YES proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 152 ms] (4) QTRS (5) DependencyPairsProof [EQUIVALENT, 237 ms] (6) QDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) QDP (9) QDPOrderProof [EQUIVALENT, 599 ms] (10) QDP (11) DependencyGraphProof [EQUIVALENT, 2 ms] (12) QDP (13) QDPOrderProof [EQUIVALENT, 441 ms] (14) QDP (15) DependencyGraphProof [EQUIVALENT, 0 ms] (16) QDP (17) QDPOrderProof [EQUIVALENT, 70 ms] (18) QDP (19) PisEmptyProof [EQUIVALENT, 0 ms] (20) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0(1(1(x1))) -> 2(1(1(x1))) 3(4(0(x1))) -> 4(5(5(x1))) 2(2(1(4(0(x1))))) -> 2(2(1(3(5(x1))))) 4(5(3(1(0(0(4(x1))))))) -> 4(5(0(1(4(5(4(x1))))))) 1(3(3(2(1(4(4(0(x1)))))))) -> 1(1(3(0(4(4(0(2(x1)))))))) 2(2(1(5(0(2(2(2(4(x1))))))))) -> 2(0(0(4(5(5(0(x1))))))) 0(2(3(0(2(0(3(5(2(4(x1)))))))))) -> 2(0(4(5(4(4(5(4(0(x1))))))))) 3(2(3(1(3(5(5(2(2(0(x1)))))))))) -> 4(0(0(0(0(2(2(2(0(x1))))))))) 3(4(1(1(0(0(5(0(5(0(x1)))))))))) -> 3(0(1(0(1(1(3(0(4(5(x1)))))))))) 4(1(3(0(0(5(1(5(0(1(x1)))))))))) -> 4(0(4(2(2(3(0(3(0(1(x1)))))))))) 3(3(0(2(0(1(4(5(5(5(0(x1))))))))))) -> 1(2(3(5(4(4(0(2(3(5(5(x1))))))))))) 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)))))))))))) 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)))))))))))) 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))))))))))) 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))))))))))))) 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)))))))))))))) 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)))))))))))))) 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))))))))))))))) 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))))))))))))))) 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)))))))))))))))) 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))))))))))))))))) 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))))))))))))))))))) 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)))))))))))))))))))) 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))))))))))))))))))))) 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)))))))))))))))))))) Q is empty. ---------------------------------------- (1) QTRS Reverse (EQUIVALENT) We applied the QTRS Reverse Processor [REVERSE]. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 1(1(0(x1))) -> 1(1(2(x1))) 0(4(3(x1))) -> 5(5(4(x1))) 0(4(1(2(2(x1))))) -> 5(3(1(2(2(x1))))) 4(0(0(1(3(5(4(x1))))))) -> 4(5(4(1(0(5(4(x1))))))) 0(4(4(1(2(3(3(1(x1)))))))) -> 2(0(4(4(0(3(1(1(x1)))))))) 4(2(2(2(0(5(1(2(2(x1))))))))) -> 0(5(5(4(0(0(2(x1))))))) 4(2(5(3(0(2(0(3(2(0(x1)))))))))) -> 0(4(5(4(4(5(4(0(2(x1))))))))) 0(2(2(5(5(3(1(3(2(3(x1)))))))))) -> 0(2(2(2(0(0(0(0(4(x1))))))))) 0(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5(4(0(3(1(1(0(1(0(3(x1)))))))))) 1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 1(0(3(0(3(2(2(4(0(4(x1)))))))))) 0(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 5(5(3(2(0(4(4(5(3(2(1(x1))))))))))) 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)))))))))))) 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)))))))))))) 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))))))))))) 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))))))))))))) 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)))))))))))))) 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)))))))))))))) 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))))))))))))))) 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))))))))))))))) 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)))))))))))))))) 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))))))))))))))))) 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))))))))))))))))))) 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)))))))))))))))))))) 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))))))))))))))))))))) 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)))))))))))))))))))) Q is empty. ---------------------------------------- (3) 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)) = 1 + x_1 POL(5(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: 4(2(2(2(0(5(1(2(2(x1))))))))) -> 0(5(5(4(0(0(2(x1))))))) 4(2(5(3(0(2(0(3(2(0(x1)))))))))) -> 0(4(5(4(4(5(4(0(2(x1))))))))) 0(2(2(5(5(3(1(3(2(3(x1)))))))))) -> 0(2(2(2(0(0(0(0(4(x1))))))))) 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))))))))))) 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))))))))))))) 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)))))))))))))) 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))))))))))))))) 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))))))))))))))))) 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))))))))))))))))))) 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)))))))))))))))))))) ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 1(1(0(x1))) -> 1(1(2(x1))) 0(4(3(x1))) -> 5(5(4(x1))) 0(4(1(2(2(x1))))) -> 5(3(1(2(2(x1))))) 4(0(0(1(3(5(4(x1))))))) -> 4(5(4(1(0(5(4(x1))))))) 0(4(4(1(2(3(3(1(x1)))))))) -> 2(0(4(4(0(3(1(1(x1)))))))) 0(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5(4(0(3(1(1(0(1(0(3(x1)))))))))) 1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 1(0(3(0(3(2(2(4(0(4(x1)))))))))) 0(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 5(5(3(2(0(4(4(5(3(2(1(x1))))))))))) 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)))))))))))) 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)))))))))))) 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)))))))))))))) 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))))))))))))))) 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)))))))))))))))) 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)))))))))))))))))))) 