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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRSRRRProof [EQUIVALENT, 202 ms] (2) QTRS (3) DependencyPairsProof [EQUIVALENT, 400 ms] (4) QDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) QDP (8) QDPOrderProof [EQUIVALENT, 184 ms] (9) QDP (10) QDPOrderProof [EQUIVALENT, 142 ms] (11) QDP (12) QDPOrderProof [EQUIVALENT, 83 ms] (13) QDP (14) DependencyGraphProof [EQUIVALENT, 0 ms] (15) QDP (16) QDPOrderProof [EQUIVALENT, 93 ms] (17) QDP (18) PisEmptyProof [EQUIVALENT, 0 ms] (19) YES (20) QDP (21) UsableRulesProof [EQUIVALENT, 0 ms] (22) QDP (23) QDPSizeChangeProof [EQUIVALENT, 0 ms] (24) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0(0(1(2(2(x1))))) -> 2(3(3(2(2(x1))))) 0(4(1(0(4(5(x1)))))) -> 5(2(5(3(4(5(x1)))))) 0(5(2(4(2(0(0(5(2(x1))))))))) -> 1(5(2(0(2(3(5(2(x1)))))))) 3(4(2(2(1(5(3(3(2(x1))))))))) -> 3(1(0(3(5(3(3(0(2(x1))))))))) 0(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 3(3(3(2(5(0(4(0(5(0(x1)))))))))) 5(0(3(5(1(4(2(1(0(2(x1)))))))))) -> 5(2(0(1(3(1(1(0(2(x1))))))))) 2(4(5(2(4(3(4(3(1(2(3(x1))))))))))) -> 5(3(1(3(3(3(2(2(4(1(x1)))))))))) 0(2(1(5(3(5(3(0(5(5(5(0(x1)))))))))))) -> 2(2(5(4(4(2(3(3(0(3(5(4(x1)))))))))))) 3(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 1(5(2(4(0(4(2(0(0(5(0(2(x1)))))))))))) 3(3(3(2(3(3(1(2(2(4(3(3(x1)))))))))))) -> 1(2(2(1(3(2(3(5(2(5(5(x1))))))))))) 5(5(4(0(3(0(2(3(3(2(3(3(x1)))))))))))) -> 5(2(2(3(0(1(1(3(4(4(5(x1))))))))))) 0(0(3(1(0(2(2(3(2(5(3(0(1(x1))))))))))))) -> 0(1(3(4(3(1(0(0(3(3(4(0(x1)))))))))))) 2(1(3(3(4(3(0(2(0(4(0(3(1(x1))))))))))))) -> 2(4(1(3(0(1(3(3(2(1(3(4(4(x1))))))))))))) 2(4(2(3(3(5(0(4(5(5(2(5(5(x1))))))))))))) -> 3(5(4(3(3(2(0(1(1(4(3(5(x1)))))))))))) 5(2(3(1(2(1(5(4(5(1(1(0(0(x1))))))))))))) -> 5(2(0(5(4(3(4(3(3(0(5(0(0(x1))))))))))))) 2(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1))))))))))))))) -> 5(3(1(3(1(4(4(2(5(4(5(4(4(1(0(x1))))))))))))))) 2(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 0(3(2(5(5(2(4(4(3(5(3(4(3(4(1(x1))))))))))))))) 2(5(3(5(2(4(2(1(3(0(5(0(3(1(3(x1))))))))))))))) -> 5(3(2(3(5(1(5(3(5(3(5(1(5(x1))))))))))))) 0(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 0(5(1(2(4(0(2(0(1(2(4(3(3(3(0(5(x1)))))))))))))))) 0(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 0(1(5(2(2(5(0(5(3(3(2(3(4(1(5(2(x1)))))))))))))))) 3(3(2(0(1(5(4(5(4(2(4(2(3(4(4(3(x1)))))))))))))))) -> 5(2(5(5(5(5(2(3(3(1(2(5(0(4(0(x1))))))))))))))) 3(0(0(2(0(3(5(3(0(2(5(3(5(5(2(3(4(x1))))))))))))))))) -> 5(3(4(3(0(4(2(2(1(0(4(1(0(0(3(4(x1)))))))))))))))) 0(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 2(4(5(5(5(4(0(2(5(0(5(4(4(5(5(0(5(0(0(x1))))))))))))))))))) 1(0(3(2(0(2(4(1(3(2(5(0(0(4(5(0(3(4(3(x1))))))))))))))))))) -> 4(2(2(0(0(0(0(4(1(4(0(5(3(5(0(2(4(3(x1)))))))))))))))))) 1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 1(5(3(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1))))))))))))))))))) 3(3(5(4(1(1(3(2(1(4(4(0(1(1(0(4(3(1(0(x1))))))))))))))))))) -> 3(4(0(2(3(1(3(0(4(4(3(4(4(4(3(2(1(0(x1)))))))))))))))))) 2(5(0(4(0(3(4(3(4(0(0(2(4(2(4(1(0(1(2(3(x1)))))))))))))))))))) -> 0(1(2(0(5(4(3(2(0(2(3(3(3(0(1(5(5(5(1(x1))))))))))))))))))) 0(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 2(4(4(3(0(5(5(5(5(2(2(0(2(5(1(1(0(1(0(0(1(x1))))))))))))))))))))) 2(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2(0(0(2(2(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1))))))))))))))))))))) 4(3(1(4(3(3(3(3(4(4(2(5(1(4(5(1(4(3(2(3(3(x1))))))))))))))))))))) -> 2(5(5(1(1(1(4(5(3(2(3(3(0(4(2(3(1(5(4(5(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)) = 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: 0(5(2(4(2(0(0(5(2(x1))))))))) -> 1(5(2(0(2(3(5(2(x1)))))))) 5(0(3(5(1(4(2(1(0(2(x1)))))))))) -> 5(2(0(1(3(1(1(0(2(x1))))))))) 2(4(5(2(4(3(4(3(1(2(3(x1))))))))))) -> 5(3(1(3(3(3(2(2(4(1(x1)))))))))) 3(3(3(2(3(3(1(2(2(4(3(3(x1)))))))))))) -> 1(2(2(1(3(2(3(5(2(5(5(x1))))))))))) 5(5(4(0(3(0(2(3(3(2(3(3(x1)))))))))))) -> 5(2(2(3(0(1(1(3(4(4(5(x1))))))))))) 0(0(3(1(0(2(2(3(2(5(3(0(1(x1))))))))))))) -> 0(1(3(4(3(1(0(0(3(3(4(0(x1)))))))))))) 2(4(2(3(3(5(0(4(5(5(2(5(5(x1))))))))))))) -> 3(5(4(3(3(2(0(1(1(4(3(5(x1)))))))))))) 2(5(3(5(2(4(2(1(3(0(5(0(3(1(3(x1))))))))))))))) -> 5(3(2(3(5(1(5(3(5(3(5(1(5(x1))))))))))))) 3(3(2(0(1(5(4(5(4(2(4(2(3(4(4(3(x1)))))))))))))))) -> 5(2(5(5(5(5(2(3(3(1(2(5(0(4(0(x1))))))))))))))) 3(0(0(2(0(3(5(3(0(2(5(3(5(5(2(3(4(x1))))))))))))))))) -> 5(3(4(3(0(4(2(2(1(0(4(1(0(0(3(4(x1)))))))))))))))) 1(0(3(2(0(2(4(1(3(2(5(0(0(4(5(0(3(4(3(x1))))))))))))))))))) -> 4(2(2(0(0(0(0(4(1(4(0(5(3(5(0(2(4(3(x1)))))))))))))))))) 3(3(5(4(1(1(3(2(1(4(4(0(1(1(0(4(3(1(0(x1))))))))))))))))))) -> 3(4(0(2(3(1(3(0(4(4(3(4(4(4(3(2(1(0(x1)))))))))))))))))) 2(5(0(4(0(3(4(3(4(0(0(2(4(2(4(1(0(1(2(3(x1)))))))))))))))))))) -> 0(1(2(0(5(4(3(2(0(2(3(3(3(0(1(5(5(5(1(x1))))))))))))))))))) 4(3(1(4(3(3(3(3(4(4(2(5(1(4(5(1(4(3(2(3(3(x1))))))))))))))))))))) -> 2(5(5(1(1(1(4(5(3(2(3(3(0(4(2(3(1(5(4(5(x1)))))))))))))))))))) ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0(0(1(2(2(x1))))) -> 2(3(3(2(2(x1))))) 0(4(1(0(4(5(x1)))))) -> 5(2(5(3(4(5(x1)))))) 