68.54/18.24 YES 70.31/18.69 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 70.31/18.69 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 70.31/18.69 70.31/18.69 70.31/18.69 Termination w.r.t. Q of the given QTRS could be proven: 70.31/18.69 70.31/18.69 (0) QTRS 70.31/18.69 (1) QTRS Reverse [EQUIVALENT, 0 ms] 70.31/18.69 (2) QTRS 70.31/18.69 (3) QTRSRRRProof [EQUIVALENT, 122 ms] 70.31/18.69 (4) QTRS 70.31/18.69 (5) DependencyPairsProof [EQUIVALENT, 230 ms] 70.31/18.69 (6) QDP 70.31/18.69 (7) DependencyGraphProof [EQUIVALENT, 10 ms] 70.31/18.69 (8) QDP 70.31/18.69 (9) QDPOrderProof [EQUIVALENT, 561 ms] 70.31/18.69 (10) QDP 70.31/18.69 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 70.31/18.69 (12) QDP 70.31/18.69 (13) QDPOrderProof [EQUIVALENT, 184 ms] 70.31/18.69 (14) QDP 70.31/18.69 (15) DependencyGraphProof [EQUIVALENT, 0 ms] 70.31/18.69 (16) QDP 70.31/18.69 (17) QDPOrderProof [EQUIVALENT, 36 ms] 70.31/18.69 (18) QDP 70.31/18.69 (19) PisEmptyProof [EQUIVALENT, 0 ms] 70.31/18.69 (20) YES 70.31/18.69 70.31/18.69 70.31/18.69 ---------------------------------------- 70.31/18.69 70.31/18.69 (0) 70.31/18.69 Obligation: 70.31/18.69 Q restricted rewrite system: 70.31/18.69 The TRS R consists of the following rules: 70.31/18.69 70.31/18.69 0(1(1(x1))) -> 2(1(1(x1))) 70.31/18.69 3(4(0(x1))) -> 4(5(5(x1))) 70.31/18.69 2(2(1(4(0(x1))))) -> 2(2(1(3(5(x1))))) 70.31/18.69 4(5(3(1(0(0(4(x1))))))) -> 4(5(0(1(4(5(4(x1))))))) 70.31/18.69 1(3(3(2(1(4(4(0(x1)))))))) -> 1(1(3(0(4(4(0(2(x1)))))))) 70.31/18.69 2(2(1(5(0(2(2(2(4(x1))))))))) -> 2(0(0(4(5(5(0(x1))))))) 70.31/18.69 0(2(3(0(2(0(3(5(2(4(x1)))))))))) -> 2(0(4(5(4(4(5(4(0(x1))))))))) 70.31/18.69 3(2(3(1(3(5(5(2(2(0(x1)))))))))) -> 4(0(0(0(0(2(2(2(0(x1))))))))) 70.31/18.69 3(4(1(1(0(0(5(0(5(0(x1)))))))))) -> 3(0(1(0(1(1(3(0(4(5(x1)))))))))) 70.31/18.69 4(1(3(0(0(5(1(5(0(1(x1)))))))))) -> 4(0(4(2(2(3(0(3(0(1(x1)))))))))) 70.31/18.69 3(3(0(2(0(1(4(5(5(5(0(x1))))))))))) -> 1(2(3(5(4(4(0(2(3(5(5(x1))))))))))) 70.31/18.69 3(0(2(0(2(1(2(5(4(3(5(4(x1)))))))))))) -> 3(2(2(0(3(0(5(4(5(1(5(4(x1)))))))))))) 70.31/18.69 3(0(5(4(5(3(0(0(0(5(1(1(x1)))))))))))) -> 3(2(4(1(4(0(3(0(2(1(1(3(x1)))))))))))) 70.31/18.69 3(3(0(4(4(1(1(1(3(1(2(1(x1)))))))))))) -> 3(3(5(3(0(1(4(4(0(0(1(x1))))))))))) 70.31/18.69 3(2(0(2(1(1(4(1(3(1(4(5(2(4(x1)))))))))))))) -> 0(2(2(4(4(5(4(1(0(0(5(0(0(x1))))))))))))) 70.31/18.69 4(1(3(3(0(5(2(3(3(3(0(5(4(3(x1)))))))))))))) -> 4(4(2(2(0(3(1(1(5(1(3(0(5(3(x1)))))))))))))) 70.31/18.69 3(4(0(2(1(1(0(4(1(1(0(2(3(5(3(x1))))))))))))))) -> 2(4(5(0(0(3(0(3(0(4(4(1(1(3(x1)))))))))))))) 70.31/18.69 5(5(1(4(1(3(4(2(3(4(3(1(2(0(5(x1))))))))))))))) -> 1(3(5(1(1(2(1(2(0(3(0(1(1(2(0(x1))))))))))))))) 70.31/18.69 0(5(0(3(3(3(2(3(1(1(1(2(5(1(2(5(x1)))))))))))))))) -> 0(5(1(4(3(3(4(4(1(1(4(1(0(1(4(x1))))))))))))))) 70.31/18.69 2(4(0(5(0(0(4(4(2(2(3(5(3(4(5(0(x1)))))))))))))))) -> 2(3(1(3(0(1(1(1(0(5(5(0(2(3(4(0(x1)))))))))))))))) 70.31/18.69 3(1(3(5(4(1(0(4(1(2(4(2(3(3(1(3(1(3(x1)))))))))))))))))) -> 0(3(3(0(0(3(3(4(4(1(4(1(5(2(2(4(3(x1))))))))))))))))) 70.31/18.69 2(5(0(5(2(1(3(1(4(5(5(0(0(2(3(0(3(2(4(3(x1)))))))))))))))))))) -> 0(2(0(0(3(1(4(2(5(2(0(5(4(4(1(1(0(4(3(x1))))))))))))))))))) 70.31/18.69 2(5(2(1(1(0(4(3(3(0(4(5(4(3(0(4(1(0(4(0(x1)))))))))))))))))))) -> 5(1(1(0(1(3(5(5(0(5(3(4(5(1(0(0(2(3(2(0(x1)))))))))))))))))))) 70.31/18.69 1(3(0(5(3(1(1(1(0(3(3(0(1(3(3(4(3(5(3(0(5(x1))))))))))))))))))))) -> 1(5(3(3(1(3(3(4(0(5(5(0(3(2(4(1(2(3(3(4(5(x1))))))))))))))))))))) 70.31/18.69 3(3(0(5(3(5(1(2(5(1(2(1(2(1(4(0(1(2(3(1(5(x1))))))))))))))))))))) -> 0(2(0(5(5(4(5(2(5(0(0(4(2(0(5(5(5(3(1(5(x1)))))))))))))))))))) 70.31/18.69 70.31/18.69 Q is empty. 70.31/18.69 70.31/18.69 ---------------------------------------- 70.31/18.69 70.31/18.69 (1) QTRS Reverse (EQUIVALENT) 70.31/18.69 We applied the QTRS Reverse Processor [REVERSE]. 70.31/18.69 ---------------------------------------- 70.31/18.69 70.31/18.69 (2) 70.31/18.69 Obligation: 70.31/18.69 Q restricted rewrite system: 70.31/18.69 The TRS R consists of the following rules: 70.31/18.69 70.31/18.69 1(1(0(x1))) -> 1(1(2(x1))) 70.31/18.69 0(4(3(x1))) -> 5(5(4(x1))) 70.31/18.69 0(4(1(2(2(x1))))) -> 5(3(1(2(2(x1))))) 70.31/18.69 4(0(0(1(3(5(4(x1))))))) -> 4(5(4(1(0(5(4(x1))))))) 70.31/18.69 0(4(4(1(2(3(3(1(x1)))))))) -> 2(0(4(4(0(3(1(1(x1)))))))) 70.31/18.69 4(2(2(2(0(5(1(2(2(x1))))))))) -> 0(5(5(4(0(0(2(x1))))))) 70.31/18.69 4(2(5(3(0(2(0(3(2(0(x1)))))))))) -> 0(4(5(4(4(5(4(0(2(x1))))))))) 70.31/18.69 0(2(2(5(5(3(1(3(2(3(x1)))))))))) -> 0(2(2(2(0(0(0(0(4(x1))))))))) 70.31/18.69 0(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5(4(0(3(1(1(0(1(0(3(x1)))))))))) 70.31/18.69 1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 1(0(3(0(3(2(2(4(0(4(x1)))))))))) 70.31/18.69 0(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 5(5(3(2(0(4(4(5(3(2(1(x1))))))))))) 70.31/18.69 4(5(3(4(5(2(1(2(0(2(0(3(x1)))))))))))) -> 4(5(1(5(4(5(0(3(0(2(2(3(x1)))))))))))) 70.31/18.69 1(1(5(0(0(0(3(5(4(5(0(3(x1)))))))))))) -> 3(1(1(2(0(3(0(4(1(4(2(3(x1)))))))))))) 70.31/18.69 1(2(1(3(1(1(1(4(4(0(3(3(x1)))))))))))) -> 1(0(0(4(4(1(0(3(5(3(3(x1))))))))))) 70.31/18.69 4(2(5(4(1(3(1(4(1(1(2(0(2(3(x1)))))))))))))) -> 0(0(5(0(0(1(4(5(4(4(2(2(0(x1))))))))))))) 70.31/18.69 3(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 3(5(0(3(1(5(1(1(3(0(2(2(4(4(x1)))))))))))))) 70.31/18.69 3(5(3(2(0(1(1(4(0(1(1(2(0(4(3(x1))))))))))))))) -> 3(1(1(4(4(0(3(0(3(0(0(5(4(2(x1)))))))))))))) 70.31/18.69 5(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 0(2(1(1(0(3(0(2(1(2(1(1(5(3(1(x1))))))))))))))) 70.31/18.69 5(2(1(5(2(1(1(1(3(2(3(3(3(0(5(0(x1)))))))))))))))) -> 4(1(0(1(4(1(1(4(4(3(3(4(1(5(0(x1))))))))))))))) 70.31/18.69 0(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 0(4(3(2(0(5(5(0(1(1(1(0(3(1(3(2(x1)))))))))))))))) 70.31/18.69 3(1(3(1(3(3(2(4(2(1(4(0(1(4(5(3(1(3(x1)))))))))))))))))) -> 3(4(2(2(5(1(4(1(4(4(3(3(0(0(3(3(0(x1))))))))))))))))) 70.31/18.69 3(4(2(3(0(3(2(0(0(5(5(4(1(3(1(2(5(0(5(2(x1)))))))))))))))))))) -> 3(4(0(1(1(4(4(5(0(2(5(2(4(1(3(0(0(2(0(x1))))))))))))))))))) 70.31/18.69 0(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0(2(3(2(0(0(1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))))))))))) 70.31/18.69 5(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5(4(3(3(2(1(4(2(3(0(5(5(0(4(3(3(1(3(3(5(1(x1))))))))))))))))))))) 70.31/18.69 5(1(3(2(1(0(4(1(2(1(2(1(5(2(1(5(3(5(0(3(3(x1))))))))))))))))))))) -> 5(1(3(5(5(5(0(2(4(0(0(5(2(5(4(5(5(0(2(0(x1)))))))))))))))))))) 70.31/18.69 70.31/18.69 Q is empty. 