1(0(x0)))))))) (0^1(1(2(4(0(1(2(4(x0)))))))) -> 4^1(5(1(4(0(2(1(x0))))))),0^1(1(2(4(0(1(2(4(x0)))))))) -> 4^1(5(1(4(0(2(1(x0)))))))) (0^1(1(2(4(2(1(3(4(x0)))))))) -> 4^1(1(4(1(2(2(3(x0))))))),0^1(1(2(4(2(1(3(4(x0)))))))) -> 4^1(1(4(1(2(2(3(x0)))))))) (0^1(1(2(4(0(1(4(4(x0)))))))) -> 4^1(1(4(1(3(0(4(x0))))))),0^1(1(2(4(0(1(4(4(x0)))))))) -> 4^1(1(4(1(3(0(4(x0)))))))) ---------------------------------------- (44) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(x1)))) -> 0^1(4(1(2(x1)))) 0^1(1(2(4(x1)))) -> 4^1(1(2(x1))) 0^1(1(2(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(0(1(0(x0))))))) -> 4^1(2(0(3(0(1(1(x0))))))) 0^1(1(2(4(0(0(3(1(x0)))))))) -> 4^1(3(0(1(5(1(0(x0))))))) 0^1(1(2(4(0(1(2(4(x0)))))))) -> 4^1(5(1(4(0(2(1(x0))))))) 0^1(1(2(4(2(1(3(4(x0)))))))) -> 4^1(1(4(1(2(2(3(x0))))))) 0^1(1(2(4(0(1(4(4(x0)))))))) -> 4^1(1(4(1(3(0(4(x0))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (45) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes. ---------------------------------------- (46) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 0^1(4(1(2(x1)))) 0^1(1(2(4(x1)))) -> 4^1(1(2(x1))) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (47) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 0^1(1(2(4(x1)))) -> 0^1(4(1(2(x1)))) at position [0] we obtained the following new rules [LPAR04]: (0^1(1(2(4(1(3(4(x0))))))) -> 0^1(4(1(4(1(2(2(3(x0)))))))),0^1(1(2(4(1(3(4(x0))))))) -> 0^1(4(1(4(1(2(2(3(x0))))))))) ---------------------------------------- (48) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 4^1(1(2(x1))) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(3(4(x0))))))) -> 0^1(4(1(4(1(2(2(3(x0)))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (49) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (50) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 4^1(1(2(x1))) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (51) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 0^1(1(2(4(x1)))) -> 4^1(1(2(x1))) at position [0] we obtained the following new rules [LPAR04]: (0^1(1(2(4(1(3(4(x0))))))) -> 4^1(1(4(1(2(2(3(x0))))))),0^1(1(2(4(1(3(4(x0))))))) -> 4^1(1(4(1(2(2(3(x0)))))))) ---------------------------------------- (52) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(3(4(x0))))))) -> 4^1(1(4(1(2(2(3(x0))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (53) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (54) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (55) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 0^1(1(2(4(x1)))) -> 1^1(4(x1)) at position [0] we obtained the following new rules [LPAR04]: (0^1(1(2(4(1(1(0(1(x0)))))))) -> 1^1(5(0(4(1(1(1(x0))))))),0^1(1(2(4(1(1(0(1(x0)))))))) -> 1^1(5(0(4(1(1(1(x0)))))))) (0^1(1(2(4(0(1(2(4(x0)))))))) -> 1^1(4(0(2(2(1(4(x0))))))),0^1(1(2(4(0(1(2(4(x0)))))))) -> 1^1(4(0(2(2(1(4(x0)))))))) (0^1(1(2(4(0(1(2(4(x0)))))))) -> 1^1(0(4(2(5(1(4(x0))))))),0^1(1(2(4(0(1(2(4(x0)))))))) -> 1^1(0(4(2(5(1(4(x0)))))))) ---------------------------------------- (56) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(1(0(1(x0)))))))) -> 1^1(5(0(4(1(1(1(x0))))))) 0^1(1(2(4(0(1(2(4(x0)))))))) -> 1^1(4(0(2(2(1(4(x0))))))) 0^1(1(2(4(0(1(2(4(x0)))))))) -> 1^1(0(4(2(5(1(4(x0))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (57) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (58) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(4(4(x1)))) -> 0^1(4(x1)) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (59) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 0^1(1(4(4(x1)))) -> 0^1(4(x1)) at position [0] we obtained the following new rules [LPAR04]: (0^1(1(4(4(1(1(0(1(x0)))))))) -> 0^1(5(0(4(1(1(1(x0))))))),0^1(1(4(4(1(1(0(1(x0)))))))) -> 0^1(5(0(4(1(1(1(x0)))))))) (0^1(1(4(4(0(1(2(4(x0)))))))) -> 0^1(4(0(2(2(1(4(x0))))))),0^1(1(4(4(0(1(2(4(x0)))))))) -> 0^1(4(0(2(2(1(4(x0)))))))) (0^1(1(4(4(0(1(2(4(x0)))))))) -> 0^1(0(4(2(5(1(4(x0))))))),0^1(1(4(4(0(1(2(4(x0)))))))) -> 