568.51/147.49 YES 588.65/152.66 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 588.65/152.66 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 588.65/152.66 588.65/152.66 588.65/152.66 Termination w.r.t. Q of the given QTRS could be proven: 588.65/152.66 588.65/152.66 (0) QTRS 588.65/152.66 (1) QTRS Reverse [EQUIVALENT, 0 ms] 588.65/152.66 (2) QTRS 588.65/152.66 (3) DependencyPairsProof [EQUIVALENT, 508 ms] 588.65/152.66 (4) QDP 588.65/152.66 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 588.65/152.66 (6) AND 588.65/152.66 (7) QDP 588.65/152.66 (8) UsableRulesProof [EQUIVALENT, 0 ms] 588.65/152.66 (9) QDP 588.65/152.66 (10) MRRProof [EQUIVALENT, 14 ms] 588.65/152.66 (11) QDP 588.65/152.66 (12) PisEmptyProof [EQUIVALENT, 0 ms] 588.65/152.66 (13) YES 588.65/152.66 (14) QDP 588.65/152.66 (15) QDPOrderProof [EQUIVALENT, 277 ms] 588.65/152.66 (16) QDP 588.65/152.66 (17) DependencyGraphProof [EQUIVALENT, 0 ms] 588.65/152.66 (18) QDP 588.65/152.66 (19) QDPOrderProof [EQUIVALENT, 133 ms] 588.65/152.66 (20) QDP 588.65/152.66 (21) DependencyGraphProof [EQUIVALENT, 0 ms] 588.65/152.66 (22) AND 588.65/152.66 (23) QDP 588.65/152.66 (24) QDPOrderProof [EQUIVALENT, 6372 ms] 588.65/152.66 (25) QDP 588.65/152.66 (26) PisEmptyProof [EQUIVALENT, 0 ms] 588.65/152.66 (27) YES 588.65/152.66 (28) QDP 588.65/152.66 (29) TransformationProof [EQUIVALENT, 0 ms] 588.65/152.66 (30) QDP 588.65/152.66 (31) DependencyGraphProof [EQUIVALENT, 0 ms] 588.65/152.66 (32) QDP 588.65/152.66 (33) TransformationProof [EQUIVALENT, 0 ms] 588.65/152.66 (34) QDP 588.65/152.66 (35) DependencyGraphProof [EQUIVALENT, 0 ms] 588.65/152.66 (36) QDP 588.65/152.66 (37) TransformationProof [EQUIVALENT, 0 ms] 588.65/152.66 (38) QDP 588.65/152.66 (39) DependencyGraphProof [EQUIVALENT, 0 ms] 588.65/152.66 (40) QDP 588.65/152.66 (41) TransformationProof [EQUIVALENT, 0 ms] 588.65/152.66 (42) QDP 588.65/152.66 (43) DependencyGraphProof [EQUIVALENT, 0 ms] 588.65/152.66 (44) QDP 588.65/152.66 (45) TransformationProof [EQUIVALENT, 0 ms] 588.65/152.66 (46) QDP 588.65/152.66 (47) DependencyGraphProof [EQUIVALENT, 0 ms] 588.65/152.66 (48) QDP 588.65/152.66 (49) TransformationProof [EQUIVALENT, 0 ms] 588.65/152.66 (50) QDP 588.65/152.66 (51) DependencyGraphProof [EQUIVALENT, 0 ms] 588.65/152.66 (52) QDP 588.65/152.66 (53) TransformationProof [EQUIVALENT, 0 ms] 588.65/152.66 (54) QDP 588.65/152.66 (55) DependencyGraphProof [EQUIVALENT, 0 ms] 588.65/152.66 (56) QDP 588.65/152.66 (57) TransformationProof [EQUIVALENT, 0 ms] 588.65/152.66 (58) QDP 588.65/152.66 (59) DependencyGraphProof [EQUIVALENT, 0 ms] 588.65/152.66 (60) QDP 588.65/152.66 (61) TransformationProof [EQUIVALENT, 0 ms] 588.65/152.66 (62) QDP 588.65/152.66 (63) DependencyGraphProof [EQUIVALENT, 0 ms] 588.65/152.66 (64) QDP 588.65/152.66 (65) TransformationProof [EQUIVALENT, 0 ms] 588.65/152.66 (66) QDP 588.65/152.66 (67) DependencyGraphProof [EQUIVALENT, 0 ms] 588.65/152.66 (68) QDP 588.65/152.66 (69) TransformationProof [EQUIVALENT, 0 ms] 588.65/152.66 (70) QDP 588.65/152.66 (71) DependencyGraphProof [EQUIVALENT, 0 ms] 588.65/152.66 (72) QDP 588.65/152.66 (73) TransformationProof [EQUIVALENT, 0 ms] 588.65/152.66 (74) QDP 588.65/152.66 (75) DependencyGraphProof [EQUIVALENT, 0 ms] 588.65/152.66 (76) QDP 588.65/152.66 (77) QDPOrderProof [EQUIVALENT, 801 ms] 588.65/152.66 (78) QDP 588.65/152.66 (79) UsableRulesProof [EQUIVALENT, 0 ms] 588.65/152.66 (80) QDP 588.65/152.66 (81) QDPSizeChangeProof [EQUIVALENT, 0 ms] 588.65/152.66 (82) YES 588.65/152.66 588.65/152.66 588.65/152.66 ---------------------------------------- 588.65/152.66 588.65/152.66 (0) 588.65/152.66 Obligation: 588.65/152.66 Q restricted rewrite system: 588.65/152.66 The TRS R consists of the following rules: 588.65/152.66 588.65/152.66 0(1(1(2(x1)))) -> 1(0(1(3(2(x1))))) 588.65/152.66 0(1(1(2(x1)))) -> 4(1(0(1(2(x1))))) 588.65/152.66 0(1(1(2(x1)))) -> 0(1(4(1(3(2(x1)))))) 588.65/152.66 0(1(1(2(x1)))) -> 0(4(1(4(1(2(x1)))))) 588.65/152.66 0(1(1(2(x1)))) -> 4(1(0(3(1(2(x1)))))) 588.65/152.66 0(1(1(2(x1)))) -> 4(1(3(1(0(2(x1)))))) 588.65/152.66 0(1(1(5(x1)))) -> 4(1(0(1(5(x1))))) 588.65/152.66 0(1(1(5(x1)))) -> 5(4(1(0(1(x1))))) 588.65/152.66 0(1(1(5(x1)))) -> 0(4(1(0(1(5(x1)))))) 588.65/152.66 0(1(1(5(x1)))) -> 0(5(4(1(0(1(x1)))))) 588.65/152.66 0(1(1(5(x1)))) -> 1(0(1(3(1(5(x1)))))) 588.65/152.66 0(1(1(5(x1)))) -> 1(4(4(0(1(5(x1)))))) 588.65/152.66 0(1(1(5(x1)))) -> 3(0(1(5(4(1(x1)))))) 588.65/152.66 0(1(1(5(x1)))) -> 3(4(1(0(1(5(x1)))))) 588.65/152.66 0(1(1(5(x1)))) -> 3(4(1(5(0(1(x1)))))) 588.65/152.66 0(1(1(5(x1)))) -> 3(5(4(1(0(1(x1)))))) 588.65/152.66 0(1(1(5(x1)))) -> 4(1(0(1(5(3(x1)))))) 588.65/152.66 0(1(1(5(x1)))) -> 4(1(0(1(5(4(x1)))))) 588.65/152.66 0(1(1(5(x1)))) -> 4(1(3(1(0(5(x1)))))) 588.65/152.66 0(1(1(5(x1)))) -> 4(1(4(1(0(5(x1)))))) 588.65/152.66 0(1(1(5(x1)))) -> 4(4(1(5(0(1(x1)))))) 588.65/152.66 0(1(1(5(x1)))) -> 5(4(1(3(1(0(x1)))))) 588.65/152.66 0(1(2(0(x1)))) -> 0(2(4(1(0(3(x1)))))) 588.65/152.66 0(1(3(5(x1)))) -> 0(3(5(4(1(x1))))) 588.65/152.66 0(1(4(5(x1)))) -> 0(3(5(4(1(x1))))) 588.65/152.66 0(1(4(5(x1)))) -> 4(4(0(1(5(3(x1)))))) 588.65/152.66 0(2(4(5(x1)))) -> 4(0(2(3(5(x1))))) 588.65/152.66 0(2(4(5(x1)))) -> 4(4(0(2(5(x1))))) 588.65/152.66 0(2(4(5(x1)))) -> 4(0(3(2(3(5(x1)))))) 588.65/152.66 0(0(2(1(5(x1))))) -> 0(0(2(5(4(1(x1)))))) 588.65/152.66 0(0(2(4(5(x1))))) -> 0(0(4(4(2(5(x1)))))) 588.65/152.66 0(1(0(4(5(x1))))) -> 0(4(0(0(1(5(x1)))))) 588.65/152.66 0(1(0(5(0(x1))))) -> 4(1(5(0(0(0(x1)))))) 588.65/152.66 0(1(1(0(5(x1))))) -> 1(0(4(0(1(5(x1)))))) 588.65/152.66 0(1(1(2(0(x1))))) -> 0(4(1(2(1(0(x1)))))) 588.65/152.66 0(1(1(2(0(x1))))) -> 4(1(2(1(0(0(x1)))))) 588.65/152.66 0(1(1(3(5(x1))))) -> 4(1(0(1(3(5(x1)))))) 588.65/152.66 0(1(1(3(5(x1))))) -> 5(4(1(0(3(1(x1)))))) 588.65/152.66 0(1(1(4(2(x1))))) -> 0(4(1(4(1(2(x1)))))) 588.65/152.66 0(1(1(4(2(x1))))) -> 4(1(3(1(2(0(x1)))))) 588.65/152.66 0(1(1(4(2(x1))))) -> 4(2(4(1(0(1(x1)))))) 588.65/152.66 0(1(1(4(5(x1))))) -> 0(5(4(1(3(1(x1)))))) 588.65/152.66 0(1(1(4(5(x1))))) -> 0(5(4(1(4(1(x1)))))) 588.65/152.66 0(1(1(4(5(x1))))) -> 2(4(1(0(1(5(x1)))))) 588.65/152.66 0(1(2(0(2(x1))))) -> 0(4(0(1(2(2(x1)))))) 588.65/152.66 0(1(2(1(5(x1))))) -> 0(1(4(1(2(5(x1)))))) 588.65/152.66 0(1(4(5(0(x1))))) -> 0(5(4(1(0(3(x1)))))) 588.65/152.66 0(1(5(1(5(x1))))) -> 5(4(1(0(1(5(x1)))))) 588.65/152.66 0(2(0(1(5(x1))))) -> 1(0(0(2(3(5(x1)))))) 588.65/152.66 0(2(0(4(5(x1))))) -> 0(0(2(4(1(5(x1)))))) 588.65/152.66 0(2(0(5(0(x1))))) -> 0(2(5(0(3(0(x1)))))) 588.65/152.66 0(2(3(1(5(x1))))) -> 0(0(1(2(3(5(x1)))))) 588.65/152.66 0(2(3(1(5(x1))))) -> 0(2(5(3(4(1(x1)))))) 588.65/152.66 0(2(3(1(5(x1))))) -> 0(3(5(2(4(1(x1)))))) 588.65/152.66 0(2(3(1(5(x1))))) -> 2(0(4(1(3(5(x1)))))) 588.65/152.66 0(2(3(1(5(x1))))) -> 2(0(4(1(5(3(x1)))))) 588.65/152.66 0(2(3(1(5(x1))))) -> 2(3(5(3(0(1(x1)))))) 588.65/152.66 0(2(3(1(5(x1))))) -> 2(5(3(4(1(0(x1)))))) 588.65/152.66 0(2(3(1(5(x1))))) -> 4(1(0(5(2(3(x1)))))) 588.65/152.66 0(2(3(1(5(x1))))) -> 4(1(3(0(2(5(x1)))))) 588.65/152.66 0(2(3(1(5(x1))))) -> 4(1(5(2(0(3(x1)))))) 588.65/152.66 0(2(5(1(2(x1))))) -> 0(2(3(2(1(5(x1)))))) 588.65/152.66 0(2(5(1(5(x1))))) -> 0(3(5(2(1(5(x1)))))) 588.65/152.66 0(2(5(1(5(x1))))) -> 0(4(1(5(2(5(x1)))))) 588.65/152.66 0(2(5(1(5(x1))))) -> 2(4(1(5(0(5(x1)))))) 588.65/152.66 0(2(5(1(5(x1))))) -> 4(1(0(5(2(5(x1)))))) 588.65/152.66 0(2(5(1(5(x1))))) -> 4(1(5(5(2(0(x1)))))) 588.65/152.66 0(3(5(1(5(x1))))) -> 5(0(3(5(4(1(x1)))))) 588.65/152.66 0(4(2(0(2(x1))))) -> 0(0(4(3(2(2(x1)))))) 588.65/152.66 0(4(2(1(5(x1))))) -> 0(2(5(4(4(1(x1)))))) 588.65/152.66 0(4(2(1(5(x1))))) -> 0(4(1(5(3(2(x1)))))) 588.65/152.66 0(4(2(1(5(x1))))) -> 2(4(1(0(0(5(x1)))))) 588.65/152.66 0(4(2(1(5(x1))))) -> 2(4(1(3(0(5(x1)))))) 588.65/152.66 0(4(2(1(5(x1))))) -> 2(4(1(5(4(0(x1)))))) 588.65/152.66 0(4(2(1(5(x1))))) -> 3(0(1(5(2(4(x1)))))) 588.65/152.66 0(4(2(1(5(x1))))) -> 3(0(5(2(4(1(x1)))))) 588.65/152.66 0(4(2(1(5(x1))))) -> 4(1(3(2(5(0(x1)))))) 588.65/152.66 0(4(2(1(5(x1))))) -> 4(4(0(1(5(2(x1)))))) 588.65/152.66 0(4(5(1(5(x1))))) -> 5(4(1(5(0(4(x1)))))) 588.65/152.66 588.65/152.66 Q is empty. 588.65/152.66 588.65/152.66 ---------------------------------------- 588.65/152.66 588.65/152.66 (1) QTRS Reverse (EQUIVALENT) 588.65/152.66 We applied the QTRS Reverse Processor [REVERSE]. 588.65/152.66 ---------------------------------------- 588.65/152.66 588.65/152.66 (2) 588.65/152.66 Obligation: 588.65/152.66 Q restricted rewrite system: 588.65/152.66 The TRS R consists of the following rules: 588.65/152.66 588.65/152.66 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.66 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.66 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.66 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.66 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.66 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.66 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.66 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.66 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.66 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.66 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.66 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.66 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.66 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.66 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.66 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.66 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.66 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.66 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.66 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.66 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.66 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.66 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.66 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.66 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.66 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.66 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.66 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.66 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.66 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.66 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.66 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.66 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.66 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.66 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.66 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.66 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.66 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.66 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.66 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.66 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.66 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.66 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.66 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.66 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.66 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.66 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.66 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.66 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.66 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.66 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.66 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.66 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.66 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.66 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.66 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.66 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.66 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.66 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.66 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.66 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.66 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.66 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.66 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.66 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.66 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.66 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.66 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.66 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.66 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.66 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.66 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.66 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.66 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.66 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.66 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.66 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.66 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.66 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.66 588.65/152.66 Q is empty. 588.65/152.66 588.65/152.66 ---------------------------------------- 588.65/152.66 588.65/152.66 (3) DependencyPairsProof (EQUIVALENT) 588.65/152.66 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 588.65/152.66 ---------------------------------------- 588.65/152.66 588.65/152.66 (4) 588.65/152.66 Obligation: 588.65/152.66 Q DP problem: 588.65/152.66 The TRS P consists of the following rules: 588.65/152.66 588.65/152.66 2^1(1(1(0(x1)))) -> 2^1(3(1(0(1(x1))))) 588.65/152.66 2^1(1(1(0(x1)))) -> 0^1(1(x1)) 588.65/152.66 2^1(1(1(0(x1)))) -> 2^1(1(0(1(4(x1))))) 588.65/152.66 2^1(1(1(0(x1)))) -> 0^1(1(4(x1))) 588.65/152.66 2^1(1(1(0(x1)))) -> 2^1(3(1(4(1(0(x1)))))) 588.65/152.66 2^1(1(1(0(x1)))) -> 2^1(1(4(1(4(0(x1)))))) 588.65/152.66 2^1(1(1(0(x1)))) -> 2^1(1(3(0(1(4(x1)))))) 588.65/152.66 2^1(1(1(0(x1)))) -> 2^1(0(1(3(1(4(x1)))))) 588.65/152.66 2^1(1(1(0(x1)))) -> 0^1(1(3(1(4(x1))))) 588.65/152.66 5^1(1(1(0(x1)))) -> 5^1(1(0(1(4(x1))))) 588.65/152.66 5^1(1(1(0(x1)))) -> 0^1(1(4(x1))) 588.65/152.66 5^1(1(1(0(x1)))) -> 0^1(1(4(5(x1)))) 588.65/152.66 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.66 5^1(1(1(0(x1)))) -> 5^1(1(0(1(4(0(x1)))))) 588.65/152.66 5^1(1(1(0(x1)))) -> 0^1(1(4(0(x1)))) 588.65/152.66 5^1(1(1(0(x1)))) -> 0^1(1(4(5(0(x1))))) 588.65/152.66 5^1(1(1(0(x1)))) -> 5^1(0(x1)) 588.65/152.66 5^1(1(1(0(x1)))) -> 5^1(1(3(1(0(1(x1)))))) 588.65/152.66 5^1(1(1(0(x1)))) -> 0^1(1(x1)) 588.65/152.66 5^1(1(1(0(x1)))) -> 5^1(1(0(4(4(1(x1)))))) 588.65/152.66 5^1(1(1(0(x1)))) -> 0^1(4(4(1(x1)))) 588.65/152.66 5^1(1(1(0(x1)))) -> 5^1(1(0(3(x1)))) 588.65/152.66 5^1(1(1(0(x1)))) -> 0^1(3(x1)) 588.65/152.66 5^1(1(1(0(x1)))) -> 5^1(1(0(1(4(3(x1)))))) 588.65/152.66 5^1(1(1(0(x1)))) -> 0^1(1(4(3(x1)))) 588.65/152.66 5^1(1(1(0(x1)))) -> 0^1(5(1(4(3(x1))))) 588.65/152.66 5^1(1(1(0(x1)))) -> 5^1(1(4(3(x1)))) 588.65/152.66 5^1(1(1(0(x1)))) -> 0^1(1(4(5(3(x1))))) 588.65/152.66 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.66 5^1(1(1(0(x1)))) -> 5^1(0(1(3(1(4(x1)))))) 588.65/152.66 5^1(1(1(0(x1)))) -> 0^1(1(3(1(4(x1))))) 588.65/152.66 5^1(1(1(0(x1)))) -> 5^1(0(1(4(1(4(x1)))))) 588.65/152.66 5^1(1(1(0(x1)))) -> 0^1(1(4(1(4(x1))))) 588.65/152.66 5^1(1(1(0(x1)))) -> 0^1(5(1(4(4(x1))))) 588.65/152.66 5^1(1(1(0(x1)))) -> 5^1(1(4(4(x1)))) 588.65/152.66 5^1(1(1(0(x1)))) -> 0^1(1(3(1(4(5(x1)))))) 588.65/152.66 0^1(2(1(0(x1)))) -> 0^1(1(4(2(0(x1))))) 588.65/152.66 0^1(2(1(0(x1)))) -> 2^1(0(x1)) 588.65/152.66 5^1(3(1(0(x1)))) -> 5^1(3(0(x1))) 588.65/152.66 5^1(4(1(0(x1)))) -> 5^1(3(0(x1))) 588.65/152.66 5^1(4(1(0(x1)))) -> 5^1(1(0(4(4(x1))))) 588.65/152.66 5^1(4(1(0(x1)))) -> 0^1(4(4(x1))) 588.65/152.66 5^1(4(2(0(x1)))) -> 5^1(3(2(0(4(x1))))) 588.65/152.66 5^1(4(2(0(x1)))) -> 2^1(0(4(x1))) 588.65/152.66 5^1(4(2(0(x1)))) -> 0^1(4(x1)) 588.65/152.66 5^1(4(2(0(x1)))) -> 5^1(2(0(4(4(x1))))) 588.65/152.66 5^1(4(2(0(x1)))) -> 2^1(0(4(4(x1)))) 588.65/152.66 5^1(4(2(0(x1)))) -> 0^1(4(4(x1))) 588.65/152.66 5^1(4(2(0(x1)))) -> 5^1(3(2(3(0(4(x1)))))) 588.65/152.66 5^1(4(2(0(x1)))) -> 2^1(3(0(4(x1)))) 588.65/152.66 5^1(1(2(0(0(x1))))) -> 5^1(2(0(0(x1)))) 588.65/152.66 5^1(4(2(0(0(x1))))) -> 5^1(2(4(4(0(0(x1)))))) 588.65/152.66 5^1(4(2(0(0(x1))))) -> 2^1(4(4(0(0(x1))))) 588.65/152.66 5^1(4(0(1(0(x1))))) -> 5^1(1(0(0(4(0(x1)))))) 588.65/152.66 5^1(4(0(1(0(x1))))) -> 0^1(0(4(0(x1)))) 588.65/152.66 5^1(4(0(1(0(x1))))) -> 0^1(4(0(x1))) 588.65/152.66 0^1(5(0(1(0(x1))))) -> 0^1(0(0(5(1(4(x1)))))) 588.65/152.66 0^1(5(0(1(0(x1))))) -> 0^1(0(5(1(4(x1))))) 588.65/152.66 0^1(5(0(1(0(x1))))) -> 0^1(5(1(4(x1)))) 588.65/152.66 0^1(5(0(1(0(x1))))) -> 5^1(1(4(x1))) 588.65/152.66 5^1(0(1(1(0(x1))))) -> 5^1(1(0(4(0(1(x1)))))) 588.65/152.66 5^1(0(1(1(0(x1))))) -> 0^1(4(0(1(x1)))) 588.65/152.66 5^1(0(1(1(0(x1))))) -> 0^1(1(x1)) 588.65/152.66 0^1(2(1(1(0(x1))))) -> 0^1(1(2(1(4(0(x1)))))) 588.65/152.66 0^1(2(1(1(0(x1))))) -> 2^1(1(4(0(x1)))) 588.65/152.66 0^1(2(1(1(0(x1))))) -> 0^1(0(1(2(1(4(x1)))))) 588.65/152.66 0^1(2(1(1(0(x1))))) -> 0^1(1(2(1(4(x1))))) 588.65/152.66 0^1(2(1(1(0(x1))))) -> 2^1(1(4(x1))) 588.65/152.66 5^1(3(1(1(0(x1))))) -> 5^1(3(1(0(1(4(x1)))))) 588.65/152.66 5^1(3(1(1(0(x1))))) -> 0^1(1(4(x1))) 588.65/152.66 5^1(3(1(1(0(x1))))) -> 0^1(1(4(5(x1)))) 588.65/152.66 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.66 2^1(4(1(1(0(x1))))) -> 2^1(1(4(1(4(0(x1)))))) 588.65/152.66 2^1(4(1(1(0(x1))))) -> 0^1(2(1(3(1(4(x1)))))) 588.65/152.66 2^1(4(1(1(0(x1))))) -> 2^1(1(3(1(4(x1))))) 588.65/152.66 2^1(4(1(1(0(x1))))) -> 0^1(1(4(2(4(x1))))) 588.65/152.66 2^1(4(1(1(0(x1))))) -> 2^1(4(x1)) 588.65/152.66 5^1(4(1(1(0(x1))))) -> 5^1(0(x1)) 588.65/152.66 5^1(4(1(1(0(x1))))) -> 5^1(1(0(1(4(2(x1)))))) 588.65/152.66 5^1(4(1(1(0(x1))))) -> 0^1(1(4(2(x1)))) 588.65/152.66 5^1(4(1(1(0(x1))))) -> 2^1(x1) 588.65/152.66 2^1(0(2(1(0(x1))))) -> 2^1(2(1(0(4(0(x1)))))) 588.65/152.66 2^1(0(2(1(0(x1))))) -> 2^1(1(0(4(0(x1))))) 588.65/152.66 2^1(0(2(1(0(x1))))) -> 0^1(4(0(x1))) 588.65/152.66 5^1(1(2(1(0(x1))))) -> 5^1(2(1(4(1(0(x1)))))) 588.65/152.66 5^1(1(2(1(0(x1))))) -> 2^1(1(4(1(0(x1))))) 588.65/152.66 0^1(5(4(1(0(x1))))) -> 0^1(1(4(5(0(x1))))) 588.65/152.66 0^1(5(4(1(0(x1))))) -> 5^1(0(x1)) 588.65/152.66 5^1(1(5(1(0(x1))))) -> 5^1(1(0(1(4(5(x1)))))) 588.65/152.66 5^1(1(5(1(0(x1))))) -> 0^1(1(4(5(x1)))) 588.65/152.66 5^1(1(5(1(0(x1))))) -> 5^1(x1) 588.65/152.66 