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))))))))))))))))))))) Q is empty. ---------------------------------------- (5) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(1(0(x1))) -> 1^1(1(2(x1))) 1^1(1(0(x1))) -> 1^1(2(x1)) 0^1(4(3(x1))) -> 5^1(5(4(x1))) 0^1(4(3(x1))) -> 5^1(4(x1)) 0^1(4(3(x1))) -> 4^1(x1) 0^1(4(1(2(2(x1))))) -> 5^1(3(1(2(2(x1))))) 0^1(4(1(2(2(x1))))) -> 3^1(1(2(2(x1)))) 4^1(0(0(1(3(5(4(x1))))))) -> 4^1(5(4(1(0(5(4(x1))))))) 4^1(0(0(1(3(5(4(x1))))))) -> 5^1(4(1(0(5(4(x1)))))) 4^1(0(0(1(3(5(4(x1))))))) -> 4^1(1(0(5(4(x1))))) 4^1(0(0(1(3(5(4(x1))))))) -> 1^1(0(5(4(x1)))) 4^1(0(0(1(3(5(4(x1))))))) -> 0^1(5(4(x1))) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 0^1(4(4(0(3(1(1(x1))))))) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 4^1(4(0(3(1(1(x1)))))) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 4^1(0(3(1(1(x1))))) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 0^1(3(1(1(x1)))) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 3^1(1(1(x1))) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 1^1(1(x1)) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5^1(4(0(3(1(1(0(1(0(3(x1)))))))))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 4^1(0(3(1(1(0(1(0(3(x1))))))))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(3(1(1(0(1(0(3(x1)))))))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 3^1(1(1(0(1(0(3(x1))))))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(1(0(1(0(3(x1)))))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(0(1(0(3(x1))))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(1(0(3(x1)))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(0(3(x1))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(3(x1)) 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 1^1(0(3(0(3(2(2(4(0(4(x1)))))))))) 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 0^1(3(0(3(2(2(4(0(4(x1))))))))) 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 3^1(0(3(2(2(4(0(4(x1)))))))) 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 0^1(3(2(2(4(0(4(x1))))))) 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 3^1(2(2(4(0(4(x1)))))) 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 4^1(0(4(x1))) 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 0^1(4(x1)) 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))))))))))) 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)))))))))) 0^1(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 3^1(2(0(4(4(5(3(2(1(x1))))))))) 0^1(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 0^1(4(4(5(3(2(1(x1))))))) 0^1(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 4^1(4(5(3(2(1(x1)))))) 0^1(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 4^1(5(3(2(1(x1))))) 0^1(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 5^1(3(2(1(x1)))) 0^1(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 3^1(2(1(x1))) 0^1(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 1^1(x1) 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)))))))))))) 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))))))))))) 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)))))))))) 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))))))))) 4^1(5(3(4(5(2(1(2(0(2(0(3(x1)))))))))))) -> 4^1(5(0(3(0(2(2(3(x1)))))))) 4^1(5(3(4(5(2(1(2(0(2(0(3(x1)))))))))))) -> 5^1(0(3(0(2(2(3(x1))))))) 4^1(5(3(4(5(2(1(2(0(2(0(3(x1)))))))))))) -> 0^1(3(0(2(2(3(x1)))))) 4^1(5(3(4(5(2(1(2(0(2(0(3(x1)))))))))))) -> 3^1(0(2(2(3(x1))))) 4^1(5(3(4(5(2(1(2(0(2(0(3(x1)))))))))))) -> 0^1(2(2(3(x1)))) 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)))))))))))) 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))))))))))) 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)))))))))) 1^1(1(5(0(0(0(3(5(4(5(0(3(x1)))))))))))) -> 0^1(3(0(4(1(4(2(3(x1)))))))) 1^1(1(5(0(0(0(3(5(4(5(0(3(x1)))))))))))) -> 3^1(0(4(1(4(2(3(x1))))))) 1^1(1(5(0(0(0(3(5(4(5(0(3(x1)))))))))))) -> 0^1(4(1(4(2(3(x1)))))) 1^1(1(5(0(0(0(3(5(4(5(0(3(x1)))))))))))) -> 4^1(1(4(2(3(x1))))) 1^1(1(5(0(0(0(3(5(4(5(0(3(x1)))))))))))) -> 1^1(4(2(3(x1)))) 1^1(1(5(0(0(0(3(5(4(5(0(3(x1)))))))))))) -> 4^1(2(3(x1))) 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)))))))))))))) 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))))))))))))) 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)))))))))))) 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))))))))))) 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)))))))))) 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))))))))) 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)))))))) 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))))))) 3^1(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 3^1(0(2(2(4(4(x1)))))) 3^1(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 0^1(2(2(4(4(x1))))) 3^1(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 4^1(4(x1)) 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))))))))))))))) 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))))))))))))) 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)))))))))))) 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))))))))))) 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)))))))))) 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))))))))) 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))))))) 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(1(5(3(1(x1))))) 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(5(3(1(x1)))) 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 5^1(3(1(x1))) 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 3^1(1(x1)) 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(x1) 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)))))))))))))))) 