0^1(0(4(2(5(1(4(x0)))))))) ---------------------------------------- (60) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(4(1(1(0(1(x0)))))))) -> 0^1(5(0(4(1(1(1(x0))))))) 0^1(1(4(4(0(1(2(4(x0)))))))) -> 0^1(4(0(2(2(1(4(x0))))))) 0^1(1(4(4(0(1(2(4(x0)))))))) -> 0^1(0(4(2(5(1(4(x0))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (61) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (62) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(4(4(x1)))) -> 1^1(4(x1)) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (63) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 0^1(1(4(4(x1)))) -> 1^1(4(x1)) at position [0] we obtained the following new rules [LPAR04]: (0^1(1(4(4(1(1(0(1(x0)))))))) -> 1^1(5(0(4(1(1(1(x0))))))),0^1(1(4(4(1(1(0(1(x0)))))))) -> 1^1(5(0(4(1(1(1(x0)))))))) (0^1(1(4(4(0(1(2(4(x0)))))))) -> 1^1(4(0(2(2(1(4(x0))))))),0^1(1(4(4(0(1(2(4(x0)))))))) -> 1^1(4(0(2(2(1(4(x0)))))))) (0^1(1(4(4(0(1(2(4(x0)))))))) -> 1^1(0(4(2(5(1(4(x0))))))),0^1(1(4(4(0(1(2(4(x0)))))))) -> 1^1(0(4(2(5(1(4(x0)))))))) ---------------------------------------- (64) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(4(1(1(0(1(x0)))))))) -> 1^1(5(0(4(1(1(1(x0))))))) 0^1(1(4(4(0(1(2(4(x0)))))))) -> 1^1(4(0(2(2(1(4(x0))))))) 0^1(1(4(4(0(1(2(4(x0)))))))) -> 1^1(0(4(2(5(1(4(x0))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (65) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (66) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (67) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 0^1(1(4(4(x1)))) -> 0^1(4(1(4(x1)))) at position [0] we obtained the following new rules [LPAR04]: (0^1(1(4(4(1(1(0(1(x0)))))))) -> 0^1(4(1(5(0(4(1(1(1(x0))))))))),0^1(1(4(4(1(1(0(1(x0)))))))) -> 0^1(4(1(5(0(4(1(1(1(x0)))))))))) (0^1(1(4(4(0(1(2(4(x0)))))))) -> 0^1(4(1(4(0(2(2(1(4(x0))))))))),0^1(1(4(4(0(1(2(4(x0)))))))) -> 0^1(4(1(4(0(2(2(1(4(x0)))))))))) (0^1(1(4(4(0(1(2(4(x0)))))))) -> 0^1(4(1(0(4(2(5(1(4(x0))))))))),0^1(1(4(4(0(1(2(4(x0)))))))) -> 0^1(4(1(0(4(2(5(1(4(x0)))))))))) ---------------------------------------- (68) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(4(1(1(0(1(x0)))))))) -> 0^1(4(1(5(0(4(1(1(1(x0))))))))) 0^1(1(4(4(0(1(2(4(x0)))))))) -> 0^1(4(1(4(0(2(2(1(4(x0))))))))) 0^1(1(4(4(0(1(2(4(x0)))))))) -> 0^1(4(1(0(4(2(5(1(4(x0))))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (69) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (70) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (71) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 0^1(1(4(4(x1)))) -> 4^1(1(4(x1))) at position [0] we obtained the following new rules [LPAR04]: (0^1(1(4(4(1(1(0(1(x0)))))))) -> 4^1(1(5(0(4(1(1(1(x0)))))))),0^1(1(4(4(1(1(0(1(x0)))))))) -> 4^1(1(5(0(4(1(1(1(x0))))))))) (0^1(1(4(4(0(1(2(4(x0)))))))) -> 4^1(1(4(0(2(2(1(4(x0)))))))),0^1(1(4(4(0(1(2(4(x0)))))))) -> 4^1(1(4(0(2(2(1(4(x0))))))))) (0^1(1(4(4(0(1(2(4(x0)))))))) -> 4^1(1(0(4(2(5(1(4(x0)))))))),0^1(1(4(4(0(1(2(4(x0)))))))) -> 4^1(1(0(4(2(5(1(4(x0))))))))) ---------------------------------------- (72) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(4(1(1(0(1(x0)))))))) -> 4^1(1(5(0(4(1(1(1(x0)))))))) 0^1(1(4(4(0(1(2(4(x0)))))))) -> 4^1(1(4(0(2(2(1(4(x0)))))))) 0^1(1(4(4(0(1(2(4(x0)))))))) -> 4^1(1(0(4(2(5(1(4(x0)))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (73) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (74) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (75) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 0^1(1(2(4(0(x1))))) -> 0^1(4(x1)) at position [0] we obtained the following new rules [LPAR04]: (0^1(1(2(4(0(1(1(0(1(x0))))))))) -> 0^1(5(0(4(1(1(1(x0))))))),0^1(1(2(4(0(1(1(0(1(x0))))))))) -> 0^1(5(0(4(1(1(1(x0)))))))) (0^1(1(2(4(0(0(1(2(4(x0))))))))) -> 0^1(4(0(2(2(1(4(x0))))))),0^1(1(2(4(0(0(1(2(4(x0))))))))) -> 0^1(4(0(2(2(1(4(x0)))))))) (0^1(1(2(4(0(0(1(2(4(x0))))))))) -> 0^1(0(4(2(5(1(4(x0))))))),0^1(1(2(4(0(0(1(2(4(x0))))))))) -> 