70.31/18.69 70.31/18.69 ---------------------------------------- 70.31/18.69 70.31/18.69 (3) QTRSRRRProof (EQUIVALENT) 70.31/18.69 Used ordering: 70.31/18.69 Polynomial interpretation [POLO]: 70.31/18.69 70.31/18.69 POL(0(x_1)) = 1 + x_1 70.31/18.69 POL(1(x_1)) = 1 + x_1 70.31/18.69 POL(2(x_1)) = 1 + x_1 70.31/18.69 POL(3(x_1)) = 1 + x_1 70.31/18.69 POL(4(x_1)) = 1 + x_1 70.31/18.69 POL(5(x_1)) = 1 + x_1 70.31/18.69 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 70.31/18.69 70.31/18.69 4(2(2(2(0(5(1(2(2(x1))))))))) -> 0(5(5(4(0(0(2(x1))))))) 70.31/18.69 4(2(5(3(0(2(0(3(2(0(x1)))))))))) -> 0(4(5(4(4(5(4(0(2(x1))))))))) 70.31/18.69 0(2(2(5(5(3(1(3(2(3(x1)))))))))) -> 0(2(2(2(0(0(0(0(4(x1))))))))) 70.31/18.69 1(2(1(3(1(1(1(4(4(0(3(3(x1)))))))))))) -> 1(0(0(4(4(1(0(3(5(3(3(x1))))))))))) 70.31/18.69 4(2(5(4(1(3(1(4(1(1(2(0(2(3(x1)))))))))))))) -> 0(0(5(0(0(1(4(5(4(4(2(2(0(x1))))))))))))) 70.31/18.69 3(5(3(2(0(1(1(4(0(1(1(2(0(4(3(x1))))))))))))))) -> 3(1(1(4(4(0(3(0(3(0(0(5(4(2(x1)))))))))))))) 70.31/18.69 5(2(1(5(2(1(1(1(3(2(3(3(3(0(5(0(x1)))))))))))))))) -> 4(1(0(1(4(1(1(4(4(3(3(4(1(5(0(x1))))))))))))))) 70.31/18.69 3(1(3(1(3(3(2(4(2(1(4(0(1(4(5(3(1(3(x1)))))))))))))))))) -> 3(4(2(2(5(1(4(1(4(4(3(3(0(0(3(3(0(x1))))))))))))))))) 70.31/18.69 3(4(2(3(0(3(2(0(0(5(5(4(1(3(1(2(5(0(5(2(x1)))))))))))))))))))) -> 3(4(0(1(1(4(4(5(0(2(5(2(4(1(3(0(0(2(0(x1))))))))))))))))))) 70.31/18.69 5(1(3(2(1(0(4(1(2(1(2(1(5(2(1(5(3(5(0(3(3(x1))))))))))))))))))))) -> 5(1(3(5(5(5(0(2(4(0(0(5(2(5(4(5(5(0(2(0(x1)))))))))))))))))))) 70.31/18.69 70.31/18.69 70.31/18.69 70.31/18.69 70.31/18.69 ---------------------------------------- 70.31/18.69 70.31/18.69 (4) 70.31/18.69 Obligation: 70.31/18.69 Q restricted rewrite system: 70.31/18.69 The TRS R consists of the following rules: 70.31/18.69 70.31/18.69 1(1(0(x1))) -> 1(1(2(x1))) 70.31/18.69 0(4(3(x1))) -> 5(5(4(x1))) 70.31/18.69 0(4(1(2(2(x1))))) -> 5(3(1(2(2(x1))))) 70.31/18.69 4(0(0(1(3(5(4(x1))))))) -> 4(5(4(1(0(5(4(x1))))))) 70.31/18.69 0(4(4(1(2(3(3(1(x1)))))))) -> 2(0(4(4(0(3(1(1(x1)))))))) 70.31/18.69 0(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5(4(0(3(1(1(0(1(0(3(x1)))))))))) 70.31/18.69 1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 1(0(3(0(3(2(2(4(0(4(x1)))))))))) 70.31/18.69 0(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 5(5(3(2(0(4(4(5(3(2(1(x1))))))))))) 70.31/18.69 4(5(3(4(5(2(1(2(0(2(0(3(x1)))))))))))) -> 4(5(1(5(4(5(0(3(0(2(2(3(x1)))))))))))) 70.31/18.69 1(1(5(0(0(0(3(5(4(5(0(3(x1)))))))))))) -> 3(1(1(2(0(3(0(4(1(4(2(3(x1)))))))))))) 70.31/18.69 3(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 3(5(0(3(1(5(1(1(3(0(2(2(4(4(x1)))))))))))))) 70.31/18.69 5(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 0(2(1(1(0(3(0(2(1(2(1(1(5(3(1(x1))))))))))))))) 70.31/18.69 0(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 0(4(3(2(0(5(5(0(1(1(1(0(3(1(3(2(x1)))))))))))))))) 70.31/18.69 0(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0(2(3(2(0(0(1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))))))))))) 70.31/18.69 5(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5(4(3(3(2(1(4(2(3(0(5(5(0(4(3(3(1(3(3(5(1(x1))))))))))))))))))))) 70.31/18.69 70.31/18.69 Q is empty. 70.31/18.69 70.31/18.69 ---------------------------------------- 70.31/18.69 70.31/18.69 (5) DependencyPairsProof (EQUIVALENT) 70.31/18.69 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 70.31/18.69 ---------------------------------------- 70.31/18.69 70.31/18.69 (6) 70.31/18.69 Obligation: 70.31/18.69 Q DP problem: 70.31/18.69 The TRS P consists of the following rules: 70.31/18.69 70.31/18.69 1^1(1(0(x1))) -> 1^1(1(2(x1))) 70.31/18.69 1^1(1(0(x1))) -> 1^1(2(x1)) 70.31/18.69 0^1(4(3(x1))) -> 5^1(5(4(x1))) 70.31/18.69 0^1(4(3(x1))) -> 5^1(4(x1)) 70.31/18.69 0^1(4(3(x1))) -> 4^1(x1) 70.31/18.69 0^1(4(1(2(2(x1))))) -> 5^1(3(1(2(2(x1))))) 70.31/18.69 0^1(4(1(2(2(x1))))) -> 3^1(1(2(2(x1)))) 70.31/18.69 4^1(0(0(1(3(5(4(x1))))))) -> 4^1(5(4(1(0(5(4(x1))))))) 70.31/18.69 4^1(0(0(1(3(5(4(x1))))))) -> 5^1(4(1(0(5(4(x1)))))) 70.31/18.69 4^1(0(0(1(3(5(4(x1))))))) -> 4^1(1(0(5(4(x1))))) 70.31/18.69 4^1(0(0(1(3(5(4(x1))))))) -> 1^1(0(5(4(x1)))) 70.31/18.69 4^1(0(0(1(3(5(4(x1))))))) -> 0^1(5(4(x1))) 70.31/18.69 0^1(4(4(1(2(3(3(1(x1)))))))) -> 0^1(4(4(0(3(1(1(x1))))))) 70.31/18.69 0^1(4(4(1(2(3(3(1(x1)))))))) -> 4^1(4(0(3(1(1(x1)))))) 70.31/18.69 0^1(4(4(1(2(3(3(1(x1)))))))) -> 4^1(0(3(1(1(x1))))) 70.31/18.69 0^1(4(4(1(2(3(3(1(x1)))))))) -> 0^1(3(1(1(x1)))) 70.31/18.69 0^1(4(4(1(2(3(3(1(x1)))))))) -> 3^1(1(1(x1))) 70.31/18.69 0^1(4(4(1(2(3(3(1(x1)))))))) -> 1^1(1(x1)) 70.31/18.69 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5^1(4(0(3(1(1(0(1(0(3(x1)))))))))) 70.31/18.69 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 4^1(0(3(1(1(0(1(0(3(x1))))))))) 70.31/18.69 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(3(1(1(0(1(0(3(x1)))))))) 70.31/18.69 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 3^1(1(1(0(1(0(3(x1))))))) 70.31/18.69 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(1(0(1(0(3(x1)))))) 70.31/18.69 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(0(1(0(3(x1))))) 70.31/18.69 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(1(0(3(x1)))) 70.31/18.69 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(0(3(x1))) 70.31/18.69 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(3(x1)) 70.31/18.69 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 1^1(0(3(0(3(2(2(4(0(4(x1)))))))))) 70.31/18.69 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 0^1(3(0(3(2(2(4(0(4(x1))))))))) 70.31/18.69 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 3^1(0(3(2(2(4(0(4(x1)))))))) 70.31/18.69 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 0^1(3(2(2(4(0(4(x1))))))) 70.31/18.69 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 3^1(2(2(4(0(4(x1)))))) 70.31/18.69 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 4^1(0(4(x1))) 70.31/18.69 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 0^1(4(x1)) 70.31/18.69 0^1(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 5^1(5(3(2(0(4(4(5(3(2(1(x1))))))))))) 70.31/18.69 0^1(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 5^1(3(2(0(4(4(5(3(2(1(x1)))))))))) 70.31/18.69 0^1(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 3^1(2(0(4(4(5(3(2(1(x1))))))))) 70.31/18.69 0^1(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 0^1(4(4(5(3(2(1(x1))))))) 70.31/18.69 0^1(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 4^1(4(5(3(2(1(x1)))))) 70.31/18.69 0^1(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 4^1(5(3(2(1(x1))))) 70.31/18.69 0^1(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 5^1(3(2(1(x1)))) 70.31/18.69 0^1(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 3^1(2(1(x1))) 70.31/18.69 0^1(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 1^1(x1) 70.31/18.69 4^1(5(3(4(5(2(1(2(0(2(0(3(x1)))))))))))) -> 4^1(5(1(5(4(5(0(3(0(2(2(3(x1)))))))))))) 70.31/18.69 4^1(5(3(4(5(2(1(2(0(2(0(3(x1)))))))))))) -> 5^1(1(5(4(5(0(3(0(2(2(3(x1))))))))))) 70.31/18.69 4^1(5(3(4(5(2(1(2(0(2(0(3(x1)))))))))))) -> 1^1(5(4(5(0(3(0(2(2(3(x1)))))))))) 70.31/18.69 4^1(5(3(4(5(2(1(2(0(2(0(3(x1)))))))))))) -> 5^1(4(5(0(3(0(2(2(3(x1))))))))) 70.31/18.69 4^1(5(3(4(5(2(1(2(0(2(0(3(x1)))))))))))) -> 4^1(5(0(3(0(2(2(3(x1)))))))) 70.31/18.69 4^1(5(3(4(5(2(1(2(0(2(0(3(x1)))))))))))) -> 5^1(0(3(0(2(2(3(x1))))))) 70.31/18.69 4^1(5(3(4(5(2(1(2(0(2(0(3(x1)))))))))))) -> 0^1(3(0(2(2(3(x1)))))) 70.31/18.69 4^1(5(3(4(5(2(1(2(0(2(0(3(x1)))))))))))) -> 3^1(0(2(2(3(x1))))) 70.31/18.69 4^1(5(3(4(5(2(1(2(0(2(0(3(x1)))))))))))) -> 0^1(2(2(3(x1)))) 70.31/18.69 1^1(1(5(0(0(0(3(5(4(5(0(3(x1)))))))))))) -> 3^1(1(1(2(0(3(0(4(1(4(2(3(x1)))))))))))) 70.31/18.69 1^1(1(5(0(0(0(3(5(4(5(0(3(x1)))))))))))) -> 1^1(1(2(0(3(0(4(1(4(2(3(x1))))))))))) 70.31/18.69 1^1(1(5(0(0(0(3(5(4(5(0(3(x1)))))))))))) -> 1^1(2(0(3(0(4(1(4(2(3(x1)))))))))) 70.31/18.69 1^1(1(5(0(0(0(3(5(4(5(0(3(x1)))))))))))) -> 0^1(3(0(4(1(4(2(3(x1)))))))) 70.31/18.69 1^1(1(5(0(0(0(3(5(4(5(0(3(x1)))))))))))) -> 3^1(0(4(1(4(2(3(x1))))))) 70.31/18.69 1^1(1(5(0(0(0(3(5(4(5(0(3(x1)))))))))))) -> 0^1(4(1(4(2(3(x1)))))) 70.31/18.69 1^1(1(5(0(0(0(3(5(4(5(0(3(x1)))))))))))) -> 4^1(1(4(2(3(x1))))) 70.31/18.69 1^1(1(5(0(0(0(3(5(4(5(0(3(x1)))))))))))) -> 1^1(4(2(3(x1)))) 