YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRSRRRProof [EQUIVALENT, 348 ms] (2) QTRS (3) DependencyPairsProof [EQUIVALENT, 56 ms] (4) QDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) QDP (7) UsableRulesProof [EQUIVALENT, 0 ms] (8) QDP (9) QDPSizeChangeProof [EQUIVALENT, 0 ms] (10) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0(0(2(0(1(1(1(1(3(3(2(3(x1)))))))))))) -> 3(2(4(0(0(5(4(4(0(4(4(4(1(3(1(4(5(x1))))))))))))))))) 0(0(2(2(1(1(2(3(1(5(1(2(x1)))))))))))) -> 0(5(4(5(4(4(5(2(0(0(5(5(4(0(2(4(5(1(x1)))))))))))))))))) 0(1(0(1(3(5(0(2(0(5(0(4(x1)))))))))))) -> 0(1(4(4(5(4(1(4(3(0(0(1(5(2(5(4(4(4(x1)))))))))))))))))) 0(1(2(3(2(3(5(1(3(3(5(3(x1)))))))))))) -> 0(4(5(0(4(2(4(1(4(4(4(5(5(1(5(x1))))))))))))))) 0(1(3(1(3(3(1(1(0(5(1(0(x1)))))))))))) -> 4(4(3(5(2(1(5(2(5(4(3(4(1(3(5(4(x1)))))))))))))))) 0(1(4(0(3(2(2(2(4(2(2(3(x1)))))))))))) -> 4(4(4(4(4(1(3(1(5(5(4(4(1(3(4(4(x1)))))))))))))))) 0(1(5(1(5(1(2(4(5(2(0(0(x1)))))))))))) -> 3(4(4(5(5(5(4(1(0(3(4(0(4(5(4(5(3(x1))))))))))))))))) 0(2(0(1(1(1(1(2(5(1(2(2(x1)))))))))))) -> 5(3(3(0(3(2(4(4(2(3(4(1(5(4(4(4(0(4(x1)))))))))))))))))) 0(2(2(0(1(1(0(1(3(0(3(5(x1)))))))))))) -> 4(2(1(3(3(4(5(2(5(4(5(5(4(4(2(3(4(1(x1)))))))))))))))))) 0(2(2(2(2(0(4(0(2(3(5(1(x1)))))))))))) -> 3(2(1(4(5(2(5(5(4(4(5(5(0(0(4(1(x1)))))))))))))))) 0(2(3(3(1(0(5(2(2(0(0(0(x1)))))))))))) -> 1(1(4(5(3(5(2(2(3(1(4(5(4(4(3(x1))))))))))))))) 0(2(5(1(5(0(2(0(2(3(2(0(x1)))))))))))) -> 5(2(3(4(4(1(4(5(4(5(0(0(1(4(5(5(1(1(x1)))))))))))))))))) 0(2(5(5(0(3(0(3(2(2(0(1(x1)))))))))))) -> 0(4(5(2(4(2(5(4(1(2(2(2(0(5(4(1(x1)))))))))))))))) 0(3(3(1(3(0(4(2(2(3(1(0(x1)))))))))))) -> 5(5(4(5(3(3(4(5(4(5(0(1(0(3(0(1(x1)))))))))))))))) 0(3(5(1(0(3(1(1(1(0(2(2(x1)))))))))))) -> 4(3(4(4(0(3(2(1(2(4(2(5(5(5(4(5(4(4(x1)))))))))))))))))) 0(3(5(2(1(3(0(3(2(3(1(0(x1)))))))))))) -> 5(2(5(3(1(3(4(0(4(0(0(4(4(3(2(x1))))))))))))))) 0(4(3(3(2(2(3(4(5(3(2(1(x1)))))))))))) -> 0(5(4(1(4(4(4(3(5(4(5(2(4(4(2(2(1(x1))))))))))))))))) 0(5(3(1(3(2(5(2(0(0(2(1(x1)))))))))))) -> 0(4(4(5(4(3(5(2(5(5(2(5(4(5(5(4(5(x1))))))))))))))))) 1(0(0(1(3(3(3(2(0(3(0(4(x1)))))))))))) -> 4(4(3(0(4(5(0(5(4(4(0(4(0(0(x1)))))))))))))) 1(0(0(2(4(2(2(2(2(4(2(0(x1)))))))))))) -> 5(1(5(3(5(4(5(0(3(3(4(1(4(4(x1)))))))))))))) 1(0(2(0(3(3(3(3(3(4(3(3(x1)))))))))))) -> 5(1(5(3(1(4(4(5(3(1(5(4(5(4(4(1(2(x1))))))))))))))))) 1(0(2(2(5(1(3(1(3(3(2(5(x1)))))))))))) -> 3(5(3(5(0(5(2(3(0(2(4(0(2(3(4(x1))))))))))))))) 1(0(3(2(0(2(2(2(1(2(3(2(x1)))))))))))) -> 3(3(2(4(4(0(3(4(0(1(4(5(2(2(4(x1))))))))))))))) 1(0(3(3(2(2(1(1(0(2(3(1(x1)))))))))))) -> 2(4(3(4(4(1(4(5(1(5(1(0(5(4(3(x1))))))))))))))) 1(1(0(0(5(2(3(2(1(1(0(5(x1)))))))))))) -> 5(4(5(4(0(1(2(1(2(3(3(0(4(1(x1)))))))))))))) 1(1(0(2(3(2(2(2(0(5(1(4(x1)))))))))))) -> 4(4(4(1(4(4(4(4(5(5(2(1(4(3(0(5(4(2(x1)))))))))))))))))) 1(1(1(3(1(2(2(4(3(0(0(2(x1)))))))))))) -> 5(2(4(3(4(4(1(1(3(1(4(5(2(4(0(2(x1)))))))))))))))) 1(2(1(1(3(0(2(2(4(1(1(3(x1)))))))))))) -> 1(4(5(4(1(1(5(1(1(2(5(4(1(1(3(4(x1)))))))))))))))) 1(2(2(2(3(2(0(2(5(3(5(3(x1)))))))))))) -> 1(4(0(5(3(3(5(3(3(4(3(4(2(1(x1)))))))))))))) 1(2(5(0(2(3(0(0(2(0(5(2(x1)))))))))))) -> 