3(4(2(2(1(5(3(3(2(x1))))))))) -> 3(1(0(3(5(3(3(0(2(x1))))))))) 0(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 3(3(3(2(5(0(4(0(5(0(x1)))))))))) 0(2(1(5(3(5(3(0(5(5(5(0(x1)))))))))))) -> 2(2(5(4(4(2(3(3(0(3(5(4(x1)))))))))))) 3(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 1(5(2(4(0(4(2(0(0(5(0(2(x1)))))))))))) 2(1(3(3(4(3(0(2(0(4(0(3(1(x1))))))))))))) -> 2(4(1(3(0(1(3(3(2(1(3(4(4(x1))))))))))))) 5(2(3(1(2(1(5(4(5(1(1(0(0(x1))))))))))))) -> 5(2(0(5(4(3(4(3(3(0(5(0(0(x1))))))))))))) 2(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1))))))))))))))) -> 5(3(1(3(1(4(4(2(5(4(5(4(4(1(0(x1))))))))))))))) 2(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 0(3(2(5(5(2(4(4(3(5(3(4(3(4(1(x1))))))))))))))) 0(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 0(5(1(2(4(0(2(0(1(2(4(3(3(3(0(5(x1)))))))))))))))) 0(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 0(1(5(2(2(5(0(5(3(3(2(3(4(1(5(2(x1)))))))))))))))) 0(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 2(4(5(5(5(4(0(2(5(0(5(4(4(5(5(0(5(0(0(x1))))))))))))))))))) 1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 1(5(3(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1))))))))))))))))))) 0(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 2(4(4(3(0(5(5(5(5(2(2(0(2(5(1(1(0(1(0(0(1(x1))))))))))))))))))))) 2(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2(0(0(2(2(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(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: 0^1(0(1(2(2(x1))))) -> 2^1(3(3(2(2(x1))))) 0^1(0(1(2(2(x1))))) -> 3^1(3(2(2(x1)))) 0^1(0(1(2(2(x1))))) -> 3^1(2(2(x1))) 0^1(4(1(0(4(5(x1)))))) -> 5^1(2(5(3(4(5(x1)))))) 0^1(4(1(0(4(5(x1)))))) -> 2^1(5(3(4(5(x1))))) 0^1(4(1(0(4(5(x1)))))) -> 5^1(3(4(5(x1)))) 0^1(4(1(0(4(5(x1)))))) -> 3^1(4(5(x1))) 3^1(4(2(2(1(5(3(3(2(x1))))))))) -> 3^1(1(0(3(5(3(3(0(2(x1))))))))) 3^1(4(2(2(1(5(3(3(2(x1))))))))) -> 1^1(0(3(5(3(3(0(2(x1)))))))) 3^1(4(2(2(1(5(3(3(2(x1))))))))) -> 0^1(3(5(3(3(0(2(x1))))))) 3^1(4(2(2(1(5(3(3(2(x1))))))))) -> 3^1(5(3(3(0(2(x1)))))) 3^1(4(2(2(1(5(3(3(2(x1))))))))) -> 5^1(3(3(0(2(x1))))) 3^1(4(2(2(1(5(3(3(2(x1))))))))) -> 3^1(3(0(2(x1)))) 3^1(4(2(2(1(5(3(3(2(x1))))))))) -> 3^1(0(2(x1))) 3^1(4(2(2(1(5(3(3(2(x1))))))))) -> 0^1(2(x1)) 0^1(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 3^1(3(3(2(5(0(4(0(5(0(x1)))))))))) 0^1(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 3^1(3(2(5(0(4(0(5(0(x1))))))))) 0^1(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 3^1(2(5(0(4(0(5(0(x1)))))))) 0^1(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 2^1(5(0(4(0(5(0(x1))))))) 0^1(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 5^1(0(4(0(5(0(x1)))))) 0^1(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 0^1(4(0(5(0(x1))))) 0^1(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 0^1(5(0(x1))) 0^1(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 5^1(0(x1)) 0^1(2(1(5(3(5(3(0(5(5(5(0(x1)))))))))))) -> 2^1(2(5(4(4(2(3(3(0(3(5(4(x1)))))))))))) 0^1(2(1(5(3(5(3(0(5(5(5(0(x1)))))))))))) -> 2^1(5(4(4(2(3(3(0(3(5(4(x1))))))))))) 0^1(2(1(5(3(5(3(0(5(5(5(0(x1)))))))))))) -> 5^1(4(4(2(3(3(0(3(5(4(x1)))))))))) 0^1(2(1(5(3(5(3(0(5(5(5(0(x1)))))))))))) -> 2^1(3(3(0(3(5(4(x1))))))) 0^1(2(1(5(3(5(3(0(5(5(5(0(x1)))))))))))) -> 3^1(3(0(3(5(4(x1)))))) 0^1(2(1(5(3(5(3(0(5(5(5(0(x1)))))))))))) -> 3^1(0(3(5(4(x1))))) 0^1(2(1(5(3(5(3(0(5(5(5(0(x1)))))))))))) -> 0^1(3(5(4(x1)))) 0^1(2(1(5(3(5(3(0(5(5(5(0(x1)))))))))))) -> 3^1(5(4(x1))) 0^1(2(1(5(3(5(3(0(5(5(5(0(x1)))))))))))) -> 5^1(4(x1)) 3^1(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 1^1(5(2(4(0(4(2(0(0(5(0(2(x1)))))))))))) 3^1(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 5^1(2(4(0(4(2(0(0(5(0(2(x1))))))))))) 3^1(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 2^1(4(0(4(2(0(0(5(0(2(x1)))))))))) 3^1(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 0^1(4(2(0(0(5(0(2(x1)))))))) 3^1(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 2^1(0(0(5(0(2(x1)))))) 3^1(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 0^1(0(5(0(2(x1))))) 3^1(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 0^1(5(0(2(x1)))) 3^1(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 5^1(0(2(x1))) 3^1(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 0^1(2(x1)) 3^1(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 2^1(x1) 2^1(1(3(3(4(3(0(2(0(4(0(3(1(x1))))))))))))) -> 2^1(4(1(3(0(1(3(3(2(1(3(4(4(x1))))))))))))) 2^1(1(3(3(4(3(0(2(0(4(0(3(1(x1))))))))))))) -> 1^1(3(0(1(3(3(2(1(3(4(4(x1))))))))))) 2^1(1(3(3(4(3(0(2(0(4(0(3(1(x1))))))))))))) -> 3^1(0(1(3(3(2(1(3(4(4(x1)))))))))) 2^1(1(3(3(4(3(0(2(0(4(0(3(1(x1))))))))))))) -> 0^1(1(3(3(2(1(3(4(4(x1))))))))) 2^1(1(3(3(4(3(0(2(0(4(0(3(1(x1))))))))))))) -> 1^1(3(3(2(1(3(4(4(x1)))))))) 2^1(1(3(3(4(3(0(2(0(4(0(3(1(x1))))))))))))) -> 3^1(3(2(1(3(4(4(x1))))))) 2^1(1(3(3(4(3(0(2(0(4(0(3(1(x1))))))))))))) -> 3^1(2(1(3(4(4(x1)))))) 2^1(1(3(3(4(3(0(2(0(4(0(3(1(x1))))))))))))) -> 2^1(1(3(4(4(x1))))) 2^1(1(3(3(4(3(0(2(0(4(0(3(1(x1))))))))))))) -> 1^1(3(4(4(x1)))) 2^1(1(3(3(4(3(0(2(0(4(0(3(1(x1))))))))))))) -> 3^1(4(4(x1))) 5^1(2(3(1(2(1(5(4(5(1(1(0(0(x1))))))))))))) -> 