5^1(1(0(2(0(x1))))) -> 5^1(3(2(0(0(1(x1)))))) 588.65/152.66 5^1(1(0(2(0(x1))))) -> 2^1(0(0(1(x1)))) 588.65/152.66 5^1(1(0(2(0(x1))))) -> 0^1(0(1(x1))) 588.65/152.66 5^1(1(0(2(0(x1))))) -> 0^1(1(x1)) 588.65/152.66 5^1(4(0(2(0(x1))))) -> 5^1(1(4(2(0(0(x1)))))) 588.65/152.66 5^1(4(0(2(0(x1))))) -> 2^1(0(0(x1))) 588.65/152.66 5^1(4(0(2(0(x1))))) -> 0^1(0(x1)) 588.65/152.66 0^1(5(0(2(0(x1))))) -> 0^1(3(0(5(2(0(x1)))))) 588.65/152.66 0^1(5(0(2(0(x1))))) -> 0^1(5(2(0(x1)))) 588.65/152.66 0^1(5(0(2(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.66 5^1(1(3(2(0(x1))))) -> 5^1(3(2(1(0(0(x1)))))) 588.65/152.66 5^1(1(3(2(0(x1))))) -> 2^1(1(0(0(x1)))) 588.65/152.66 5^1(1(3(2(0(x1))))) -> 0^1(0(x1)) 588.65/152.66 5^1(1(3(2(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.66 5^1(1(3(2(0(x1))))) -> 2^1(5(3(0(x1)))) 588.65/152.66 5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1))) 588.65/152.66 5^1(1(3(2(0(x1))))) -> 5^1(3(1(4(0(2(x1)))))) 588.65/152.66 5^1(1(3(2(0(x1))))) -> 0^1(2(x1)) 588.65/152.66 5^1(1(3(2(0(x1))))) -> 2^1(x1) 588.65/152.66 5^1(1(3(2(0(x1))))) -> 5^1(1(4(0(2(x1))))) 588.65/152.66 5^1(1(3(2(0(x1))))) -> 0^1(3(5(3(2(x1))))) 588.65/152.66 5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1))) 588.65/152.66 5^1(1(3(2(0(x1))))) -> 0^1(1(4(3(5(2(x1)))))) 588.65/152.66 5^1(1(3(2(0(x1))))) -> 5^1(2(x1)) 588.65/152.66 5^1(1(3(2(0(x1))))) -> 2^1(5(0(1(4(x1))))) 588.65/152.66 5^1(1(3(2(0(x1))))) -> 5^1(0(1(4(x1)))) 588.65/152.66 5^1(1(3(2(0(x1))))) -> 0^1(1(4(x1))) 588.65/152.66 5^1(1(3(2(0(x1))))) -> 5^1(2(0(3(1(4(x1)))))) 588.65/152.66 5^1(1(3(2(0(x1))))) -> 2^1(0(3(1(4(x1))))) 588.65/152.66 5^1(1(3(2(0(x1))))) -> 0^1(3(1(4(x1)))) 588.65/152.66 5^1(1(3(2(0(x1))))) -> 0^1(2(5(1(4(x1))))) 588.65/152.66 5^1(1(3(2(0(x1))))) -> 2^1(5(1(4(x1)))) 588.65/152.66 5^1(1(3(2(0(x1))))) -> 5^1(1(4(x1))) 588.65/152.66 2^1(1(5(2(0(x1))))) -> 5^1(1(2(3(2(0(x1)))))) 588.65/152.66 2^1(1(5(2(0(x1))))) -> 2^1(3(2(0(x1)))) 588.65/152.66 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) 588.65/152.66 5^1(1(5(2(0(x1))))) -> 2^1(5(3(0(x1)))) 588.65/152.66 5^1(1(5(2(0(x1))))) -> 5^1(3(0(x1))) 588.65/152.67 5^1(1(5(2(0(x1))))) -> 5^1(2(5(1(4(0(x1)))))) 588.65/152.67 5^1(1(5(2(0(x1))))) -> 2^1(5(1(4(0(x1))))) 588.65/152.67 5^1(1(5(2(0(x1))))) -> 5^1(1(4(0(x1)))) 588.65/152.67 5^1(1(5(2(0(x1))))) -> 5^1(0(5(1(4(2(x1)))))) 588.65/152.67 5^1(1(5(2(0(x1))))) -> 0^1(5(1(4(2(x1))))) 588.65/152.67 5^1(1(5(2(0(x1))))) -> 5^1(1(4(2(x1)))) 588.65/152.67 5^1(1(5(2(0(x1))))) -> 2^1(x1) 588.65/152.67 5^1(1(5(2(0(x1))))) -> 5^1(2(5(0(1(4(x1)))))) 588.65/152.67 5^1(1(5(2(0(x1))))) -> 2^1(5(0(1(4(x1))))) 588.65/152.67 5^1(1(5(2(0(x1))))) -> 5^1(0(1(4(x1)))) 588.65/152.67 5^1(1(5(2(0(x1))))) -> 0^1(1(4(x1))) 588.65/152.67 5^1(1(5(2(0(x1))))) -> 0^1(2(5(5(1(4(x1)))))) 588.65/152.67 5^1(1(5(2(0(x1))))) -> 2^1(5(5(1(4(x1))))) 588.65/152.67 5^1(1(5(2(0(x1))))) -> 5^1(5(1(4(x1)))) 588.65/152.67 5^1(1(5(2(0(x1))))) -> 5^1(1(4(x1))) 588.65/152.67 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.67 5^1(1(5(3(0(x1))))) -> 0^1(5(x1)) 588.65/152.67 5^1(1(5(3(0(x1))))) -> 5^1(x1) 588.65/152.67 2^1(0(2(4(0(x1))))) -> 2^1(2(3(4(0(0(x1)))))) 588.65/152.67 2^1(0(2(4(0(x1))))) -> 2^1(3(4(0(0(x1))))) 588.65/152.67 2^1(0(2(4(0(x1))))) -> 0^1(0(x1)) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 2^1(0(x1)) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 2^1(3(5(1(4(0(x1)))))) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 5^1(1(4(0(x1)))) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 5^1(0(0(1(4(2(x1)))))) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 0^1(0(1(4(2(x1))))) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 0^1(1(4(2(x1)))) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 2^1(x1) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 5^1(0(3(1(4(2(x1)))))) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 0^1(3(1(4(2(x1))))) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 0^1(4(5(1(4(2(x1)))))) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 5^1(1(4(2(x1)))) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 2^1(5(1(0(3(x1))))) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 5^1(1(0(3(x1)))) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 0^1(3(x1)) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 2^1(5(0(3(x1)))) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 5^1(0(3(x1))) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 0^1(5(2(3(1(4(x1)))))) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 5^1(2(3(1(4(x1))))) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 2^1(3(1(4(x1)))) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 2^1(5(1(0(4(4(x1)))))) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 5^1(1(0(4(4(x1))))) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 0^1(4(4(x1))) 588.65/152.67 5^1(1(5(4(0(x1))))) -> 0^1(5(1(4(5(x1))))) 588.65/152.67 5^1(1(5(4(0(x1))))) -> 5^1(1(4(5(x1)))) 588.65/152.67 5^1(1(5(4(0(x1))))) -> 5^1(x1) 588.65/152.67 588.65/152.67 The TRS R consists of the following rules: 588.65/152.67 588.65/152.67 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.67 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.67 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.67 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.67 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.67 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.67 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.67 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.67 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.67 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.67 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.67 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.67 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.67 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.67 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.67 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.67 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.67 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.67 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.67 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.67 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.67 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.67 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.67 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.67 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.67 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.67 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.67 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.67 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.67 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.67 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.67 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.67 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.67 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.67 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.67 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.67 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.67 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.67 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.67 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.67 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.67 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.67 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.67 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.67 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.67 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.67 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.67 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.67 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.67 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.67 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.67 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.67 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.67 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.67 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.67 588.65/152.67 Q is empty. 588.65/152.67 We have to consider all minimal (P,Q,R)-chains. 588.65/152.67 ---------------------------------------- 588.65/152.67 588.65/152.67 (5) DependencyGraphProof (EQUIVALENT) 588.65/152.67 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 137 less nodes. 588.65/152.67 ---------------------------------------- 588.65/152.67 588.65/152.67 (6) 588.65/152.67 Complex Obligation (AND) 588.65/152.67 588.65/152.67 ---------------------------------------- 588.65/152.67 588.65/152.67 (7) 588.65/152.67 Obligation: 588.65/152.67 Q DP problem: 588.65/152.67 The TRS P consists of the following rules: 588.65/152.67 588.65/152.67 2^1(4(1(1(0(x1))))) -> 2^1(4(x1)) 588.65/152.67 588.65/152.67 The TRS R consists of the following rules: 588.65/152.67 588.65/152.67 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.67 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.67 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.67 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.67 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.67 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.67 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.67 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.67 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.67 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.67 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.67 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.67 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.67 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.67 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.67 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.67 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.67 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.67 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.67 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.67 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.67 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.67 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.67 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.67 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.67 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.67 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.67 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.67 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.67 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.67 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.67 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.67 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.67 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.67 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.67 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.67 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.67 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.67 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.67 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.67 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.67 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.67 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.67 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.67 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.67 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.67 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.67 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.67 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.67 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.67 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.67 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.67 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.67 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.67 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.67 588.65/152.67 Q is empty. 588.65/152.67 We have to consider all minimal (P,Q,R)-chains. 588.65/152.67 ---------------------------------------- 588.65/152.67 588.65/152.67 (8) UsableRulesProof (EQUIVALENT) 588.65/152.67 We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. 588.65/152.67 ---------------------------------------- 588.65/152.67 588.65/152.67 (9) 588.65/152.67 Obligation: 588.65/152.67 Q DP problem: 588.65/152.67 The TRS P consists of the following rules: 588.65/152.67 588.65/152.67 2^1(4(1(1(0(x1))))) -> 2^1(4(x1)) 588.65/152.67 588.65/152.67 R is empty. 588.65/152.67 Q is empty. 588.65/152.67 We have to consider all minimal (P,Q,R)-chains. 588.65/152.67 ---------------------------------------- 588.65/152.67 588.65/152.67 (10) MRRProof (EQUIVALENT) 588.65/152.67 By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. 588.65/152.67 588.65/152.67 Strictly oriented dependency pairs: 588.65/152.67 588.65/152.67 2^1(4(1(1(0(x1))))) -> 2^1(4(x1)) 588.65/152.67 588.65/152.67 588.65/152.67 Used ordering: Polynomial interpretation [POLO]: 588.65/152.67 588.65/152.67 POL(0(x_1)) = 2*x_1 588.65/152.67 POL(1(x_1)) = 1 + x_1 588.65/152.67 POL(2^1(x_1)) = x_1 588.65/152.67 POL(4(x_1)) = x_1 588.65/152.67 588.65/152.67 588.65/152.67 ---------------------------------------- 588.65/152.67 588.65/152.67 (11) 588.65/152.67 Obligation: 588.65/152.67 Q DP problem: 588.65/152.67 P is empty. 588.65/152.67 R is empty. 588.65/152.67 Q is empty. 588.65/152.67 We have to consider all minimal (P,Q,R)-chains. 588.65/152.67 ---------------------------------------- 588.65/152.67 588.65/152.67 (12) PisEmptyProof (EQUIVALENT) 588.65/152.67 The TRS P is empty. Hence, there is no (P,Q,R) chain. 588.65/152.67 ---------------------------------------- 588.65/152.67 588.65/152.67 (13) 588.65/152.67 YES 588.65/152.67 588.65/152.67 ---------------------------------------- 588.65/152.67 588.65/152.67 (14) 588.65/152.67 Obligation: 588.65/152.67 Q DP problem: 588.65/152.67 The TRS P consists of the following rules: 588.65/152.67 588.65/152.67 2^1(0(2(4(0(x1))))) -> 0^1(0(x1)) 588.65/152.67 0^1(2(1(0(x1)))) -> 2^1(0(x1)) 588.65/152.67 0^1(5(4(1(0(x1))))) -> 5^1(0(x1)) 588.65/152.67 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.67 5^1(1(1(0(x1)))) -> 5^1(0(x1)) 588.65/152.67 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.67 5^1(3(1(0(x1)))) -> 5^1(3(0(x1))) 588.65/152.67 5^1(3(1(1(0(x1))))) -> 5^1(3(1(0(1(4(x1)))))) 588.65/152.67 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.67 5^1(4(1(0(x1)))) -> 5^1(3(0(x1))) 588.65/152.67 5^1(1(2(0(0(x1))))) -> 5^1(2(0(0(x1)))) 588.65/152.67 5^1(4(1(1(0(x1))))) -> 5^1(0(x1)) 588.65/152.67 5^1(4(1(1(0(x1))))) -> 2^1(x1) 588.65/152.67 5^1(1(5(1(0(x1))))) -> 5^1(x1) 588.65/152.67 5^1(4(0(2(0(x1))))) -> 2^1(0(0(x1))) 588.65/152.67 5^1(4(0(2(0(x1))))) -> 0^1(0(x1)) 588.65/152.67 0^1(5(0(2(0(x1))))) -> 0^1(5(2(0(x1)))) 588.65/152.67 0^1(5(0(2(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.67 5^1(1(3(2(0(x1))))) -> 5^1(3(2(1(0(0(x1)))))) 588.65/152.67 5^1(1(3(2(0(x1))))) -> 0^1(0(x1)) 588.65/152.67 5^1(1(3(2(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.67 5^1(1(3(2(0(x1))))) -> 2^1(5(3(0(x1)))) 588.65/152.67 5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1))) 588.65/152.67 5^1(1(3(2(0(x1))))) -> 0^1(2(x1)) 588.65/152.67 5^1(1(3(2(0(x1))))) -> 2^1(x1) 588.65/152.67 5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1))) 588.65/152.67 5^1(1(3(2(0(x1))))) -> 5^1(2(x1)) 588.65/152.67 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) 588.65/152.67 5^1(1(5(2(0(x1))))) -> 2^1(5(3(0(x1)))) 588.65/152.67 5^1(1(5(2(0(x1))))) -> 5^1(3(0(x1))) 588.65/152.67 5^1(1(5(2(0(x1))))) -> 2^1(x1) 588.65/152.67 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.67 5^1(1(5(3(0(x1))))) -> 0^1(5(x1)) 588.65/152.67 5^1(1(5(3(0(x1))))) -> 5^1(x1) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 2^1(0(x1)) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 2^1(x1) 588.65/152.67 5^1(1(5(4(0(x1))))) -> 5^1(x1) 588.65/152.67 588.65/152.67 The TRS R consists of the following rules: 588.65/152.67 588.65/152.67 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.67 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.67 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.67 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.67 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.67 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.67 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.67 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.67 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.67 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.67 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.67 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.67 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.67 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.67 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.67 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.67 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.67 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.67 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.67 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.67 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.67 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.67 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.67 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.67 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.67 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.67 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.67 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.67 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.67 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.67 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.67 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.67 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.67 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.67 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.67 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.67 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.67 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.67 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.67 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.67 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.67 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.67 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.67 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.67 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.67 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.67 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.67 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.67 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.67 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.67 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.67 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.67 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.67 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.67 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.67 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.67 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.67 588.65/152.67 Q is empty. 