0^1(0(4(2(5(1(4(x0)))))))) ---------------------------------------- (76) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(0(1(1(0(1(x0))))))))) -> 0^1(5(0(4(1(1(1(x0))))))) 0^1(1(2(4(0(0(1(2(4(x0))))))))) -> 0^1(4(0(2(2(1(4(x0))))))) 0^1(1(2(4(0(0(1(2(4(x0))))))))) -> 0^1(0(4(2(5(1(4(x0))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (77) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (78) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (79) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 4^1(0(1(2(4(x1))))) -> 1^1(4(x1)) at position [0] we obtained the following new rules [LPAR04]: (4^1(0(1(2(4(1(1(0(1(x0))))))))) -> 1^1(5(0(4(1(1(1(x0))))))),4^1(0(1(2(4(1(1(0(1(x0))))))))) -> 1^1(5(0(4(1(1(1(x0)))))))) (4^1(0(1(2(4(0(1(2(4(x0))))))))) -> 1^1(4(0(2(2(1(4(x0))))))),4^1(0(1(2(4(0(1(2(4(x0))))))))) -> 1^1(4(0(2(2(1(4(x0)))))))) (4^1(0(1(2(4(0(1(2(4(x0))))))))) -> 1^1(0(4(2(5(1(4(x0))))))),4^1(0(1(2(4(0(1(2(4(x0))))))))) -> 1^1(0(4(2(5(1(4(x0)))))))) ---------------------------------------- (80) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 4^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(4(0(x1))))) -> 4^1(1(x1)) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(4(0(x1))))) -> 4^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 4^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 4^1(x1) 0^1(1(0(1(4(x1))))) -> 4^1(0(1(x1))) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(0(1(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(2(2(4(x1))))) -> 4^1(1(2(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(4(1(x1))) 0^1(1(4(2(4(x1))))) -> 4^1(1(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 4^1(1(4(x1))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 4^1(0(1(2(4(1(1(0(1(x0))))))))) -> 1^1(5(0(4(1(1(1(x0))))))) 4^1(0(1(2(4(0(1(2(4(x0))))))))) -> 1^1(4(0(2(2(1(4(x0))))))) 4^1(0(1(2(4(0(1(2(4(x0))))))))) -> 1^1(0(4(2(5(1(4(x0))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (81) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 18 less nodes. ---------------------------------------- (82) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (83) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 0^1(1(4(4(0(x1))))) -> 0^1(4(4(1(x1)))) at position [0] we obtained the following new rules [LPAR04]: (0^1(1(4(4(0(1(0(1(x0)))))))) -> 0^1(4(5(0(4(1(1(1(x0)))))))),0^1(1(4(4(0(1(0(1(x0)))))))) -> 0^1(4(5(0(4(1(1(1(x0))))))))) (0^1(1(4(4(0(0(1(0(x0)))))))) -> 0^1(4(4(2(0(3(0(1(1(x0))))))))),0^1(1(4(4(0(0(1(0(x0)))))))) -> 0^1(4(4(2(0(3(0(1(1(x0)))))))))) (0^1(1(4(4(0(0(0(3(1(x0))))))))) -> 0^1(4(4(3(0(1(5(1(0(x0))))))))),0^1(1(4(4(0(0(0(3(1(x0))))))))) -> 0^1(4(4(3(0(1(5(1(0(x0)))))))))) (0^1(1(4(4(0(0(1(2(4(x0))))))))) -> 0^1(4(4(5(1(4(0(2(1(x0))))))))),0^1(1(4(4(0(0(1(2(4(x0))))))))) -> 0^1(4(4(5(1(4(0(2(1(x0)))))))))) (0^1(1(4(4(0(2(1(3(4(x0))))))))) -> 0^1(4(4(1(4(1(2(2(3(x0))))))))),0^1(1(4(4(0(2(1(3(4(x0))))))))) -> 0^1(4(4(1(4(1(2(2(3(x0)))))))))) (0^1(1(4(4(0(0(1(4(4(x0))))))))) -> 0^1(4(4(1(4(1(3(0(4(x0))))))))),0^1(1(4(4(0(0(1(4(4(x0))))))))) -> 0^1(4(4(1(4(1(3(0(4(x0)))))))))) ---------------------------------------- (84) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(4(0(1(0(1(x0)))))))) -> 0^1(4(5(0(4(1(1(1(x0)))))))) 0^1(1(4(4(0(0(1(0(x0)))))))) -> 0^1(4(4(2(0(3(0(1(1(x0))))))))) 0^1(1(4(4(0(0(0(3(1(x0))))))))) -> 0^1(4(4(3(0(1(5(1(0(x0))))))))) 0^1(1(4(4(0(0(1(2(4(x0))))))))) -> 0^1(4(4(5(1(4(0(2(1(x0))))))))) 0^1(1(4(4(0(2(1(3(4(x0))))))))) -> 0^1(4(4(1(4(1(2(2(3(x0))))))))) 0^1(1(4(4(0(0(1(4(4(x0))))))))) -> 0^1(4(4(1(4(1(3(0(4(x0))))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (85) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes. ---------------------------------------- (86) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (87) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 0^1(1(4(4(0(x1))))) -> 1^1(0(4(x1))) at position [0] we obtained the following new rules [LPAR04]: (0^1(1(4(4(0(1(1(0(1(x0))))))))) -> 1^1(0(5(0(4(1(1(1(x0)))))))),0^1(1(4(4(0(1(1(0(1(x0))))))))) -> 1^1(0(5(0(4(1(1(1(x0))))))))) (0^1(1(4(4(0(0(1(2(4(x0))))))))) -> 1^1(0(4(0(2(2(1(4(x0)))))))),0^1(1(4(4(0(0(1(2(4(x0))))))))) -> 1^1(0(4(0(2(2(1(4(x0))))))))) (0^1(1(4(4(0(0(1(2(4(x0))))))))) -> 1^1(0(0(4(2(5(1(4(x0)))))))),0^1(1(4(4(0(0(1(2(4(x0))))))))) -> 1^1(0(0(4(2(5(1(4(x0))))))))) ---------------------------------------- (88) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(4(0(1(1(0(1(x0))))))))) -> 1^1(0(5(0(4(1(1(1(x0)))))))) 0^1(1(4(4(0(0(1(2(4(x0))))))))) -> 1^1(0(4(0(2(2(1(4(x0)))))))) 0^1(1(4(4(0(0(1(2(4(x0))))))))) -> 1^1(0(0(4(2(5(1(4(x0)))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (89) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (90) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (91) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 0^1(1(4(4(0(x1))))) -> 0^1(4(x1)) at position [0] we obtained the following new rules [LPAR04]: (0^1(1(4(4(0(1(1(0(1(x0))))))))) -> 0^1(5(0(4(1(1(1(x0))))))),0^1(1(4(4(0(1(1(0(1(x0))))))))) -> 0^1(5(0(4(1(1(1(x0)))))))) (0^1(1(4(4(0(0(1(2(4(x0))))))))) -> 0^1(4(0(2(2(1(4(x0))))))),0^1(1(4(4(0(0(1(2(4(x0))))))))) -> 0^1(4(0(2(2(1(4(x0)))))))) (0^1(1(4(4(0(0(1(2(4(x0))))))))) -> 0^1(0(4(2(5(1(4(x0))))))),0^1(1(4(4(0(0(1(2(4(x0))))))))) -> 0^1(0(4(2(5(1(4(x0)))))))) ---------------------------------------- (92) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(4(0(1(1(0(1(x0))))))))) -> 0^1(5(0(4(1(1(1(x0))))))) 0^1(1(4(4(0(0(1(2(4(x0))))))))) -> 0^1(4(0(2(2(1(4(x0))))))) 0^1(1(4(4(0(0(1(2(4(x0))))))))) -> 0^1(0(4(2(5(1(4(x0))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (93) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (94) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (95) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 0^1(1(4(0(1(x1))))) -> 0^1(4(x1)) at position [0] we obtained the following new rules [LPAR04]: (0^1(1(4(0(1(1(1(0(1(x0))))))))) -> 0^1(5(0(4(1(1(1(x0))))))),0^1(1(4(0(1(1(1(0(1(x0))))))))) -> 0^1(5(0(4(1(1(1(x0)))))))) (0^1(1(4(0(1(0(1(2(4(x0))))))))) -> 0^1(4(0(2(2(1(4(x0))))))),0^1(1(4(0(1(0(1(2(4(x0))))))))) -> 0^1(4(0(2(2(1(4(x0)))))))) (0^1(1(4(0(1(0(1(2(4(x0))))))))) -> 0^1(0(4(2(5(1(4(x0))))))),0^1(1(4(0(1(0(1(2(4(x0))))))))) -> 0^1(0(4(2(5(1(4(x0)))))))) ---------------------------------------- (96) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(0(1(1(1(0(1(x0))))))))) -> 0^1(5(0(4(1(1(1(x0))))))) 0^1(1(4(0(1(0(1(2(4(x0))))))))) -> 0^1(4(0(2(2(1(4(x0))))))) 0^1(1(4(0(1(0(1(2(4(x0))))))))) -> 0^1(0(4(2(5(1(4(x0))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (97) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (98) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (99) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 0^1(1(2(4(1(x1))))) -> 1^1(4(x1)) at position [0] we obtained the following new rules [LPAR04]: (0^1(1(2(4(1(1(1(0(1(x0))))))))) -> 1^1(5(0(4(1(1(1(x0))))))),0^1(1(2(4(1(1(1(0(1(x0))))))))) -> 1^1(5(0(4(1(1(1(x0)))))))) (0^1(1(2(4(1(0(1(2(4(x0))))))))) -> 1^1(4(0(2(2(1(4(x0))))))),0^1(1(2(4(1(0(1(2(4(x0))))))))) -> 1^1(4(0(2(2(1(4(x0)))))))) (0^1(1(2(4(1(0(1(2(4(x0))))))))) -> 1^1(0(4(2(5(1(4(x0))))))),0^1(1(2(4(1(0(1(2(4(x0))))))))) -> 1^1(0(4(2(5(1(4(x0)))))))) ---------------------------------------- (100) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(1(1(1(0(1(x0))))))))) -> 1^1(5(0(4(1(1(1(x0))))))) 0^1(1(2(4(1(0(1(2(4(x0))))))))) -> 1^1(4(0(2(2(1(4(x0))))))) 0^1(1(2(4(1(0(1(2(4(x0))))))))) -> 1^1(0(4(2(5(1(4(x0))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (101) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (102) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (103) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 0^1(1(1(2(4(x1))))) -> 1^1(4(x1)) at position [0] we obtained the following new rules [LPAR04]: (0^1(1(1(2(4(1(1(0(1(x0))))))))) -> 1^1(5(0(4(1(1(1(x0))))))),0^1(1(1(2(4(1(1(0(1(x0))))))))) -> 