5(2(2(1(4(0(4(5(4(1(0(1(5(4(1(3(5(4(x1)))))))))))))))))) 1(3(0(0(2(3(0(4(2(0(2(5(x1)))))))))))) -> 0(2(1(4(2(1(4(5(0(5(5(4(5(5(4(1(4(5(x1)))))))))))))))))) 1(3(1(1(3(0(3(3(3(5(1(5(x1)))))))))))) -> 5(3(3(3(4(4(2(5(0(0(4(4(4(1(1(x1))))))))))))))) 1(3(1(2(0(2(0(3(3(1(1(1(x1)))))))))))) -> 1(5(4(4(5(4(4(4(3(0(1(5(4(1(x1)))))))))))))) 1(3(3(1(0(0(0(2(5(1(1(1(x1)))))))))))) -> 2(1(2(4(4(4(4(1(2(1(3(5(4(5(0(5(2(x1))))))))))))))))) 1(5(1(1(1(5(1(3(2(3(1(5(x1)))))))))))) -> 1(3(4(4(4(5(3(3(3(5(2(1(5(4(1(5(2(5(x1)))))))))))))))))) 1(5(2(2(1(1(5(0(3(3(0(4(x1)))))))))))) -> 4(2(4(5(4(5(4(3(5(5(4(4(4(1(1(2(x1)))))))))))))))) 1(5(3(3(4(0(3(1(1(2(1(0(x1)))))))))))) -> 4(4(4(2(4(2(2(4(4(5(4(5(4(0(0(5(2(x1))))))))))))))))) 2(0(2(4(5(1(2(2(5(3(5(0(x1)))))))))))) -> 0(0(0(5(4(1(5(4(1(4(3(4(5(3(4(5(2(x1))))))))))))))))) 2(0(3(2(2(0(2(2(3(0(0(1(x1)))))))))))) -> 5(4(3(4(4(1(5(4(2(4(1(5(4(5(3(0(0(x1))))))))))))))))) 2(0(3(3(2(2(3(1(2(3(0(3(x1)))))))))))) -> 5(4(5(5(1(5(4(4(4(5(4(0(2(1(2(0(2(x1))))))))))))))))) 2(1(0(0(0(3(3(3(3(0(1(1(x1)))))))))))) -> 5(3(2(2(4(2(2(4(0(4(5(3(4(2(5(x1))))))))))))))) 2(1(0(2(1(3(1(2(5(1(3(5(x1)))))))))))) -> 2(5(1(3(5(1(4(4(3(1(0(1(4(5(x1)))))))))))))) 2(1(0(2(2(1(1(3(1(1(1(3(x1)))))))))))) -> 5(5(4(5(5(2(2(0(1(0(4(4(2(5(2(0(x1)))))))))))))))) 2(1(1(0(0(5(5(0(0(2(5(5(x1)))))))))))) -> 2(2(5(2(3(2(5(4(1(1(4(4(5(5(x1)))))))))))))) 2(1(3(0(0(1(0(3(3(1(0(4(x1)))))))))))) -> 4(4(5(3(3(4(2(3(5(4(2(4(4(5(3(3(x1)))))))))))))))) 2(1(5(1(1(5(0(2(5(1(3(0(x1)))))))))))) -> 3(5(4(2(3(4(4(3(3(4(0(5(2(0(5(4(x1)))))))))))))))) 2(2(0(3(2(0(1(0(2(3(3(2(x1)))))))))))) -> 5(5(5(0(2(2(4(1(4(4(3(3(0(0(2(x1))))))))))))))) 2(2(1(0(2(5(1(2(1(5(4(5(x1)))))))))))) -> 4(4(3(5(4(5(4(0(2(2(0(1(3(4(4(3(5(x1))))))))))))))))) 2(2(1(1(0(2(0(0(2(3(0(1(x1)))))))))))) -> 5(4(1(1(4(3(5(5(0(1(4(3(5(4(1(1(4(x1))))))))))))))))) 2(2(2(0(3(1(0(2(5(0(0(1(x1)))))))))))) -> 5(3(0(5(1(4(4(5(2(4(4(2(4(4(0(5(x1)))))))))))))))) 2(2(2(0(4(2(0(1(3(1(2(2(x1)))))))))))) -> 4(4(5(4(0(4(2(5(4(3(0(4(4(4(4(5(3(2(x1)))))))))))))))))) 2(2(2(4(1(1(0(1(1(1(4(0(x1)))))))))))) -> 3(3(5(4(1(4(4(3(5(4(4(4(1(4(4(1(4(4(x1)))))))))))))))))) 2(2(3(0(4(2(3(1(1(0(4(5(x1)))))))))))) -> 4(1(3(2(5(5(5(3(3(5(2(4(1(5(x1)))))))))))))) 2(2(3(4(3(3(5(1(2(4(2(2(x1)))))))))))) -> 3(4(3(5(3(4(1(4(0(4(5(5(0(2(x1)))))))))))))) 2(2(3(5(0(2(2(2(3(3(2(2(x1)))))))))))) -> 3(4(0(5(3(2(0(4(5(4(0(1(5(4(1(2(4(x1))))))))))))))))) 2(3(0(2(0(3(0(0(3(2(2(3(x1)))))))))))) -> 4(5(4(0(2(3(4(5(5(4(1(3(3(1(5(1(1(x1))))))))))))))))) 2(3(1(1(3(0(3(0(4(0(5(3(x1)))))))))))) -> 2(0(1(2(4(3(4(1(4(4(3(4(1(4(4(0(3(x1))))))))))))))))) 2(3(1(2(1(1(2(0(4(5(1(3(x1)))))))))))) -> 4(0(4(1(4(1(4(5(3(3(3(4(1(5(0(4(1(x1))))))))))))))))) 2(3(1(3(5(1(4(5(1(1(5(1(x1)))))))))))) -> 5(0(4(0(4(1(0(4(1(5(5(3(4(4(3(x1))))))))))))))) 2(3(2(1(2(3(1(5(5(2(3(0(x1)))))))))))) -> 2(4(5(2(4(1(4(0(1(4(3(2(3(4(1(x1))))))))))))))) 2(3(2(2(3(2(0(3(4(5(2(1(x1)))))))))))) -> 3(4(1(5(0(0(2(5(2(0(5(3(4(2(x1)))))))))))))) 2(3(3(1(1(0(0(2(2(1(2(5(x1)))))))))))) -> 5(3(4(4(1(5(2(1(1(5(0(4(5(3(x1)))))))))))))) 2(3(3(2(4(3(0(2(3(3(5(1(x1)))))))))))) -> 0(0(4(3(4(4(3(0(4(1(0(1(4(4(x1)))))))))))))) 2(4(0(0(0(3(1(1(2(3(2(3(x1)))))))))))) -> 