62.66/16.83 YES 66.60/17.78 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 66.60/17.78 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 66.60/17.78 66.60/17.78 66.60/17.78 Termination w.r.t. Q of the given QTRS could be proven: 66.60/17.78 66.60/17.78 (0) QTRS 66.60/17.78 (1) QTRSRRRProof [EQUIVALENT, 297 ms] 66.60/17.78 (2) QTRS 66.60/17.78 (3) DependencyPairsProof [EQUIVALENT, 456 ms] 66.60/17.78 (4) QDP 66.60/17.78 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 66.60/17.78 (6) QDP 66.60/17.78 (7) QDPOrderProof [EQUIVALENT, 566 ms] 66.60/17.78 (8) QDP 66.60/17.78 (9) DependencyGraphProof [EQUIVALENT, 0 ms] 66.60/17.78 (10) QDP 66.60/17.78 (11) QDPOrderProof [EQUIVALENT, 0 ms] 66.60/17.78 (12) QDP 66.60/17.78 (13) PisEmptyProof [EQUIVALENT, 0 ms] 66.60/17.78 (14) YES 66.60/17.78 66.60/17.78 66.60/17.78 ---------------------------------------- 66.60/17.78 66.60/17.78 (0) 66.60/17.78 Obligation: 66.60/17.78 Q restricted rewrite system: 66.60/17.78 The TRS R consists of the following rules: 66.60/17.78 66.60/17.78 0(1(2(1(2(2(x1)))))) -> 0(3(4(3(5(2(x1)))))) 66.60/17.78 4(0(1(0(3(0(x1)))))) -> 2(5(0(5(5(0(x1)))))) 66.60/17.78 2(1(3(0(3(3(5(x1))))))) -> 2(5(2(2(5(5(2(x1))))))) 66.60/17.78 3(3(0(0(0(5(2(x1))))))) -> 5(3(3(1(2(5(2(x1))))))) 66.60/17.78 3(3(3(0(1(1(0(x1))))))) -> 5(3(3(0(2(1(0(x1))))))) 66.60/17.78 4(0(2(3(1(4(1(x1))))))) -> 1(4(3(2(5(4(x1)))))) 66.60/17.78 4(1(5(0(0(5(2(1(3(x1))))))))) -> 4(5(0(3(5(2(2(2(3(x1))))))))) 66.60/17.78 5(2(5(1(5(0(3(0(4(2(x1)))))))))) -> 3(3(2(5(1(3(3(5(0(2(x1)))))))))) 66.60/17.78 2(1(4(2(2(0(3(0(1(0(5(x1))))))))))) -> 2(2(5(4(2(4(1(4(3(1(x1)))))))))) 66.60/17.78 3(3(0(1(3(4(2(1(1(2(1(x1))))))))))) -> 5(2(4(4(3(5(3(5(5(1(1(x1))))))))))) 66.60/17.78 5(3(2(3(5(0(4(0(2(3(4(5(x1)))))))))))) -> 3(5(2(1(1(5(2(1(0(5(4(x1))))))))))) 66.60/17.78 5(2(3(5(5(0(0(0(3(4(0(2(5(x1))))))))))))) -> 5(5(5(2(4(3(5(1(1(2(5(0(x1)))))))))))) 66.60/17.78 5(5(3(4(3(2(4(0(5(2(1(2(2(x1))))))))))))) -> 5(5(4(4(0(1(0(1(3(1(4(2(x1)))))))))))) 66.60/17.78 1(4(5(1(5(0(1(1(1(0(3(2(3(3(1(x1))))))))))))))) -> 1(1(3(2(3(4(3(5(4(4(3(5(3(1(x1)))))))))))))) 66.60/17.78 3(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 3(5(2(1(1(3(3(0(2(2(1(2(0(3(3(x1))))))))))))))) 66.60/17.78 