588.65/152.67 We have to consider all minimal (P,Q,R)-chains. 588.65/152.67 ---------------------------------------- 588.65/152.67 588.65/152.67 (15) QDPOrderProof (EQUIVALENT) 588.65/152.67 We use the reduction pair processor [LPAR04,JAR06]. 588.65/152.67 588.65/152.67 588.65/152.67 The following pairs can be oriented strictly and are deleted. 588.65/152.67 588.65/152.67 0^1(2(1(0(x1)))) -> 2^1(0(x1)) 588.65/152.67 5^1(4(1(1(0(x1))))) -> 2^1(x1) 588.65/152.67 5^1(4(0(2(0(x1))))) -> 2^1(0(0(x1))) 588.65/152.67 5^1(4(0(2(0(x1))))) -> 0^1(0(x1)) 588.65/152.67 5^1(1(3(2(0(x1))))) -> 0^1(0(x1)) 588.65/152.67 5^1(1(3(2(0(x1))))) -> 2^1(5(3(0(x1)))) 588.65/152.67 5^1(1(3(2(0(x1))))) -> 2^1(x1) 588.65/152.67 5^1(1(5(2(0(x1))))) -> 2^1(5(3(0(x1)))) 588.65/152.67 5^1(1(5(2(0(x1))))) -> 2^1(x1) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 2^1(0(x1)) 588.65/152.67 5^1(1(2(4(0(x1))))) -> 2^1(x1) 588.65/152.67 The remaining pairs can at least be oriented weakly. 588.65/152.67 Used ordering: Polynomial interpretation [POLO]: 588.65/152.67 588.65/152.67 POL(0(x_1)) = 0 588.65/152.67 POL(0^1(x_1)) = x_1 588.65/152.67 POL(1(x_1)) = 0 588.65/152.67 POL(2(x_1)) = 1 588.65/152.67 POL(2^1(x_1)) = 0 588.65/152.67 POL(3(x_1)) = 0 588.65/152.67 POL(4(x_1)) = 0 588.65/152.67 POL(5(x_1)) = 1 588.65/152.67 POL(5^1(x_1)) = 1 588.65/152.67 588.65/152.67 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 588.65/152.67 588.65/152.67 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.67 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.67 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.67 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.67 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.67 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.67 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.67 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.67 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.68 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.68 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.68 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.68 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.68 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.68 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.68 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.68 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.68 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.68 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.68 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.68 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.68 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.68 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.68 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.68 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.68 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.68 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.68 588.65/152.68 588.65/152.68 ---------------------------------------- 588.65/152.68 588.65/152.68 (16) 588.65/152.68 Obligation: 588.65/152.68 Q DP problem: 588.65/152.68 The TRS P consists of the following rules: 588.65/152.68 588.65/152.68 2^1(0(2(4(0(x1))))) -> 0^1(0(x1)) 588.65/152.68 0^1(5(4(1(0(x1))))) -> 5^1(0(x1)) 588.65/152.68 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.68 5^1(1(1(0(x1)))) -> 5^1(0(x1)) 588.65/152.68 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.68 5^1(3(1(0(x1)))) -> 5^1(3(0(x1))) 588.65/152.68 5^1(3(1(1(0(x1))))) -> 5^1(3(1(0(1(4(x1)))))) 588.65/152.68 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.68 5^1(4(1(0(x1)))) -> 5^1(3(0(x1))) 588.65/152.68 5^1(1(2(0(0(x1))))) -> 5^1(2(0(0(x1)))) 588.65/152.68 5^1(4(1(1(0(x1))))) -> 5^1(0(x1)) 588.65/152.68 5^1(1(5(1(0(x1))))) -> 5^1(x1) 588.65/152.68 0^1(5(0(2(0(x1))))) -> 0^1(5(2(0(x1)))) 588.65/152.68 0^1(5(0(2(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(3(2(1(0(0(x1)))))) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1))) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 0^1(2(x1)) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1))) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(2(x1)) 588.65/152.68 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) 588.65/152.68 5^1(1(5(2(0(x1))))) -> 5^1(3(0(x1))) 588.65/152.68 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.68 5^1(1(5(3(0(x1))))) -> 0^1(5(x1)) 588.65/152.68 5^1(1(5(3(0(x1))))) -> 5^1(x1) 588.65/152.68 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.68 5^1(1(5(4(0(x1))))) -> 5^1(x1) 588.65/152.68 588.65/152.68 The TRS R consists of the following rules: 588.65/152.68 588.65/152.68 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.68 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.68 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.68 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.68 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.68 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.68 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.68 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.68 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.68 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.68 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.68 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.68 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.68 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.68 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.68 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.68 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.68 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.68 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.68 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.68 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.68 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.68 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.68 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.68 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.68 588.65/152.68 Q is empty. 588.65/152.68 We have to consider all minimal (P,Q,R)-chains. 588.65/152.68 ---------------------------------------- 588.65/152.68 588.65/152.68 (17) DependencyGraphProof (EQUIVALENT) 588.65/152.68 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 588.65/152.68 ---------------------------------------- 588.65/152.68 588.65/152.68 (18) 588.65/152.68 Obligation: 588.65/152.68 Q DP problem: 588.65/152.68 The TRS P consists of the following rules: 588.65/152.68 588.65/152.68 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.68 5^1(1(1(0(x1)))) -> 5^1(0(x1)) 588.65/152.68 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.68 5^1(3(1(0(x1)))) -> 5^1(3(0(x1))) 588.65/152.68 5^1(3(1(1(0(x1))))) -> 5^1(3(1(0(1(4(x1)))))) 588.65/152.68 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.68 5^1(4(1(0(x1)))) -> 5^1(3(0(x1))) 588.65/152.68 5^1(1(2(0(0(x1))))) -> 5^1(2(0(0(x1)))) 588.65/152.68 5^1(4(1(1(0(x1))))) -> 5^1(0(x1)) 588.65/152.68 5^1(1(5(1(0(x1))))) -> 5^1(x1) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(3(2(1(0(0(x1)))))) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1))) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 0^1(2(x1)) 588.65/152.68 0^1(5(4(1(0(x1))))) -> 5^1(0(x1)) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1))) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(2(x1)) 588.65/152.68 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) 588.65/152.68 5^1(1(5(2(0(x1))))) -> 5^1(3(0(x1))) 588.65/152.68 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.68 5^1(1(5(3(0(x1))))) -> 0^1(5(x1)) 588.65/152.68 0^1(5(0(2(0(x1))))) -> 0^1(5(2(0(x1)))) 588.65/152.68 0^1(5(0(2(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.68 5^1(1(5(3(0(x1))))) -> 5^1(x1) 588.65/152.68 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.68 5^1(1(5(4(0(x1))))) -> 5^1(x1) 588.65/152.68 588.65/152.68 The TRS R consists of the following rules: 588.65/152.68 588.65/152.68 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.68 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.68 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.68 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.68 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.68 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.68 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.68 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.68 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.68 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.68 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.68 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.68 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.68 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.68 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.68 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.68 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.68 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.68 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.68 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.68 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.68 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.68 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.68 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.68 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.68 588.65/152.68 Q is empty. 588.65/152.68 We have to consider all minimal (P,Q,R)-chains. 588.65/152.68 ---------------------------------------- 588.65/152.68 588.65/152.68 (19) QDPOrderProof (EQUIVALENT) 588.65/152.68 We use the reduction pair processor [LPAR04,JAR06]. 588.65/152.68 588.65/152.68 588.65/152.68 The following pairs can be oriented strictly and are deleted. 588.65/152.68 588.65/152.68 5^1(1(5(1(0(x1))))) -> 5^1(x1) 588.65/152.68 0^1(5(4(1(0(x1))))) -> 5^1(0(x1)) 588.65/152.68 5^1(1(5(2(0(x1))))) -> 5^1(3(0(x1))) 588.65/152.68 0^1(5(0(2(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.68 5^1(1(5(3(0(x1))))) -> 5^1(x1) 588.65/152.68 5^1(1(5(4(0(x1))))) -> 5^1(x1) 588.65/152.68 The remaining pairs can at least be oriented weakly. 588.65/152.68 Used ordering: Polynomial interpretation [POLO]: 588.65/152.68 588.65/152.68 POL(0(x_1)) = x_1 588.65/152.68 POL(0^1(x_1)) = x_1 588.65/152.68 POL(1(x_1)) = x_1 588.65/152.68 POL(2(x_1)) = x_1 588.65/152.68 POL(3(x_1)) = x_1 588.65/152.68 POL(4(x_1)) = x_1 588.65/152.68 POL(5(x_1)) = 1 + x_1 588.65/152.68 POL(5^1(x_1)) = x_1 588.65/152.68 588.65/152.68 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 588.65/152.68 588.65/152.68 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.68 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.68 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.68 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.68 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.68 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.68 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.68 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.68 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.68 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.68 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.68 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.68 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.68 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.68 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.68 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.68 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.68 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.68 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.68 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.68 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.68 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.68 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.68 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.68 588.65/152.68 588.65/152.68 ---------------------------------------- 588.65/152.68 588.65/152.68 (20) 588.65/152.68 Obligation: 588.65/152.68 Q DP problem: 588.65/152.68 The TRS P consists of the following rules: 588.65/152.68 588.65/152.68 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.68 5^1(1(1(0(x1)))) -> 5^1(0(x1)) 588.65/152.68 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.68 5^1(3(1(0(x1)))) -> 5^1(3(0(x1))) 588.65/152.68 5^1(3(1(1(0(x1))))) -> 5^1(3(1(0(1(4(x1)))))) 588.65/152.68 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.68 5^1(4(1(0(x1)))) -> 5^1(3(0(x1))) 588.65/152.68 5^1(1(2(0(0(x1))))) -> 5^1(2(0(0(x1)))) 588.65/152.68 5^1(4(1(1(0(x1))))) -> 5^1(0(x1)) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(3(2(1(0(0(x1)))))) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1))) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 0^1(2(x1)) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1))) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(2(x1)) 588.65/152.68 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) 588.65/152.68 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.68 5^1(1(5(3(0(x1))))) -> 0^1(5(x1)) 588.65/152.68 0^1(5(0(2(0(x1))))) -> 0^1(5(2(0(x1)))) 588.65/152.68 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.68 588.65/152.68 The TRS R consists of the following rules: 588.65/152.68 588.65/152.68 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.68 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.68 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.68 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.68 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.68 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.68 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.68 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.68 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.68 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.68 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.68 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.68 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.68 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.68 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.68 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.68 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.68 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.68 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.68 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.68 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.68 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.68 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.68 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.68 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.68 588.65/152.68 Q is empty. 588.65/152.68 We have to consider all minimal (P,Q,R)-chains. 588.65/152.68 ---------------------------------------- 588.65/152.68 588.65/152.68 (21) DependencyGraphProof (EQUIVALENT) 588.65/152.68 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes. 588.65/152.68 ---------------------------------------- 588.65/152.68 588.65/152.68 (22) 588.65/152.68 Complex Obligation (AND) 588.65/152.68 588.65/152.68 ---------------------------------------- 588.65/152.68 588.65/152.68 (23) 588.65/152.68 Obligation: 588.65/152.68 Q DP problem: 588.65/152.68 The TRS P consists of the following rules: 588.65/152.68 588.65/152.68 0^1(5(0(2(0(x1))))) -> 0^1(5(2(0(x1)))) 588.65/152.68 588.65/152.68 The TRS R consists of the following rules: 588.65/152.68 588.65/152.68 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.68 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.68 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.68 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.68 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.68 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.68 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.68 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.68 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.68 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.68 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.68 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.68 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.68 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.68 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.68 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.68 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.68 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.68 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.68 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.68 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.68 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.68 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.68 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.68 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.68 588.65/152.68 Q is empty. 588.65/152.68 We have to consider all minimal (P,Q,R)-chains. 