5^1(2(0(5(4(3(4(3(3(0(5(0(0(x1))))))))))))) 5^1(2(3(1(2(1(5(4(5(1(1(0(0(x1))))))))))))) -> 2^1(0(5(4(3(4(3(3(0(5(0(0(x1)))))))))))) 5^1(2(3(1(2(1(5(4(5(1(1(0(0(x1))))))))))))) -> 0^1(5(4(3(4(3(3(0(5(0(0(x1))))))))))) 5^1(2(3(1(2(1(5(4(5(1(1(0(0(x1))))))))))))) -> 5^1(4(3(4(3(3(0(5(0(0(x1)))))))))) 5^1(2(3(1(2(1(5(4(5(1(1(0(0(x1))))))))))))) -> 3^1(4(3(3(0(5(0(0(x1)))))))) 5^1(2(3(1(2(1(5(4(5(1(1(0(0(x1))))))))))))) -> 3^1(3(0(5(0(0(x1)))))) 5^1(2(3(1(2(1(5(4(5(1(1(0(0(x1))))))))))))) -> 3^1(0(5(0(0(x1))))) 5^1(2(3(1(2(1(5(4(5(1(1(0(0(x1))))))))))))) -> 0^1(5(0(0(x1)))) 5^1(2(3(1(2(1(5(4(5(1(1(0(0(x1))))))))))))) -> 5^1(0(0(x1))) 2^1(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1))))))))))))))) -> 5^1(3(1(3(1(4(4(2(5(4(5(4(4(1(0(x1))))))))))))))) 2^1(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1))))))))))))))) -> 3^1(1(3(1(4(4(2(5(4(5(4(4(1(0(x1)))))))))))))) 2^1(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1))))))))))))))) -> 1^1(3(1(4(4(2(5(4(5(4(4(1(0(x1))))))))))))) 2^1(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1))))))))))))))) -> 3^1(1(4(4(2(5(4(5(4(4(1(0(x1)))))))))))) 2^1(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1))))))))))))))) -> 1^1(4(4(2(5(4(5(4(4(1(0(x1))))))))))) 2^1(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1))))))))))))))) -> 2^1(5(4(5(4(4(1(0(x1)))))))) 2^1(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1))))))))))))))) -> 5^1(4(5(4(4(1(0(x1))))))) 2^1(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1))))))))))))))) -> 5^1(4(4(1(0(x1))))) 2^1(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1))))))))))))))) -> 1^1(0(x1)) 2^1(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 0^1(3(2(5(5(2(4(4(3(5(3(4(3(4(1(x1))))))))))))))) 2^1(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 3^1(2(5(5(2(4(4(3(5(3(4(3(4(1(x1)))))))))))))) 2^1(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 2^1(5(5(2(4(4(3(5(3(4(3(4(1(x1))))))))))))) 2^1(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 5^1(5(2(4(4(3(5(3(4(3(4(1(x1)))))))))))) 2^1(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 5^1(2(4(4(3(5(3(4(3(4(1(x1))))))))))) 2^1(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 2^1(4(4(3(5(3(4(3(4(1(x1)))))))))) 2^1(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 3^1(5(3(4(3(4(1(x1))))))) 2^1(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 5^1(3(4(3(4(1(x1)))))) 2^1(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 3^1(4(3(4(1(x1))))) 2^1(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 3^1(4(1(x1))) 2^1(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 1^1(x1) 0^1(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 0^1(5(1(2(4(0(2(0(1(2(4(3(3(3(0(5(x1)))))))))))))))) 0^1(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 5^1(1(2(4(0(2(0(1(2(4(3(3(3(0(5(x1))))))))))))))) 0^1(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 1^1(2(4(0(2(0(1(2(4(3(3(3(0(5(x1)))))))))))))) 0^1(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 2^1(4(0(2(0(1(2(4(3(3(3(0(5(x1))))))))))))) 0^1(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 0^1(2(0(1(2(4(3(3(3(0(5(x1))))))))))) 0^1(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 2^1(0(1(2(4(3(3(3(0(5(x1)))))))))) 0^1(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 0^1(1(2(4(3(3(3(0(5(x1))))))))) 0^1(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 1^1(2(4(3(3(3(0(5(x1)))))))) 0^1(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 2^1(4(3(3(3(0(5(x1))))))) 0^1(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 3^1(3(3(0(5(x1))))) 0^1(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 3^1(3(0(5(x1)))) 0^1(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 3^1(0(5(x1))) 0^1(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 0^1(5(x1)) 0^1(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 5^1(x1) 0^1(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 0^1(1(5(2(2(5(0(5(3(3(2(3(4(1(5(2(x1)))))))))))))))) 0^1(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 1^1(5(2(2(5(0(5(3(3(2(3(4(1(5(2(x1))))))))))))))) 0^1(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 5^1(2(2(5(0(5(3(3(2(3(4(1(5(2(x1)))))))))))))) 0^1(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 2^1(2(5(0(5(3(3(2(3(4(1(5(2(x1))))))))))))) 0^1(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 2^1(5(0(5(3(3(2(3(4(1(5(2(x1)))))))))))) 0^1(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 5^1(0(5(3(3(2(3(4(1(5(2(x1))))))))))) 0^1(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 0^1(5(3(3(2(3(4(1(5(2(x1)))))))))) 0^1(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 5^1(3(3(2(3(4(1(5(2(x1))))))))) 0^1(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 3^1(3(2(3(4(1(5(2(x1)))))))) 0^1(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 3^1(2(3(4(1(5(2(x1))))))) 0^1(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 2^1(3(4(1(5(2(x1)))))) 0^1(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 3^1(4(1(5(2(x1))))) 0^1(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 1^1(5(2(x1))) 0^1(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 5^1(2(x1)) 