1^1(5(0(4(1(1(1(x0)))))))) (0^1(1(1(2(4(0(1(2(4(x0))))))))) -> 1^1(4(0(2(2(1(4(x0))))))),0^1(1(1(2(4(0(1(2(4(x0))))))))) -> 1^1(4(0(2(2(1(4(x0)))))))) (0^1(1(1(2(4(0(1(2(4(x0))))))))) -> 1^1(0(4(2(5(1(4(x0))))))),0^1(1(1(2(4(0(1(2(4(x0))))))))) -> 1^1(0(4(2(5(1(4(x0)))))))) ---------------------------------------- (104) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 0^1(1(1(2(4(1(1(0(1(x0))))))))) -> 1^1(5(0(4(1(1(1(x0))))))) 0^1(1(1(2(4(0(1(2(4(x0))))))))) -> 1^1(4(0(2(2(1(4(x0))))))) 0^1(1(1(2(4(0(1(2(4(x0))))))))) -> 1^1(0(4(2(5(1(4(x0))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (105) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (106) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (107) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 1^1(0(1(0(x1)))) -> 0^1(1(1(x1))) at position [0] we obtained the following new rules [LPAR04]: (1^1(0(1(0(0(1(0(x0))))))) -> 0^1(1(2(0(3(0(1(1(x0)))))))),1^1(0(1(0(0(1(0(x0))))))) -> 0^1(1(2(0(3(0(1(1(x0))))))))) (1^1(0(1(0(0(0(3(1(x0)))))))) -> 0^1(1(3(0(1(5(1(0(x0)))))))),1^1(0(1(0(0(0(3(1(x0)))))))) -> 0^1(1(3(0(1(5(1(0(x0))))))))) (1^1(0(1(0(0(1(2(4(x0)))))))) -> 0^1(1(5(1(4(0(2(1(x0)))))))),1^1(0(1(0(0(1(2(4(x0)))))))) -> 0^1(1(5(1(4(0(2(1(x0))))))))) (1^1(0(1(0(2(1(3(4(x0)))))))) -> 0^1(1(1(4(1(2(2(3(x0)))))))),1^1(0(1(0(2(1(3(4(x0)))))))) -> 0^1(1(1(4(1(2(2(3(x0))))))))) (1^1(0(1(0(0(1(4(4(x0)))))))) -> 0^1(1(1(4(1(3(0(4(x0)))))))),1^1(0(1(0(0(1(4(4(x0)))))))) -> 0^1(1(1(4(1(3(0(4(x0))))))))) ---------------------------------------- (108) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 1^1(0(1(0(0(1(0(x0))))))) -> 0^1(1(2(0(3(0(1(1(x0)))))))) 1^1(0(1(0(0(0(3(1(x0)))))))) -> 0^1(1(3(0(1(5(1(0(x0)))))))) 1^1(0(1(0(0(1(2(4(x0)))))))) -> 0^1(1(5(1(4(0(2(1(x0)))))))) 1^1(0(1(0(2(1(3(4(x0)))))))) -> 0^1(1(1(4(1(2(2(3(x0)))))))) 1^1(0(1(0(0(1(4(4(x0)))))))) -> 0^1(1(1(4(1(3(0(4(x0)))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (109) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes. ---------------------------------------- (110) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (111) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 0^1(1(2(2(4(x1))))) -> 1^1(4(1(2(x1)))) at position [0] we obtained the following new rules [LPAR04]: (0^1(1(2(2(4(1(3(4(x0)))))))) -> 1^1(4(1(4(1(2(2(3(x0)))))))),0^1(1(2(2(4(1(3(4(x0)))))))) -> 1^1(4(1(4(1(2(2(3(x0))))))))) ---------------------------------------- (112) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(2(4(1(3(4(x0)))))))) -> 1^1(4(1(4(1(2(2(3(x0)))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (113) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (114) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (115) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 0^1(1(4(2(4(x1))))) -> 1^1(0(4(x1))) at position [0] we obtained the following new rules [LPAR04]: (0^1(1(4(2(4(1(1(0(1(x0))))))))) -> 1^1(0(5(0(4(1(1(1(x0)))))))),0^1(1(4(2(4(1(1(0(1(x0))))))))) -> 1^1(0(5(0(4(1(1(1(x0))))))))) (0^1(1(4(2(4(0(1(2(4(x0))))))))) -> 1^1(0(4(0(2(2(1(4(x0)))))))),0^1(1(4(2(4(0(1(2(4(x0))))))))) -> 1^1(0(4(0(2(2(1(4(x0))))))))) (0^1(1(4(2(4(0(1(2(4(x0))))))))) -> 1^1(0(0(4(2(5(1(4(x0)))))))),0^1(1(4(2(4(0(1(2(4(x0))))))))) -> 1^1(0(0(4(2(5(1(4(x0))))))))) ---------------------------------------- (116) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(1(1(0(1(x0))))))))) -> 1^1(0(5(0(4(1(1(1(x0)))))))) 0^1(1(4(2(4(0(1(2(4(x0))))))))) -> 1^1(0(4(0(2(2(1(4(x0)))))))) 0^1(1(4(2(4(0(1(2(4(x0))))))))) -> 1^1(0(0(4(2(5(1(4(x0)))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (117) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (118) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (119) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 0^1(1(4(2(4(x1))))) -> 0^1(4(x1)) at position [0] we obtained the following new rules [LPAR04]: (0^1(1(4(2(4(1(1(0(1(x0))))))))) -> 0^1(5(0(4(1(1(1(x0))))))),0^1(1(4(2(4(1(1(0(1(x0))))))))) -> 