588.65/152.68 ---------------------------------------- 588.65/152.68 588.65/152.68 (24) QDPOrderProof (EQUIVALENT) 588.65/152.68 We use the reduction pair processor [LPAR04,JAR06]. 588.65/152.68 588.65/152.68 588.65/152.68 The following pairs can be oriented strictly and are deleted. 588.65/152.68 588.65/152.68 0^1(5(0(2(0(x1))))) -> 0^1(5(2(0(x1)))) 588.65/152.68 The remaining pairs can at least be oriented weakly. 588.65/152.68 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 588.65/152.68 588.65/152.68 <<< 588.65/152.68 POL(0^1(x_1)) = [[-I]] + [[0A, -I, -I]] * x_1 588.65/152.68 >>> 588.65/152.68 588.65/152.68 <<< 588.65/152.68 POL(5(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 588.65/152.68 >>> 588.65/152.68 588.65/152.68 <<< 588.65/152.68 POL(0(x_1)) = [[1A], [0A], [0A]] + [[0A, 0A, 0A], [0A, -I, -I], [0A, 0A, 0A]] * x_1 588.65/152.68 >>> 588.65/152.68 588.65/152.68 <<< 588.65/152.68 POL(2(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, 0A], [0A, -I, 0A], [-I, -I, -I]] * x_1 588.65/152.68 >>> 588.65/152.68 588.65/152.68 <<< 588.65/152.68 POL(1(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, -I], [0A, 0A, 0A], [-I, 0A, -I]] * x_1 588.65/152.68 >>> 588.65/152.68 588.65/152.68 <<< 588.65/152.68 POL(3(x_1)) = [[0A], [-I], [0A]] + [[-I, -I, -I], [0A, -I, -I], [-I, -I, -I]] * x_1 588.65/152.68 >>> 588.65/152.68 588.65/152.68 <<< 588.65/152.68 POL(4(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 588.65/152.68 >>> 588.65/152.68 588.65/152.68 588.65/152.68 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 588.65/152.68 588.65/152.68 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.68 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.68 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.68 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.68 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.68 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.68 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.68 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.68 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.68 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.68 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.68 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.68 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.68 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.68 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.68 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.68 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.68 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.68 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.68 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.68 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.68 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.68 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.68 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.68 588.65/152.68 588.65/152.68 ---------------------------------------- 588.65/152.68 588.65/152.68 (25) 588.65/152.68 Obligation: 588.65/152.68 Q DP problem: 588.65/152.68 P is empty. 588.65/152.68 The TRS R consists of the following rules: 588.65/152.68 588.65/152.68 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.68 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.68 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.68 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.68 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.68 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.68 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.68 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.68 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.68 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.68 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.68 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.68 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.68 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.68 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.68 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.68 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.68 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.68 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.68 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.68 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.68 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.68 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.68 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.68 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.68 588.65/152.68 Q is empty. 588.65/152.68 We have to consider all minimal (P,Q,R)-chains. 588.65/152.68 ---------------------------------------- 588.65/152.68 588.65/152.68 (26) PisEmptyProof (EQUIVALENT) 588.65/152.68 The TRS P is empty. Hence, there is no (P,Q,R) chain. 588.65/152.68 ---------------------------------------- 588.65/152.68 588.65/152.68 (27) 588.65/152.68 YES 588.65/152.68 588.65/152.68 ---------------------------------------- 588.65/152.68 588.65/152.68 (28) 588.65/152.68 Obligation: 588.65/152.68 Q DP problem: 588.65/152.68 The TRS P consists of the following rules: 588.65/152.68 588.65/152.68 5^1(1(1(0(x1)))) -> 5^1(0(x1)) 588.65/152.68 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.68 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.68 5^1(3(1(0(x1)))) -> 5^1(3(0(x1))) 588.65/152.68 5^1(3(1(1(0(x1))))) -> 5^1(3(1(0(1(4(x1)))))) 588.65/152.68 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.68 5^1(4(1(0(x1)))) -> 5^1(3(0(x1))) 588.65/152.68 5^1(1(2(0(0(x1))))) -> 5^1(2(0(0(x1)))) 588.65/152.68 5^1(4(1(1(0(x1))))) -> 5^1(0(x1)) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(3(2(1(0(0(x1)))))) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1))) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1))) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(2(x1)) 588.65/152.68 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) 588.65/152.68 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.68 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.68 588.65/152.68 The TRS R consists of the following rules: 588.65/152.68 588.65/152.68 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.68 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.68 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.68 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.68 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.68 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.68 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.68 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.68 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.68 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.68 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.68 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.68 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.68 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.68 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.68 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.68 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.68 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.68 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.68 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.68 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.68 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.68 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.68 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.68 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.68 588.65/152.68 Q is empty. 588.65/152.68 We have to consider all minimal (P,Q,R)-chains. 588.65/152.68 ---------------------------------------- 588.65/152.68 588.65/152.68 (29) TransformationProof (EQUIVALENT) 588.65/152.68 By narrowing [LPAR04] the rule 5^1(1(1(0(x1)))) -> 5^1(0(x1)) at position [0] we obtained the following new rules [LPAR04]: 588.65/152.68 588.65/152.68 (5^1(1(1(0(2(1(0(x0))))))) -> 5^1(3(0(1(4(2(0(x0))))))),5^1(1(1(0(2(1(0(x0))))))) -> 5^1(3(0(1(4(2(0(x0)))))))) 588.65/152.68 (5^1(1(1(0(5(0(1(0(x0)))))))) -> 5^1(0(0(0(5(1(4(x0))))))),5^1(1(1(0(5(0(1(0(x0)))))))) -> 5^1(0(0(0(5(1(4(x0)))))))) 588.65/152.68 (5^1(1(1(0(2(1(1(0(x0)))))))) -> 5^1(0(1(2(1(4(0(x0))))))),5^1(1(1(0(2(1(1(0(x0)))))))) -> 5^1(0(1(2(1(4(0(x0)))))))) 588.65/152.68 (5^1(1(1(0(2(1(1(0(x0)))))))) -> 5^1(0(0(1(2(1(4(x0))))))),5^1(1(1(0(2(1(1(0(x0)))))))) -> 5^1(0(0(1(2(1(4(x0)))))))) 588.65/152.68 (5^1(1(1(0(5(4(1(0(x0)))))))) -> 5^1(3(0(1(4(5(0(x0))))))),5^1(1(1(0(5(4(1(0(x0)))))))) -> 5^1(3(0(1(4(5(0(x0)))))))) 588.65/152.68 (5^1(1(1(0(5(0(2(0(x0)))))))) -> 5^1(0(3(0(5(2(0(x0))))))),5^1(1(1(0(5(0(2(0(x0)))))))) -> 5^1(0(3(0(5(2(0(x0)))))))) 588.65/152.68 588.65/152.68 588.65/152.68 ---------------------------------------- 588.65/152.68 588.65/152.68 (30) 588.65/152.68 Obligation: 588.65/152.68 Q DP problem: 588.65/152.68 The TRS P consists of the following rules: 588.65/152.68 588.65/152.68 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.68 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.68 5^1(3(1(0(x1)))) -> 5^1(3(0(x1))) 588.65/152.68 5^1(3(1(1(0(x1))))) -> 5^1(3(1(0(1(4(x1)))))) 588.65/152.68 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.68 5^1(4(1(0(x1)))) -> 5^1(3(0(x1))) 588.65/152.68 5^1(1(2(0(0(x1))))) -> 5^1(2(0(0(x1)))) 588.65/152.68 5^1(4(1(1(0(x1))))) -> 5^1(0(x1)) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(3(2(1(0(0(x1)))))) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1))) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1))) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(2(x1)) 588.65/152.68 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) 588.65/152.68 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.68 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.68 5^1(1(1(0(2(1(0(x0))))))) -> 5^1(3(0(1(4(2(0(x0))))))) 588.65/152.68 5^1(1(1(0(5(0(1(0(x0)))))))) -> 5^1(0(0(0(5(1(4(x0))))))) 588.65/152.68 5^1(1(1(0(2(1(1(0(x0)))))))) -> 5^1(0(1(2(1(4(0(x0))))))) 588.65/152.68 5^1(1(1(0(2(1(1(0(x0)))))))) -> 5^1(0(0(1(2(1(4(x0))))))) 588.65/152.68 5^1(1(1(0(5(4(1(0(x0)))))))) -> 5^1(3(0(1(4(5(0(x0))))))) 588.65/152.68 5^1(1(1(0(5(0(2(0(x0)))))))) -> 5^1(0(3(0(5(2(0(x0))))))) 588.65/152.68 588.65/152.68 The TRS R consists of the following rules: 588.65/152.68 588.65/152.68 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.68 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.68 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.68 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.68 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.68 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.68 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.68 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.68 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.68 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.68 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.68 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.68 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.68 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.68 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.68 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.68 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.68 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.68 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.68 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.68 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.68 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.68 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.68 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.68 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.68 588.65/152.68 Q is empty. 588.65/152.68 We have to consider all minimal (P,Q,R)-chains. 588.65/152.68 ---------------------------------------- 588.65/152.68 588.65/152.68 (31) DependencyGraphProof (EQUIVALENT) 588.65/152.68 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes. 588.65/152.68 ---------------------------------------- 588.65/152.68 588.65/152.68 (32) 588.65/152.68 Obligation: 588.65/152.68 Q DP problem: 588.65/152.68 The TRS P consists of the following rules: 588.65/152.68 588.65/152.68 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.68 5^1(3(1(0(x1)))) -> 5^1(3(0(x1))) 588.65/152.68 5^1(3(1(1(0(x1))))) -> 5^1(3(1(0(1(4(x1)))))) 588.65/152.68 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.68 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.68 5^1(4(1(0(x1)))) -> 5^1(3(0(x1))) 588.65/152.68 5^1(1(2(0(0(x1))))) -> 5^1(2(0(0(x1)))) 588.65/152.68 5^1(4(1(1(0(x1))))) -> 5^1(0(x1)) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(3(2(1(0(0(x1)))))) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1))) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1))) 588.65/152.68 5^1(1(3(2(0(x1))))) -> 5^1(2(x1)) 588.65/152.68 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) 588.65/152.68 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.68 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.68 588.65/152.68 The TRS R consists of the following rules: 588.65/152.68 588.65/152.68 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.68 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.68 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.68 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.68 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.68 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.68 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.68 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.68 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.68 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.68 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.68 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.68 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.68 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.68 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.68 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.68 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.68 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.68 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.68 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.68 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.68 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.68 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.68 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.68 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.68 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.68 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.68 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.68 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.68 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.68 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.68 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.68 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.69 588.65/152.69 Q is empty. 588.65/152.69 We have to consider all minimal (P,Q,R)-chains. 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (33) TransformationProof (EQUIVALENT) 588.65/152.69 By narrowing [LPAR04] the rule 5^1(3(1(0(x1)))) -> 5^1(3(0(x1))) at position [0,0] we obtained the following new rules [LPAR04]: 588.65/152.69 588.65/152.69 (5^1(3(1(0(2(1(0(x0))))))) -> 5^1(3(3(0(1(4(2(0(x0)))))))),5^1(3(1(0(2(1(0(x0))))))) -> 5^1(3(3(0(1(4(2(0(x0))))))))) 588.65/152.69 (5^1(3(1(0(5(0(1(0(x0)))))))) -> 5^1(3(0(0(0(5(1(4(x0)))))))),5^1(3(1(0(5(0(1(0(x0)))))))) -> 5^1(3(0(0(0(5(1(4(x0))))))))) 588.65/152.69 (5^1(3(1(0(2(1(1(0(x0)))))))) -> 5^1(3(0(1(2(1(4(0(x0)))))))),5^1(3(1(0(2(1(1(0(x0)))))))) -> 5^1(3(0(1(2(1(4(0(x0))))))))) 588.65/152.69 (5^1(3(1(0(2(1(1(0(x0)))))))) -> 5^1(3(0(0(1(2(1(4(x0)))))))),5^1(3(1(0(2(1(1(0(x0)))))))) -> 5^1(3(0(0(1(2(1(4(x0))))))))) 588.65/152.69 (5^1(3(1(0(5(4(1(0(x0)))))))) -> 5^1(3(3(0(1(4(5(0(x0)))))))),5^1(3(1(0(5(4(1(0(x0)))))))) -> 5^1(3(3(0(1(4(5(0(x0))))))))) 588.65/152.69 (5^1(3(1(0(5(0(2(0(x0)))))))) -> 5^1(3(0(3(0(5(2(0(x0)))))))),5^1(3(1(0(5(0(2(0(x0)))))))) -> 5^1(3(0(3(0(5(2(0(x0))))))))) 588.65/152.69 588.65/152.69 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (34) 588.65/152.69 Obligation: 588.65/152.69 Q DP problem: 588.65/152.69 The TRS P consists of the following rules: 588.65/152.69 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.69 5^1(3(1(1(0(x1))))) -> 5^1(3(1(0(1(4(x1)))))) 588.65/152.69 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.69 5^1(4(1(0(x1)))) -> 5^1(3(0(x1))) 588.65/152.69 5^1(1(2(0(0(x1))))) -> 5^1(2(0(0(x1)))) 588.65/152.69 5^1(4(1(1(0(x1))))) -> 5^1(0(x1)) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(2(1(0(0(x1)))))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(2(x1)) 588.65/152.69 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) 588.65/152.69 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.69 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.69 5^1(3(1(0(2(1(0(x0))))))) -> 5^1(3(3(0(1(4(2(0(x0)))))))) 588.65/152.69 5^1(3(1(0(5(0(1(0(x0)))))))) -> 5^1(3(0(0(0(5(1(4(x0)))))))) 588.65/152.69 5^1(3(1(0(2(1(1(0(x0)))))))) -> 5^1(3(0(1(2(1(4(0(x0)))))))) 588.65/152.69 5^1(3(1(0(2(1(1(0(x0)))))))) -> 5^1(3(0(0(1(2(1(4(x0)))))))) 588.65/152.69 5^1(3(1(0(5(4(1(0(x0)))))))) -> 5^1(3(3(0(1(4(5(0(x0)))))))) 588.65/152.69 5^1(3(1(0(5(0(2(0(x0)))))))) -> 5^1(3(0(3(0(5(2(0(x0)))))))) 588.65/152.69 588.65/152.69 The TRS R consists of the following rules: 588.65/152.69 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.69 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.69 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.69 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.69 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.69 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.69 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.69 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.69 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.69 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.69 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.69 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.69 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.69 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.69 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.69 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.69 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.69 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.69 588.65/152.69 Q is empty. 