70.31/18.69 1^1(1(5(0(0(0(3(5(4(5(0(3(x1)))))))))))) -> 4^1(2(3(x1))) 70.31/18.69 3^1(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 3^1(5(0(3(1(5(1(1(3(0(2(2(4(4(x1)))))))))))))) 70.31/18.69 3^1(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 5^1(0(3(1(5(1(1(3(0(2(2(4(4(x1))))))))))))) 70.31/18.69 3^1(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 0^1(3(1(5(1(1(3(0(2(2(4(4(x1)))))))))))) 70.31/18.69 3^1(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 3^1(1(5(1(1(3(0(2(2(4(4(x1))))))))))) 70.31/18.69 3^1(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 1^1(5(1(1(3(0(2(2(4(4(x1)))))))))) 70.31/18.69 3^1(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 5^1(1(1(3(0(2(2(4(4(x1))))))))) 70.31/18.69 3^1(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 1^1(1(3(0(2(2(4(4(x1)))))))) 70.31/18.69 3^1(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 1^1(3(0(2(2(4(4(x1))))))) 70.31/18.69 3^1(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 3^1(0(2(2(4(4(x1)))))) 70.31/18.69 3^1(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 0^1(2(2(4(4(x1))))) 70.31/18.69 3^1(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 4^1(4(x1)) 70.31/18.69 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 0^1(2(1(1(0(3(0(2(1(2(1(1(5(3(1(x1))))))))))))))) 70.31/18.69 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(1(0(3(0(2(1(2(1(1(5(3(1(x1))))))))))))) 70.31/18.69 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(0(3(0(2(1(2(1(1(5(3(1(x1)))))))))))) 70.31/18.69 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 0^1(3(0(2(1(2(1(1(5(3(1(x1))))))))))) 70.31/18.69 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 3^1(0(2(1(2(1(1(5(3(1(x1)))))))))) 70.31/18.69 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 0^1(2(1(2(1(1(5(3(1(x1))))))))) 70.31/18.69 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(2(1(1(5(3(1(x1))))))) 70.31/18.69 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(1(5(3(1(x1))))) 70.31/18.69 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(5(3(1(x1)))) 70.31/18.69 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 5^1(3(1(x1))) 70.31/18.69 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 3^1(1(x1)) 70.31/18.69 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(x1) 70.31/18.69 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 0^1(4(3(2(0(5(5(0(1(1(1(0(3(1(3(2(x1)))))))))))))))) 70.31/18.69 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 4^1(3(2(0(5(5(0(1(1(1(0(3(1(3(2(x1))))))))))))))) 70.31/18.69 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 3^1(2(0(5(5(0(1(1(1(0(3(1(3(2(x1)))))))))))))) 70.31/18.69 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 0^1(5(5(0(1(1(1(0(3(1(3(2(x1)))))))))))) 70.31/18.69 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 5^1(5(0(1(1(1(0(3(1(3(2(x1))))))))))) 70.31/18.69 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 5^1(0(1(1(1(0(3(1(3(2(x1)))))))))) 70.31/18.69 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 0^1(1(1(1(0(3(1(3(2(x1))))))))) 70.31/18.69 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 1^1(1(1(0(3(1(3(2(x1)))))))) 70.31/18.69 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 1^1(1(0(3(1(3(2(x1))))))) 70.31/18.69 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 1^1(0(3(1(3(2(x1)))))) 70.31/18.69 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 0^1(3(1(3(2(x1))))) 70.31/18.69 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 3^1(1(3(2(x1)))) 70.31/18.69 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 1^1(3(2(x1))) 70.31/18.69 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 3^1(2(x1)) 70.31/18.69 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0^1(2(3(2(0(0(1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))))))))))) 70.31/18.69 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 3^1(2(0(0(1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))))))))) 70.31/18.69 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0^1(0(1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))))))) 70.31/18.69 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0^1(1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1))))))))))))))) 70.31/18.69 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 1^1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))))) 70.31/18.69 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 5^1(4(3(5(0(5(5(3(1(0(1(1(5(x1))))))))))))) 70.31/18.69 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 4^1(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))) 70.31/18.69 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 3^1(5(0(5(5(3(1(0(1(1(5(x1))))))))))) 70.31/18.69 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 5^1(0(5(5(3(1(0(1(1(5(x1)))))))))) 70.31/18.69 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0^1(5(5(3(1(0(1(1(5(x1))))))))) 70.31/18.69 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 5^1(5(3(1(0(1(1(5(x1)))))))) 70.31/18.69 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 5^1(3(1(0(1(1(5(x1))))))) 70.31/18.69 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 3^1(1(0(1(1(5(x1)))))) 70.31/18.69 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 1^1(0(1(1(5(x1))))) 70.31/18.69 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0^1(1(1(5(x1)))) 70.31/18.69 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 1^1(1(5(x1))) 70.31/18.69 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 1^1(5(x1)) 70.31/18.69 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 5^1(x1) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5^1(4(3(3(2(1(4(2(3(0(5(5(0(4(3(3(1(3(3(5(1(x1))))))))))))))))))))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 4^1(3(3(2(1(4(2(3(0(5(5(0(4(3(3(1(3(3(5(1(x1)))))))))))))))))))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 3^1(3(2(1(4(2(3(0(5(5(0(4(3(3(1(3(3(5(1(x1))))))))))))))))))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 3^1(2(1(4(2(3(0(5(5(0(4(3(3(1(3(3(5(1(x1)))))))))))))))))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 1^1(4(2(3(0(5(5(0(4(3(3(1(3(3(5(1(x1)))))))))))))))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 4^1(2(3(0(5(5(0(4(3(3(1(3(3(5(1(x1))))))))))))))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 3^1(0(5(5(0(4(3(3(1(3(3(5(1(x1))))))))))))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 0^1(5(5(0(4(3(3(1(3(3(5(1(x1)))))))))))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5^1(5(0(4(3(3(1(3(3(5(1(x1))))))))))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5^1(0(4(3(3(1(3(3(5(1(x1)))))))))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 0^1(4(3(3(1(3(3(5(1(x1))))))))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 4^1(3(3(1(3(3(5(1(x1)))))))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 3^1(3(1(3(3(5(1(x1))))))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 3^1(1(3(3(5(1(x1)))))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 1^1(3(3(5(1(x1))))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 3^1(3(5(1(x1)))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 3^1(5(1(x1))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5^1(1(x1)) 70.31/18.69 70.31/18.69 The TRS R consists of the following rules: 70.31/18.69 70.31/18.69 1(1(0(x1))) -> 1(1(2(x1))) 70.31/18.69 0(4(3(x1))) -> 5(5(4(x1))) 70.31/18.69 0(4(1(2(2(x1))))) -> 5(3(1(2(2(x1))))) 70.31/18.69 