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))))))))))))))) 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)))))))))))))) 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)))))))))))) 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))))))))))) 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)))))))))) 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))))))))) 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)))))))) 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))))))) 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)))))) 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))))) 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 3^1(1(3(2(x1)))) 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 1^1(3(2(x1))) 0^1(5(4(3(5(3(2(2(4(4(0(0(5(0(4(2(x1)))))))))))))))) -> 3^1(2(x1)) 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)))))))))))))))))))) 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)))))))))))))))))) 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)))))))))))))))) 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))))))))))))))) 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)))))))))))))) 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))))))))))))) 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)))))))))))) 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))))))))))) 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)))))))))) 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))))))))) 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)))))))) 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))))))) 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)))))) 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))))) 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)))) 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))) 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)) 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) 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))))))))))))))))))))) 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)))))))))))))))))))) 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))))))))))))))))))) 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)))))))))))))))))) 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)))))))))))))))) 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))))))))))))))) 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))))))))))))) 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)))))))))))) 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))))))))))) 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)))))))))) 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))))))))) 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)))))))) 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))))))) 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)))))) 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))))) 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)))) 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))) 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)) The TRS R consists of the following rules: 1(1(0(x1))) -> 1(1(2(x1))) 0(4(3(x1))) -> 5(5(4(x1))) 0(4(1(2(2(x1))))) -> 5(3(1(2(2(x1))))) 4(0(0(1(3(5(4(x1))))))) -> 4(5(4(1(0(5(4(x1))))))) 0(4(4(1(2(3(3(1(x1)))))))) -> 2(0(4(4(0(3(1(1(x1)))))))) 0(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5(4(0(3(1(1(0(1(0(3(x1)))))))))) 1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 1(0(3(0(3(2(2(4(0(4(x1)))))))))) 0(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 5(5(3(2(0(4(4(5(3(2(1(x1))))))))))) 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)))))))))))) 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)))))))))))) 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)))))))))))))) 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))))))))))))))) 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)))))))))))))))) 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)))))))))))))))))))) 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))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 68 less nodes. ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 4^1(0(4(x1))) 4^1(0(0(1(3(5(4(x1))))))) -> 4^1(5(4(1(0(5(4(x1))))))) 4^1(0(0(1(3(5(4(x1))))))) -> 5^1(4(1(0(5(4(x1)))))) 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(1(5(3(1(x1))))) 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 0^1(4(x1)) 0^1(4(3(x1))) -> 5^1(5(4(x1))) 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(5(3(1(x1)))) 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 5^1(3(1(x1))) 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 3^1(1(x1)) 3^1(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 4^1(4(x1)) 4^1(0(0(1(3(5(4(x1))))))) -> 4^1(1(0(5(4(x1))))) 4^1(0(0(1(3(5(4(x1))))))) -> 1^1(0(5(4(x1)))) 4^1(0(0(1(3(5(4(x1))))))) -> 0^1(5(4(x1))) 0^1(4(3(x1))) -> 5^1(4(x1)) 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(x1) 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))))))))))))) 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)))))))))))) 0^1(4(3(x1))) -> 4^1(x1) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 0^1(4(4(0(3(1(1(x1))))))) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 4^1(4(0(3(1(1(x1)))))) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 4^1(0(3(1(1(x1))))) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 0^1(3(1(1(x1)))) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 3^1(1(1(x1))) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 1^1(1(x1)) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5^1(4(0(3(1(1(0(1(0(3(x1)))))))))) 