0^1(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 2^1(x1) 0^1(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 2^1(4(5(5(5(4(0(2(5(0(5(4(4(5(5(0(5(0(0(x1))))))))))))))))))) 0^1(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 5^1(5(5(4(0(2(5(0(5(4(4(5(5(0(5(0(0(x1))))))))))))))))) 0^1(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 5^1(5(4(0(2(5(0(5(4(4(5(5(0(5(0(0(x1)))))))))))))))) 0^1(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 5^1(4(0(2(5(0(5(4(4(5(5(0(5(0(0(x1))))))))))))))) 0^1(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 0^1(2(5(0(5(4(4(5(5(0(5(0(0(x1))))))))))))) 0^1(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 2^1(5(0(5(4(4(5(5(0(5(0(0(x1)))))))))))) 0^1(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 5^1(0(5(4(4(5(5(0(5(0(0(x1))))))))))) 0^1(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 0^1(5(4(4(5(5(0(5(0(0(x1)))))))))) 0^1(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 5^1(4(4(5(5(0(5(0(0(x1))))))))) 0^1(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 5^1(5(0(5(0(0(x1)))))) 0^1(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 5^1(0(5(0(0(x1))))) 0^1(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 0^1(5(0(0(x1)))) 0^1(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 5^1(0(0(x1))) 0^1(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 0^1(0(x1)) 1^1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 1^1(5(3(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1))))))))))))))))))) 1^1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 5^1(3(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1)))))))))))))))))) 1^1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 3^1(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1))))))))))))))))) 1^1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 2^1(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1)))))))))))))))) 1^1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 5^1(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1))))))))))))))) 1^1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 0^1(5(2(3(0(0(3(1(5(3(4(4(3(0(x1)))))))))))))) 1^1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 5^1(2(3(0(0(3(1(5(3(4(4(3(0(x1))))))))))))) 1^1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 2^1(3(0(0(3(1(5(3(4(4(3(0(x1)))))))))))) 1^1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 3^1(0(0(3(1(5(3(4(4(3(0(x1))))))))))) 1^1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 0^1(0(3(1(5(3(4(4(3(0(x1)))))))))) 1^1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 0^1(3(1(5(3(4(4(3(0(x1))))))))) 1^1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 3^1(1(5(3(4(4(3(0(x1)))))))) 1^1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 1^1(5(3(4(4(3(0(x1))))))) 1^1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 5^1(3(4(4(3(0(x1)))))) 1^1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 3^1(4(4(3(0(x1))))) 1^1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 3^1(0(x1)) 0^1(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 2^1(4(4(3(0(5(5(5(5(2(2(0(2(5(1(1(0(1(0(0(1(x1))))))))))))))))))))) 0^1(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 3^1(0(5(5(5(5(2(2(0(2(5(1(1(0(1(0(0(1(x1)))))))))))))))))) 0^1(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 0^1(5(5(5(5(2(2(0(2(5(1(1(0(1(0(0(1(x1))))))))))))))))) 0^1(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 5^1(5(5(5(2(2(0(2(5(1(1(0(1(0(0(1(x1)))))))))))))))) 0^1(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 5^1(5(5(2(2(0(2(5(1(1(0(1(0(0(1(x1))))))))))))))) 0^1(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 5^1(5(2(2(0(2(5(1(1(0(1(0(0(1(x1)))))))))))))) 0^1(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 5^1(2(2(0(2(5(1(1(0(1(0(0(1(x1))))))))))))) 0^1(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 2^1(2(0(2(5(1(1(0(1(0(0(1(x1)))))))))))) 0^1(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 2^1(0(2(5(1(1(0(1(0(0(1(x1))))))))))) 0^1(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 0^1(2(5(1(1(0(1(0(0(1(x1)))))))))) 0^1(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 2^1(5(1(1(0(1(0(0(1(x1))))))))) 0^1(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 5^1(1(1(0(1(0(0(1(x1)))))))) 0^1(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 1^1(1(0(1(0(0(1(x1))))))) 0^1(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 1^1(0(1(0(0(1(x1)))))) 0^1(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 0^1(1(0(0(1(x1))))) 0^1(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 1^1(0(0(1(x1)))) 0^1(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 0^1(0(1(x1))) 0^1(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 0^1(1(x1)) 0^1(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 1^1(x1) 2^1(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2^1(0(0(2(2(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1))))))))))))))))))))) 2^1(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 0^1(0(2(2(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1)))))))))))))))))))) 2^1(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 