0(0(3(5(4(2(5(4(3(3(4(4(1(0(3(x1))))))))))))))) 2(5(1(3(0(0(4(5(3(1(3(4(x1)))))))))))) -> 0(4(1(5(4(0(5(4(0(2(4(0(1(2(x1)))))))))))))) 2(5(1(5(1(4(3(3(0(4(1(1(x1)))))))))))) -> 4(3(5(5(5(3(1(3(4(0(3(5(4(2(x1)))))))))))))) 3(0(0(0(2(2(4(1(2(1(1(1(x1)))))))))))) -> 4(0(5(2(5(4(3(2(5(4(3(5(4(2(5(4(3(x1))))))))))))))))) 3(0(0(3(1(1(3(3(1(2(0(3(x1)))))))))))) -> 5(0(1(4(5(3(0(5(0(0(4(4(2(4(4(2(5(3(x1)))))))))))))))))) 3(0(1(4(2(0(3(0(5(1(4(2(x1)))))))))))) -> 0(5(4(5(2(1(5(5(4(3(5(4(5(5(4(3(2(x1))))))))))))))))) 3(0(2(3(3(0(5(3(4(5(0(0(x1)))))))))))) -> 5(4(4(0(4(5(3(3(5(3(5(0(4(4(4(x1))))))))))))))) 3(0(3(0(0(3(0(3(5(1(3(2(x1)))))))))))) -> 3(4(0(5(4(4(4(5(4(1(0(2(1(4(1(5(3(4(x1)))))))))))))))))) 3(0(3(3(4(0(5(1(3(2(3(1(x1)))))))))))) -> 3(4(5(2(1(1(3(4(4(4(3(5(5(0(4(4(5(x1))))))))))))))))) 3(1(0(0(5(4(3(2(2(3(3(2(x1)))))))))))) -> 5(2(1(4(1(5(3(3(4(4(0(5(2(5(1(x1))))))))))))))) 3(1(0(5(1(1(3(0(5(1(2(1(x1)))))))))))) -> 4(0(4(4(5(4(4(3(4(5(4(4(2(1(0(1(x1)))))))))))))))) 3(1(1(1(2(0(1(0(0(5(0(4(x1)))))))))))) -> 5(0(1(5(5(5(0(1(5(4(4(3(2(0(4(x1))))))))))))))) 3(1(1(1(3(0(3(1(0(3(4(0(x1)))))))))))) -> 0(5(3(1(4(4(1(5(5(5(5(4(1(5(x1)))))))))))))) 3(1(1(2(0(2(5(5(3(3(1(3(x1)))))))))))) -> 4(2(1(5(1(5(4(0(3(3(0(4(2(4(x1)))))))))))))) 3(1(5(0(0(1(0(4(2(1(5(3(x1)))))))))))) -> 1(5(0(0(4(1(4(5(4(5(4(1(4(4(0(4(x1)))))))))))))))) 3(2(0(0(3(0(2(3(1(3(3(2(x1)))))))))))) -> 1(4(5(4(4(0(3(1(5(4(3(3(0(4(4(5(4(x1))))))))))))))))) 3(2(2(5(1(5(0(3(4(2(4(3(x1)))))))))))) -> 1(5(4(5(2(4(1(5(4(4(4(2(4(4(5(4(0(4(x1)))))))))))))))))) 3(3(0(2(4(3(0(1(5(5(3(0(x1)))))))))))) -> 5(1(4(3(4(5(3(4(4(0(1(3(1(4(0(x1))))))))))))))) 3(3(1(1(2(3(5(5(2(4(0(0(x1)))))))))))) -> 0(1(5(5(4(5(5(2(4(4(4(1(2(5(4(5(4(x1))))))))))))))))) 3(3(1(2(2(1(3(3(0(2(0(4(x1)))))))))))) -> 2(5(4(3(1(4(4(4(0(4(2(3(0(2(x1)))))))))))))) 3(3(1(2(3(0(2(0(3(0(1(0(x1)))))))))))) -> 3(0(1(3(0(4(5(5(1(0(5(2(4(1(2(4(x1)))))))))))))))) 3(3(1(2(5(1(2(0(5(5(0(0(x1)))))))))))) -> 3(4(4(5(4(1(3(1(4(4(0(5(4(3(2(0(2(x1))))))))))))))))) 3(3(2(3(0(1(3(3(5(1(1(2(x1)))))))))))) -> 2(5(5(5(4(4(2(5(4(1(5(5(4(1(4(5(2(x1))))))))))))))))) 3(5(1(2(1(0(3(2(2(2(0(1(x1)))))))))))) -> 1(5(5(4(1(2(5(5(4(4(4(4(2(1(2(5(4(5(x1)))))))))))))))))) 3(5(5(0(2(3(5(0(5(0(4(0(x1)))))))))))) -> 0(0(4(5(4(5(2(1(3(0(5(4(4(2(x1)))))))))))))) 4(0(0(0(2(5(0(2(0(5(5(0(x1)))))))))))) -> 4(5(3(4(3(4(2(4(4(3(0(4(1(2(5(x1))))))))))))))) 4(0(0(3(1(3(5(0(3(3(2(4(x1)))))))))))) -> 4(0(2(4(0(1(2(4(4(0(4(1(4(2(4(4(x1)))))))))))))))) 4(0(1(3(1(2(3(3(1(1(2(2(x1)))))))))))) -> 1(4(5(3(0(1(4(3(2(4(1(5(3(4(0(x1))))))))))))))) 4(1(3(1(1(1(1(2(1(0(4(3(x1)))))))))))) -> 0(5(4(2(4(4(5(4(2(4(0(0(4(4(4(4(5(x1))))))))))))))))) 4(1(3(3(0(0(3(3(3(3(3(0(x1)))))))))))) -> 4(0(2(1(4(0(5(1(0(4(2(4(5(2(x1)))))))))))))) 4(1(5(0(2(2(1(3(0(0(3(1(x1)))))))))))) -> 4(5(4(1(4(1(4(4(5(5(4(4(3(4(4(3(2(4(x1)))))))))))))))))) 4(3(3(2(0(1(1(2(0(1(5(5(x1)))))))))))) -> 3(0(2(4(1(4(4(4(5(4(4(4(5(0(4(x1))))))))))))))) 4(4(3(0(0(2(2(5(5(3(1(1(x1)))))))))))) -> 5(5(4(4(0(4(2(5(4(2(4(1(4(1(1(x1))))))))))))))) Q is empty. ---------------------------------------- (1) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 + x_1 POL(1(x_1)) = 1 + x_1 POL(2(x_1)) = 1 + x_1 POL(3(x_1)) = 1 + x_1 POL(4(x_1)) = x_1 POL(5(x_1)) = x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 0(0(2(0(1(1(1(1(3(3(2(3(x1)))))))))))) -> 3(2(4(0(0(5(4(4(0(4(4(4(1(3(1(4(5(x1))))))))))))))))) 0(0(2(2(1(1(2(3(1(5(1(2(x1)))))))))))) -> 0(5(4(5(4(4(5(2(0(0(5(5(4(0(2(4(5(1(x1)))))))))))))))))) 0(1(0(1(3(5(0(2(0(5(0(4(x1)))))))))))) -> 0(1(4(4(5(4(1(4(3(0(0(1(5(2(5(4(4(4(x1)))))))))))))))))) 0(1(2(3(2(3(5(1(3(3(5(3(x1)))))))))))) -> 0(4(5(0(4(2(4(1(4(4(4(5(5(1(5(x1))))))))))))))) 0(1(3(1(3(3(1(1(0(5(1(0(x1)))))))))))) -> 4(4(3(5(2(1(5(2(5(4(3(4(1(3(5(4(x1)))))))))))))))) 0(1(4(0(3(2(2(2(4(2(2(3(x1)))))))))))) -> 4(4(4(4(4(1(3(1(5(5(4(4(1(3(4(4(x1)))))))))))))))) 0(1(5(1(5(1(2(4(5(2(0(0(x1)))))))))))) -> 3(4(4(5(5(5(4(1(0(3(4(0(4(5(4(5(3(x1))))))))))))))))) 0(2(0(1(1(1(1(2(5(1(2(2(x1)))))))))))) -> 5(3(3(0(3(2(4(4(2(3(4(1(5(4(4(4(0(4(x1)))))))))))))))))) 0(2(2(0(1(1(0(1(3(0(3(5(x1)))))))))))) -> 4(2(1(3(3(4(5(2(5(4(5(5(4(4(2(3(4(1(x1)))))))))))))))))) 0(2(2(2(2(0(4(0(2(3(5(1(x1)))))))))))) -> 3(2(1(4(5(2(5(5(4(4(5(5(0(0(4(1(x1)))))))))))))))) 0(2(3(3(1(0(5(2(2(0(0(0(x1)))))))))))) -> 1(1(4(5(3(5(2(2(3(1(4(5(4(4(3(x1))))))))))))))) 0(2(5(1(5(0(2(0(2(3(2(0(x1)))))))))))) -> 5(2(3(4(4(1(4(5(4(5(0(0(1(4(5(5(1(1(x1)))))))))))))))))) 0(2(5(5(0(3(0(3(2(2(0(1(x1)))))))))))) -> 0(4(5(2(4(2(5(4(1(2(2(2(0(5(4(1(x1)))))))))))))))) 0(3(3(1(3(0(4(2(2(3(1(0(x1)))))))))))) -> 5(5(4(5(3(3(4(5(4(5(0(1(0(3(0(1(x1)))))))))))))))) 0(3(5(1(0(3(1(1(1(0(2(2(x1)))))))))))) -> 4(3(4(4(0(3(2(1(2(4(2(5(5(5(4(5(4(4(x1)))))))))))))))))) 0(3(5(2(1(3(0(3(2(3(1(0(x1)))))))))))) -> 5(2(5(3(1(3(4(0(4(0(0(4(4(3(2(x1))))))))))))))) 0(4(3(3(2(2(3(4(5(3(2(1(x1)))))))))))) -> 0(5(4(1(4(4(4(3(5(4(5(2(4(4(2(2(1(x1))))))))))))))))) 0(5(3(1(3(2(5(2(0(0(2(1(x1)))))))))))) -> 0(4(4(5(4(3(5(2(5(5(2(5(4(5(5(4(5(x1))))))))))))))))) 1(0(0(1(3(3(3(2(0(3(0(4(x1)))))))))))) -> 4(4(3(0(4(5(0(5(4(4(0(4(0(0(x1)))))))))))))) 1(0(0(2(4(2(2(2(2(4(2(0(x1)))))))))))) -> 5(1(5(3(5(4(5(0(3(3(4(1(4(4(x1)))))))))))))) 1(0(2(0(3(3(3(3(3(4(3(3(x1)))))))))))) -> 5(1(5(3(1(4(4(5(3(1(5(4(5(4(4(1(2(x1))))))))))))))))) 1(0(3(2(0(2(2(2(1(2(3(2(x1)))))))))))) -> 3(3(2(4(4(0(3(4(0(1(4(5(2(2(4(x1))))))))))))))) 1(0(3(3(2(2(1(1(0(2(3(1(x1)))))))))))) -> 2(4(3(4(4(1(4(5(1(5(1(0(5(4(3(x1))))))))))))))) 1(1(0(0(5(2(3(2(1(1(0(5(x1)))))))))))) -> 5(4(5(4(0(1(2(1(2(3(3(0(4(1(x1)))))))))))))) 1(1(0(2(3(2(2(2(0(5(1(4(x1)))))))))))) -> 4(4(4(1(4(4(4(4(5(5(2(1(4(3(0(5(4(2(x1)))))))))))))))))) 1(1(1(3(1(2(2(4(3(0(0(2(x1)))))))))))) -> 5(2(4(3(4(4(1(1(3(1(4(5(2(4(0(2(x1)))))))))))))))) 1(2(1(1(3(0(2(2(4(1(1(3(x1)))))))))))) -> 1(4(5(4(1(1(5(1(1(2(5(4(1(1(3(4(x1)))))))))))))))) 1(2(2(2(3(2(0(2(5(3(5(3(x1)))))))))))) -> 1(4(0(5(3(3(5(3(3(4(3(4(2(1(x1)))))))))))))) 1(2(5(0(2(3(0(0(2(0(5(2(x1)))))))))))) -> 5(2(2(1(4(0(4(5(4(1(0(1(5(4(1(3(5(4(x1)))))))))))))))))) 1(3(0(0(2(3(0(4(2(0(2(5(x1)))))))))))) -> 0(2(1(4(2(1(4(5(0(5(5(4(5(5(4(1(4(5(x1)))))))))))))))))) 1(3(1(1(3(0(3(3(3(5(1(5(x1)))))))))))) -> 5(3(3(3(4(4(2(5(0(0(4(4(4(1(1(x1))))))))))))))) 1(3(1(2(0(2(0(3(3(1(1(1(x1)))))))))))) -> 