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))))))))))) 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)))))))))) 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))))))))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 4^1(0(3(1(1(0(1(0(3(x1))))))))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(3(1(1(0(1(0(3(x1)))))))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 3^1(1(1(0(1(0(3(x1))))))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(1(0(1(0(3(x1)))))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(0(1(0(3(x1))))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(1(0(3(x1)))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(0(3(x1))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(3(x1)) 0^1(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 1^1(x1) 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)))))))))))))))) 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)))))))))))) 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))))))))))) 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)))))))) 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))))))) 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)))))) 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))))) 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)))) 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))) 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)) 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)))))))))) 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))))))))) 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)))))))) 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)))))))))))))))) 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))))))))))))))) 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)))))))))))))) 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))))))))))))) 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)))))))))))) 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))))))))))) 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)))))))))) 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))))))))) 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)))))))) 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))))))) 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)))))) 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))))) 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)))) 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))) 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)) 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) The TRS R consists of the following rules: 1(1(0(x1))) -> 1(1(2(x1))) 0(4(3(x1))) -> 5(5(4(x1))) 0(4(1(2(2(x1))))) -> 5(3(1(2(2(x1))))) 4(0(0(1(3(5(4(x1))))))) -> 4(5(4(1(0(5(4(x1))))))) 0(4(4(1(2(3(3(1(x1)))))))) -> 2(0(4(4(0(3(1(1(x1)))))))) 0(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5(4(0(3(1(1(0(1(0(3(x1)))))))))) 1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 1(0(3(0(3(2(2(4(0(4(x1)))))))))) 0(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 5(5(3(2(0(4(4(5(3(2(1(x1))))))))))) 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)))))))))))) 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)))))))))))) 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)))))))))))))) 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))))))))))))))) 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)))))))))))))))) 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)))))))))))))))))))) 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))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 3^1(4(5(0(3(3(3(2(5(0(3(3(1(4(x1)))))))))))))) -> 4^1(4(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 0 POL(0^1(x_1)) = 0 POL(1(x_1)) = 0 POL(1^1(x_1)) = 0 POL(2(x_1)) = 0 POL(3(x_1)) = x_1 POL(3^1(x_1)) = x_1 POL(4(x_1)) = 1 POL(4^1(x_1)) = 0 POL(5(x_1)) = 0 POL(5^1(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0(4(3(x1))) -> 5(5(4(x1))) 0(4(1(2(2(x1))))) -> 5(3(1(2(2(x1))))) 0(4(4(1(2(3(3(1(x1)))))))) -> 2(0(4(4(0(3(1(1(x1)))))))) 0(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5(4(0(3(1(1(0(1(0(3(x1)))))))))) 0(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 5(5(3(2(0(4(4(5(3(2(1(x1))))))))))) 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)))))))))))))))) 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)))))))))))))))))))) 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))))))))))))))) 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))))))))))))))))))))) 1(1(0(x1))) -> 1(1(2(x1))) 1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 1(0(3(0(3(2(2(4(0(4(x1)))))))))) 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)))))))))))) 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)))))))))))))) ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 4^1(0(4(x1))) 4^1(0(0(1(3(5(4(x1))))))) -> 4^1(5(4(1(0(5(4(x1))))))) 4^1(0(0(1(3(5(4(x1))))))) -> 5^1(4(1(0(5(4(x1)))))) 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(1(5(3(1(x1))))) 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 0^1(4(x1)) 0^1(4(3(x1))) -> 5^1(5(4(x1))) 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(5(3(1(x1)))) 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 5^1(3(1(x1))) 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 3^1(1(x1)) 4^1(0(0(1(3(5(4(x1))))))) -> 4^1(1(0(5(4(x1))))) 4^1(0(0(1(3(5(4(x1))))))) -> 1^1(0(5(4(x1)))) 4^1(0(0(1(3(5(4(x1))))))) -> 0^1(5(4(x1))) 0^1(4(3(x1))) -> 5^1(4(x1)) 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(x1) 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))))))))))))) 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)))))))))))) 