0^1(2(2(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1))))))))))))))))))) 2^1(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2^1(2(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1)))))))))))))))))) 2^1(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2^1(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1))))))))))))))))) 2^1(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 1^1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1)))))))))))))))) 2^1(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2^1(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1))))))))))))))) 2^1(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 1^1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1)))))))))))))) 2^1(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2^1(2(2(0(5(3(3(2(5(2(2(5(2(x1))))))))))))) 2^1(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2^1(2(0(5(3(3(2(5(2(2(5(2(x1)))))))))))) 2^1(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2^1(0(5(3(3(2(5(2(2(5(2(x1))))))))))) 2^1(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 0^1(5(3(3(2(5(2(2(5(2(x1)))))))))) 2^1(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 5^1(3(3(2(5(2(2(5(2(x1))))))))) 2^1(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 3^1(3(2(5(2(2(5(2(x1)))))))) 2^1(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 3^1(2(5(2(2(5(2(x1))))))) 2^1(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2^1(5(2(2(5(2(x1)))))) 2^1(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 5^1(2(2(5(2(x1))))) 2^1(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2^1(2(5(2(x1)))) 2^1(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2^1(5(2(x1))) 2^1(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 5^1(2(x1)) 2^1(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2^1(x1) The TRS R consists of the following rules: 0(0(1(2(2(x1))))) -> 2(3(3(2(2(x1))))) 0(4(1(0(4(5(x1)))))) -> 5(2(5(3(4(5(x1)))))) 3(4(2(2(1(5(3(3(2(x1))))))))) -> 3(1(0(3(5(3(3(0(2(x1))))))))) 0(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 3(3(3(2(5(0(4(0(5(0(x1)))))))))) 0(2(1(5(3(5(3(0(5(5(5(0(x1)))))))))))) -> 2(2(5(4(4(2(3(3(0(3(5(4(x1)))))))))))) 3(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 1(5(2(4(0(4(2(0(0(5(0(2(x1)))))))))))) 2(1(3(3(4(3(0(2(0(4(0(3(1(x1))))))))))))) -> 2(4(1(3(0(1(3(3(2(1(3(4(4(x1))))))))))))) 5(2(3(1(2(1(5(4(5(1(1(0(0(x1))))))))))))) -> 5(2(0(5(4(3(4(3(3(0(5(0(0(x1))))))))))))) 2(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1))))))))))))))) -> 5(3(1(3(1(4(4(2(5(4(5(4(4(1(0(x1))))))))))))))) 2(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 0(3(2(5(5(2(4(4(3(5(3(4(3(4(1(x1))))))))))))))) 0(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 0(5(1(2(4(0(2(0(1(2(4(3(3(3(0(5(x1)))))))))))))))) 0(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 0(1(5(2(2(5(0(5(3(3(2(3(4(1(5(2(x1)))))))))))))))) 0(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 2(4(5(5(5(4(0(2(5(0(5(4(4(5(5(0(5(0(0(x1))))))))))))))))))) 1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 1(5(3(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1))))))))))))))))))) 0(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 2(4(4(3(0(5(5(5(5(2(2(0(2(5(1(1(0(1(0(0(1(x1))))))))))))))))))))) 2(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2(0(0(2(2(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(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 2 SCCs with 172 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 3^1(4(3(4(1(x1))))) 3^1(4(2(2(1(5(3(3(2(x1))))))))) -> 0^1(3(5(3(3(0(2(x1))))))) 0^1(0(1(2(2(x1))))) -> 2^1(3(3(2(2(x1))))) 2^1(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 3^1(4(1(x1))) 3^1(4(2(2(1(5(3(3(2(x1))))))))) -> 0^1(2(x1)) 0^1(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 2^1(5(0(4(0(5(0(x1))))))) 0^1(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 0^1(5(0(x1))) The TRS R consists of the following rules: 0(0(1(2(2(x1))))) -> 2(3(3(2(2(x1))))) 0(4(1(0(4(5(x1)))))) -> 5(2(5(3(4(5(x1)))))) 3(4(2(2(1(5(3(3(2(x1))))))))) -> 3(1(0(3(5(3(3(0(2(x1))))))))) 0(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 3(3(3(2(5(0(4(0(5(0(x1)))))))))) 0(2(1(5(3(5(3(0(5(5(5(0(x1)))))))))))) -> 2(2(5(4(4(2(3(3(0(3(5(4(x1)))))))))))) 3(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 1(5(2(4(0(4(2(0(0(5(0(2(x1)))))))))))) 2(1(3(3(4(3(0(2(0(4(0(3(1(x1))))))))))))) -> 2(4(1(3(0(1(3(3(2(1(3(4(4(x1))))))))))))) 5(2(3(1(2(1(5(4(5(1(1(0(0(x1))))))))))))) -> 5(2(0(5(4(3(4(3(3(0(5(0(0(x1))))))))))))) 2(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1))))))))))))))) -> 5(3(1(3(1(4(4(2(5(4(5(4(4(1(0(x1))))))))))))))) 2(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 0(3(2(5(5(2(4(4(3(5(3(4(3(4(1(x1))))))))))))))) 0(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 0(5(1(2(4(0(2(0(1(2(4(3(3(3(0(5(x1)))))))))))))))) 0(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 0(1(5(2(2(5(0(5(3(3(2(3(4(1(5(2(x1)))))))))))))))) 0(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 2(4(5(5(5(4(0(2(5(0(5(4(4(5(5(0(5(0(0(x1))))))))))))))))))) 1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 