1(5(4(4(5(4(4(4(3(0(1(5(4(1(x1)))))))))))))) 1(3(3(1(0(0(0(2(5(1(1(1(x1)))))))))))) -> 2(1(2(4(4(4(4(1(2(1(3(5(4(5(0(5(2(x1))))))))))))))))) 1(5(2(2(1(1(5(0(3(3(0(4(x1)))))))))))) -> 4(2(4(5(4(5(4(3(5(5(4(4(4(1(1(2(x1)))))))))))))))) 1(5(3(3(4(0(3(1(1(2(1(0(x1)))))))))))) -> 4(4(4(2(4(2(2(4(4(5(4(5(4(0(0(5(2(x1))))))))))))))))) 2(0(3(2(2(0(2(2(3(0(0(1(x1)))))))))))) -> 5(4(3(4(4(1(5(4(2(4(1(5(4(5(3(0(0(x1))))))))))))))))) 2(0(3(3(2(2(3(1(2(3(0(3(x1)))))))))))) -> 5(4(5(5(1(5(4(4(4(5(4(0(2(1(2(0(2(x1))))))))))))))))) 2(1(0(0(0(3(3(3(3(0(1(1(x1)))))))))))) -> 5(3(2(2(4(2(2(4(0(4(5(3(4(2(5(x1))))))))))))))) 2(1(0(2(1(3(1(2(5(1(3(5(x1)))))))))))) -> 2(5(1(3(5(1(4(4(3(1(0(1(4(5(x1)))))))))))))) 2(1(0(2(2(1(1(3(1(1(1(3(x1)))))))))))) -> 5(5(4(5(5(2(2(0(1(0(4(4(2(5(2(0(x1)))))))))))))))) 2(1(1(0(0(5(5(0(0(2(5(5(x1)))))))))))) -> 2(2(5(2(3(2(5(4(1(1(4(4(5(5(x1)))))))))))))) 2(1(3(0(0(1(0(3(3(1(0(4(x1)))))))))))) -> 4(4(5(3(3(4(2(3(5(4(2(4(4(5(3(3(x1)))))))))))))))) 2(1(5(1(1(5(0(2(5(1(3(0(x1)))))))))))) -> 3(5(4(2(3(4(4(3(3(4(0(5(2(0(5(4(x1)))))))))))))))) 2(2(0(3(2(0(1(0(2(3(3(2(x1)))))))))))) -> 5(5(5(0(2(2(4(1(4(4(3(3(0(0(2(x1))))))))))))))) 2(2(1(1(0(2(0(0(2(3(0(1(x1)))))))))))) -> 5(4(1(1(4(3(5(5(0(1(4(3(5(4(1(1(4(x1))))))))))))))))) 2(2(2(0(3(1(0(2(5(0(0(1(x1)))))))))))) -> 5(3(0(5(1(4(4(5(2(4(4(2(4(4(0(5(x1)))))))))))))))) 2(2(2(0(4(2(0(1(3(1(2(2(x1)))))))))))) -> 4(4(5(4(0(4(2(5(4(3(0(4(4(4(4(5(3(2(x1)))))))))))))))))) 2(2(2(4(1(1(0(1(1(1(4(0(x1)))))))))))) -> 3(3(5(4(1(4(4(3(5(4(4(4(1(4(4(1(4(4(x1)))))))))))))))))) 2(2(3(0(4(2(3(1(1(0(4(5(x1)))))))))))) -> 4(1(3(2(5(5(5(3(3(5(2(4(1(5(x1)))))))))))))) 2(2(3(4(3(3(5(1(2(4(2(2(x1)))))))))))) -> 3(4(3(5(3(4(1(4(0(4(5(5(0(2(x1)))))))))))))) 2(2(3(5(0(2(2(2(3(3(2(2(x1)))))))))))) -> 3(4(0(5(3(2(0(4(5(4(0(1(5(4(1(2(4(x1))))))))))))))))) 2(3(0(2(0(3(0(0(3(2(2(3(x1)))))))))))) -> 4(5(4(0(2(3(4(5(5(4(1(3(3(1(5(1(1(x1))))))))))))))))) 2(3(1(2(1(1(2(0(4(5(1(3(x1)))))))))))) -> 4(0(4(1(4(1(4(5(3(3(3(4(1(5(0(4(1(x1))))))))))))))))) 2(3(1(3(5(1(4(5(1(1(5(1(x1)))))))))))) -> 5(0(4(0(4(1(0(4(1(5(5(3(4(4(3(x1))))))))))))))) 2(3(2(1(2(3(1(5(5(2(3(0(x1)))))))))))) -> 2(4(5(2(4(1(4(0(1(4(3(2(3(4(1(x1))))))))))))))) 2(3(2(2(3(2(0(3(4(5(2(1(x1)))))))))))) -> 3(4(1(5(0(0(2(5(2(0(5(3(4(2(x1)))))))))))))) 2(3(3(1(1(0(0(2(2(1(2(5(x1)))))))))))) -> 5(3(4(4(1(5(2(1(1(5(0(4(5(3(x1)))))))))))))) 2(3(3(2(4(3(0(2(3(3(5(1(x1)))))))))))) -> 0(0(4(3(4(4(3(0(4(1(0(1(4(4(x1)))))))))))))) 2(4(0(0(0(3(1(1(2(3(2(3(x1)))))))))))) -> 0(0(3(5(4(2(5(4(3(3(4(4(1(0(3(x1))))))))))))))) 2(5(1(5(1(4(3(3(0(4(1(1(x1)))))))))))) -> 4(3(5(5(5(3(1(3(4(0(3(5(4(2(x1)))))))))))))) 3(0(0(0(2(2(4(1(2(1(1(1(x1)))))))))))) -> 4(0(5(2(5(4(3(2(5(4(3(5(4(2(5(4(3(x1))))))))))))))))) 3(0(0(3(1(1(3(3(1(2(0(3(x1)))))))))))) -> 5(0(1(4(5(3(0(5(0(0(4(4(2(4(4(2(5(3(x1)))))))))))))))))) 3(0(1(4(2(0(3(0(5(1(4(2(x1)))))))))))) -> 0(5(4(5(2(1(5(5(4(3(5(4(5(5(4(3(2(x1))))))))))))))))) 3(0(2(3(3(0(5(3(4(5(0(0(x1)))))))))))) -> 5(4(4(0(4(5(3(3(5(3(5(0(4(4(4(x1))))))))))))))) 3(0(3(0(0(3(0(3(5(1(3(2(x1)))))))))))) -> 3(4(0(5(4(4(4(5(4(1(0(2(1(4(1(5(3(4(x1)))))))))))))))))) 3(0(3(3(4(0(5(1(3(2(3(1(x1)))))))))))) -> 