4(0(0(1(3(5(4(x1))))))) -> 4(5(4(1(0(5(4(x1))))))) 70.31/18.69 0(4(4(1(2(3(3(1(x1)))))))) -> 2(0(4(4(0(3(1(1(x1)))))))) 70.31/18.69 0(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5(4(0(3(1(1(0(1(0(3(x1)))))))))) 70.31/18.69 1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 1(0(3(0(3(2(2(4(0(4(x1)))))))))) 70.31/18.69 0(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 5(5(3(2(0(4(4(5(3(2(1(x1))))))))))) 70.31/18.69 4(5(3(4(5(2(1(2(0(2(0(3(x1)))))))))))) -> 4(5(1(5(4(5(0(3(0(2(2(3(x1)))))))))))) 70.31/18.69 1(1(5(0(0(0(3(5(4(5(0(3(x1)))))))))))) -> 3(1(1(2(0(3(0(4(1(4(2(3(x1)))))))))))) 70.31/18.69 3(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 3(5(0(3(1(5(1(1(3(0(2(2(4(4(x1)))))))))))))) 70.31/18.69 5(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 0(2(1(1(0(3(0(2(1(2(1(1(5(3(1(x1))))))))))))))) 70.31/18.69 0(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 0(4(3(2(0(5(5(0(1(1(1(0(3(1(3(2(x1)))))))))))))))) 70.31/18.69 0(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0(2(3(2(0(0(1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))))))))))) 70.31/18.69 5(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5(4(3(3(2(1(4(2(3(0(5(5(0(4(3(3(1(3(3(5(1(x1))))))))))))))))))))) 70.31/18.69 70.31/18.69 Q is empty. 70.31/18.69 We have to consider all minimal (P,Q,R)-chains. 70.31/18.69 ---------------------------------------- 70.31/18.69 70.31/18.69 (7) DependencyGraphProof (EQUIVALENT) 70.31/18.69 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 68 less nodes. 70.31/18.69 ---------------------------------------- 70.31/18.69 70.31/18.69 (8) 70.31/18.69 Obligation: 70.31/18.69 Q DP problem: 70.31/18.69 The TRS P consists of the following rules: 70.31/18.69 70.31/18.69 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 4^1(0(4(x1))) 70.31/18.69 4^1(0(0(1(3(5(4(x1))))))) -> 4^1(5(4(1(0(5(4(x1))))))) 70.31/18.69 4^1(0(0(1(3(5(4(x1))))))) -> 5^1(4(1(0(5(4(x1)))))) 70.31/18.69 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(1(5(3(1(x1))))) 70.31/18.69 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 0^1(4(x1)) 70.31/18.69 0^1(4(3(x1))) -> 5^1(5(4(x1))) 70.31/18.69 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(5(3(1(x1)))) 70.31/18.69 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 5^1(3(1(x1))) 70.31/18.69 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 3^1(1(x1)) 70.31/18.69 3^1(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 4^1(4(x1)) 70.31/18.69 4^1(0(0(1(3(5(4(x1))))))) -> 4^1(1(0(5(4(x1))))) 70.31/18.69 4^1(0(0(1(3(5(4(x1))))))) -> 1^1(0(5(4(x1)))) 70.31/18.69 4^1(0(0(1(3(5(4(x1))))))) -> 0^1(5(4(x1))) 70.31/18.69 0^1(4(3(x1))) -> 5^1(4(x1)) 70.31/18.69 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(x1) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 3^1(0(5(5(0(4(3(3(1(3(3(5(1(x1))))))))))))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 0^1(5(5(0(4(3(3(1(3(3(5(1(x1)))))))))))) 70.31/18.69 0^1(4(3(x1))) -> 4^1(x1) 70.31/18.69 0^1(4(4(1(2(3(3(1(x1)))))))) -> 0^1(4(4(0(3(1(1(x1))))))) 70.31/18.69 0^1(4(4(1(2(3(3(1(x1)))))))) -> 4^1(4(0(3(1(1(x1)))))) 70.31/18.69 0^1(4(4(1(2(3(3(1(x1)))))))) -> 4^1(0(3(1(1(x1))))) 70.31/18.69 0^1(4(4(1(2(3(3(1(x1)))))))) -> 0^1(3(1(1(x1)))) 70.31/18.69 0^1(4(4(1(2(3(3(1(x1)))))))) -> 3^1(1(1(x1))) 70.31/18.69 0^1(4(4(1(2(3(3(1(x1)))))))) -> 1^1(1(x1)) 70.31/18.69 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5^1(4(0(3(1(1(0(1(0(3(x1)))))))))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5^1(5(0(4(3(3(1(3(3(5(1(x1))))))))))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5^1(0(4(3(3(1(3(3(5(1(x1)))))))))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 0^1(4(3(3(1(3(3(5(1(x1))))))))) 70.31/18.69 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 4^1(0(3(1(1(0(1(0(3(x1))))))))) 70.31/18.69 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(3(1(1(0(1(0(3(x1)))))))) 70.31/18.69 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 3^1(1(1(0(1(0(3(x1))))))) 70.31/18.69 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(1(0(1(0(3(x1)))))) 70.31/18.69 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(0(1(0(3(x1))))) 70.31/18.69 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(1(0(3(x1)))) 70.31/18.69 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(0(3(x1))) 70.31/18.69 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(3(x1)) 70.31/18.69 0^1(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 1^1(x1) 70.31/18.69 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 0^1(4(3(2(0(5(5(0(1(1(1(0(3(1(3(2(x1)))))))))))))))) 70.31/18.69 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 0^1(5(5(0(1(1(1(0(3(1(3(2(x1)))))))))))) 70.31/18.69 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 5^1(5(0(1(1(1(0(3(1(3(2(x1))))))))))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 4^1(3(3(1(3(3(5(1(x1)))))))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 3^1(3(1(3(3(5(1(x1))))))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 3^1(1(3(3(5(1(x1)))))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 1^1(3(3(5(1(x1))))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 3^1(3(5(1(x1)))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 3^1(5(1(x1))) 70.31/18.69 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5^1(1(x1)) 70.31/18.69 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 5^1(0(1(1(1(0(3(1(3(2(x1)))))))))) 70.31/18.69 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 0^1(1(1(1(0(3(1(3(2(x1))))))))) 70.31/18.69 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 1^1(1(1(0(3(1(3(2(x1)))))))) 70.31/18.69 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0^1(0(1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0^1(1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1))))))))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 1^1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 5^1(4(3(5(0(5(5(3(1(0(1(1(5(x1))))))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 4^1(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 3^1(5(0(5(5(3(1(0(1(1(5(x1))))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 5^1(0(5(5(3(1(0(1(1(5(x1)))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0^1(5(5(3(1(0(1(1(5(x1))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 5^1(5(3(1(0(1(1(5(x1)))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 5^1(3(1(0(1(1(5(x1))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 3^1(1(0(1(1(5(x1)))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 1^1(0(1(1(5(x1))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0^1(1(1(5(x1)))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 1^1(1(5(x1))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 1^1(5(x1)) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 5^1(x1) 70.31/18.71 70.31/18.71 The TRS R consists of the following rules: 70.31/18.71 70.31/18.71 1(1(0(x1))) -> 1(1(2(x1))) 70.31/18.71 0(4(3(x1))) -> 5(5(4(x1))) 70.31/18.71 0(4(1(2(2(x1))))) -> 5(3(1(2(2(x1))))) 70.31/18.71 4(0(0(1(3(5(4(x1))))))) -> 4(5(4(1(0(5(4(x1))))))) 70.31/18.71 0(4(4(1(2(3(3(1(x1)))))))) -> 2(0(4(4(0(3(1(1(x1)))))))) 70.31/18.71 0(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5(4(0(3(1(1(0(1(0(3(x1)))))))))) 70.31/18.71 1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 1(0(3(0(3(2(2(4(0(4(x1)))))))))) 70.31/18.71 0(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 5(5(3(2(0(4(4(5(3(2(1(x1))))))))))) 70.31/18.71 4(5(3(4(5(2(1(2(0(2(0(3(x1)))))))))))) -> 4(5(1(5(4(5(0(3(0(2(2(3(x1)))))))))))) 70.31/18.71 1(1(5(0(0(0(3(5(4(5(0(3(x1)))))))))))) -> 3(1(1(2(0(3(0(4(1(4(2(3(x1)))))))))))) 70.31/18.71 3(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 3(5(0(3(1(5(1(1(3(0(2(2(4(4(x1)))))))))))))) 70.31/18.71 5(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 0(2(1(1(0(3(0(2(1(2(1(1(5(3(1(x1))))))))))))))) 70.31/18.71 0(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 0(4(3(2(0(5(5(0(1(1(1(0(3(1(3(2(x1)))))))))))))))) 70.31/18.71 0(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0(2(3(2(0(0(1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))))))))))) 70.31/18.71 5(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5(4(3(3(2(1(4(2(3(0(5(5(0(4(3(3(1(3(3(5(1(x1))))))))))))))))))))) 70.31/18.71 70.31/18.71 Q is empty. 