0^1(4(3(x1))) -> 4^1(x1) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 0^1(4(4(0(3(1(1(x1))))))) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 4^1(4(0(3(1(1(x1)))))) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 4^1(0(3(1(1(x1))))) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 0^1(3(1(1(x1)))) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 3^1(1(1(x1))) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 1^1(1(x1)) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5^1(4(0(3(1(1(0(1(0(3(x1)))))))))) 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))))))))))) 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)))))))))) 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))))))))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 4^1(0(3(1(1(0(1(0(3(x1))))))))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(3(1(1(0(1(0(3(x1)))))))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 3^1(1(1(0(1(0(3(x1))))))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(1(0(1(0(3(x1)))))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(0(1(0(3(x1))))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(1(0(3(x1)))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(0(3(x1))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(3(x1)) 0^1(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 1^1(x1) 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)))))))))))))))) 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)))))))))))) 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))))))))))) 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)))))))) 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))))))) 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)))))) 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))))) 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)))) 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))) 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)) 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)))))))))) 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))))))))) 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)))))))) 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)))))))))))))))) 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))))))))))))))) 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)))))))))))))) 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))))))))))))) 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)))))))))))) 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))))))))))) 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)))))))))) 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))))))))) 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)))))))) 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))))))) 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)))))) 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))))) 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)))) 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))) 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)) 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) The TRS R consists of the following rules: 1(1(0(x1))) -> 1(1(2(x1))) 0(4(3(x1))) -> 5(5(4(x1))) 0(4(1(2(2(x1))))) -> 5(3(1(2(2(x1))))) 4(0(0(1(3(5(4(x1))))))) -> 4(5(4(1(0(5(4(x1))))))) 0(4(4(1(2(3(3(1(x1)))))))) -> 2(0(4(4(0(3(1(1(x1)))))))) 0(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5(4(0(3(1(1(0(1(0(3(x1)))))))))) 1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 1(0(3(0(3(2(2(4(0(4(x1)))))))))) 0(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 5(5(3(2(0(4(4(5(3(2(1(x1))))))))))) 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)))))))))))) 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)))))))))))) 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)))))))))))))) 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))))))))))))))) 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)))))))))))))))) 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)))))))))))))))))))) 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))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 10 less nodes. ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: 4^1(0(0(1(3(5(4(x1))))))) -> 4^1(5(4(1(0(5(4(x1))))))) 4^1(0(0(1(3(5(4(x1))))))) -> 5^1(4(1(0(5(4(x1)))))) 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(1(5(3(1(x1))))) 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 4^1(0(4(x1))) 4^1(0(0(1(3(5(4(x1))))))) -> 4^1(1(0(5(4(x1))))) 4^1(0(0(1(3(5(4(x1))))))) -> 1^1(0(5(4(x1)))) 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 0^1(4(x1)) 0^1(4(3(x1))) -> 5^1(5(4(x1))) 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(5(3(1(x1)))) 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 5^1(3(1(x1))) 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(x1) 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)))))))))))) 0^1(4(3(x1))) -> 5^1(4(x1)) 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))))))))))) 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)))))))))) 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))))))))) 0^1(4(3(x1))) -> 4^1(x1) 4^1(0(0(1(3(5(4(x1))))))) -> 0^1(5(4(x1))) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 0^1(4(4(0(3(1(1(x1))))))) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 4^1(4(0(3(1(1(x1)))))) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 4^1(0(3(1(1(x1))))) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 0^1(3(1(1(x1)))) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 1^1(1(x1)) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5^1(4(0(3(1(1(0(1(0(3(x1)))))))))) 