3(4(5(2(1(1(3(4(4(4(3(5(5(0(4(4(5(x1))))))))))))))))) 3(1(0(0(5(4(3(2(2(3(3(2(x1)))))))))))) -> 5(2(1(4(1(5(3(3(4(4(0(5(2(5(1(x1))))))))))))))) 3(1(0(5(1(1(3(0(5(1(2(1(x1)))))))))))) -> 4(0(4(4(5(4(4(3(4(5(4(4(2(1(0(1(x1)))))))))))))))) 3(1(1(1(2(0(1(0(0(5(0(4(x1)))))))))))) -> 5(0(1(5(5(5(0(1(5(4(4(3(2(0(4(x1))))))))))))))) 3(1(1(1(3(0(3(1(0(3(4(0(x1)))))))))))) -> 0(5(3(1(4(4(1(5(5(5(5(4(1(5(x1)))))))))))))) 3(1(1(2(0(2(5(5(3(3(1(3(x1)))))))))))) -> 4(2(1(5(1(5(4(0(3(3(0(4(2(4(x1)))))))))))))) 3(1(5(0(0(1(0(4(2(1(5(3(x1)))))))))))) -> 1(5(0(0(4(1(4(5(4(5(4(1(4(4(0(4(x1)))))))))))))))) 3(2(0(0(3(0(2(3(1(3(3(2(x1)))))))))))) -> 1(4(5(4(4(0(3(1(5(4(3(3(0(4(4(5(4(x1))))))))))))))))) 3(2(2(5(1(5(0(3(4(2(4(3(x1)))))))))))) -> 1(5(4(5(2(4(1(5(4(4(4(2(4(4(5(4(0(4(x1)))))))))))))))))) 3(3(0(2(4(3(0(1(5(5(3(0(x1)))))))))))) -> 5(1(4(3(4(5(3(4(4(0(1(3(1(4(0(x1))))))))))))))) 3(3(1(1(2(3(5(5(2(4(0(0(x1)))))))))))) -> 0(1(5(5(4(5(5(2(4(4(4(1(2(5(4(5(4(x1))))))))))))))))) 3(3(1(2(2(1(3(3(0(2(0(4(x1)))))))))))) -> 2(5(4(3(1(4(4(4(0(4(2(3(0(2(x1)))))))))))))) 3(3(1(2(3(0(2(0(3(0(1(0(x1)))))))))))) -> 3(0(1(3(0(4(5(5(1(0(5(2(4(1(2(4(x1)))))))))))))))) 3(3(2(3(0(1(3(3(5(1(1(2(x1)))))))))))) -> 2(5(5(5(4(4(2(5(4(1(5(5(4(1(4(5(2(x1))))))))))))))))) 3(5(1(2(1(0(3(2(2(2(0(1(x1)))))))))))) -> 1(5(5(4(1(2(5(5(4(4(4(4(2(1(2(5(4(5(x1)))))))))))))))))) 4(0(0(0(2(5(0(2(0(5(5(0(x1)))))))))))) -> 4(5(3(4(3(4(2(4(4(3(0(4(1(2(5(x1))))))))))))))) 4(0(0(3(1(3(5(0(3(3(2(4(x1)))))))))))) -> 4(0(2(4(0(1(2(4(4(0(4(1(4(2(4(4(x1)))))))))))))))) 4(0(1(3(1(2(3(3(1(1(2(2(x1)))))))))))) -> 1(4(5(3(0(1(4(3(2(4(1(5(3(4(0(x1))))))))))))))) 4(1(3(1(1(1(1(2(1(0(4(3(x1)))))))))))) -> 0(5(4(2(4(4(5(4(2(4(0(0(4(4(4(4(5(x1))))))))))))))))) 4(1(3(3(0(0(3(3(3(3(3(0(x1)))))))))))) -> 4(0(2(1(4(0(5(1(0(4(2(4(5(2(x1)))))))))))))) 4(1(5(0(2(2(1(3(0(0(3(1(x1)))))))))))) -> 4(5(4(1(4(1(4(4(5(5(4(4(3(4(4(3(2(4(x1)))))))))))))))))) 4(3(3(2(0(1(1(2(0(1(5(5(x1)))))))))))) -> 3(0(2(4(1(4(4(4(5(4(4(4(5(0(4(x1))))))))))))))) 4(4(3(0(0(2(2(5(5(3(1(1(x1)))))))))))) -> 5(5(4(4(0(4(2(5(4(2(4(1(4(1(1(x1))))))))))))))) ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 1(0(2(2(5(1(3(1(3(3(2(5(x1)))))))))))) -> 3(5(3(5(0(5(2(3(0(2(4(0(2(3(4(x1))))))))))))))) 1(5(1(1(1(5(1(3(2(3(1(5(x1)))))))))))) -> 1(3(4(4(4(5(3(3(3(5(2(1(5(4(1(5(2(5(x1)))))))))))))))))) 2(0(2(4(5(1(2(2(5(3(5(0(x1)))))))))))) -> 0(0(0(5(4(1(5(4(1(4(3(4(5(3(4(5(2(x1))))))))))))))))) 2(2(1(0(2(5(1(2(1(5(4(5(x1)))))))))))) -> 4(4(3(5(4(5(4(0(2(2(0(1(3(4(4(3(5(x1))))))))))))))))) 2(3(1(1(3(0(3(0(4(0(5(3(x1)))))))))))) -> 2(0(1(2(4(3(4(1(4(4(3(4(1(4(4(0(3(x1))))))))))))))))) 2(5(1(3(0(0(4(5(3(1(3(4(x1)))))))))))) -> 0(4(1(5(4(0(5(4(0(2(4(0(1(2(x1)))))))))))))) 3(3(1(2(5(1(2(0(5(5(0(0(x1)))))))))))) -> 3(4(4(5(4(1(3(1(4(4(0(5(4(3(2(0(2(x1))))))))))))))))) 3(5(5(0(2(3(5(0(5(0(4(0(x1)))))))))))) -> 0(0(4(5(4(5(2(1(3(0(5(4(4(2(x1)))))))))))))) Q is empty. ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (4) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(2(2(5(1(3(1(3(3(2(5(x1)))))))))))) -> 3^1(5(3(5(0(5(2(3(0(2(4(0(2(3(4(x1))))))))))))))) 1^1(0(2(2(5(1(3(1(3(3(2(5(x1)))))))))))) -> 3^1(5(0(5(2(3(0(2(4(0(2(3(4(x1))))))))))))) 1^1(0(2(2(5(1(3(1(3(3(2(5(x1)))))))))))) -> 2^1(3(0(2(4(0(2(3(4(x1))))))))) 1^1(0(2(2(5(1(3(1(3(3(2(5(x1)))))))))))) -> 3^1(0(2(4(0(2(3(4(x1)))))))) 1^1(0(2(2(5(1(3(1(3(3(2(5(x1)))))))))))) -> 2^1(4(0(2(3(4(x1)))))) 1^1(0(2(2(5(1(3(1(3(3(2(5(x1)))))))))))) -> 2^1(3(4(x1))) 1^1(0(2(2(5(1(3(1(3(3(2(5(x1)))))))))))) -> 3^1(4(x1)) 1^1(5(1(1(1(5(1(3(2(3(1(5(x1)))))))))))) -> 1^1(3(4(4(4(5(3(3(3(5(2(1(5(4(1(5(2(5(x1)))))))))))))))))) 1^1(5(1(1(1(5(1(3(2(3(1(5(x1)))))))))))) -> 3^1(4(4(4(5(3(3(3(5(2(1(5(4(1(5(2(5(x1))))))))))))))))) 1^1(5(1(1(1(5(1(3(2(3(1(5(x1)))))))))))) -> 3^1(3(3(5(2(1(5(4(1(5(2(5(x1)))))))))))) 1^1(5(1(1(1(5(1(3(2(3(1(5(x1)))))))))))) -> 3^1(3(5(2(1(5(4(1(5(2(5(x1))))))))))) 1^1(5(1(1(1(5(1(3(2(3(1(5(x1)))))))))))) -> 3^1(5(2(1(5(4(1(5(2(5(x1)))))))))) 1^1(5(1(1(1(5(1(3(2(3(1(5(x1)))))))))))) -> 2^1(1(5(4(1(5(2(5(x1)))))))) 1^1(5(1(1(1(5(1(3(2(3(1(5(x1)))))))))))) -> 1^1(5(4(1(5(2(5(x1))))))) 1^1(5(1(1(1(5(1(3(2(3(1(5(x1)))))))))))) -> 1^1(5(2(5(x1)))) 1^1(5(1(1(1(5(1(3(2(3(1(5(x1)))))))))))) -> 2^1(5(x1)) 2^1(0(2(4(5(1(2(2(5(3(5(0(x1)))))))))))) -> 1^1(5(4(1(4(3(4(5(3(4(5(2(x1)))))))))))) 2^1(0(2(4(5(1(2(2(5(3(5(0(x1)))))))))))) -> 1^1(4(3(4(5(3(4(5(2(x1))))))))) 2^1(0(2(4(5(1(2(2(5(3(5(0(x1)))))))))))) -> 3^1(4(5(3(4(5(2(x1))))))) 2^1(0(2(4(5(1(2(2(5(3(5(0(x1)))))))))))) -> 3^1(4(5(2(x1)))) 2^1(0(2(4(5(1(2(2(5(3(5(0(x1)))))))))))) -> 2^1(x1) 2^1(2(1(0(2(5(1(2(1(5(4(5(x1)))))))))))) -> 3^1(5(4(5(4(0(2(2(0(1(3(4(4(3(5(x1))))))))))))))) 2^1(2(1(0(2(5(1(2(1(5(4(5(x1)))))))))))) -> 2^1(2(0(1(3(4(4(3(5(x1))))))))) 2^1(2(1(0(2(5(1(2(1(5(4(5(x1)))))))))))) -> 2^1(0(1(3(4(4(3(5(x1)))))))) 2^1(2(1(0(2(5(1(2(1(5(4(5(x1)))))))))))) -> 1^1(3(4(4(3(5(x1)))))) 2^1(2(1(0(2(5(1(2(1(5(4(5(x1)))))))))))) -> 3^1(4(4(3(5(x1))))) 2^1(2(1(0(2(5(1(2(1(5(4(5(x1)))))))))))) -> 3^1(5(x1)) 2^1(3(1(1(3(0(3(0(4(0(5(3(x1)))))))))))) -> 2^1(0(1(2(4(3(4(1(4(4(3(4(1(4(4(0(3(x1))))))))))))))))) 2^1(3(1(1(3(0(3(0(4(0(5(3(x1)))))))))))) -> 1^1(2(4(3(4(1(4(4(3(4(1(4(4(0(3(x1))))))))))))))) 2^1(3(1(1(3(0(3(0(4(0(5(3(x1)))))))))))) -> 2^1(4(3(4(1(4(4(3(4(1(4(4(0(3(x1)))))))))))))) 2^1(3(1(1(3(0(3(0(4(0(5(3(x1)))))))))))) -> 3^1(4(1(4(4(3(4(1(4(4(0(3(x1)))))))))))) 2^1(3(1(1(3(0(3(0(4(0(5(3(x1)))))))))))) -> 1^1(4(4(3(4(1(4(4(0(3(x1)))))))))) 2^1(3(1(1(3(0(3(0(4(0(5(3(x1)))))))))))) -> 3^1(4(1(4(4(0(3(x1))))))) 2^1(3(1(1(3(0(3(0(4(0(5(3(x1)))))))))))) -> 1^1(4(4(0(3(x1))))) 2^1(5(1(3(0(0(4(5(3(1(3(4(x1)))))))))))) -> 1^1(5(4(0(5(4(0(2(4(0(1(2(x1)))))))))))) 2^1(5(1(3(0(0(4(5(3(1(3(4(x1)))))))))))) -> 2^1(4(0(1(2(x1))))) 2^1(5(1(3(0(0(4(5(3(1(3(4(x1)))))))))))) -> 1^1(2(x1)) 2^1(5(1(3(0(0(4(5(3(1(3(4(x1)))))))))))) -> 2^1(x1) 3^1(3(1(2(5(1(2(0(5(5(0(0(x1)))))))))))) -> 3^1(4(4(5(4(1(3(1(4(4(0(5(4(3(2(0(2(x1))))))))))))))))) 3^1(3(1(2(5(1(2(0(5(5(0(0(x1)))))))))))) -> 1^1(3(1(4(4(0(5(4(3(2(0(2(x1)))))))))))) 3^1(3(1(2(5(1(2(0(5(5(0(0(x1)))))))))))) -> 3^1(1(4(4(0(5(4(3(2(0(2(x1))))))))))) 3^1(3(1(2(5(1(2(0(5(5(0(0(x1)))))))))))) -> 1^1(4(4(0(5(4(3(2(0(2(x1)))))))))) 3^1(3(1(2(5(1(2(0(5(5(0(0(x1)))))))))))) -> 3^1(2(0(2(x1)))) 3^1(3(1(2(5(1(2(0(5(5(0(0(x1)))))))))))) -> 