70.31/18.71 We have to consider all minimal (P,Q,R)-chains. 70.31/18.71 ---------------------------------------- 70.31/18.71 70.31/18.71 (9) QDPOrderProof (EQUIVALENT) 70.31/18.71 We use the reduction pair processor [LPAR04,JAR06]. 70.31/18.71 70.31/18.71 70.31/18.71 The following pairs can be oriented strictly and are deleted. 70.31/18.71 70.31/18.71 3^1(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 4^1(4(x1)) 70.31/18.71 The remaining pairs can at least be oriented weakly. 70.31/18.71 Used ordering: Polynomial interpretation [POLO]: 70.31/18.71 70.31/18.71 POL(0(x_1)) = 0 70.31/18.71 POL(0^1(x_1)) = 0 70.31/18.71 POL(1(x_1)) = 0 70.31/18.71 POL(1^1(x_1)) = 0 70.31/18.71 POL(2(x_1)) = 0 70.31/18.71 POL(3(x_1)) = x_1 70.31/18.71 POL(3^1(x_1)) = x_1 70.31/18.71 POL(4(x_1)) = 1 70.31/18.71 POL(4^1(x_1)) = 0 70.31/18.71 POL(5(x_1)) = 0 70.31/18.71 POL(5^1(x_1)) = 0 70.31/18.71 70.31/18.71 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 70.31/18.71 70.31/18.71 0(4(3(x1))) -> 5(5(4(x1))) 70.31/18.71 0(4(1(2(2(x1))))) -> 5(3(1(2(2(x1))))) 70.31/18.71 0(4(4(1(2(3(3(1(x1)))))))) -> 2(0(4(4(0(3(1(1(x1)))))))) 70.31/18.71 0(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5(4(0(3(1(1(0(1(0(3(x1)))))))))) 70.31/18.71 0(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 5(5(3(2(0(4(4(5(3(2(1(x1))))))))))) 70.31/18.71 0(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 0(4(3(2(0(5(5(0(1(1(1(0(3(1(3(2(x1)))))))))))))))) 70.31/18.71 0(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0(2(3(2(0(0(1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))))))))))) 70.31/18.71 5(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 0(2(1(1(0(3(0(2(1(2(1(1(5(3(1(x1))))))))))))))) 70.31/18.71 5(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5(4(3(3(2(1(4(2(3(0(5(5(0(4(3(3(1(3(3(5(1(x1))))))))))))))))))))) 70.31/18.71 1(1(0(x1))) -> 1(1(2(x1))) 70.31/18.71 1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 1(0(3(0(3(2(2(4(0(4(x1)))))))))) 70.31/18.71 1(1(5(0(0(0(3(5(4(5(0(3(x1)))))))))))) -> 3(1(1(2(0(3(0(4(1(4(2(3(x1)))))))))))) 70.31/18.71 3(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 3(5(0(3(1(5(1(1(3(0(2(2(4(4(x1)))))))))))))) 70.31/18.71 70.31/18.71 70.31/18.71 ---------------------------------------- 70.31/18.71 70.31/18.71 (10) 70.31/18.71 Obligation: 70.31/18.71 Q DP problem: 70.31/18.71 The TRS P consists of the following rules: 70.31/18.71 70.31/18.71 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 4^1(0(4(x1))) 70.31/18.71 4^1(0(0(1(3(5(4(x1))))))) -> 4^1(5(4(1(0(5(4(x1))))))) 70.31/18.71 4^1(0(0(1(3(5(4(x1))))))) -> 5^1(4(1(0(5(4(x1)))))) 70.31/18.71 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(1(5(3(1(x1))))) 70.31/18.71 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 0^1(4(x1)) 70.31/18.71 0^1(4(3(x1))) -> 5^1(5(4(x1))) 70.31/18.71 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(5(3(1(x1)))) 70.31/18.71 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 5^1(3(1(x1))) 70.31/18.71 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 3^1(1(x1)) 70.31/18.71 4^1(0(0(1(3(5(4(x1))))))) -> 4^1(1(0(5(4(x1))))) 70.31/18.71 4^1(0(0(1(3(5(4(x1))))))) -> 1^1(0(5(4(x1)))) 70.31/18.71 4^1(0(0(1(3(5(4(x1))))))) -> 0^1(5(4(x1))) 70.31/18.71 0^1(4(3(x1))) -> 5^1(4(x1)) 70.31/18.71 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(x1) 70.31/18.71 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 3^1(0(5(5(0(4(3(3(1(3(3(5(1(x1))))))))))))) 70.31/18.71 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 0^1(5(5(0(4(3(3(1(3(3(5(1(x1)))))))))))) 70.31/18.71 0^1(4(3(x1))) -> 4^1(x1) 70.31/18.71 0^1(4(4(1(2(3(3(1(x1)))))))) -> 0^1(4(4(0(3(1(1(x1))))))) 70.31/18.71 0^1(4(4(1(2(3(3(1(x1)))))))) -> 4^1(4(0(3(1(1(x1)))))) 70.31/18.71 0^1(4(4(1(2(3(3(1(x1)))))))) -> 4^1(0(3(1(1(x1))))) 70.31/18.71 0^1(4(4(1(2(3(3(1(x1)))))))) -> 0^1(3(1(1(x1)))) 70.31/18.71 0^1(4(4(1(2(3(3(1(x1)))))))) -> 3^1(1(1(x1))) 70.31/18.71 0^1(4(4(1(2(3(3(1(x1)))))))) -> 1^1(1(x1)) 70.31/18.71 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5^1(4(0(3(1(1(0(1(0(3(x1)))))))))) 70.31/18.71 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5^1(5(0(4(3(3(1(3(3(5(1(x1))))))))))) 70.31/18.71 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5^1(0(4(3(3(1(3(3(5(1(x1)))))))))) 70.31/18.71 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 0^1(4(3(3(1(3(3(5(1(x1))))))))) 70.31/18.71 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 4^1(0(3(1(1(0(1(0(3(x1))))))))) 70.31/18.71 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(3(1(1(0(1(0(3(x1)))))))) 70.31/18.71 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 3^1(1(1(0(1(0(3(x1))))))) 70.31/18.71 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(1(0(1(0(3(x1)))))) 70.31/18.71 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(0(1(0(3(x1))))) 70.31/18.71 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(1(0(3(x1)))) 70.31/18.71 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(0(3(x1))) 70.31/18.71 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(3(x1)) 70.31/18.71 0^1(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 1^1(x1) 70.31/18.71 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 0^1(4(3(2(0(5(5(0(1(1(1(0(3(1(3(2(x1)))))))))))))))) 70.31/18.71 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 0^1(5(5(0(1(1(1(0(3(1(3(2(x1)))))))))))) 70.31/18.71 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 5^1(5(0(1(1(1(0(3(1(3(2(x1))))))))))) 70.31/18.71 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 4^1(3(3(1(3(3(5(1(x1)))))))) 70.31/18.71 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 3^1(3(1(3(3(5(1(x1))))))) 70.31/18.71 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 3^1(1(3(3(5(1(x1)))))) 70.31/18.71 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 1^1(3(3(5(1(x1))))) 70.31/18.71 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 3^1(3(5(1(x1)))) 70.31/18.71 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 3^1(5(1(x1))) 70.31/18.71 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5^1(1(x1)) 70.31/18.71 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 5^1(0(1(1(1(0(3(1(3(2(x1)))))))))) 70.31/18.71 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 0^1(1(1(1(0(3(1(3(2(x1))))))))) 70.31/18.71 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 1^1(1(1(0(3(1(3(2(x1)))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0^1(0(1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0^1(1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1))))))))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 