3(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 3(5(1(2(5(3(5(4(5(3(1(1(4(2(2(3(x1)))))))))))))))) 66.60/17.78 5(0(2(5(5(1(3(2(5(2(0(4(4(0(4(1(x1)))))))))))))))) -> 5(1(1(0(0(2(2(2(1(3(4(3(2(5(4(x1))))))))))))))) 66.60/17.78 3(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 3(4(0(5(0(0(1(0(4(0(4(3(4(5(0(2(3(1(x1)))))))))))))))))) 66.60/17.78 4(5(3(3(2(5(5(1(3(5(1(5(0(0(2(5(5(1(x1)))))))))))))))))) -> 4(5(5(2(2(3(0(3(3(0(4(1(3(1(1(1(x1)))))))))))))))) 66.60/17.78 0(0(5(3(2(0(5(1(3(3(5(2(0(2(5(3(4(4(5(x1))))))))))))))))))) -> 0(0(5(3(5(5(0(1(3(5(0(1(3(1(3(4(2(4(x1)))))))))))))))))) 66.60/17.78 2(5(1(2(2(3(2(3(2(1(4(2(0(5(5(1(0(3(3(x1))))))))))))))))))) -> 2(3(0(4(5(0(4(3(1(3(2(0(4(1(2(4(1(3(x1)))))))))))))))))) 66.60/17.78 3(2(1(3(3(1(5(1(2(0(2(0(4(1(4(2(5(5(1(5(x1)))))))))))))))))))) -> 5(1(0(1(3(1(1(4(2(0(0(2(3(2(5(3(5(1(2(x1))))))))))))))))))) 66.60/17.78 3(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 3(2(5(5(0(5(4(2(2(5(2(1(0(1(5(0(3(3(5(2(x1)))))))))))))))))))) 66.60/17.78 0(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 2(1(0(0(3(4(0(3(4(5(3(1(4(2(1(5(3(3(5(5(0(x1))))))))))))))))))))) 66.60/17.78 1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 3(3(3(1(0(4(5(2(1(2(4(5(5(5(0(3(5(3(4(4(0(x1))))))))))))))))))))) 66.60/17.78 66.60/17.78 Q is empty. 66.60/17.78 66.60/17.78 ---------------------------------------- 66.60/17.78 66.60/17.78 (1) QTRSRRRProof (EQUIVALENT) 66.60/17.78 Used ordering: 66.60/17.78 Polynomial interpretation [POLO]: 66.60/17.78 66.60/17.78 POL(0(x_1)) = 1 + x_1 66.60/17.78 POL(1(x_1)) = 1 + x_1 66.60/17.78 POL(2(x_1)) = 1 + x_1 66.60/17.78 POL(3(x_1)) = 1 + x_1 66.60/17.78 POL(4(x_1)) = 1 + x_1 66.60/17.78 POL(5(x_1)) = 1 + x_1 66.60/17.78 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 66.60/17.78 66.60/17.78 4(0(2(3(1(4(1(x1))))))) -> 1(4(3(2(5(4(x1)))))) 66.60/17.78 2(1(4(2(2(0(3(0(1(0(5(x1))))))))))) -> 2(2(5(4(2(4(1(4(3(1(x1)))))))))) 66.60/17.78 5(3(2(3(5(0(4(0(2(3(4(5(x1)))))))))))) -> 3(5(2(1(1(5(2(1(0(5(4(x1))))))))))) 66.60/17.78 5(2(3(5(5(0(0(0(3(4(0(2(5(x1))))))))))))) -> 5(5(5(2(4(3(5(1(1(2(5(0(x1)))))))))))) 66.60/17.78 5(5(3(4(3(2(4(0(5(2(1(2(2(x1))))))))))))) -> 5(5(4(4(0(1(0(1(3(1(4(2(x1)))))))))))) 66.60/17.78 1(4(5(1(5(0(1(1(1(0(3(2(3(3(1(x1))))))))))))))) -> 