2^1(0(2(x1))) 3^1(3(1(2(5(1(2(0(5(5(0(0(x1)))))))))))) -> 2^1(x1) 3^1(5(5(0(2(3(5(0(5(0(4(0(x1)))))))))))) -> 2^1(1(3(0(5(4(4(2(x1)))))))) 3^1(5(5(0(2(3(5(0(5(0(4(0(x1)))))))))))) -> 1^1(3(0(5(4(4(2(x1))))))) 3^1(5(5(0(2(3(5(0(5(0(4(0(x1)))))))))))) -> 3^1(0(5(4(4(2(x1)))))) 3^1(5(5(0(2(3(5(0(5(0(4(0(x1)))))))))))) -> 2^1(x1) The TRS R consists of the following rules: 1(0(2(2(5(1(3(1(3(3(2(5(x1)))))))))))) -> 3(5(3(5(0(5(2(3(0(2(4(0(2(3(4(x1))))))))))))))) 1(5(1(1(1(5(1(3(2(3(1(5(x1)))))))))))) -> 1(3(4(4(4(5(3(3(3(5(2(1(5(4(1(5(2(5(x1)))))))))))))))))) 2(0(2(4(5(1(2(2(5(3(5(0(x1)))))))))))) -> 0(0(0(5(4(1(5(4(1(4(3(4(5(3(4(5(2(x1))))))))))))))))) 2(2(1(0(2(5(1(2(1(5(4(5(x1)))))))))))) -> 4(4(3(5(4(5(4(0(2(2(0(1(3(4(4(3(5(x1))))))))))))))))) 2(3(1(1(3(0(3(0(4(0(5(3(x1)))))))))))) -> 2(0(1(2(4(3(4(1(4(4(3(4(1(4(4(0(3(x1))))))))))))))))) 2(5(1(3(0(0(4(5(3(1(3(4(x1)))))))))))) -> 0(4(1(5(4(0(5(4(0(2(4(0(1(2(x1)))))))))))))) 3(3(1(2(5(1(2(0(5(5(0(0(x1)))))))))))) -> 3(4(4(5(4(1(3(1(4(4(0(5(4(3(2(0(2(x1))))))))))))))))) 3(5(5(0(2(3(5(0(5(0(4(0(x1)))))))))))) -> 0(0(4(5(4(5(2(1(3(0(5(4(4(2(x1)))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 45 less nodes. ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(2(1(0(2(5(1(2(1(5(4(5(x1)))))))))))) -> 3^1(5(x1)) 3^1(5(5(0(2(3(5(0(5(0(4(0(x1)))))))))))) -> 2^1(x1) 2^1(0(2(4(5(1(2(2(5(3(5(0(x1)))))))))))) -> 2^1(x1) 2^1(5(1(3(0(0(4(5(3(1(3(4(x1)))))))))))) -> 2^1(x1) The TRS R consists of the following rules: 1(0(2(2(5(1(3(1(3(3(2(5(x1)))))))))))) -> 3(5(3(5(0(5(2(3(0(2(4(0(2(3(4(x1))))))))))))))) 1(5(1(1(1(5(1(3(2(3(1(5(x1)))))))))))) -> 1(3(4(4(4(5(3(3(3(5(2(1(5(4(1(5(2(5(x1)))))))))))))))))) 2(0(2(4(5(1(2(2(5(3(5(0(x1)))))))))))) -> 0(0(0(5(4(1(5(4(1(4(3(4(5(3(4(5(2(x1))))))))))))))))) 2(2(1(0(2(5(1(2(1(5(4(5(x1)))))))))))) -> 4(4(3(5(4(5(4(0(2(2(0(1(3(4(4(3(5(x1))))))))))))))))) 2(3(1(1(3(0(3(0(4(0(5(3(x1)))))))))))) -> 2(0(1(2(4(3(4(1(4(4(3(4(1(4(4(0(3(x1))))))))))))))))) 2(5(1(3(0(0(4(5(3(1(3(4(x1)))))))))))) -> 0(4(1(5(4(0(5(4(0(2(4(0(1(2(x1)))))))))))))) 3(3(1(2(5(1(2(0(5(5(0(0(x1)))))))))))) -> 3(4(4(5(4(1(3(1(4(4(0(5(4(3(2(0(2(x1))))))))))))))))) 3(5(5(0(2(3(5(0(5(0(4(0(x1)))))))))))) -> 0(0(4(5(4(5(2(1(3(0(5(4(4(2(x1)))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (7) UsableRulesProof (EQUIVALENT) 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. ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(2(1(0(2(5(1(2(1(5(4(5(x1)))))))))))) -> 3^1(5(x1)) 3^1(5(5(0(2(3(5(0(5(0(4(0(x1)))))))))))) -> 2^1(x1) 2^1(0(2(4(5(1(2(2(5(3(5(0(x1)))))))))))) -> 2^1(x1) 2^1(5(1(3(0(0(4(5(3(1(3(4(x1)))))))))))) -> 2^1(x1) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *3^1(5(5(0(2(3(5(0(5(0(4(0(x1)))))))))))) -> 2^1(x1) The graph contains the following edges 1 > 1 *2^1(2(1(0(2(5(1(2(1(5(4(5(x1)))))))))))) -> 3^1(5(x1)) The graph contains the following edges 1 > 1 *2^1(0(2(4(5(1(2(2(5(3(5(0(x1)))))))))))) -> 2^1(x1) The graph contains the following edges 1 > 1 *2^1(5(1(3(0(0(4(5(3(1(3(4(x1)))))))))))) -> 2^1(x1) The graph contains the following edges 1 > 1 ---------------------------------------- (10) YES