0^1(5(0(4(1(1(1(x0)))))))) (0^1(1(4(2(4(0(1(2(4(x0))))))))) -> 0^1(4(0(2(2(1(4(x0))))))),0^1(1(4(2(4(0(1(2(4(x0))))))))) -> 0^1(4(0(2(2(1(4(x0)))))))) (0^1(1(4(2(4(0(1(2(4(x0))))))))) -> 0^1(0(4(2(5(1(4(x0))))))),0^1(1(4(2(4(0(1(2(4(x0))))))))) -> 0^1(0(4(2(5(1(4(x0)))))))) ---------------------------------------- (120) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(1(1(0(1(x0))))))))) -> 0^1(5(0(4(1(1(1(x0))))))) 0^1(1(4(2(4(0(1(2(4(x0))))))))) -> 0^1(4(0(2(2(1(4(x0))))))) 0^1(1(4(2(4(0(1(2(4(x0))))))))) -> 0^1(0(4(2(5(1(4(x0))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (121) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (122) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (123) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 0^1(1(4(2(4(x1))))) -> 1^1(4(x1)) at position [0] we obtained the following new rules [LPAR04]: (0^1(1(4(2(4(1(1(0(1(x0))))))))) -> 1^1(5(0(4(1(1(1(x0))))))),0^1(1(4(2(4(1(1(0(1(x0))))))))) -> 1^1(5(0(4(1(1(1(x0)))))))) (0^1(1(4(2(4(0(1(2(4(x0))))))))) -> 1^1(4(0(2(2(1(4(x0))))))),0^1(1(4(2(4(0(1(2(4(x0))))))))) -> 1^1(4(0(2(2(1(4(x0)))))))) (0^1(1(4(2(4(0(1(2(4(x0))))))))) -> 1^1(0(4(2(5(1(4(x0))))))),0^1(1(4(2(4(0(1(2(4(x0))))))))) -> 1^1(0(4(2(5(1(4(x0)))))))) ---------------------------------------- (124) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(2(4(1(1(0(1(x0))))))))) -> 1^1(5(0(4(1(1(1(x0))))))) 0^1(1(4(2(4(0(1(2(4(x0))))))))) -> 1^1(4(0(2(2(1(4(x0))))))) 0^1(1(4(2(4(0(1(2(4(x0))))))))) -> 1^1(0(4(2(5(1(4(x0))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (125) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (126) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (127) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 0^1(1(0(4(4(x1))))) -> 0^1(4(x1)) at position [0] we obtained the following new rules [LPAR04]: (0^1(1(0(4(4(1(1(0(1(x0))))))))) -> 0^1(5(0(4(1(1(1(x0))))))),0^1(1(0(4(4(1(1(0(1(x0))))))))) -> 0^1(5(0(4(1(1(1(x0)))))))) (0^1(1(0(4(4(0(1(2(4(x0))))))))) -> 0^1(4(0(2(2(1(4(x0))))))),0^1(1(0(4(4(0(1(2(4(x0))))))))) -> 0^1(4(0(2(2(1(4(x0)))))))) (0^1(1(0(4(4(0(1(2(4(x0))))))))) -> 0^1(0(4(2(5(1(4(x0))))))),0^1(1(0(4(4(0(1(2(4(x0))))))))) -> 0^1(0(4(2(5(1(4(x0)))))))) ---------------------------------------- (128) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 0^1(1(0(4(4(1(1(0(1(x0))))))))) -> 0^1(5(0(4(1(1(1(x0))))))) 0^1(1(0(4(4(0(1(2(4(x0))))))))) -> 0^1(4(0(2(2(1(4(x0))))))) 0^1(1(0(4(4(0(1(2(4(x0))))))))) -> 0^1(0(4(2(5(1(4(x0))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (129) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (130) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (131) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 0^1(1(2(4(4(x1))))) -> 0^1(4(x1)) at position [0] we obtained the following new rules [LPAR04]: (0^1(1(2(4(4(1(1(0(1(x0))))))))) -> 0^1(5(0(4(1(1(1(x0))))))),0^1(1(2(4(4(1(1(0(1(x0))))))))) -> 0^1(5(0(4(1(1(1(x0)))))))) (0^1(1(2(4(4(0(1(2(4(x0))))))))) -> 0^1(4(0(2(2(1(4(x0))))))),0^1(1(2(4(4(0(1(2(4(x0))))))))) -> 0^1(4(0(2(2(1(4(x0)))))))) (0^1(1(2(4(4(0(1(2(4(x0))))))))) -> 0^1(0(4(2(5(1(4(x0))))))),0^1(1(2(4(4(0(1(2(4(x0))))))))) -> 0^1(0(4(2(5(1(4(x0)))))))) ---------------------------------------- (132) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 0^1(1(2(4(4(1(1(0(1(x0))))))))) -> 0^1(5(0(4(1(1(1(x0))))))) 0^1(1(2(4(4(0(1(2(4(x0))))))))) -> 0^1(4(0(2(2(1(4(x0))))))) 0^1(1(2(4(4(0(1(2(4(x0))))))))) -> 0^1(0(4(2(5(1(4(x0))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (133) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (134) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (135) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 0^1(1(4(5(4(x1))))) -> 0^1(4(1(4(x1)))) at position [0] we obtained the following new rules [LPAR04]: (0^1(1(4(5(4(1(1(0(1(x0))))))))) -> 0^1(4(1(5(0(4(1(1(1(x0))))))))),0^1(1(4(5(4(1(1(0(1(x0))))))))) -> 0^1(4(1(5(0(4(1(1(1(x0)))))))))) (0^1(1(4(5(4(0(1(2(4(x0))))))))) -> 