1^1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 5^1(4(3(5(0(5(5(3(1(0(1(1(5(x1))))))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 4^1(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 3^1(5(0(5(5(3(1(0(1(1(5(x1))))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 5^1(0(5(5(3(1(0(1(1(5(x1)))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0^1(5(5(3(1(0(1(1(5(x1))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 5^1(5(3(1(0(1(1(5(x1)))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 5^1(3(1(0(1(1(5(x1))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 3^1(1(0(1(1(5(x1)))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 1^1(0(1(1(5(x1))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0^1(1(1(5(x1)))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 1^1(1(5(x1))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 1^1(5(x1)) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 5^1(x1) 70.31/18.71 70.31/18.71 The TRS R consists of the following rules: 70.31/18.71 70.31/18.71 1(1(0(x1))) -> 1(1(2(x1))) 70.31/18.71 0(4(3(x1))) -> 5(5(4(x1))) 70.31/18.71 0(4(1(2(2(x1))))) -> 5(3(1(2(2(x1))))) 70.31/18.71 4(0(0(1(3(5(4(x1))))))) -> 4(5(4(1(0(5(4(x1))))))) 70.31/18.71 0(4(4(1(2(3(3(1(x1)))))))) -> 2(0(4(4(0(3(1(1(x1)))))))) 70.31/18.71 0(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5(4(0(3(1(1(0(1(0(3(x1)))))))))) 70.31/18.71 1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 1(0(3(0(3(2(2(4(0(4(x1)))))))))) 70.31/18.71 0(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 5(5(3(2(0(4(4(5(3(2(1(x1))))))))))) 70.31/18.71 4(5(3(4(5(2(1(2(0(2(0(3(x1)))))))))))) -> 4(5(1(5(4(5(0(3(0(2(2(3(x1)))))))))))) 70.31/18.71 1(1(5(0(0(0(3(5(4(5(0(3(x1)))))))))))) -> 3(1(1(2(0(3(0(4(1(4(2(3(x1)))))))))))) 70.31/18.71 3(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 3(5(0(3(1(5(1(1(3(0(2(2(4(4(x1)))))))))))))) 70.31/18.71 5(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 0(2(1(1(0(3(0(2(1(2(1(1(5(3(1(x1))))))))))))))) 70.31/18.71 0(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 0(4(3(2(0(5(5(0(1(1(1(0(3(1(3(2(x1)))))))))))))))) 70.31/18.71 0(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0(2(3(2(0(0(1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))))))))))) 70.31/18.71 5(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5(4(3(3(2(1(4(2(3(0(5(5(0(4(3(3(1(3(3(5(1(x1))))))))))))))))))))) 70.31/18.71 70.31/18.71 Q is empty. 70.31/18.71 We have to consider all minimal (P,Q,R)-chains. 70.31/18.71 ---------------------------------------- 70.31/18.71 70.31/18.71 (11) DependencyGraphProof (EQUIVALENT) 70.31/18.71 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 10 less nodes. 70.31/18.71 ---------------------------------------- 70.31/18.71 70.31/18.71 (12) 70.31/18.71 Obligation: 70.31/18.71 Q DP problem: 70.31/18.71 The TRS P consists of the following rules: 70.31/18.71 70.31/18.71 4^1(0(0(1(3(5(4(x1))))))) -> 4^1(5(4(1(0(5(4(x1))))))) 70.31/18.71 4^1(0(0(1(3(5(4(x1))))))) -> 5^1(4(1(0(5(4(x1)))))) 70.31/18.71 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(1(5(3(1(x1))))) 70.31/18.71 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 4^1(0(4(x1))) 70.31/18.71 4^1(0(0(1(3(5(4(x1))))))) -> 4^1(1(0(5(4(x1))))) 70.31/18.71 4^1(0(0(1(3(5(4(x1))))))) -> 1^1(0(5(4(x1)))) 70.31/18.71 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 0^1(4(x1)) 70.31/18.71 0^1(4(3(x1))) -> 5^1(5(4(x1))) 70.31/18.71 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(5(3(1(x1)))) 70.31/18.71 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 5^1(3(1(x1))) 70.31/18.71 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(x1) 70.31/18.71 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 0^1(5(5(0(4(3(3(1(3(3(5(1(x1)))))))))))) 70.31/18.71 0^1(4(3(x1))) -> 5^1(4(x1)) 70.31/18.71 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5^1(5(0(4(3(3(1(3(3(5(1(x1))))))))))) 70.31/18.71 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5^1(0(4(3(3(1(3(3(5(1(x1)))))))))) 70.31/18.71 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 0^1(4(3(3(1(3(3(5(1(x1))))))))) 70.31/18.71 0^1(4(3(x1))) -> 4^1(x1) 70.31/18.71 4^1(0(0(1(3(5(4(x1))))))) -> 0^1(5(4(x1))) 70.31/18.71 0^1(4(4(1(2(3(3(1(x1)))))))) -> 0^1(4(4(0(3(1(1(x1))))))) 70.31/18.71 0^1(4(4(1(2(3(3(1(x1)))))))) -> 4^1(4(0(3(1(1(x1)))))) 70.31/18.71 0^1(4(4(1(2(3(3(1(x1)))))))) -> 4^1(0(3(1(1(x1))))) 70.31/18.71 0^1(4(4(1(2(3(3(1(x1)))))))) -> 0^1(3(1(1(x1)))) 70.31/18.71 0^1(4(4(1(2(3(3(1(x1)))))))) -> 1^1(1(x1)) 70.31/18.71 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5^1(4(0(3(1(1(0(1(0(3(x1)))))))))) 70.31/18.71 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 4^1(3(3(1(3(3(5(1(x1)))))))) 70.31/18.71 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 1^1(3(3(5(1(x1))))) 70.31/18.71 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5^1(1(x1)) 70.31/18.71 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 4^1(0(3(1(1(0(1(0(3(x1))))))))) 70.31/18.71 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(3(1(1(0(1(0(3(x1)))))))) 70.31/18.71 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(1(0(1(0(3(x1)))))) 70.31/18.71 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(0(1(0(3(x1))))) 70.31/18.71 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(1(0(3(x1)))) 70.31/18.71 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(0(3(x1))) 70.31/18.71 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(3(x1)) 70.31/18.71 0^1(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 1^1(x1) 70.31/18.71 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 0^1(4(3(2(0(5(5(0(1(1(1(0(3(1(3(2(x1)))))))))))))))) 70.31/18.71 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 0^1(5(5(0(1(1(1(0(3(1(3(2(x1)))))))))))) 70.31/18.71 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 5^1(5(0(1(1(1(0(3(1(3(2(x1))))))))))) 70.31/18.71 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 5^1(0(1(1(1(0(3(1(3(2(x1)))))))))) 70.31/18.71 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 0^1(1(1(1(0(3(1(3(2(x1))))))))) 70.31/18.71 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 1^1(1(1(0(3(1(3(2(x1)))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0^1(0(1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0^1(1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1))))))))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 1^1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 5^1(4(3(5(0(5(5(3(1(0(1(1(5(x1))))))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 4^1(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 5^1(0(5(5(3(1(0(1(1(5(x1)))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0^1(5(5(3(1(0(1(1(5(x1))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 5^1(5(3(1(0(1(1(5(x1)))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 5^1(3(1(0(1(1(5(x1))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 1^1(0(1(1(5(x1))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0^1(1(1(5(x1)))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 1^1(1(5(x1))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 1^1(5(x1)) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 5^1(x1) 70.31/18.71 70.31/18.71 The TRS R consists of the following rules: 70.31/18.71 70.31/18.71 1(1(0(x1))) -> 1(1(2(x1))) 70.31/18.71 