1(5(3(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1))))))))))))))))))) 0(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 2(4(4(3(0(5(5(5(5(2(2(0(2(5(1(1(0(1(0(0(1(x1))))))))))))))))))))) 2(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2(0(0(2(2(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (8) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(0(1(2(2(x1))))) -> 2^1(3(3(2(2(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)) = 1 POL(1(x_1)) = 0 POL(2(x_1)) = 0 POL(2^1(x_1)) = x_1 POL(3(x_1)) = 0 POL(3^1(x_1)) = 1 POL(4(x_1)) = 0 POL(5(x_1)) = 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 1(5(3(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1))))))))))))))))))) 3(4(2(2(1(5(3(3(2(x1))))))))) -> 3(1(0(3(5(3(3(0(2(x1))))))))) 3(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 1(5(2(4(0(4(2(0(0(5(0(2(x1)))))))))))) 5(2(3(1(2(1(5(4(5(1(1(0(0(x1))))))))))))) -> 5(2(0(5(4(3(4(3(3(0(5(0(0(x1))))))))))))) ---------------------------------------- (9) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 3^1(4(3(4(1(x1))))) 3^1(4(2(2(1(5(3(3(2(x1))))))))) -> 0^1(3(5(3(3(0(2(x1))))))) 2^1(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 3^1(4(1(x1))) 3^1(4(2(2(1(5(3(3(2(x1))))))))) -> 0^1(2(x1)) 0^1(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 2^1(5(0(4(0(5(0(x1))))))) 0^1(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 0^1(5(0(x1))) The TRS R consists of the following rules: 0(0(1(2(2(x1))))) -> 2(3(3(2(2(x1))))) 0(4(1(0(4(5(x1)))))) -> 5(2(5(3(4(5(x1)))))) 3(4(2(2(1(5(3(3(2(x1))))))))) -> 3(1(0(3(5(3(3(0(2(x1))))))))) 0(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 3(3(3(2(5(0(4(0(5(0(x1)))))))))) 0(2(1(5(3(5(3(0(5(5(5(0(x1)))))))))))) -> 2(2(5(4(4(2(3(3(0(3(5(4(x1)))))))))))) 3(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 1(5(2(4(0(4(2(0(0(5(0(2(x1)))))))))))) 2(1(3(3(4(3(0(2(0(4(0(3(1(x1))))))))))))) -> 2(4(1(3(0(1(3(3(2(1(3(4(4(x1))))))))))))) 5(2(3(1(2(1(5(4(5(1(1(0(0(x1))))))))))))) -> 5(2(0(5(4(3(4(3(3(0(5(0(0(x1))))))))))))) 2(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1))))))))))))))) -> 5(3(1(3(1(4(4(2(5(4(5(4(4(1(0(x1))))))))))))))) 2(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 0(3(2(5(5(2(4(4(3(5(3(4(3(4(1(x1))))))))))))))) 0(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 0(5(1(2(4(0(2(0(1(2(4(3(3(3(0(5(x1)))))))))))))))) 0(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 0(1(5(2(2(5(0(5(3(3(2(3(4(1(5(2(x1)))))))))))))))) 0(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 2(4(5(5(5(4(0(2(5(0(5(4(4(5(5(0(5(0(0(x1))))))))))))))))))) 1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 1(5(3(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1))))))))))))))))))) 0(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 2(4(4(3(0(5(5(5(5(2(2(0(2(5(1(1(0(1(0(0(1(x1))))))))))))))))))))) 2(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2(0(0(2(2(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (10) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 3^1(4(2(2(1(5(3(3(2(x1))))))))) -> 0^1(3(5(3(3(0(2(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)) = x_1 POL(1(x_1)) = 0 POL(2(x_1)) = 1 POL(2^1(x_1)) = 1 POL(3(x_1)) = 0 POL(3^1(x_1)) = 1 POL(4(x_1)) = 0 POL(5(x_1)) = 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 1(5(3(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1))))))))))))))))))) 3(4(2(2(1(5(3(3(2(x1))))))))) -> 3(1(0(3(5(3(3(0(2(x1))))))))) 2(1(3(3(4(3(0(2(0(4(0(3(1(x1))))))))))))) -> 2(4(1(3(0(1(3(3(2(1(3(4(4(x1))))))))))))) 2(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1))))))))))))))) -> 5(3(1(3(1(4(4(2(5(4(5(4(4(1(0(x1))))))))))))))) 2(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 0(3(2(5(5(2(4(4(3(5(3(4(3(4(1(x1))))))))))))))) 2(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2(0(0(2(2(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1))))))))))))))))))))) 3(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 1(5(2(4(0(4(2(0(0(5(0(2(x1)))))))))))) 5(2(3(1(2(1(5(4(5(1(1(0(0(x1))))))))))))) -> 5(2(0(5(4(3(4(3(3(0(5(0(0(x1))))))))))))) ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 3^1(4(3(4(1(x1))))) 2^1(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 3^1(4(1(x1))) 3^1(4(2(2(1(5(3(3(2(x1))))))))) -> 0^1(2(x1)) 0^1(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 2^1(5(0(4(0(5(0(x1))))))) 0^1(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 0^1(5(0(x1))) The TRS R consists of the following rules: 0(0(1(2(2(x1))))) -> 2(3(3(2(2(x1))))) 0(4(1(0(4(5(x1)))))) -> 5(2(5(3(4(5(x1)))))) 3(4(2(2(1(5(3(3(2(x1))))))))) -> 3(1(0(3(5(3(3(0(2(x1))))))))) 0(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 3(3(3(2(5(0(4(0(5(0(x1)))))))))) 0(2(1(5(3(5(3(0(5(5(5(0(x1)))))))))))) -> 2(2(5(4(4(2(3(3(0(3(5(4(x1)))))))))))) 3(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 1(5(2(4(0(4(2(0(0(5(0(2(x1)))))))))))) 2(1(3(3(4(3(0(2(0(4(0(3(1(x1))))))))))))) -> 2(4(1(3(0(1(3(3(2(1(3(4(4(x1))))))))))))) 5(2(3(1(2(1(5(4(5(1(1(0(0(x1))))))))))))) -> 5(2(0(5(4(3(4(3(3(0(5(0(0(x1))))))))))))) 2(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1))))))))))))))) -> 