588.65/152.69 We have to consider all minimal (P,Q,R)-chains. 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (35) DependencyGraphProof (EQUIVALENT) 588.65/152.69 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 7 less nodes. 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (36) 588.65/152.69 Obligation: 588.65/152.69 Q DP problem: 588.65/152.69 The TRS P consists of the following rules: 588.65/152.69 588.65/152.69 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.69 5^1(4(1(0(x1)))) -> 5^1(3(0(x1))) 588.65/152.69 5^1(1(2(0(0(x1))))) -> 5^1(2(0(0(x1)))) 588.65/152.69 5^1(4(1(1(0(x1))))) -> 5^1(0(x1)) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(2(1(0(0(x1)))))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(2(x1)) 588.65/152.69 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) 588.65/152.69 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.69 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.69 588.65/152.69 The TRS R consists of the following rules: 588.65/152.69 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.69 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.69 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.69 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.69 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.69 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.69 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.69 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.69 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.69 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.69 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.69 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.69 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.69 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.69 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.69 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.69 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.69 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.69 588.65/152.69 Q is empty. 588.65/152.69 We have to consider all minimal (P,Q,R)-chains. 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (37) TransformationProof (EQUIVALENT) 588.65/152.69 By narrowing [LPAR04] the rule 5^1(4(1(0(x1)))) -> 5^1(3(0(x1))) at position [0,0] we obtained the following new rules [LPAR04]: 588.65/152.69 588.65/152.69 (5^1(4(1(0(2(1(0(x0))))))) -> 5^1(3(3(0(1(4(2(0(x0)))))))),5^1(4(1(0(2(1(0(x0))))))) -> 5^1(3(3(0(1(4(2(0(x0))))))))) 588.65/152.69 (5^1(4(1(0(5(0(1(0(x0)))))))) -> 5^1(3(0(0(0(5(1(4(x0)))))))),5^1(4(1(0(5(0(1(0(x0)))))))) -> 5^1(3(0(0(0(5(1(4(x0))))))))) 588.65/152.69 (5^1(4(1(0(2(1(1(0(x0)))))))) -> 5^1(3(0(1(2(1(4(0(x0)))))))),5^1(4(1(0(2(1(1(0(x0)))))))) -> 5^1(3(0(1(2(1(4(0(x0))))))))) 588.65/152.69 (5^1(4(1(0(2(1(1(0(x0)))))))) -> 5^1(3(0(0(1(2(1(4(x0)))))))),5^1(4(1(0(2(1(1(0(x0)))))))) -> 5^1(3(0(0(1(2(1(4(x0))))))))) 588.65/152.69 (5^1(4(1(0(5(4(1(0(x0)))))))) -> 5^1(3(3(0(1(4(5(0(x0)))))))),5^1(4(1(0(5(4(1(0(x0)))))))) -> 5^1(3(3(0(1(4(5(0(x0))))))))) 588.65/152.69 (5^1(4(1(0(5(0(2(0(x0)))))))) -> 5^1(3(0(3(0(5(2(0(x0)))))))),5^1(4(1(0(5(0(2(0(x0)))))))) -> 5^1(3(0(3(0(5(2(0(x0))))))))) 588.65/152.69 588.65/152.69 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (38) 588.65/152.69 Obligation: 588.65/152.69 Q DP problem: 588.65/152.69 The TRS P consists of the following rules: 588.65/152.69 588.65/152.69 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.69 5^1(1(2(0(0(x1))))) -> 5^1(2(0(0(x1)))) 588.65/152.69 5^1(4(1(1(0(x1))))) -> 5^1(0(x1)) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(2(1(0(0(x1)))))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(2(x1)) 588.65/152.69 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) 588.65/152.69 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.69 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.69 5^1(4(1(0(2(1(0(x0))))))) -> 5^1(3(3(0(1(4(2(0(x0)))))))) 588.65/152.69 5^1(4(1(0(5(0(1(0(x0)))))))) -> 5^1(3(0(0(0(5(1(4(x0)))))))) 588.65/152.69 5^1(4(1(0(2(1(1(0(x0)))))))) -> 5^1(3(0(1(2(1(4(0(x0)))))))) 588.65/152.69 5^1(4(1(0(2(1(1(0(x0)))))))) -> 5^1(3(0(0(1(2(1(4(x0)))))))) 588.65/152.69 5^1(4(1(0(5(4(1(0(x0)))))))) -> 5^1(3(3(0(1(4(5(0(x0)))))))) 588.65/152.69 5^1(4(1(0(5(0(2(0(x0)))))))) -> 5^1(3(0(3(0(5(2(0(x0)))))))) 588.65/152.69 588.65/152.69 The TRS R consists of the following rules: 588.65/152.69 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.69 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.69 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.69 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.69 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.69 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.69 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.69 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.69 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.69 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.69 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.69 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.69 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.69 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.69 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.69 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.69 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.69 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.69 588.65/152.69 Q is empty. 588.65/152.69 We have to consider all minimal (P,Q,R)-chains. 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (39) DependencyGraphProof (EQUIVALENT) 588.65/152.69 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes. 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (40) 588.65/152.69 Obligation: 588.65/152.69 Q DP problem: 588.65/152.69 The TRS P consists of the following rules: 588.65/152.69 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.69 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.69 5^1(1(2(0(0(x1))))) -> 5^1(2(0(0(x1)))) 588.65/152.69 5^1(4(1(1(0(x1))))) -> 5^1(0(x1)) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(2(1(0(0(x1)))))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(2(x1)) 588.65/152.69 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) 588.65/152.69 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.69 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.69 588.65/152.69 The TRS R consists of the following rules: 588.65/152.69 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.69 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.69 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.69 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.69 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.69 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.69 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.69 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.69 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.69 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.69 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.69 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.69 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.69 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.69 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.69 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.69 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.69 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.69 588.65/152.69 Q is empty. 588.65/152.69 We have to consider all minimal (P,Q,R)-chains. 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (41) TransformationProof (EQUIVALENT) 588.65/152.69 By narrowing [LPAR04] the rule 5^1(1(2(0(0(x1))))) -> 5^1(2(0(0(x1)))) at position [0] we obtained the following new rules [LPAR04]: 588.65/152.69 588.65/152.69 (5^1(1(2(0(0(2(1(0(x0)))))))) -> 5^1(2(0(3(0(1(4(2(0(x0))))))))),5^1(1(2(0(0(2(1(0(x0)))))))) -> 5^1(2(0(3(0(1(4(2(0(x0)))))))))) 588.65/152.69 (5^1(1(2(0(0(5(0(1(0(x0))))))))) -> 5^1(2(0(0(0(0(5(1(4(x0))))))))),5^1(1(2(0(0(5(0(1(0(x0))))))))) -> 5^1(2(0(0(0(0(5(1(4(x0)))))))))) 588.65/152.69 (5^1(1(2(0(0(2(1(1(0(x0))))))))) -> 5^1(2(0(0(1(2(1(4(0(x0))))))))),5^1(1(2(0(0(2(1(1(0(x0))))))))) -> 5^1(2(0(0(1(2(1(4(0(x0)))))))))) 588.65/152.69 (5^1(1(2(0(0(2(1(1(0(x0))))))))) -> 5^1(2(0(0(0(1(2(1(4(x0))))))))),5^1(1(2(0(0(2(1(1(0(x0))))))))) -> 5^1(2(0(0(0(1(2(1(4(x0)))))))))) 588.65/152.69 (5^1(1(2(0(0(5(4(1(0(x0))))))))) -> 5^1(2(0(3(0(1(4(5(0(x0))))))))),5^1(1(2(0(0(5(4(1(0(x0))))))))) -> 5^1(2(0(3(0(1(4(5(0(x0)))))))))) 588.65/152.69 (5^1(1(2(0(0(5(0(2(0(x0))))))))) -> 5^1(2(0(0(3(0(5(2(0(x0))))))))),5^1(1(2(0(0(5(0(2(0(x0))))))))) -> 5^1(2(0(0(3(0(5(2(0(x0)))))))))) 588.65/152.69 588.65/152.69 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (42) 588.65/152.69 Obligation: 588.65/152.69 Q DP problem: 588.65/152.69 The TRS P consists of the following rules: 588.65/152.69 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.69 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.69 5^1(4(1(1(0(x1))))) -> 5^1(0(x1)) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(2(1(0(0(x1)))))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(2(x1)) 588.65/152.69 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) 588.65/152.69 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.69 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.69 5^1(1(2(0(0(2(1(0(x0)))))))) -> 5^1(2(0(3(0(1(4(2(0(x0))))))))) 588.65/152.69 5^1(1(2(0(0(5(0(1(0(x0))))))))) -> 5^1(2(0(0(0(0(5(1(4(x0))))))))) 588.65/152.69 5^1(1(2(0(0(2(1(1(0(x0))))))))) -> 5^1(2(0(0(1(2(1(4(0(x0))))))))) 588.65/152.69 5^1(1(2(0(0(2(1(1(0(x0))))))))) -> 5^1(2(0(0(0(1(2(1(4(x0))))))))) 588.65/152.69 5^1(1(2(0(0(5(4(1(0(x0))))))))) -> 5^1(2(0(3(0(1(4(5(0(x0))))))))) 588.65/152.69 5^1(1(2(0(0(5(0(2(0(x0))))))))) -> 5^1(2(0(0(3(0(5(2(0(x0))))))))) 588.65/152.69 588.65/152.69 The TRS R consists of the following rules: 588.65/152.69 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.69 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.69 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.69 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.69 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.69 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.69 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.69 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.69 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.69 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.69 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.69 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.69 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.69 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.69 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.69 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.69 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.69 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.69 588.65/152.69 Q is empty. 588.65/152.69 We have to consider all minimal (P,Q,R)-chains. 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (43) DependencyGraphProof (EQUIVALENT) 588.65/152.69 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes. 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (44) 588.65/152.69 Obligation: 588.65/152.69 Q DP problem: 588.65/152.69 The TRS P consists of the following rules: 588.65/152.69 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.69 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.69 5^1(4(1(1(0(x1))))) -> 5^1(0(x1)) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(2(1(0(0(x1)))))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(2(x1)) 588.65/152.69 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) 588.65/152.69 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.69 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.69 588.65/152.69 The TRS R consists of the following rules: 588.65/152.69 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.69 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.69 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.69 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.69 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.69 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.69 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.69 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.69 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.69 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.69 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.69 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.69 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.69 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.69 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.69 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.69 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.69 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.69 588.65/152.69 Q is empty. 588.65/152.69 We have to consider all minimal (P,Q,R)-chains. 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (45) TransformationProof (EQUIVALENT) 588.65/152.69 By narrowing [LPAR04] the rule 5^1(4(1(1(0(x1))))) -> 5^1(0(x1)) at position [0] we obtained the following new rules [LPAR04]: 588.65/152.69 588.65/152.69 (5^1(4(1(1(0(2(1(0(x0)))))))) -> 5^1(3(0(1(4(2(0(x0))))))),5^1(4(1(1(0(2(1(0(x0)))))))) -> 5^1(3(0(1(4(2(0(x0)))))))) 588.65/152.69 (5^1(4(1(1(0(5(0(1(0(x0))))))))) -> 5^1(0(0(0(5(1(4(x0))))))),5^1(4(1(1(0(5(0(1(0(x0))))))))) -> 5^1(0(0(0(5(1(4(x0)))))))) 588.65/152.69 (5^1(4(1(1(0(2(1(1(0(x0))))))))) -> 5^1(0(1(2(1(4(0(x0))))))),5^1(4(1(1(0(2(1(1(0(x0))))))))) -> 5^1(0(1(2(1(4(0(x0)))))))) 588.65/152.69 (5^1(4(1(1(0(2(1(1(0(x0))))))))) -> 5^1(0(0(1(2(1(4(x0))))))),5^1(4(1(1(0(2(1(1(0(x0))))))))) -> 5^1(0(0(1(2(1(4(x0)))))))) 588.65/152.69 (5^1(4(1(1(0(5(4(1(0(x0))))))))) -> 5^1(3(0(1(4(5(0(x0))))))),5^1(4(1(1(0(5(4(1(0(x0))))))))) -> 5^1(3(0(1(4(5(0(x0)))))))) 588.65/152.69 (5^1(4(1(1(0(5(0(2(0(x0))))))))) -> 5^1(0(3(0(5(2(0(x0))))))),5^1(4(1(1(0(5(0(2(0(x0))))))))) -> 5^1(0(3(0(5(2(0(x0)))))))) 588.65/152.69 588.65/152.69 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (46) 588.65/152.69 Obligation: 588.65/152.69 Q DP problem: 588.65/152.69 The TRS P consists of the following rules: 588.65/152.69 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.69 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(2(1(0(0(x1)))))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(2(x1)) 588.65/152.69 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) 588.65/152.69 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.69 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.69 5^1(4(1(1(0(2(1(0(x0)))))))) -> 5^1(3(0(1(4(2(0(x0))))))) 588.65/152.69 5^1(4(1(1(0(5(0(1(0(x0))))))))) -> 5^1(0(0(0(5(1(4(x0))))))) 588.65/152.69 5^1(4(1(1(0(2(1(1(0(x0))))))))) -> 5^1(0(1(2(1(4(0(x0))))))) 588.65/152.69 5^1(4(1(1(0(2(1(1(0(x0))))))))) -> 5^1(0(0(1(2(1(4(x0))))))) 588.65/152.69 5^1(4(1(1(0(5(4(1(0(x0))))))))) -> 5^1(3(0(1(4(5(0(x0))))))) 588.65/152.69 5^1(4(1(1(0(5(0(2(0(x0))))))))) -> 5^1(0(3(0(5(2(0(x0))))))) 588.65/152.69 588.65/152.69 The TRS R consists of the following rules: 588.65/152.69 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.69 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.69 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.69 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.69 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.69 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.69 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.69 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.69 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.69 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.69 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.69 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.69 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.69 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.69 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.69 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.69 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.69 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.69 588.65/152.69 Q is empty. 588.65/152.69 We have to consider all minimal (P,Q,R)-chains. 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (47) DependencyGraphProof (EQUIVALENT) 588.65/152.69 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes. 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (48) 588.65/152.69 Obligation: 588.65/152.69 Q DP problem: 588.65/152.69 The TRS P consists of the following rules: 588.65/152.69 588.65/152.69 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(2(1(0(0(x1)))))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(2(x1)) 588.65/152.69 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) 588.65/152.69 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.69 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.69 588.65/152.69 The TRS R consists of the following rules: 588.65/152.69 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.69 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.69 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.69 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.69 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.69 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.69 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.69 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.69 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.69 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.69 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.69 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.69 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.69 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.69 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.69 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.69 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.69 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.69 588.65/152.69 Q is empty. 