0(4(3(x1))) -> 5(5(4(x1))) 70.31/18.71 0(4(1(2(2(x1))))) -> 5(3(1(2(2(x1))))) 70.31/18.71 4(0(0(1(3(5(4(x1))))))) -> 4(5(4(1(0(5(4(x1))))))) 70.31/18.71 0(4(4(1(2(3(3(1(x1)))))))) -> 2(0(4(4(0(3(1(1(x1)))))))) 70.31/18.71 0(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5(4(0(3(1(1(0(1(0(3(x1)))))))))) 70.31/18.71 1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 1(0(3(0(3(2(2(4(0(4(x1)))))))))) 70.31/18.71 0(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 5(5(3(2(0(4(4(5(3(2(1(x1))))))))))) 70.31/18.71 4(5(3(4(5(2(1(2(0(2(0(3(x1)))))))))))) -> 4(5(1(5(4(5(0(3(0(2(2(3(x1)))))))))))) 70.31/18.71 1(1(5(0(0(0(3(5(4(5(0(3(x1)))))))))))) -> 3(1(1(2(0(3(0(4(1(4(2(3(x1)))))))))))) 70.31/18.71 3(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 3(5(0(3(1(5(1(1(3(0(2(2(4(4(x1)))))))))))))) 70.31/18.71 5(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 0(2(1(1(0(3(0(2(1(2(1(1(5(3(1(x1))))))))))))))) 70.31/18.71 0(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 0(4(3(2(0(5(5(0(1(1(1(0(3(1(3(2(x1)))))))))))))))) 70.31/18.71 0(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0(2(3(2(0(0(1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))))))))))) 70.31/18.71 5(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5(4(3(3(2(1(4(2(3(0(5(5(0(4(3(3(1(3(3(5(1(x1))))))))))))))))))))) 70.31/18.71 70.31/18.71 Q is empty. 70.31/18.71 We have to consider all minimal (P,Q,R)-chains. 70.31/18.71 ---------------------------------------- 70.31/18.71 70.31/18.71 (13) QDPOrderProof (EQUIVALENT) 70.31/18.71 We use the reduction pair processor [LPAR04,JAR06]. 70.31/18.71 70.31/18.71 70.31/18.71 The following pairs can be oriented strictly and are deleted. 70.31/18.71 70.31/18.71 4^1(0(0(1(3(5(4(x1))))))) -> 5^1(4(1(0(5(4(x1)))))) 70.31/18.71 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(1(5(3(1(x1))))) 70.31/18.71 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 4^1(0(4(x1))) 70.31/18.71 4^1(0(0(1(3(5(4(x1))))))) -> 4^1(1(0(5(4(x1))))) 70.31/18.71 4^1(0(0(1(3(5(4(x1))))))) -> 1^1(0(5(4(x1)))) 70.31/18.71 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 0^1(4(x1)) 70.31/18.71 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(5(3(1(x1)))) 70.31/18.71 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 5^1(3(1(x1))) 70.31/18.71 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(x1) 70.31/18.71 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 0^1(5(5(0(4(3(3(1(3(3(5(1(x1)))))))))))) 70.31/18.71 0^1(4(3(x1))) -> 5^1(4(x1)) 70.31/18.71 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5^1(5(0(4(3(3(1(3(3(5(1(x1))))))))))) 70.31/18.71 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5^1(0(4(3(3(1(3(3(5(1(x1)))))))))) 70.31/18.71 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 0^1(4(3(3(1(3(3(5(1(x1))))))))) 70.31/18.71 0^1(4(3(x1))) -> 4^1(x1) 70.31/18.71 4^1(0(0(1(3(5(4(x1))))))) -> 0^1(5(4(x1))) 70.31/18.71 0^1(4(4(1(2(3(3(1(x1)))))))) -> 0^1(4(4(0(3(1(1(x1))))))) 70.31/18.71 0^1(4(4(1(2(3(3(1(x1)))))))) -> 4^1(4(0(3(1(1(x1)))))) 70.31/18.71 0^1(4(4(1(2(3(3(1(x1)))))))) -> 4^1(0(3(1(1(x1))))) 70.31/18.71 0^1(4(4(1(2(3(3(1(x1)))))))) -> 0^1(3(1(1(x1)))) 70.31/18.71 0^1(4(4(1(2(3(3(1(x1)))))))) -> 1^1(1(x1)) 70.31/18.71 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 4^1(3(3(1(3(3(5(1(x1)))))))) 70.31/18.71 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 1^1(3(3(5(1(x1))))) 70.31/18.71 5^1(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5^1(1(x1)) 70.31/18.71 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 4^1(0(3(1(1(0(1(0(3(x1))))))))) 70.31/18.71 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(3(1(1(0(1(0(3(x1)))))))) 70.31/18.71 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(1(0(1(0(3(x1)))))) 70.31/18.71 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(0(1(0(3(x1))))) 70.31/18.71 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(1(0(3(x1)))) 70.31/18.71 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(0(3(x1))) 70.31/18.71 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(3(x1)) 70.31/18.71 0^1(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 1^1(x1) 70.31/18.71 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 0^1(5(5(0(1(1(1(0(3(1(3(2(x1)))))))))))) 70.31/18.71 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 5^1(5(0(1(1(1(0(3(1(3(2(x1))))))))))) 70.31/18.71 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 5^1(0(1(1(1(0(3(1(3(2(x1)))))))))) 70.31/18.71 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 0^1(1(1(1(0(3(1(3(2(x1))))))))) 70.31/18.71 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 1^1(1(1(0(3(1(3(2(x1)))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0^1(0(1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0^1(1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1))))))))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 1^1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 5^1(4(3(5(0(5(5(3(1(0(1(1(5(x1))))))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 4^1(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 5^1(0(5(5(3(1(0(1(1(5(x1)))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0^1(5(5(3(1(0(1(1(5(x1))))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 5^1(5(3(1(0(1(1(5(x1)))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 5^1(3(1(0(1(1(5(x1))))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 1^1(0(1(1(5(x1))))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0^1(1(1(5(x1)))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 1^1(1(5(x1))) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 1^1(5(x1)) 70.31/18.71 0^1(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 5^1(x1) 70.31/18.71 The remaining pairs can at least be oriented weakly. 70.31/18.71 Used ordering: Polynomial interpretation [POLO]: 70.31/18.71 70.31/18.71 POL(0(x_1)) = 1 + x_1 70.31/18.71 POL(0^1(x_1)) = 1 + x_1 70.31/18.71 POL(1(x_1)) = 1 + x_1 70.31/18.71 POL(1^1(x_1)) = x_1 70.31/18.71 POL(2(x_1)) = 1 + x_1 70.31/18.71 POL(3(x_1)) = 1 + x_1 70.31/18.71 POL(4(x_1)) = 1 + x_1 70.31/18.71 POL(4^1(x_1)) = 1 + x_1 70.31/18.71 POL(5(x_1)) = 1 + x_1 70.31/18.71 POL(5^1(x_1)) = 1 + x_1 70.31/18.71 70.31/18.71 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 70.31/18.71 70.31/18.71 4(0(0(1(3(5(4(x1))))))) -> 4(5(4(1(0(5(4(x1))))))) 70.31/18.71 4(5(3(4(5(2(1(2(0(2(0(3(x1)))))))))))) -> 4(5(1(5(4(5(0(3(0(2(2(3(x1)))))))))))) 70.31/18.71 5(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 0(2(1(1(0(3(0(2(1(2(1(1(5(3(1(x1))))))))))))))) 70.31/18.71 5(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5(4(3(3(2(1(4(2(3(0(5(5(0(4(3(3(1(3(3(5(1(x1))))))))))))))))))))) 70.31/18.71 0(4(3(x1))) -> 5(5(4(x1))) 70.31/18.71 0(4(1(2(2(x1))))) -> 5(3(1(2(2(x1))))) 70.31/18.71 0(4(4(1(2(3(3(1(x1)))))))) -> 2(0(4(4(0(3(1(1(x1)))))))) 70.31/18.71 0(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5(4(0(3(1(1(0(1(0(3(x1)))))))))) 70.31/18.71 0(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 5(5(3(2(0(4(4(5(3(2(1(x1))))))))))) 70.31/18.71 0(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 0(4(3(2(0(5(5(0(1(1(1(0(3(1(3(2(x1)))))))))))))))) 70.31/18.71 0(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0(2(3(2(0(0(1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))))))))))) 70.31/18.71 1(1(0(x1))) -> 1(1(2(x1))) 70.31/18.71 1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 