5(3(1(3(1(4(4(2(5(4(5(4(4(1(0(x1))))))))))))))) 2(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 0(3(2(5(5(2(4(4(3(5(3(4(3(4(1(x1))))))))))))))) 0(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 0(5(1(2(4(0(2(0(1(2(4(3(3(3(0(5(x1)))))))))))))))) 0(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 0(1(5(2(2(5(0(5(3(3(2(3(4(1(5(2(x1)))))))))))))))) 0(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 2(4(5(5(5(4(0(2(5(0(5(4(4(5(5(0(5(0(0(x1))))))))))))))))))) 1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 1(5(3(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1))))))))))))))))))) 0(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 2(4(4(3(0(5(5(5(5(2(2(0(2(5(1(1(0(1(0(0(1(x1))))))))))))))))))))) 2(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2(0(0(2(2(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (12) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 3^1(4(2(2(1(5(3(3(2(x1))))))))) -> 0^1(2(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(2(x_1)) = 1 POL(2^1(x_1)) = 0 POL(3(x_1)) = 0 POL(3^1(x_1)) = x_1 POL(4(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: 1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 1(5(3(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1))))))))))))))))))) 3(4(2(2(1(5(3(3(2(x1))))))))) -> 3(1(0(3(5(3(3(0(2(x1))))))))) 3(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 1(5(2(4(0(4(2(0(0(5(0(2(x1)))))))))))) ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 3^1(4(3(4(1(x1))))) 2^1(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 3^1(4(1(x1))) 0^1(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 2^1(5(0(4(0(5(0(x1))))))) 0^1(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 0^1(5(0(x1))) The TRS R consists of the following rules: 0(0(1(2(2(x1))))) -> 2(3(3(2(2(x1))))) 0(4(1(0(4(5(x1)))))) -> 5(2(5(3(4(5(x1)))))) 3(4(2(2(1(5(3(3(2(x1))))))))) -> 3(1(0(3(5(3(3(0(2(x1))))))))) 0(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 3(3(3(2(5(0(4(0(5(0(x1)))))))))) 0(2(1(5(3(5(3(0(5(5(5(0(x1)))))))))))) -> 2(2(5(4(4(2(3(3(0(3(5(4(x1)))))))))))) 3(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 1(5(2(4(0(4(2(0(0(5(0(2(x1)))))))))))) 2(1(3(3(4(3(0(2(0(4(0(3(1(x1))))))))))))) -> 2(4(1(3(0(1(3(3(2(1(3(4(4(x1))))))))))))) 5(2(3(1(2(1(5(4(5(1(1(0(0(x1))))))))))))) -> 5(2(0(5(4(3(4(3(3(0(5(0(0(x1))))))))))))) 2(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1))))))))))))))) -> 5(3(1(3(1(4(4(2(5(4(5(4(4(1(0(x1))))))))))))))) 2(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 0(3(2(5(5(2(4(4(3(5(3(4(3(4(1(x1))))))))))))))) 0(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 0(5(1(2(4(0(2(0(1(2(4(3(3(3(0(5(x1)))))))))))))))) 0(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 0(1(5(2(2(5(0(5(3(3(2(3(4(1(5(2(x1)))))))))))))))) 0(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 2(4(5(5(5(4(0(2(5(0(5(4(4(5(5(0(5(0(0(x1))))))))))))))))))) 1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 1(5(3(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1))))))))))))))))))) 0(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 2(4(4(3(0(5(5(5(5(2(2(0(2(5(1(1(0(1(0(0(1(x1))))))))))))))))))))) 2(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2(0(0(2(2(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 0^1(5(0(x1))) The TRS R consists of the following rules: 0(0(1(2(2(x1))))) -> 2(3(3(2(2(x1))))) 0(4(1(0(4(5(x1)))))) -> 5(2(5(3(4(5(x1)))))) 3(4(2(2(1(5(3(3(2(x1))))))))) -> 3(1(0(3(5(3(3(0(2(x1))))))))) 0(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 3(3(3(2(5(0(4(0(5(0(x1)))))))))) 0(2(1(5(3(5(3(0(5(5(5(0(x1)))))))))))) -> 2(2(5(4(4(2(3(3(0(3(5(4(x1)))))))))))) 3(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 1(5(2(4(0(4(2(0(0(5(0(2(x1)))))))))))) 2(1(3(3(4(3(0(2(0(4(0(3(1(x1))))))))))))) -> 2(4(1(3(0(1(3(3(2(1(3(4(4(x1))))))))))))) 5(2(3(1(2(1(5(4(5(1(1(0(0(x1))))))))))))) -> 5(2(0(5(4(3(4(3(3(0(5(0(0(x1))))))))))))) 2(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1))))))))))))))) -> 5(3(1(3(1(4(4(2(5(4(5(4(4(1(0(x1))))))))))))))) 2(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 0(3(2(5(5(2(4(4(3(5(3(4(3(4(1(x1))))))))))))))) 0(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 0(5(1(2(4(0(2(0(1(2(4(3(3(3(0(5(x1)))))))))))))))) 0(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 0(1(5(2(2(5(0(5(3(3(2(3(4(1(5(2(x1)))))))))))))))) 0(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 2(4(5(5(5(4(0(2(5(0(5(4(4(5(5(0(5(0(0(x1))))))))))))))))))) 1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 1(5(3(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1))))))))))))))))))) 0(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 2(4(4(3(0(5(5(5(5(2(2(0(2(5(1(1(0(1(0(0(1(x1))))))))))))))))))))) 2(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2(0(0(2(2(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 0^1(5(0(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)) = 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 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0(0(1(2(2(x1))))) -> 2(3(3(2(2(x1))))) 0(4(1(0(4(5(x1)))))) -> 5(2(5(3(4(5(x1)))))) 