1(1(3(2(3(4(3(5(4(4(3(5(3(1(x1)))))))))))))) 66.60/17.78 5(0(2(5(5(1(3(2(5(2(0(4(4(0(4(1(x1)))))))))))))))) -> 5(1(1(0(0(2(2(2(1(3(4(3(2(5(4(x1))))))))))))))) 66.60/17.78 4(5(3(3(2(5(5(1(3(5(1(5(0(0(2(5(5(1(x1)))))))))))))))))) -> 4(5(5(2(2(3(0(3(3(0(4(1(3(1(1(1(x1)))))))))))))))) 66.60/17.78 0(0(5(3(2(0(5(1(3(3(5(2(0(2(5(3(4(4(5(x1))))))))))))))))))) -> 0(0(5(3(5(5(0(1(3(5(0(1(3(1(3(4(2(4(x1)))))))))))))))))) 66.60/17.78 2(5(1(2(2(3(2(3(2(1(4(2(0(5(5(1(0(3(3(x1))))))))))))))))))) -> 2(3(0(4(5(0(4(3(1(3(2(0(4(1(2(4(1(3(x1)))))))))))))))))) 66.60/17.78 3(2(1(3(3(1(5(1(2(0(2(0(4(1(4(2(5(5(1(5(x1)))))))))))))))))))) -> 5(1(0(1(3(1(1(4(2(0(0(2(3(2(5(3(5(1(2(x1))))))))))))))))))) 66.60/17.78 66.60/17.78 66.60/17.78 66.60/17.78 66.60/17.78 ---------------------------------------- 66.60/17.78 66.60/17.78 (2) 66.60/17.78 Obligation: 66.60/17.78 Q restricted rewrite system: 66.60/17.78 The TRS R consists of the following rules: 66.60/17.78 66.60/17.78 0(1(2(1(2(2(x1)))))) -> 0(3(4(3(5(2(x1)))))) 66.60/17.78 4(0(1(0(3(0(x1)))))) -> 2(5(0(5(5(0(x1)))))) 66.60/17.78 2(1(3(0(3(3(5(x1))))))) -> 2(5(2(2(5(5(2(x1))))))) 66.60/17.78 3(3(0(0(0(5(2(x1))))))) -> 5(3(3(1(2(5(2(x1))))))) 66.60/17.78 3(3(3(0(1(1(0(x1))))))) -> 5(3(3(0(2(1(0(x1))))))) 66.60/17.78 4(1(5(0(0(5(2(1(3(x1))))))))) -> 4(5(0(3(5(2(2(2(3(x1))))))))) 66.60/17.78 5(2(5(1(5(0(3(0(4(2(x1)))))))))) -> 3(3(2(5(1(3(3(5(0(2(x1)))))))))) 66.60/17.78 3(3(0(1(3(4(2(1(1(2(1(x1))))))))))) -> 5(2(4(4(3(5(3(5(5(1(1(x1))))))))))) 66.60/17.78 3(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 3(5(2(1(1(3(3(0(2(2(1(2(0(3(3(x1))))))))))))))) 66.60/17.78 3(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 3(5(1(2(5(3(5(4(5(3(1(1(4(2(2(3(x1)))))))))))))))) 66.60/17.78 3(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 3(4(0(5(0(0(1(0(4(0(4(3(4(5(0(2(3(1(x1)))))))))))))))))) 66.60/17.78 3(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 3(2(5(5(0(5(4(2(2(5(2(1(0(1(5(0(3(3(5(2(x1)))))))))))))))))))) 66.60/17.78 0(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 2(1(0(0(3(4(0(3(4(5(3(1(4(2(1(5(3(3(5(5(0(x1))))))))))))))))))))) 66.60/17.78 1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 3(3(3(1(0(4(5(2(1(2(4(5(5(5(0(3(5(3(4(4(0(x1))))))))))))))))))))) 66.60/17.78 66.60/17.78 Q is empty. 