588.65/152.69 We have to consider all minimal (P,Q,R)-chains. 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (49) TransformationProof (EQUIVALENT) 588.65/152.69 By narrowing [LPAR04] the rule 5^1(1(3(2(0(x1))))) -> 5^1(3(2(1(0(0(x1)))))) at position [0,0] we obtained the following new rules [LPAR04]: 588.65/152.69 588.65/152.69 (5^1(1(3(2(0(2(1(0(x0)))))))) -> 5^1(3(2(1(0(3(0(1(4(2(0(x0))))))))))),5^1(1(3(2(0(2(1(0(x0)))))))) -> 5^1(3(2(1(0(3(0(1(4(2(0(x0)))))))))))) 588.65/152.69 (5^1(1(3(2(0(5(0(1(0(x0))))))))) -> 5^1(3(2(1(0(0(0(0(5(1(4(x0))))))))))),5^1(1(3(2(0(5(0(1(0(x0))))))))) -> 5^1(3(2(1(0(0(0(0(5(1(4(x0)))))))))))) 588.65/152.69 (5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(3(2(1(0(0(1(2(1(4(0(x0))))))))))),5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(3(2(1(0(0(1(2(1(4(0(x0)))))))))))) 588.65/152.69 (5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(3(2(1(0(0(0(1(2(1(4(x0))))))))))),5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(3(2(1(0(0(0(1(2(1(4(x0)))))))))))) 588.65/152.69 (5^1(1(3(2(0(5(4(1(0(x0))))))))) -> 5^1(3(2(1(0(3(0(1(4(5(0(x0))))))))))),5^1(1(3(2(0(5(4(1(0(x0))))))))) -> 5^1(3(2(1(0(3(0(1(4(5(0(x0)))))))))))) 588.65/152.69 (5^1(1(3(2(0(5(0(2(0(x0))))))))) -> 5^1(3(2(1(0(0(3(0(5(2(0(x0))))))))))),5^1(1(3(2(0(5(0(2(0(x0))))))))) -> 5^1(3(2(1(0(0(3(0(5(2(0(x0)))))))))))) 588.65/152.69 588.65/152.69 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (50) 588.65/152.69 Obligation: 588.65/152.69 Q DP problem: 588.65/152.69 The TRS P consists of the following rules: 588.65/152.69 588.65/152.69 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(2(x1)) 588.65/152.69 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) 588.65/152.69 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.69 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.69 5^1(1(3(2(0(2(1(0(x0)))))))) -> 5^1(3(2(1(0(3(0(1(4(2(0(x0))))))))))) 588.65/152.69 5^1(1(3(2(0(5(0(1(0(x0))))))))) -> 5^1(3(2(1(0(0(0(0(5(1(4(x0))))))))))) 588.65/152.69 5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(3(2(1(0(0(1(2(1(4(0(x0))))))))))) 588.65/152.69 5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(3(2(1(0(0(0(1(2(1(4(x0))))))))))) 588.65/152.69 5^1(1(3(2(0(5(4(1(0(x0))))))))) -> 5^1(3(2(1(0(3(0(1(4(5(0(x0))))))))))) 588.65/152.69 5^1(1(3(2(0(5(0(2(0(x0))))))))) -> 5^1(3(2(1(0(0(3(0(5(2(0(x0))))))))))) 588.65/152.69 588.65/152.69 The TRS R consists of the following rules: 588.65/152.69 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.69 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.69 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.69 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.69 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.69 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.69 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.69 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.69 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.69 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.69 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.69 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.69 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.69 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.69 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.69 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.69 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.69 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.69 588.65/152.69 Q is empty. 588.65/152.69 We have to consider all minimal (P,Q,R)-chains. 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (51) DependencyGraphProof (EQUIVALENT) 588.65/152.69 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes. 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (52) 588.65/152.69 Obligation: 588.65/152.69 Q DP problem: 588.65/152.69 The TRS P consists of the following rules: 588.65/152.69 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.69 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(2(x1)) 588.65/152.69 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) 588.65/152.69 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.69 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.69 588.65/152.69 The TRS R consists of the following rules: 588.65/152.69 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.69 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.69 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.69 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.69 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.69 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.69 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.69 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.69 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.69 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.69 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.69 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.69 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.69 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.69 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.69 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.69 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.69 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.69 588.65/152.69 Q is empty. 588.65/152.69 We have to consider all minimal (P,Q,R)-chains. 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (53) TransformationProof (EQUIVALENT) 588.65/152.69 By narrowing [LPAR04] the rule 5^1(1(3(2(0(x1))))) -> 5^1(2(0(x1))) at position [0] we obtained the following new rules [LPAR04]: 588.65/152.69 588.65/152.69 (5^1(1(3(2(0(2(1(0(x0)))))))) -> 5^1(2(2(1(0(4(0(x0))))))),5^1(1(3(2(0(2(1(0(x0)))))))) -> 5^1(2(2(1(0(4(0(x0)))))))) 588.65/152.69 (5^1(1(3(2(0(2(4(0(x0)))))))) -> 5^1(2(2(3(4(0(0(x0))))))),5^1(1(3(2(0(2(4(0(x0)))))))) -> 5^1(2(2(3(4(0(0(x0)))))))) 588.65/152.69 (5^1(1(3(2(0(2(1(0(x0)))))))) -> 5^1(2(3(0(1(4(2(0(x0)))))))),5^1(1(3(2(0(2(1(0(x0)))))))) -> 5^1(2(3(0(1(4(2(0(x0))))))))) 588.65/152.69 (5^1(1(3(2(0(5(0(1(0(x0))))))))) -> 5^1(2(0(0(0(5(1(4(x0)))))))),5^1(1(3(2(0(5(0(1(0(x0))))))))) -> 5^1(2(0(0(0(5(1(4(x0))))))))) 588.65/152.69 (5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(2(0(1(2(1(4(0(x0)))))))),5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(2(0(1(2(1(4(0(x0))))))))) 588.65/152.69 (5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(2(0(0(1(2(1(4(x0)))))))),5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(2(0(0(1(2(1(4(x0))))))))) 588.65/152.69 (5^1(1(3(2(0(5(4(1(0(x0))))))))) -> 5^1(2(3(0(1(4(5(0(x0)))))))),5^1(1(3(2(0(5(4(1(0(x0))))))))) -> 5^1(2(3(0(1(4(5(0(x0))))))))) 588.65/152.69 (5^1(1(3(2(0(5(0(2(0(x0))))))))) -> 5^1(2(0(3(0(5(2(0(x0)))))))),5^1(1(3(2(0(5(0(2(0(x0))))))))) -> 5^1(2(0(3(0(5(2(0(x0))))))))) 588.65/152.69 588.65/152.69 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (54) 588.65/152.69 Obligation: 588.65/152.69 Q DP problem: 588.65/152.69 The TRS P consists of the following rules: 588.65/152.69 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.69 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(2(x1)) 588.65/152.69 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) 588.65/152.69 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.69 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.69 5^1(1(3(2(0(2(1(0(x0)))))))) -> 5^1(2(2(1(0(4(0(x0))))))) 588.65/152.69 5^1(1(3(2(0(2(4(0(x0)))))))) -> 5^1(2(2(3(4(0(0(x0))))))) 588.65/152.69 5^1(1(3(2(0(2(1(0(x0)))))))) -> 5^1(2(3(0(1(4(2(0(x0)))))))) 588.65/152.69 5^1(1(3(2(0(5(0(1(0(x0))))))))) -> 5^1(2(0(0(0(5(1(4(x0)))))))) 588.65/152.69 5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(2(0(1(2(1(4(0(x0)))))))) 588.65/152.69 5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(2(0(0(1(2(1(4(x0)))))))) 588.65/152.69 5^1(1(3(2(0(5(4(1(0(x0))))))))) -> 5^1(2(3(0(1(4(5(0(x0)))))))) 588.65/152.69 5^1(1(3(2(0(5(0(2(0(x0))))))))) -> 5^1(2(0(3(0(5(2(0(x0)))))))) 588.65/152.69 588.65/152.69 The TRS R consists of the following rules: 588.65/152.69 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.69 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.69 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.69 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.69 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.69 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.69 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.69 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.69 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.69 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.69 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.69 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.69 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.69 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.69 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.69 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.69 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.69 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.69 588.65/152.69 Q is empty. 588.65/152.69 We have to consider all minimal (P,Q,R)-chains. 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (55) DependencyGraphProof (EQUIVALENT) 588.65/152.69 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 8 less nodes. 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (56) 588.65/152.69 Obligation: 588.65/152.69 Q DP problem: 588.65/152.69 The TRS P consists of the following rules: 588.65/152.69 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.69 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(2(x1)) 588.65/152.69 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) 588.65/152.69 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.69 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.69 588.65/152.69 The TRS R consists of the following rules: 588.65/152.69 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.69 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.69 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.69 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.69 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.69 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.69 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.69 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.69 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.69 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.69 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.69 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.69 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.69 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.69 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.69 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.69 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.69 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.69 588.65/152.69 Q is empty. 588.65/152.69 We have to consider all minimal (P,Q,R)-chains. 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (57) TransformationProof (EQUIVALENT) 588.65/152.69 By narrowing [LPAR04] the rule 5^1(1(3(2(0(x1))))) -> 5^1(3(0(x1))) at position [0,0] we obtained the following new rules [LPAR04]: 588.65/152.69 588.65/152.69 (5^1(1(3(2(0(2(1(0(x0)))))))) -> 5^1(3(3(0(1(4(2(0(x0)))))))),5^1(1(3(2(0(2(1(0(x0)))))))) -> 5^1(3(3(0(1(4(2(0(x0))))))))) 588.65/152.69 (5^1(1(3(2(0(5(0(1(0(x0))))))))) -> 5^1(3(0(0(0(5(1(4(x0)))))))),5^1(1(3(2(0(5(0(1(0(x0))))))))) -> 5^1(3(0(0(0(5(1(4(x0))))))))) 588.65/152.69 (5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(3(0(1(2(1(4(0(x0)))))))),5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(3(0(1(2(1(4(0(x0))))))))) 588.65/152.69 (5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(3(0(0(1(2(1(4(x0)))))))),5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(3(0(0(1(2(1(4(x0))))))))) 588.65/152.69 (5^1(1(3(2(0(5(4(1(0(x0))))))))) -> 5^1(3(3(0(1(4(5(0(x0)))))))),5^1(1(3(2(0(5(4(1(0(x0))))))))) -> 5^1(3(3(0(1(4(5(0(x0))))))))) 588.65/152.69 (5^1(1(3(2(0(5(0(2(0(x0))))))))) -> 5^1(3(0(3(0(5(2(0(x0)))))))),5^1(1(3(2(0(5(0(2(0(x0))))))))) -> 5^1(3(0(3(0(5(2(0(x0))))))))) 588.65/152.69 588.65/152.69 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (58) 588.65/152.69 Obligation: 588.65/152.69 Q DP problem: 588.65/152.69 The TRS P consists of the following rules: 588.65/152.69 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.69 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(2(x1)) 588.65/152.69 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) 588.65/152.69 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.69 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.69 5^1(1(3(2(0(2(1(0(x0)))))))) -> 5^1(3(3(0(1(4(2(0(x0)))))))) 588.65/152.69 5^1(1(3(2(0(5(0(1(0(x0))))))))) -> 5^1(3(0(0(0(5(1(4(x0)))))))) 588.65/152.69 5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(3(0(1(2(1(4(0(x0)))))))) 588.65/152.69 5^1(1(3(2(0(2(1(1(0(x0))))))))) -> 5^1(3(0(0(1(2(1(4(x0)))))))) 588.65/152.69 5^1(1(3(2(0(5(4(1(0(x0))))))))) -> 5^1(3(3(0(1(4(5(0(x0)))))))) 588.65/152.69 5^1(1(3(2(0(5(0(2(0(x0))))))))) -> 5^1(3(0(3(0(5(2(0(x0)))))))) 588.65/152.69 588.65/152.69 The TRS R consists of the following rules: 588.65/152.69 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.69 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.69 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.69 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.69 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.69 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.69 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.69 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.69 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.69 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.69 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.69 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.69 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.69 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.69 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.69 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.69 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.69 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.69 588.65/152.69 Q is empty. 588.65/152.69 We have to consider all minimal (P,Q,R)-chains. 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (59) DependencyGraphProof (EQUIVALENT) 588.65/152.69 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes. 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (60) 588.65/152.69 Obligation: 588.65/152.69 Q DP problem: 588.65/152.69 The TRS P consists of the following rules: 588.65/152.69 588.65/152.69 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1))) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(2(x1)) 588.65/152.69 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) 588.65/152.69 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.69 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.69 588.65/152.69 The TRS R consists of the following rules: 588.65/152.69 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.69 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.69 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.69 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.69 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.69 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.69 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.69 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.69 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.69 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.69 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.69 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.69 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.69 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.69 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.69 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.69 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.69 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.69 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.69 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.69 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.69 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.69 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.69 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.69 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.69 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.69 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.69 588.65/152.69 Q is empty. 588.65/152.69 We have to consider all minimal (P,Q,R)-chains. 