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)))))))) 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))))) 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)) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 4^1(0(3(1(1(0(1(0(3(x1))))))))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(3(1(1(0(1(0(3(x1)))))))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(1(0(1(0(3(x1)))))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(0(1(0(3(x1))))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(1(0(3(x1)))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(0(3(x1))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(3(x1)) 0^1(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 1^1(x1) 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)))))))))))))))) 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)))))))))))) 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))))))))))) 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)))))))))) 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))))))))) 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)))))))) 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)))))))))))))))) 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))))))))))))))) 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)))))))))))))) 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))))))))))))) 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)))))))))))) 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)))))))))) 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))))))))) 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)))))))) 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))))))) 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))))) 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)))) 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))) 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)) 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) The TRS R consists of the following rules: 1(1(0(x1))) -> 1(1(2(x1))) 0(4(3(x1))) -> 5(5(4(x1))) 0(4(1(2(2(x1))))) -> 5(3(1(2(2(x1))))) 4(0(0(1(3(5(4(x1))))))) -> 4(5(4(1(0(5(4(x1))))))) 0(4(4(1(2(3(3(1(x1)))))))) -> 2(0(4(4(0(3(1(1(x1)))))))) 0(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5(4(0(3(1(1(0(1(0(3(x1)))))))))) 1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 1(0(3(0(3(2(2(4(0(4(x1)))))))))) 0(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 5(5(3(2(0(4(4(5(3(2(1(x1))))))))))) 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)))))))))))) 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)))))))))))) 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)))))))))))))) 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))))))))))))))) 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)))))))))))))))) 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)))))))))))))))))))) 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))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 4^1(0(0(1(3(5(4(x1))))))) -> 5^1(4(1(0(5(4(x1)))))) 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(1(5(3(1(x1))))) 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 4^1(0(4(x1))) 4^1(0(0(1(3(5(4(x1))))))) -> 4^1(1(0(5(4(x1))))) 4^1(0(0(1(3(5(4(x1))))))) -> 1^1(0(5(4(x1)))) 1^1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 0^1(4(x1)) 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(5(3(1(x1)))) 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 5^1(3(1(x1))) 5^1(0(2(1(3(4(3(2(4(3(1(4(1(5(5(x1))))))))))))))) -> 1^1(x1) 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)))))))))))) 0^1(4(3(x1))) -> 5^1(4(x1)) 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))))))))))) 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)))))))))) 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))))))))) 0^1(4(3(x1))) -> 4^1(x1) 4^1(0(0(1(3(5(4(x1))))))) -> 0^1(5(4(x1))) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 0^1(4(4(0(3(1(1(x1))))))) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 4^1(4(0(3(1(1(x1)))))) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 4^1(0(3(1(1(x1))))) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 0^1(3(1(1(x1)))) 0^1(4(4(1(2(3(3(1(x1)))))))) -> 1^1(1(x1)) 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)))))))) 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))))) 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)) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 4^1(0(3(1(1(0(1(0(3(x1))))))))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(3(1(1(0(1(0(3(x1)))))))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(1(0(1(0(3(x1)))))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(0(1(0(3(x1))))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(1(0(3(x1)))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 1^1(0(3(x1))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 0^1(3(x1)) 0^1(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 1^1(x1) 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)))))))))))) 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))))))))))) 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)))))))))) 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))))))))) 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)))))))) 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)))))))))))))))) 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))))))))))))))) 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)))))))))))))) 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))))))))))))) 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)))))))))))) 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)))))))))) 