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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) RootLabelingProof [EQUIVALENT, 11 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 1038 ms] (4) QTRS (5) QTRSRRRProof [EQUIVALENT, 18 ms] (6) QTRS (7) QTRSRRRProof [EQUIVALENT, 26 ms] (8) QTRS (9) RisEmptyProof [EQUIVALENT, 0 ms] (10) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0(0(1(1(0(1(1(1(1(0(0(1(0(1(0(0(1(1(0(1(0(0(0(1(0(1(0(0(0(0(1(1(1(1(0(0(0(0(0(0(1(1(1(1(0(0(1(1(1(0(0(1(0(1(0(1(1(0(0(0(1(1(0(0(0(1(0(0(1(1(1(0(0(1(0(0(1(0(0(1(1(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 0(1(0(0(0(0(0(0(0(1(1(1(1(0(0(1(0(0(0(1(1(0(1(1(1(1(0(1(0(0(1(0(0(1(1(1(0(1(0(1(0(0(0(0(1(1(0(0(1(1(0(0(1(0(0(0(0(0(1(1(1(1(0(1(1(0(0(0(1(1(0(0(0(1(1(0(1(1(0(1(1(1(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 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0(0(1(1(1(1(0(1(1(1(0(1(1(1(0(0(1(1(0(0(0(1(1(0(0(0(0(0(1(1(0(1(0(1(0(0(1(0(0(1(0(0(1(1(1(1(0(1(0(1(1(0(1(0(1(1(1(1(0(1(0(0(1(1(0(0(0(1(0(0(0(1(0(1(1(0(1(0(1(1(0(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 0(1(0(0(0(0(0(0(0(1(1(1(1(0(0(1(0(0(0(1(1(0(1(1(1(1(0(1(0(0(1(0(0(1(1(1(0(1(0(1(0(0(0(0(1(1(0(0(1(1(0(0(1(0(0(0(0(1(1(1(0(1(1(1(1(1(1(1(0(1(0(0(1(0(0(0(1(1(1(0(0(1(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 0(0(1(1(1(1(0(1(1(1(0(1(1(1(0(0(1(1(0(0(0(1(1(0(0(0(0(0(1(1(0(1(0(1(0(0(1(0(0(1(0(0(1(1(1(1(0(1(0(1(1(0(1(0(1(1(1(1(0(1(0(0(1(1(0(0(0(1(0(0(0(1(0(1(1(0(1(0(1(1(0(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 0(0(1(1(0(1(1(1(1(0(0(1(0(1(0(0(1(1(0(1(0(0(0(1(0(1(0(0(0(0(1(1(1(1(0(0(0(0(0(0(1(1(1(1(0(0(1(1(1(0(0(1(0(1(0(1(1(0(0(0(1(1(0(0(0(1(0(0(1(1(1(0(0(1(0(0(1(1(1(1(1(1(0(0(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) Q is empty. ---------------------------------------- (1) RootLabelingProof (EQUIVALENT) We used plain root labeling [ROOTLAB] with the following heuristic: LabelAll: All function symbols get labeled As Q is empty the root labeling was sound AND complete. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 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0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) Q is empty. ---------------------------------------- (3) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0_{0_1}(x_1)) = 1 + x_1 POL(0_{1_1}(x_1)) = x_1 POL(1_{0_1}(x_1)) = 9 + x_1 POL(1_{1_1}(x_1)) = 3 + x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 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0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) Q is empty. ---------------------------------------- (5) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0_{0_1}(x_1)) = x_1 POL(0_{1_1}(x_1)) = x_1 POL(1_{0_1}(x_1)) = 2 + x_1 POL(1_{1_1}(x_1)) = 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_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) Q is empty. ---------------------------------------- (7) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0_{0_1}(x_1)) = x_1 POL(0_{1_1}(x_1)) = x_1 POL(1_{0_1}(x_1)) = 1 + x_1 POL(1_{1_1}(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_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{0_1}(0_{1_1}(1_{1_1}(1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) ---------------------------------------- (8) Obligation: Q restricted rewrite system: R is empty. Q is empty. ---------------------------------------- (9) RisEmptyProof (EQUIVALENT) The TRS R is empty. Hence, termination is trivially proven. ---------------------------------------- (10) YES