0^1(4(1(4(0(2(2(1(4(x0))))))))),0^1(1(4(5(4(0(1(2(4(x0))))))))) -> 0^1(4(1(4(0(2(2(1(4(x0)))))))))) (0^1(1(4(5(4(0(1(2(4(x0))))))))) -> 0^1(4(1(0(4(2(5(1(4(x0))))))))),0^1(1(4(5(4(0(1(2(4(x0))))))))) -> 0^1(4(1(0(4(2(5(1(4(x0)))))))))) ---------------------------------------- (136) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) 0^1(1(4(5(4(1(1(0(1(x0))))))))) -> 0^1(4(1(5(0(4(1(1(1(x0))))))))) 0^1(1(4(5(4(0(1(2(4(x0))))))))) -> 0^1(4(1(4(0(2(2(1(4(x0))))))))) 0^1(1(4(5(4(0(1(2(4(x0))))))))) -> 0^1(4(1(0(4(2(5(1(4(x0))))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (137) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (138) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (139) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 0^1(1(4(5(4(x1))))) -> 1^1(4(x1)) at position [0] we obtained the following new rules [LPAR04]: (0^1(1(4(5(4(1(1(0(1(x0))))))))) -> 1^1(5(0(4(1(1(1(x0))))))),0^1(1(4(5(4(1(1(0(1(x0))))))))) -> 1^1(5(0(4(1(1(1(x0)))))))) (0^1(1(4(5(4(0(1(2(4(x0))))))))) -> 1^1(4(0(2(2(1(4(x0))))))),0^1(1(4(5(4(0(1(2(4(x0))))))))) -> 1^1(4(0(2(2(1(4(x0)))))))) (0^1(1(4(5(4(0(1(2(4(x0))))))))) -> 1^1(0(4(2(5(1(4(x0))))))),0^1(1(4(5(4(0(1(2(4(x0))))))))) -> 1^1(0(4(2(5(1(4(x0)))))))) ---------------------------------------- (140) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) 0^1(1(4(5(4(1(1(0(1(x0))))))))) -> 1^1(5(0(4(1(1(1(x0))))))) 0^1(1(4(5(4(0(1(2(4(x0))))))))) -> 1^1(4(0(2(2(1(4(x0))))))) 0^1(1(4(5(4(0(1(2(4(x0))))))))) -> 1^1(0(4(2(5(1(4(x0))))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (141) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (142) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (143) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 1^1(0(1(0(x1)))) -> 1^1(x1) 1^1(0(0(3(1(x1))))) -> 1^1(0(x1)) 1^1(0(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(0(x1)))) -> 1^1(0(x1)) 1^1(0(1(2(4(x1))))) -> 1^1(x1) 0^1(0(1(0(x1)))) -> 1^1(x1) 0^1(1(4(0(x1)))) -> 1^1(x1) 0^1(1(2(0(0(x1))))) -> 1^1(0(x1)) 0^1(1(2(4(0(x1))))) -> 1^1(x1) 0^1(1(4(4(0(x1))))) -> 1^1(x1) 0^1(1(0(1(4(x1))))) -> 0^1(1(x1)) 0^1(1(0(1(4(x1))))) -> 1^1(x1) 0^1(1(0(2(4(x1))))) -> 1^1(x1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 + x_1 POL(0^1(x_1)) = 1 + x_1 POL(1(x_1)) = x_1 POL(1^1(x_1)) = 1 + x_1 POL(2(x_1)) = x_1 POL(3(x_1)) = x_1 POL(4(x_1)) = x_1 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) ---------------------------------------- (144) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(2(4(x1)))) -> 1^1(x1) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 0^1(1(2(4(0(x1))))) -> 1^1(4(0(x1))) 0^1(1(2(2(4(x1))))) -> 1^1(x1) 0^1(1(4(2(4(x1))))) -> 1^1(x1) 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(4(x1))))) -> 1^1(x1) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (145) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes. ---------------------------------------- (146) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (147) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(1(2(4(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(0^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(1(x_1)) = [[-I], [-I], [0A]] + [[0A, -I, -I], [0A, -I, 0A], [-I, -I, -I]] * x_1 >>> <<< POL(2(x_1)) = [[0A], [-I], [-I]] + [[-I, -I, -I], [0A, -I, -I], [-I, -I, 0A]] * x_1 >>> <<< POL(4(x_1)) = [[0A], [-I], [0A]] + [[0A, -I, -I], [0A, 0A, -I], [0A, 0A, 0A]] * x_1 >>> <<< POL(0(x_1)) = [[0A], [1A], [0A]] + [[0A, 0A, -I], [0A, 0A, -I], [0A, 1A, 0A]] * x_1 >>> <<< POL(3(x_1)) = [[0A], [-I], [-I]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 >>> <<< POL(5(x_1)) = [[0A], [-I], [-I]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) ---------------------------------------- (148) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (149) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(1(2(4(4(x1))))) -> 0^1(1(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( 