1(0(3(0(3(2(2(4(0(4(x1)))))))))) 70.31/18.71 1(1(5(0(0(0(3(5(4(5(0(3(x1)))))))))))) -> 3(1(1(2(0(3(0(4(1(4(2(3(x1)))))))))))) 70.31/18.71 3(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 3(5(0(3(1(5(1(1(3(0(2(2(4(4(x1)))))))))))))) 70.31/18.71 70.31/18.71 70.31/18.71 ---------------------------------------- 70.31/18.71 70.31/18.71 (14) 70.31/18.71 Obligation: 70.31/18.71 Q DP problem: 70.31/18.71 The TRS P consists of the following rules: 70.31/18.71 70.31/18.71 4^1(0(0(1(3(5(4(x1))))))) -> 4^1(5(4(1(0(5(4(x1))))))) 70.31/18.71 0^1(4(3(x1))) -> 5^1(5(4(x1))) 70.31/18.71 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5^1(4(0(3(1(1(0(1(0(3(x1)))))))))) 70.31/18.71 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 0^1(4(3(2(0(5(5(0(1(1(1(0(3(1(3(2(x1)))))))))))))))) 70.31/18.71 70.31/18.71 The TRS R consists of the following rules: 70.31/18.71 70.31/18.71 1(1(0(x1))) -> 1(1(2(x1))) 70.31/18.71 0(4(3(x1))) -> 5(5(4(x1))) 70.31/18.71 0(4(1(2(2(x1))))) -> 5(3(1(2(2(x1))))) 70.31/18.71 4(0(0(1(3(5(4(x1))))))) -> 4(5(4(1(0(5(4(x1))))))) 70.31/18.71 0(4(4(1(2(3(3(1(x1)))))))) -> 2(0(4(4(0(3(1(1(x1)))))))) 70.31/18.71 0(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5(4(0(3(1(1(0(1(0(3(x1)))))))))) 70.31/18.71 1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 1(0(3(0(3(2(2(4(0(4(x1)))))))))) 70.31/18.71 0(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 5(5(3(2(0(4(4(5(3(2(1(x1))))))))))) 70.31/18.71 4(5(3(4(5(2(1(2(0(2(0(3(x1)))))))))))) -> 4(5(1(5(4(5(0(3(0(2(2(3(x1)))))))))))) 70.31/18.71 1(1(5(0(0(0(3(5(4(5(0(3(x1)))))))))))) -> 3(1(1(2(0(3(0(4(1(4(2(3(x1)))))))))))) 70.31/18.71 3(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 3(5(0(3(1(5(1(1(3(0(2(2(4(4(x1)))))))))))))) 70.31/18.71 5(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 0(2(1(1(0(3(0(2(1(2(1(1(5(3(1(x1))))))))))))))) 70.31/18.71 0(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 0(4(3(2(0(5(5(0(1(1(1(0(3(1(3(2(x1)))))))))))))))) 70.31/18.71 0(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0(2(3(2(0(0(1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))))))))))) 70.31/18.71 5(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5(4(3(3(2(1(4(2(3(0(5(5(0(4(3(3(1(3(3(5(1(x1))))))))))))))))))))) 70.31/18.71 70.31/18.71 Q is empty. 70.31/18.71 We have to consider all minimal (P,Q,R)-chains. 70.31/18.71 ---------------------------------------- 70.31/18.71 70.31/18.71 (15) DependencyGraphProof (EQUIVALENT) 70.31/18.71 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 70.31/18.71 ---------------------------------------- 70.31/18.71 70.31/18.71 (16) 70.31/18.71 Obligation: 70.31/18.71 Q DP problem: 70.31/18.71 The TRS P consists of the following rules: 70.31/18.71 70.31/18.71 4^1(0(0(1(3(5(4(x1))))))) -> 4^1(5(4(1(0(5(4(x1))))))) 70.31/18.71 70.31/18.71 The TRS R consists of the following rules: 70.31/18.71 70.31/18.71 1(1(0(x1))) -> 1(1(2(x1))) 70.31/18.71 0(4(3(x1))) -> 5(5(4(x1))) 70.31/18.71 0(4(1(2(2(x1))))) -> 5(3(1(2(2(x1))))) 70.31/18.71 4(0(0(1(3(5(4(x1))))))) -> 4(5(4(1(0(5(4(x1))))))) 70.31/18.71 0(4(4(1(2(3(3(1(x1)))))))) -> 2(0(4(4(0(3(1(1(x1)))))))) 70.31/18.71 0(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5(4(0(3(1(1(0(1(0(3(x1)))))))))) 70.31/18.71 1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 1(0(3(0(3(2(2(4(0(4(x1)))))))))) 70.31/18.71 0(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 5(5(3(2(0(4(4(5(3(2(1(x1))))))))))) 70.31/18.71 4(5(3(4(5(2(1(2(0(2(0(3(x1)))))))))))) -> 4(5(1(5(4(5(0(3(0(2(2(3(x1)))))))))))) 70.31/18.71 1(1(5(0(0(0(3(5(4(5(0(3(x1)))))))))))) -> 3(1(1(2(0(3(0(4(1(4(2(3(x1)))))))))))) 70.31/18.71 3(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 3(5(0(3(1(5(1(1(3(0(2(2(4(4(x1)))))))))))))) 70.31/18.71 5(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 0(2(1(1(0(3(0(2(1(2(1(1(5(3(1(x1))))))))))))))) 70.31/18.71 0(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 0(4(3(2(0(5(5(0(1(1(1(0(3(1(3(2(x1)))))))))))))))) 70.31/18.71 0(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0(2(3(2(0(0(1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))))))))))) 70.31/18.71 5(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5(4(3(3(2(1(4(2(3(0(5(5(0(4(3(3(1(3(3(5(1(x1))))))))))))))))))))) 70.31/18.71 70.31/18.71 Q is empty. 70.31/18.71 We have to consider all minimal (P,Q,R)-chains. 70.31/18.71 ---------------------------------------- 70.31/18.71 70.31/18.71 (17) QDPOrderProof (EQUIVALENT) 70.31/18.71 We use the reduction pair processor [LPAR04,JAR06]. 70.31/18.71 70.31/18.71 70.31/18.71 The following pairs can be oriented strictly and are deleted. 70.31/18.71 70.31/18.71 4^1(0(0(1(3(5(4(x1))))))) -> 4^1(5(4(1(0(5(4(x1))))))) 70.31/18.71 The remaining pairs can at least be oriented weakly. 70.31/18.71 Used ordering: Polynomial interpretation [POLO]: 70.31/18.71 70.31/18.71 POL(0(x_1)) = x_1 70.31/18.71 POL(1(x_1)) = 1 70.31/18.71 POL(2(x_1)) = 0 70.31/18.71 POL(3(x_1)) = x_1 70.31/18.71 POL(4(x_1)) = 1 + x_1 70.31/18.71 POL(4^1(x_1)) = x_1 70.31/18.71 POL(5(x_1)) = 0 70.31/18.71 70.31/18.71 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 70.31/18.71 70.31/18.71 5(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 0(2(1(1(0(3(0(2(1(2(1(1(5(3(1(x1))))))))))))))) 70.31/18.71 5(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5(4(3(3(2(1(4(2(3(0(5(5(0(4(3(3(1(3(3(5(1(x1))))))))))))))))))))) 70.31/18.71 70.31/18.71 70.31/18.71 ---------------------------------------- 70.31/18.71 70.31/18.71 (18) 70.31/18.71 Obligation: 70.31/18.71 Q DP problem: 70.31/18.71 P is empty. 70.31/18.71 The TRS R consists of the following rules: 70.31/18.71 70.31/18.71 1(1(0(x1))) -> 1(1(2(x1))) 70.31/18.71 0(4(3(x1))) -> 5(5(4(x1))) 70.31/18.71 0(4(1(2(2(x1))))) -> 5(3(1(2(2(x1))))) 70.31/18.71 4(0(0(1(3(5(4(x1))))))) -> 4(5(4(1(0(5(4(x1))))))) 70.31/18.71 0(4(4(1(2(3(3(1(x1)))))))) -> 2(0(4(4(0(3(1(1(x1)))))))) 70.31/18.71 0(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5(4(0(3(1(1(0(1(0(3(x1)))))))))) 70.31/18.71 1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 1(0(3(0(3(2(2(4(0(4(x1)))))))))) 70.31/18.71 0(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 5(5(3(2(0(4(4(5(3(2(1(x1))))))))))) 70.31/18.71 4(5(3(4(5(2(1(2(0(2(0(3(x1)))))))))))) -> 4(5(1(5(4(5(0(3(0(2(2(3(x1)))))))))))) 70.31/18.71 1(1(5(0(0(0(3(5(4(5(0(3(x1)))))))))))) -> 3(1(1(2(0(3(0(4(1(4(2(3(x1)))))))))))) 70.31/18.71 3(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 3(5(0(3(1(5(1(1(3(0(2(2(4(4(x1)))))))))))))) 70.31/18.71 5(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 0(2(1(1(0(3(0(2(1(2(1(1(5(3(1(x1))))))))))))))) 70.31/18.71 0(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 0(4(3(2(0(5(5(0(1(1(1(0(3(1(3(2(x1)))))))))))))))) 70.31/18.71 0(4(0(1(4(0(3(4(5(4(0(3(3(4(0(1(1(2(5(2(x1)))))))))))))))))))) -> 0(2(3(2(0(0(1(5(4(3(5(0(5(5(3(1(0(1(1(5(x1)))))))))))))))))))) 70.31/18.71 5(0(3(5(3(4(3(3(1(0(3(3(0(1(1(1(3(5(0(3(1(x1))))))))))))))))))))) -> 5(4(3(3(2(1(4(2(3(0(5(5(0(4(3(3(1(3(3(5(1(x1))))))))))))))))))))) 70.31/18.71 70.31/18.71 Q is empty. 70.31/18.71 We have to consider all minimal (P,Q,R)-chains. 70.31/18.71 ---------------------------------------- 70.31/18.71 70.31/18.71 (19) PisEmptyProof (EQUIVALENT) 70.31/18.71 The TRS P is empty. Hence, there is no (P,Q,R) chain. 70.31/18.71 ---------------------------------------- 70.31/18.71 70.31/18.71 (20) 70.31/18.71 YES 70.67/18.81 EOF