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (61) TransformationProof (EQUIVALENT) 588.65/152.69 By narrowing [LPAR04] the rule 5^1(1(3(2(0(x1))))) -> 5^1(3(2(x1))) at position [0,0] we obtained the following new rules [LPAR04]: 588.65/152.69 588.65/152.69 (5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(3(1(0(1(x0))))))),5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(3(1(0(1(x0)))))))) 588.65/152.69 (5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(1(0(1(4(x0))))))),5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(1(0(1(4(x0)))))))) 588.65/152.69 (5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(3(1(4(1(0(x0)))))))),5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(3(1(4(1(0(x0))))))))) 588.65/152.69 (5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(1(4(1(4(0(x0)))))))),5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(1(4(1(4(0(x0))))))))) 588.65/152.69 (5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(1(3(0(1(4(x0)))))))),5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(1(3(0(1(4(x0))))))))) 588.65/152.69 (5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(0(1(3(1(4(x0)))))))),5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(0(1(3(1(4(x0))))))))) 588.65/152.69 (5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(3(2(1(4(1(4(0(x0)))))))),5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(3(2(1(4(1(4(0(x0))))))))) 588.65/152.69 (5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(3(0(2(1(3(1(4(x0)))))))),5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(3(0(2(1(3(1(4(x0))))))))) 588.65/152.69 (5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(3(1(0(1(4(2(4(x0)))))))),5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(3(1(0(1(4(2(4(x0))))))))) 588.65/152.69 (5^1(1(3(2(0(0(2(1(0(x0))))))))) -> 5^1(3(2(2(1(0(4(0(x0)))))))),5^1(1(3(2(0(0(2(1(0(x0))))))))) -> 5^1(3(2(2(1(0(4(0(x0))))))))) 588.65/152.69 (5^1(1(3(2(0(1(5(2(0(x0))))))))) -> 5^1(3(5(1(2(3(2(0(x0)))))))),5^1(1(3(2(0(1(5(2(0(x0))))))))) -> 5^1(3(5(1(2(3(2(0(x0))))))))) 588.65/152.69 (5^1(1(3(2(0(0(2(4(0(x0))))))))) -> 5^1(3(2(2(3(4(0(0(x0)))))))),5^1(1(3(2(0(0(2(4(0(x0))))))))) -> 5^1(3(2(2(3(4(0(0(x0))))))))) 588.65/152.69 588.65/152.69 588.65/152.69 ---------------------------------------- 588.65/152.69 588.65/152.69 (62) 588.65/152.69 Obligation: 588.65/152.69 Q DP problem: 588.65/152.69 The TRS P consists of the following rules: 588.65/152.69 588.65/152.69 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.69 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.69 5^1(1(3(2(0(x1))))) -> 5^1(2(x1)) 588.65/152.69 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) 588.65/152.69 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.69 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.69 5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(3(1(0(1(x0))))))) 588.65/152.69 5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(1(0(1(4(x0))))))) 588.65/152.69 5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(3(1(4(1(0(x0)))))))) 588.65/152.69 5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(1(4(1(4(0(x0)))))))) 588.65/152.69 5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(1(3(0(1(4(x0)))))))) 588.65/152.69 5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(3(2(0(1(3(1(4(x0)))))))) 588.65/152.69 5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(3(2(1(4(1(4(0(x0)))))))) 588.65/152.69 5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(3(0(2(1(3(1(4(x0)))))))) 588.65/152.69 5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(3(1(0(1(4(2(4(x0)))))))) 588.65/152.69 5^1(1(3(2(0(0(2(1(0(x0))))))))) -> 5^1(3(2(2(1(0(4(0(x0)))))))) 588.65/152.69 5^1(1(3(2(0(1(5(2(0(x0))))))))) -> 5^1(3(5(1(2(3(2(0(x0)))))))) 588.65/152.69 5^1(1(3(2(0(0(2(4(0(x0))))))))) -> 5^1(3(2(2(3(4(0(0(x0)))))))) 588.65/152.69 588.65/152.69 The TRS R consists of the following rules: 588.65/152.69 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.69 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.69 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.70 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.70 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.70 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.70 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.70 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.70 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.70 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.70 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.70 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.70 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.70 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.70 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.70 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.70 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.70 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.70 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.70 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.70 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.70 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.70 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.70 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.70 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.70 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.70 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.70 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.70 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.70 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.70 588.65/152.70 Q is empty. 588.65/152.70 We have to consider all minimal (P,Q,R)-chains. 588.65/152.70 ---------------------------------------- 588.65/152.70 588.65/152.70 (63) DependencyGraphProof (EQUIVALENT) 588.65/152.70 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 12 less nodes. 588.65/152.70 ---------------------------------------- 588.65/152.70 588.65/152.70 (64) 588.65/152.70 Obligation: 588.65/152.70 Q DP problem: 588.65/152.70 The TRS P consists of the following rules: 588.65/152.70 588.65/152.70 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.70 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.70 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.70 5^1(1(3(2(0(x1))))) -> 5^1(2(x1)) 588.65/152.70 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) 588.65/152.70 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.70 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.70 588.65/152.70 The TRS R consists of the following rules: 588.65/152.70 588.65/152.70 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.70 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.70 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.70 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.70 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.70 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.70 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.70 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.70 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.70 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.70 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.70 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.70 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.70 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.70 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.70 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.70 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.70 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.70 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.70 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.70 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.70 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.70 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.70 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.70 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.70 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.70 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.70 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.70 588.65/152.70 Q is empty. 588.65/152.70 We have to consider all minimal (P,Q,R)-chains. 588.65/152.70 ---------------------------------------- 588.65/152.70 588.65/152.70 (65) TransformationProof (EQUIVALENT) 588.65/152.70 By narrowing [LPAR04] the rule 5^1(1(3(2(0(x1))))) -> 5^1(2(x1)) at position [0] we obtained the following new rules [LPAR04]: 588.65/152.70 588.65/152.70 (5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(3(1(0(1(x0)))))),5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(3(1(0(1(x0))))))) 588.65/152.70 (5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(1(0(1(4(x0)))))),5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(1(0(1(4(x0))))))) 588.65/152.70 (5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(3(1(4(1(0(x0))))))),5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(3(1(4(1(0(x0)))))))) 588.65/152.70 (5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(1(4(1(4(0(x0))))))),5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(1(4(1(4(0(x0)))))))) 588.65/152.70 (5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(1(3(0(1(4(x0))))))),5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(1(3(0(1(4(x0)))))))) 588.65/152.70 (5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(0(1(3(1(4(x0))))))),5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(0(1(3(1(4(x0)))))))) 588.65/152.70 (5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(2(1(4(1(4(0(x0))))))),5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(2(1(4(1(4(0(x0)))))))) 588.65/152.70 (5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(0(2(1(3(1(4(x0))))))),5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(0(2(1(3(1(4(x0)))))))) 588.65/152.70 (5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(1(0(1(4(2(4(x0))))))),5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(1(0(1(4(2(4(x0)))))))) 588.65/152.70 (5^1(1(3(2(0(0(2(1(0(x0))))))))) -> 5^1(2(2(1(0(4(0(x0))))))),5^1(1(3(2(0(0(2(1(0(x0))))))))) -> 5^1(2(2(1(0(4(0(x0)))))))) 588.65/152.70 (5^1(1(3(2(0(1(5(2(0(x0))))))))) -> 5^1(5(1(2(3(2(0(x0))))))),5^1(1(3(2(0(1(5(2(0(x0))))))))) -> 5^1(5(1(2(3(2(0(x0)))))))) 588.65/152.70 (5^1(1(3(2(0(0(2(4(0(x0))))))))) -> 5^1(2(2(3(4(0(0(x0))))))),5^1(1(3(2(0(0(2(4(0(x0))))))))) -> 5^1(2(2(3(4(0(0(x0)))))))) 588.65/152.70 588.65/152.70 588.65/152.70 ---------------------------------------- 588.65/152.70 588.65/152.70 (66) 588.65/152.70 Obligation: 588.65/152.70 Q DP problem: 588.65/152.70 The TRS P consists of the following rules: 588.65/152.70 588.65/152.70 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.70 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.70 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.70 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) 588.65/152.70 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.70 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.70 5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(3(1(0(1(x0)))))) 588.65/152.70 5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(1(0(1(4(x0)))))) 588.65/152.70 5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(3(1(4(1(0(x0))))))) 588.65/152.70 5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(1(4(1(4(0(x0))))))) 588.65/152.70 5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(1(3(0(1(4(x0))))))) 588.65/152.70 5^1(1(3(2(0(1(1(0(x0)))))))) -> 5^1(2(0(1(3(1(4(x0))))))) 588.65/152.70 5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(2(1(4(1(4(0(x0))))))) 588.65/152.70 5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(0(2(1(3(1(4(x0))))))) 588.65/152.70 5^1(1(3(2(0(4(1(1(0(x0))))))))) -> 5^1(1(0(1(4(2(4(x0))))))) 588.65/152.70 5^1(1(3(2(0(0(2(1(0(x0))))))))) -> 5^1(2(2(1(0(4(0(x0))))))) 588.65/152.70 5^1(1(3(2(0(1(5(2(0(x0))))))))) -> 5^1(5(1(2(3(2(0(x0))))))) 588.65/152.70 5^1(1(3(2(0(0(2(4(0(x0))))))))) -> 5^1(2(2(3(4(0(0(x0))))))) 588.65/152.70 588.65/152.70 The TRS R consists of the following rules: 588.65/152.70 588.65/152.70 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.70 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.70 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.70 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.70 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.70 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.70 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.70 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.70 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.70 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.70 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.70 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.70 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.70 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.70 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.70 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.70 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.70 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.70 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.70 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.70 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.70 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.70 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.70 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.70 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.70 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.70 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.70 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.70 588.65/152.70 Q is empty. 588.65/152.70 We have to consider all minimal (P,Q,R)-chains. 588.65/152.70 ---------------------------------------- 588.65/152.70 588.65/152.70 (67) DependencyGraphProof (EQUIVALENT) 588.65/152.70 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 12 less nodes. 588.65/152.70 ---------------------------------------- 588.65/152.70 588.65/152.70 (68) 588.65/152.70 Obligation: 588.65/152.70 Q DP problem: 588.65/152.70 The TRS P consists of the following rules: 588.65/152.70 588.65/152.70 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.70 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.70 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.70 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) 588.65/152.70 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.70 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.70 588.65/152.70 The TRS R consists of the following rules: 588.65/152.70 588.65/152.70 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.70 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.70 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.70 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.70 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.70 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.70 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.70 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.70 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.70 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.70 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.70 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.70 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.70 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.70 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.70 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.70 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.70 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.70 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.70 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.70 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.70 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.70 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.70 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.70 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.70 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.70 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.70 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.70 588.65/152.70 Q is empty. 588.65/152.70 We have to consider all minimal (P,Q,R)-chains. 588.65/152.70 ---------------------------------------- 588.65/152.70 588.65/152.70 (69) TransformationProof (EQUIVALENT) 588.65/152.70 By narrowing [LPAR04] the rule 5^1(1(5(2(0(x1))))) -> 5^1(1(2(5(3(0(x1)))))) at position [0,0] we obtained the following new rules [LPAR04]: 588.65/152.70 588.65/152.70 (5^1(1(5(2(0(2(1(0(x0)))))))) -> 5^1(1(2(5(3(3(0(1(4(2(0(x0))))))))))),5^1(1(5(2(0(2(1(0(x0)))))))) -> 5^1(1(2(5(3(3(0(1(4(2(0(x0)))))))))))) 588.65/152.70 (5^1(1(5(2(0(5(0(1(0(x0))))))))) -> 5^1(1(2(5(3(0(0(0(5(1(4(x0))))))))))),5^1(1(5(2(0(5(0(1(0(x0))))))))) -> 5^1(1(2(5(3(0(0(0(5(1(4(x0)))))))))))) 588.65/152.70 (5^1(1(5(2(0(2(1(1(0(x0))))))))) -> 5^1(1(2(5(3(0(1(2(1(4(0(x0))))))))))),5^1(1(5(2(0(2(1(1(0(x0))))))))) -> 5^1(1(2(5(3(0(1(2(1(4(0(x0)))))))))))) 588.65/152.70 (5^1(1(5(2(0(2(1(1(0(x0))))))))) -> 5^1(1(2(5(3(0(0(1(2(1(4(x0))))))))))),5^1(1(5(2(0(2(1(1(0(x0))))))))) -> 5^1(1(2(5(3(0(0(1(2(1(4(x0)))))))))))) 588.65/152.70 (5^1(1(5(2(0(5(4(1(0(x0))))))))) -> 5^1(1(2(5(3(3(0(1(4(5(0(x0))))))))))),5^1(1(5(2(0(5(4(1(0(x0))))))))) -> 5^1(1(2(5(3(3(0(1(4(5(0(x0)))))))))))) 588.65/152.70 (5^1(1(5(2(0(5(0(2(0(x0))))))))) -> 5^1(1(2(5(3(0(3(0(5(2(0(x0))))))))))),5^1(1(5(2(0(5(0(2(0(x0))))))))) -> 5^1(1(2(5(3(0(3(0(5(2(0(x0)))))))))))) 588.65/152.70 588.65/152.70 588.65/152.70 ---------------------------------------- 588.65/152.70 588.65/152.70 (70) 588.65/152.70 Obligation: 588.65/152.70 Q DP problem: 588.65/152.70 The TRS P consists of the following rules: 588.65/152.70 588.65/152.70 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.70 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.70 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.70 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.70 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.70 5^1(1(5(2(0(2(1(0(x0)))))))) -> 5^1(1(2(5(3(3(0(1(4(2(0(x0))))))))))) 588.65/152.70 5^1(1(5(2(0(5(0(1(0(x0))))))))) -> 5^1(1(2(5(3(0(0(0(5(1(4(x0))))))))))) 588.65/152.70 5^1(1(5(2(0(2(1(1(0(x0))))))))) -> 5^1(1(2(5(3(0(1(2(1(4(0(x0))))))))))) 588.65/152.70 5^1(1(5(2(0(2(1(1(0(x0))))))))) -> 5^1(1(2(5(3(0(0(1(2(1(4(x0))))))))))) 588.65/152.70 5^1(1(5(2(0(5(4(1(0(x0))))))))) -> 5^1(1(2(5(3(3(0(1(4(5(0(x0))))))))))) 588.65/152.70 5^1(1(5(2(0(5(0(2(0(x0))))))))) -> 5^1(1(2(5(3(0(3(0(5(2(0(x0))))))))))) 588.65/152.70 588.65/152.70 The TRS R consists of the following rules: 588.65/152.70 588.65/152.70 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.70 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.70 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.70 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.70 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.70 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.70 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.70 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.70 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.70 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.70 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.70 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.70 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.70 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.70 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.70 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.70 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.70 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.70 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.70 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.70 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.70 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.70 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.70 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.70 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.70 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.70 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.70 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.70 588.65/152.70 Q is empty. 