0^1_1(x_1) ) = max{0, 2x_1 - 2} POL( 1_1(x_1) ) = max{0, x_1 - 2} POL( 0_1(x_1) ) = max{0, -2} POL( 2_1(x_1) ) = max{0, 2x_1 - 2} POL( 3_1(x_1) ) = max{0, -2} POL( 5_1(x_1) ) = max{0, -2} POL( 4_1(x_1) ) = 2x_1 + 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) ---------------------------------------- (150) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: 0(1(0(0(x1)))) -> 0(2(0(2(1(0(x1)))))) 0(0(1(0(x1)))) -> 0(0(0(2(1(x1))))) 1(0(1(0(x1)))) -> 2(0(3(0(1(1(x1)))))) 0(1(4(0(x1)))) -> 0(4(1(3(0(x1))))) 0(1(4(0(x1)))) -> 5(0(0(4(1(x1))))) 0(1(4(0(x1)))) -> 0(4(0(5(1(x1))))) 0(1(4(0(x1)))) -> 0(5(4(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 0(4(5(1(5(0(x1)))))) 0(1(4(0(x1)))) -> 5(0(4(0(5(1(x1)))))) 0(1(2(4(x1)))) -> 4(0(5(2(1(x1))))) 0(1(2(4(x1)))) -> 0(5(2(4(1(x1))))) 0(1(2(4(x1)))) -> 0(2(5(4(1(x1))))) 0(1(2(4(x1)))) -> 4(1(3(0(2(x1))))) 0(1(2(4(x1)))) -> 4(1(5(0(2(x1))))) 0(1(2(4(x1)))) -> 4(0(2(1(2(x1))))) 0(1(2(4(x1)))) -> 5(0(4(1(2(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(x1))))) 0(1(2(4(x1)))) -> 0(2(2(1(4(x1))))) 0(1(2(4(x1)))) -> 0(1(2(3(4(x1))))) 0(1(2(4(x1)))) -> 0(2(1(5(4(x1))))) 0(1(2(4(x1)))) -> 4(1(2(0(5(x1))))) 0(1(2(4(x1)))) -> 0(2(1(4(5(x1))))) 0(1(2(4(x1)))) -> 4(0(2(5(1(2(x1)))))) 0(1(2(4(x1)))) -> 0(2(5(2(1(4(x1)))))) 0(1(2(4(x1)))) -> 4(1(2(0(3(4(x1)))))) 0(0(3(4(x1)))) -> 4(0(2(0(3(x1))))) 0(0(3(4(x1)))) -> 0(4(0(2(3(x1))))) 0(1(4(4(x1)))) -> 4(1(5(0(4(x1))))) 0(1(4(4(x1)))) -> 4(0(2(1(4(x1))))) 0(1(4(4(x1)))) -> 5(0(4(1(4(x1))))) 0(1(4(4(x1)))) -> 4(0(5(1(4(x1))))) 0(1(4(4(x1)))) -> 0(4(5(1(4(x1))))) 0(1(2(0(0(x1))))) -> 0(2(0(2(1(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 0(1(2(4(0(x1))))) -> 0(4(3(0(2(1(x1)))))) 0(1(2(4(0(x1))))) -> 2(0(1(3(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(5(0(4(4(1(x1)))))) 0(1(4(4(0(x1))))) -> 4(0(2(1(0(4(x1)))))) 0(1(4(4(0(x1))))) -> 0(4(1(4(0(5(x1)))))) 4(1(1(0(1(x1))))) -> 5(0(4(1(1(1(x1)))))) 0(1(4(0(1(x1))))) -> 0(1(1(5(0(4(x1)))))) 0(0(3(1(1(x1))))) -> 0(2(0(1(3(1(x1)))))) 1(0(0(3(1(x1))))) -> 3(0(1(5(1(0(x1)))))) 0(0(1(4(1(x1))))) -> 0(0(4(1(1(5(x1)))))) 0(1(2(4(1(x1))))) -> 0(2(1(2(1(4(x1)))))) 0(1(2(4(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(0(3(4(1(x1))))) -> 0(4(0(1(2(3(x1)))))) 0(1(2(0(4(x1))))) -> 0(4(1(2(0(2(x1)))))) 0(1(0(1(4(x1))))) -> 0(1(3(4(0(1(x1)))))) 0(0(3(1(4(x1))))) -> 0(5(4(0(1(3(x1)))))) 0(0(3(1(4(x1))))) -> 0(2(0(1(3(4(x1)))))) 0(1(0(2(4(x1))))) -> 0(2(0(3(4(1(x1)))))) 0(1(0(2(4(x1))))) -> 4(0(2(0(5(1(x1)))))) 0(1(0(2(4(x1))))) -> 0(4(1(2(0(3(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(4(1(2(x1)))))) 1(0(1(2(4(x1))))) -> 5(1(4(0(2(1(x1)))))) 4(0(1(2(4(x1))))) -> 4(0(2(2(1(4(x1)))))) 4(0(1(2(4(x1))))) -> 0(4(2(5(1(4(x1)))))) 0(1(1(2(4(x1))))) -> 0(5(1(2(1(4(x1)))))) 0(1(2(2(4(x1))))) -> 4(2(0(5(2(1(x1)))))) 0(1(2(2(4(x1))))) -> 5(0(2(2(4(1(x1)))))) 0(1(2(2(4(x1))))) -> 0(4(2(3(1(2(x1)))))) 0(1(2(2(4(x1))))) -> 0(2(1(4(1(2(x1)))))) 0(0(3(2(4(x1))))) -> 0(0(4(5(2(3(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(4(1(x1)))))) 0(1(4(2(4(x1))))) -> 4(4(2(1(0(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(4(0(2(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 2(0(4(3(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 5(0(2(4(1(4(x1)))))) 0(1(4(2(4(x1))))) -> 0(2(5(4(1(4(x1)))))) 1(2(1(3(4(x1))))) -> 1(4(1(2(2(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(0(4(5(0(3(x1)))))) 0(0(5(3(4(x1))))) -> 0(4(1(3(0(5(x1)))))) 0(1(0(4(4(x1))))) -> 0(4(1(3(0(4(x1)))))) 1(0(1(4(4(x1))))) -> 1(4(1(3(0(4(x1)))))) 0(1(2(4(4(x1))))) -> 4(2(4(3(0(1(x1)))))) 0(1(2(4(4(x1))))) -> 2(4(1(3(0(4(x1)))))) 0(0(3(4(4(x1))))) -> 4(4(0(3(0(2(x1)))))) 0(0(3(4(4(x1))))) -> 0(5(4(0(3(4(x1)))))) 0(1(4(4(4(x1))))) -> 4(0(1(3(4(4(x1)))))) 0(1(2(5(4(x1))))) -> 0(5(4(1(2(3(x1)))))) 0(1(4(5(4(x1))))) -> 0(5(0(4(1(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (151) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (152) YES