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))))))))) 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)))))))) 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))))))) 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))))) 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)))) 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))) 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)) 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) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 + x_1 POL(0^1(x_1)) = 1 + x_1 POL(1(x_1)) = 1 + x_1 POL(1^1(x_1)) = x_1 POL(2(x_1)) = 1 + x_1 POL(3(x_1)) = 1 + x_1 POL(4(x_1)) = 1 + x_1 POL(4^1(x_1)) = 1 + x_1 POL(5(x_1)) = 1 + x_1 POL(5^1(x_1)) = 1 + x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 4(0(0(1(3(5(4(x1))))))) -> 4(5(4(1(0(5(4(x1))))))) 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)))))))))))) 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))))))))))))))) 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))))))))))))))))))))) 0(4(3(x1))) -> 5(5(4(x1))) 0(4(1(2(2(x1))))) -> 5(3(1(2(2(x1))))) 0(4(4(1(2(3(3(1(x1)))))))) -> 2(0(4(4(0(3(1(1(x1)))))))) 0(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5(4(0(3(1(1(0(1(0(3(x1)))))))))) 0(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 5(5(3(2(0(4(4(5(3(2(1(x1))))))))))) 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)))))))))))))))) 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)))))))))))))))))))) 1(1(0(x1))) -> 1(1(2(x1))) 1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 1(0(3(0(3(2(2(4(0(4(x1)))))))))) 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)))))))))))) 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)))))))))))))) ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: 4^1(0(0(1(3(5(4(x1))))))) -> 4^1(5(4(1(0(5(4(x1))))))) 0^1(4(3(x1))) -> 5^1(5(4(x1))) 0^1(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5^1(4(0(3(1(1(0(1(0(3(x1)))))))))) 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)))))))))))))))) The TRS R consists of the following rules: 1(1(0(x1))) -> 1(1(2(x1))) 0(4(3(x1))) -> 5(5(4(x1))) 0(4(1(2(2(x1))))) -> 5(3(1(2(2(x1))))) 4(0(0(1(3(5(4(x1))))))) -> 4(5(4(1(0(5(4(x1))))))) 0(4(4(1(2(3(3(1(x1)))))))) -> 2(0(4(4(0(3(1(1(x1)))))))) 0(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5(4(0(3(1(1(0(1(0(3(x1)))))))))) 1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 1(0(3(0(3(2(2(4(0(4(x1)))))))))) 0(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 5(5(3(2(0(4(4(5(3(2(1(x1))))))))))) 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)))))))))))) 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)))))))))))) 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)))))))))))))) 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))))))))))))))) 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)))))))))))))))) 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)))))))))))))))))))) 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))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (15) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: 4^1(0(0(1(3(5(4(x1))))))) -> 4^1(5(4(1(0(5(4(x1))))))) The TRS R consists of the following rules: 1(1(0(x1))) -> 1(1(2(x1))) 0(4(3(x1))) -> 5(5(4(x1))) 0(4(1(2(2(x1))))) -> 5(3(1(2(2(x1))))) 4(0(0(1(3(5(4(x1))))))) -> 4(5(4(1(0(5(4(x1))))))) 0(4(4(1(2(3(3(1(x1)))))))) -> 2(0(4(4(0(3(1(1(x1)))))))) 0(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5(4(0(3(1(1(0(1(0(3(x1)))))))))) 1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 1(0(3(0(3(2(2(4(0(4(x1)))))))))) 0(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 5(5(3(2(0(4(4(5(3(2(1(x1))))))))))) 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)))))))))))) 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)))))))))))) 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)))))))))))))) 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))))))))))))))) 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)))))))))))))))) 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)))))))))))))))))))) 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))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (17) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 4^1(0(0(1(3(5(4(x1))))))) -> 4^1(5(4(1(0(5(4(x1))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(1(x_1)) = 1 POL(2(x_1)) = 0 POL(3(x_1)) = x_1 POL(4(x_1)) = 1 + x_1 POL(4^1(x_1)) = x_1 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 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))))))))))))))) 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))))))))))))))))))))) ---------------------------------------- (18) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: 1(1(0(x1))) -> 1(1(2(x1))) 0(4(3(x1))) -> 5(5(4(x1))) 0(4(1(2(2(x1))))) -> 5(3(1(2(2(x1))))) 4(0(0(1(3(5(4(x1))))))) -> 4(5(4(1(0(5(4(x1))))))) 0(4(4(1(2(3(3(1(x1)))))))) -> 2(0(4(4(0(3(1(1(x1)))))))) 0(5(0(5(0(0(1(1(4(3(x1)))))))))) -> 5(4(0(3(1(1(0(1(0(3(x1)))))))))) 1(0(5(1(5(0(0(3(1(4(x1)))))))))) -> 1(0(3(0(3(2(2(4(0(4(x1)))))))))) 0(5(5(5(4(1(0(2(0(3(3(x1))))))))))) -> 5(5(3(2(0(4(4(5(3(2(1(x1))))))))))) 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)))))))))))) 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)))))))))))) 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)))))))))))))) 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))))))))))))))) 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)))))))))))))))) 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)))))))))))))))))))) 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))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (20) YES