588.65/152.70 We have to consider all minimal (P,Q,R)-chains. 588.65/152.70 ---------------------------------------- 588.65/152.70 588.65/152.70 (71) DependencyGraphProof (EQUIVALENT) 588.65/152.70 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes. 588.65/152.70 ---------------------------------------- 588.65/152.70 588.65/152.70 (72) 588.65/152.70 Obligation: 588.65/152.70 Q DP problem: 588.65/152.70 The TRS P consists of the following rules: 588.65/152.70 588.65/152.70 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.70 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.70 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.70 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.70 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) 588.65/152.70 588.65/152.70 The TRS R consists of the following rules: 588.65/152.70 588.65/152.70 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.70 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.70 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.70 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.70 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.70 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.70 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.70 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.70 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.70 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.70 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.70 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.70 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.70 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.70 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.70 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.70 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.70 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.70 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.70 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.70 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.70 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.70 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.70 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.70 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.70 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.70 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.70 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.70 588.65/152.70 Q is empty. 588.65/152.70 We have to consider all minimal (P,Q,R)-chains. 588.65/152.70 ---------------------------------------- 588.65/152.70 588.65/152.70 (73) TransformationProof (EQUIVALENT) 588.65/152.70 By narrowing [LPAR04] the rule 5^1(1(2(4(0(x1))))) -> 5^1(2(0(x1))) at position [0] we obtained the following new rules [LPAR04]: 588.65/152.70 588.65/152.70 (5^1(1(2(4(0(2(1(0(x0)))))))) -> 5^1(2(2(1(0(4(0(x0))))))),5^1(1(2(4(0(2(1(0(x0)))))))) -> 5^1(2(2(1(0(4(0(x0)))))))) 588.65/152.70 (5^1(1(2(4(0(2(4(0(x0)))))))) -> 5^1(2(2(3(4(0(0(x0))))))),5^1(1(2(4(0(2(4(0(x0)))))))) -> 5^1(2(2(3(4(0(0(x0)))))))) 588.65/152.70 (5^1(1(2(4(0(2(1(0(x0)))))))) -> 5^1(2(3(0(1(4(2(0(x0)))))))),5^1(1(2(4(0(2(1(0(x0)))))))) -> 5^1(2(3(0(1(4(2(0(x0))))))))) 588.65/152.70 (5^1(1(2(4(0(5(0(1(0(x0))))))))) -> 5^1(2(0(0(0(5(1(4(x0)))))))),5^1(1(2(4(0(5(0(1(0(x0))))))))) -> 5^1(2(0(0(0(5(1(4(x0))))))))) 588.65/152.70 (5^1(1(2(4(0(2(1(1(0(x0))))))))) -> 5^1(2(0(1(2(1(4(0(x0)))))))),5^1(1(2(4(0(2(1(1(0(x0))))))))) -> 5^1(2(0(1(2(1(4(0(x0))))))))) 588.65/152.70 (5^1(1(2(4(0(2(1(1(0(x0))))))))) -> 5^1(2(0(0(1(2(1(4(x0)))))))),5^1(1(2(4(0(2(1(1(0(x0))))))))) -> 5^1(2(0(0(1(2(1(4(x0))))))))) 588.65/152.70 (5^1(1(2(4(0(5(4(1(0(x0))))))))) -> 5^1(2(3(0(1(4(5(0(x0)))))))),5^1(1(2(4(0(5(4(1(0(x0))))))))) -> 5^1(2(3(0(1(4(5(0(x0))))))))) 588.65/152.70 (5^1(1(2(4(0(5(0(2(0(x0))))))))) -> 5^1(2(0(3(0(5(2(0(x0)))))))),5^1(1(2(4(0(5(0(2(0(x0))))))))) -> 5^1(2(0(3(0(5(2(0(x0))))))))) 588.65/152.70 588.65/152.70 588.65/152.70 ---------------------------------------- 588.65/152.70 588.65/152.70 (74) 588.65/152.70 Obligation: 588.65/152.70 Q DP problem: 588.65/152.70 The TRS P consists of the following rules: 588.65/152.70 588.65/152.70 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.70 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.70 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.70 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.70 5^1(1(2(4(0(2(1(0(x0)))))))) -> 5^1(2(2(1(0(4(0(x0))))))) 588.65/152.70 5^1(1(2(4(0(2(4(0(x0)))))))) -> 5^1(2(2(3(4(0(0(x0))))))) 588.65/152.70 5^1(1(2(4(0(2(1(0(x0)))))))) -> 5^1(2(3(0(1(4(2(0(x0)))))))) 588.65/152.70 5^1(1(2(4(0(5(0(1(0(x0))))))))) -> 5^1(2(0(0(0(5(1(4(x0)))))))) 588.65/152.70 5^1(1(2(4(0(2(1(1(0(x0))))))))) -> 5^1(2(0(1(2(1(4(0(x0)))))))) 588.65/152.70 5^1(1(2(4(0(2(1(1(0(x0))))))))) -> 5^1(2(0(0(1(2(1(4(x0)))))))) 588.65/152.70 5^1(1(2(4(0(5(4(1(0(x0))))))))) -> 5^1(2(3(0(1(4(5(0(x0)))))))) 588.65/152.70 5^1(1(2(4(0(5(0(2(0(x0))))))))) -> 5^1(2(0(3(0(5(2(0(x0)))))))) 588.65/152.70 588.65/152.70 The TRS R consists of the following rules: 588.65/152.70 588.65/152.70 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.70 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.70 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.70 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.70 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.70 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.70 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.70 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.70 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.70 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.70 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.70 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.70 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.70 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.70 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.70 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.70 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.70 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.70 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.70 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.70 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.70 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.70 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.70 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.70 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.70 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.70 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.70 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.70 588.65/152.70 Q is empty. 588.65/152.70 We have to consider all minimal (P,Q,R)-chains. 588.65/152.70 ---------------------------------------- 588.65/152.70 588.65/152.70 (75) DependencyGraphProof (EQUIVALENT) 588.65/152.70 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 8 less nodes. 588.65/152.70 ---------------------------------------- 588.65/152.70 588.65/152.70 (76) 588.65/152.70 Obligation: 588.65/152.70 Q DP problem: 588.65/152.70 The TRS P consists of the following rules: 588.65/152.70 588.65/152.70 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.70 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.70 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.70 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.70 588.65/152.70 The TRS R consists of the following rules: 588.65/152.70 588.65/152.70 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.70 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.70 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.70 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.70 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.70 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.70 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.70 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.70 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.70 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.70 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.70 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.70 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.70 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.70 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.70 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.70 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.70 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.70 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.70 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.70 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.70 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.70 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.70 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.70 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.70 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.70 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.70 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.70 588.65/152.70 Q is empty. 588.65/152.70 We have to consider all minimal (P,Q,R)-chains. 588.65/152.70 ---------------------------------------- 588.65/152.70 588.65/152.70 (77) QDPOrderProof (EQUIVALENT) 588.65/152.70 We use the reduction pair processor [LPAR04,JAR06]. 588.65/152.70 588.65/152.70 588.65/152.70 The following pairs can be oriented strictly and are deleted. 588.65/152.70 588.65/152.70 5^1(1(1(0(x1)))) -> 5^1(x1) 588.65/152.70 5^1(1(1(0(x1)))) -> 5^1(3(x1)) 588.65/152.70 5^1(1(5(3(0(x1))))) -> 5^1(3(0(5(x1)))) 588.65/152.70 The remaining pairs can at least be oriented weakly. 588.65/152.70 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 588.65/152.70 588.65/152.70 POL( 0_1(x_1) ) = max{0, x_1 - 1} 588.65/152.70 POL( 3_1(x_1) ) = max{0, x_1 - 1} 588.65/152.70 POL( 5^1_1(x_1) ) = 2x_1 + 2 588.65/152.70 POL( 5_1(x_1) ) = 1 588.65/152.70 POL( 1_1(x_1) ) = x_1 + 1 588.65/152.70 POL( 4_1(x_1) ) = max{0, -2} 588.65/152.70 POL( 2_1(x_1) ) = 0 588.65/152.70 588.65/152.70 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 588.65/152.70 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.70 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.70 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.70 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.70 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.70 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.70 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.70 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.70 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.70 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.70 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.70 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.70 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.70 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.70 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.70 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.70 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.70 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.70 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.70 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.70 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.70 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.70 588.65/152.70 588.65/152.70 ---------------------------------------- 588.65/152.70 588.65/152.70 (78) 588.65/152.70 Obligation: 588.65/152.70 Q DP problem: 588.65/152.70 The TRS P consists of the following rules: 588.65/152.70 588.65/152.70 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.70 588.65/152.70 The TRS R consists of the following rules: 588.65/152.70 588.65/152.70 2(1(1(0(x1)))) -> 2(3(1(0(1(x1))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(1(0(1(4(x1))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(3(1(4(1(0(x1)))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(1(3(0(1(4(x1)))))) 588.65/152.70 2(1(1(0(x1)))) -> 2(0(1(3(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(1(4(x1))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(1(4(5(x1))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(1(4(0(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(1(4(5(0(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(3(1(0(1(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(4(4(1(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(4(5(1(0(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(1(0(1(4(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(5(1(4(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(1(4(5(3(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 3(5(1(0(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 4(5(1(0(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(0(1(3(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 5(0(1(4(1(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 1(0(5(1(4(4(x1)))))) 588.65/152.70 5(1(1(0(x1)))) -> 0(1(3(1(4(5(x1)))))) 588.65/152.70 0(2(1(0(x1)))) -> 3(0(1(4(2(0(x1)))))) 588.65/152.70 5(3(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.70 5(4(1(0(x1)))) -> 1(4(5(3(0(x1))))) 588.65/152.70 5(4(1(0(x1)))) -> 3(5(1(0(4(4(x1)))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(3(2(0(4(x1))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(2(0(4(4(x1))))) 588.65/152.70 5(4(2(0(x1)))) -> 5(3(2(3(0(4(x1)))))) 588.65/152.70 5(1(2(0(0(x1))))) -> 1(4(5(2(0(0(x1)))))) 588.65/152.70 5(4(2(0(0(x1))))) -> 5(2(4(4(0(0(x1)))))) 588.65/152.70 5(4(0(1(0(x1))))) -> 5(1(0(0(4(0(x1)))))) 588.65/152.70 0(5(0(1(0(x1))))) -> 0(0(0(5(1(4(x1)))))) 588.65/152.70 5(0(1(1(0(x1))))) -> 5(1(0(4(0(1(x1)))))) 588.65/152.70 0(2(1(1(0(x1))))) -> 0(1(2(1(4(0(x1)))))) 588.65/152.70 0(2(1(1(0(x1))))) -> 0(0(1(2(1(4(x1)))))) 588.65/152.70 5(3(1(1(0(x1))))) -> 5(3(1(0(1(4(x1)))))) 588.65/152.70 5(3(1(1(0(x1))))) -> 1(3(0(1(4(5(x1)))))) 588.65/152.70 2(4(1(1(0(x1))))) -> 2(1(4(1(4(0(x1)))))) 588.65/152.70 2(4(1(1(0(x1))))) -> 0(2(1(3(1(4(x1)))))) 588.65/152.70 2(4(1(1(0(x1))))) -> 1(0(1(4(2(4(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 1(3(1(4(5(0(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 1(4(1(4(5(0(x1)))))) 588.65/152.70 5(4(1(1(0(x1))))) -> 5(1(0(1(4(2(x1)))))) 588.65/152.70 2(0(2(1(0(x1))))) -> 2(2(1(0(4(0(x1)))))) 588.65/152.70 5(1(2(1(0(x1))))) -> 5(2(1(4(1(0(x1)))))) 588.65/152.70 0(5(4(1(0(x1))))) -> 3(0(1(4(5(0(x1)))))) 588.65/152.70 5(1(5(1(0(x1))))) -> 5(1(0(1(4(5(x1)))))) 588.65/152.70 5(1(0(2(0(x1))))) -> 5(3(2(0(0(1(x1)))))) 588.65/152.70 5(4(0(2(0(x1))))) -> 5(1(4(2(0(0(x1)))))) 588.65/152.70 0(5(0(2(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(3(2(1(0(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(4(3(5(2(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(4(2(5(3(0(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(3(1(4(0(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(5(1(4(0(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 1(0(3(5(3(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 0(1(4(3(5(2(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(2(5(0(1(4(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 5(2(0(3(1(4(x1)))))) 588.65/152.70 5(1(3(2(0(x1))))) -> 3(0(2(5(1(4(x1)))))) 588.65/152.70 2(1(5(2(0(x1))))) -> 5(1(2(3(2(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(1(2(5(3(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(2(5(1(4(0(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(0(5(1(4(2(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 588.65/152.70 5(1(5(2(0(x1))))) -> 0(2(5(5(1(4(x1)))))) 588.65/152.70 5(1(5(3(0(x1))))) -> 1(4(5(3(0(5(x1)))))) 588.65/152.70 2(0(2(4(0(x1))))) -> 2(2(3(4(0(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 1(4(4(5(2(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 2(3(5(1(4(0(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 5(0(0(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 5(0(3(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 0(4(5(1(4(2(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 4(2(5(1(0(3(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 1(4(2(5(0(3(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 0(5(2(3(1(4(x1)))))) 588.65/152.70 5(1(2(4(0(x1))))) -> 2(5(1(0(4(4(x1)))))) 588.65/152.70 5(1(5(4(0(x1))))) -> 4(0(5(1(4(5(x1)))))) 588.65/152.70 588.65/152.70 Q is empty. 588.65/152.70 We have to consider all minimal (P,Q,R)-chains. 588.65/152.70 ---------------------------------------- 588.65/152.70 588.65/152.70 (79) UsableRulesProof (EQUIVALENT) 588.65/152.70 We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. 588.65/152.70 ---------------------------------------- 588.65/152.70 588.65/152.70 (80) 588.65/152.70 Obligation: 588.65/152.70 Q DP problem: 588.65/152.70 The TRS P consists of the following rules: 588.65/152.70 588.65/152.70 5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.70 588.65/152.70 R is empty. 588.65/152.70 Q is empty. 588.65/152.70 We have to consider all minimal (P,Q,R)-chains. 588.65/152.70 ---------------------------------------- 588.65/152.70 588.65/152.70 (81) QDPSizeChangeProof (EQUIVALENT) 588.65/152.70 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 588.65/152.70 588.65/152.70 From the DPs we obtained the following set of size-change graphs: 588.65/152.70 *5^1(3(1(1(0(x1))))) -> 5^1(x1) 588.65/152.70 The graph contains the following edges 1 > 1 588.65/152.70 588.65/152.70 588.65/152.70 ---------------------------------------- 588.65/152.70 588.65/152.70 (82) 588.65/152.70 YES 588.94/152.77 EOF