0(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 3(3(3(2(5(0(4(0(5(0(x1)))))))))) 0(2(1(5(3(5(3(0(5(5(5(0(x1)))))))))))) -> 2(2(5(4(4(2(3(3(0(3(5(4(x1)))))))))))) 0(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 0(1(5(2(2(5(0(5(3(3(2(3(4(1(5(2(x1)))))))))))))))) 0(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 0(5(1(2(4(0(2(0(1(2(4(3(3(3(0(5(x1)))))))))))))))) 0(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 2(4(5(5(5(4(0(2(5(0(5(4(4(5(5(0(5(0(0(x1))))))))))))))))))) 0(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 2(4(4(3(0(5(5(5(5(2(2(0(2(5(1(1(0(1(0(0(1(x1))))))))))))))))))))) 5(2(3(1(2(1(5(4(5(1(1(0(0(x1))))))))))))) -> 5(2(0(5(4(3(4(3(3(0(5(0(0(x1))))))))))))) 2(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1))))))))))))))) -> 5(3(1(3(1(4(4(2(5(4(5(4(4(1(0(x1))))))))))))))) 2(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 0(3(2(5(5(2(4(4(3(5(3(4(3(4(1(x1))))))))))))))) 2(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2(0(0(2(2(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1))))))))))))))))))))) 3(4(2(2(1(5(3(3(2(x1))))))))) -> 3(1(0(3(5(3(3(0(2(x1))))))))) 3(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 1(5(2(4(0(4(2(0(0(5(0(2(x1)))))))))))) 1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 1(5(3(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1))))))))))))))))))) 2(1(3(3(4(3(0(2(0(4(0(3(1(x1))))))))))))) -> 2(4(1(3(0(1(3(3(2(1(3(4(4(x1))))))))))))) ---------------------------------------- (17) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: 0(0(1(2(2(x1))))) -> 2(3(3(2(2(x1))))) 0(4(1(0(4(5(x1)))))) -> 5(2(5(3(4(5(x1)))))) 3(4(2(2(1(5(3(3(2(x1))))))))) -> 3(1(0(3(5(3(3(0(2(x1))))))))) 0(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 3(3(3(2(5(0(4(0(5(0(x1)))))))))) 0(2(1(5(3(5(3(0(5(5(5(0(x1)))))))))))) -> 2(2(5(4(4(2(3(3(0(3(5(4(x1)))))))))))) 3(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 1(5(2(4(0(4(2(0(0(5(0(2(x1)))))))))))) 2(1(3(3(4(3(0(2(0(4(0(3(1(x1))))))))))))) -> 2(4(1(3(0(1(3(3(2(1(3(4(4(x1))))))))))))) 5(2(3(1(2(1(5(4(5(1(1(0(0(x1))))))))))))) -> 5(2(0(5(4(3(4(3(3(0(5(0(0(x1))))))))))))) 2(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1))))))))))))))) -> 5(3(1(3(1(4(4(2(5(4(5(4(4(1(0(x1))))))))))))))) 2(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 0(3(2(5(5(2(4(4(3(5(3(4(3(4(1(x1))))))))))))))) 0(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 0(5(1(2(4(0(2(0(1(2(4(3(3(3(0(5(x1)))))))))))))))) 0(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 0(1(5(2(2(5(0(5(3(3(2(3(4(1(5(2(x1)))))))))))))))) 0(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 2(4(5(5(5(4(0(2(5(0(5(4(4(5(5(0(5(0(0(x1))))))))))))))))))) 1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 1(5(3(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1))))))))))))))))))) 0(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 2(4(4(3(0(5(5(5(5(2(2(0(2(5(1(1(0(1(0(0(1(x1))))))))))))))))))))) 2(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2(0(0(2(2(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (18) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (19) YES ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2^1(x1) The TRS R consists of the following rules: 0(0(1(2(2(x1))))) -> 2(3(3(2(2(x1))))) 0(4(1(0(4(5(x1)))))) -> 5(2(5(3(4(5(x1)))))) 3(4(2(2(1(5(3(3(2(x1))))))))) -> 3(1(0(3(5(3(3(0(2(x1))))))))) 0(5(5(3(1(3(3(0(2(0(x1)))))))))) -> 3(3(3(2(5(0(4(0(5(0(x1)))))))))) 0(2(1(5(3(5(3(0(5(5(5(0(x1)))))))))))) -> 2(2(5(4(4(2(3(3(0(3(5(4(x1)))))))))))) 3(2(0(0(4(0(0(0(2(5(1(0(x1)))))))))))) -> 1(5(2(4(0(4(2(0(0(5(0(2(x1)))))))))))) 2(1(3(3(4(3(0(2(0(4(0(3(1(x1))))))))))))) -> 2(4(1(3(0(1(3(3(2(1(3(4(4(x1))))))))))))) 5(2(3(1(2(1(5(4(5(1(1(0(0(x1))))))))))))) -> 5(2(0(5(4(3(4(3(3(0(5(0(0(x1))))))))))))) 2(5(0(2(2(0(3(3(1(4(2(5(0(0(0(x1))))))))))))))) -> 5(3(1(3(1(4(4(2(5(4(5(4(4(1(0(x1))))))))))))))) 2(5(1(2(0(0(5(1(2(2(1(0(2(1(5(x1))))))))))))))) -> 0(3(2(5(5(2(4(4(3(5(3(4(3(4(1(x1))))))))))))))) 0(3(5(5(0(3(2(4(1(1(4(5(1(5(5(3(x1)))))))))))))))) -> 0(5(1(2(4(0(2(0(1(2(4(3(3(3(0(5(x1)))))))))))))))) 0(4(4(1(2(2(1(3(5(3(1(3(4(1(1(0(x1)))))))))))))))) -> 0(1(5(2(2(5(0(5(3(3(2(3(4(1(5(2(x1)))))))))))))))) 0(1(1(3(5(1(1(1(3(0(0(4(1(2(2(3(1(5(0(x1))))))))))))))))))) -> 2(4(5(5(5(4(0(2(5(0(5(4(4(5(5(0(5(0(0(x1))))))))))))))))))) 1(2(1(3(4(2(4(4(1(2(5(2(3(3(3(5(1(0(0(x1))))))))))))))))))) -> 1(5(3(2(5(0(5(2(3(0(0(3(1(5(3(4(4(3(0(x1))))))))))))))))))) 0(3(4(2(2(0(3(4(1(1(5(0(3(5(2(1(3(3(4(1(0(x1))))))))))))))))))))) -> 2(4(4(3(0(5(5(5(5(2(2(0(2(5(1(1(0(1(0(0(1(x1))))))))))))))))))))) 2(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2(0(0(2(2(1(2(1(2(2(2(0(5(3(3(2(5(2(2(5(2(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) 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. ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2^1(x1) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) 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: *2^1(4(5(2(1(3(5(5(1(1(1(0(0(4(5(0(1(2(0(2(5(x1))))))))))))))))))))) -> 2^1(x1) The graph contains the following edges 1 > 1 ---------------------------------------- (24) YES