66.60/17.78 66.60/17.78 ---------------------------------------- 66.60/17.78 66.60/17.78 (3) DependencyPairsProof (EQUIVALENT) 66.60/17.78 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 66.60/17.78 ---------------------------------------- 66.60/17.78 66.60/17.78 (4) 66.60/17.78 Obligation: 66.60/17.78 Q DP problem: 66.60/17.78 The TRS P consists of the following rules: 66.60/17.78 66.60/17.78 0^1(1(2(1(2(2(x1)))))) -> 0^1(3(4(3(5(2(x1)))))) 66.60/17.78 0^1(1(2(1(2(2(x1)))))) -> 3^1(4(3(5(2(x1))))) 66.60/17.78 0^1(1(2(1(2(2(x1)))))) -> 4^1(3(5(2(x1)))) 66.60/17.78 0^1(1(2(1(2(2(x1)))))) -> 3^1(5(2(x1))) 66.60/17.78 0^1(1(2(1(2(2(x1)))))) -> 5^1(2(x1)) 66.60/17.78 4^1(0(1(0(3(0(x1)))))) -> 2^1(5(0(5(5(0(x1)))))) 66.60/17.78 4^1(0(1(0(3(0(x1)))))) -> 5^1(0(5(5(0(x1))))) 66.60/17.78 4^1(0(1(0(3(0(x1)))))) -> 0^1(5(5(0(x1)))) 66.60/17.78 4^1(0(1(0(3(0(x1)))))) -> 5^1(5(0(x1))) 66.60/17.78 4^1(0(1(0(3(0(x1)))))) -> 5^1(0(x1)) 66.60/17.78 2^1(1(3(0(3(3(5(x1))))))) -> 2^1(5(2(2(5(5(2(x1))))))) 66.60/17.78 2^1(1(3(0(3(3(5(x1))))))) -> 5^1(2(2(5(5(2(x1)))))) 66.60/17.78 2^1(1(3(0(3(3(5(x1))))))) -> 2^1(2(5(5(2(x1))))) 66.60/17.78 2^1(1(3(0(3(3(5(x1))))))) -> 2^1(5(5(2(x1)))) 66.60/17.78 2^1(1(3(0(3(3(5(x1))))))) -> 5^1(5(2(x1))) 66.60/17.78 2^1(1(3(0(3(3(5(x1))))))) -> 5^1(2(x1)) 66.60/17.78 2^1(1(3(0(3(3(5(x1))))))) -> 2^1(x1) 66.60/17.78 3^1(3(0(0(0(5(2(x1))))))) -> 5^1(3(3(1(2(5(2(x1))))))) 66.60/17.78 3^1(3(0(0(0(5(2(x1))))))) -> 3^1(3(1(2(5(2(x1)))))) 66.60/17.78 3^1(3(0(0(0(5(2(x1))))))) -> 3^1(1(2(5(2(x1))))) 66.60/17.78 3^1(3(0(0(0(5(2(x1))))))) -> 1^1(2(5(2(x1)))) 66.60/17.78 3^1(3(0(0(0(5(2(x1))))))) -> 2^1(5(2(x1))) 66.60/17.78 3^1(3(3(0(1(1(0(x1))))))) -> 5^1(3(3(0(2(1(0(x1))))))) 66.60/17.78 3^1(3(3(0(1(1(0(x1))))))) -> 3^1(3(0(2(1(0(x1)))))) 66.60/17.78 3^1(3(3(0(1(1(0(x1))))))) -> 3^1(0(2(1(0(x1))))) 66.60/17.78 3^1(3(3(0(1(1(0(x1))))))) -> 0^1(2(1(0(x1)))) 66.60/17.78 3^1(3(3(0(1(1(0(x1))))))) -> 2^1(1(0(x1))) 66.60/17.78 4^1(1(5(0(0(5(2(1(3(x1))))))))) -> 4^1(5(0(3(5(2(2(2(3(x1))))))))) 66.60/17.78 4^1(1(5(0(0(5(2(1(3(x1))))))))) -> 5^1(0(3(5(2(2(2(3(x1)))))))) 66.60/17.78 4^1(1(5(0(0(5(2(1(3(x1))))))))) -> 0^1(3(5(2(2(2(3(x1))))))) 66.60/17.78 4^1(1(5(0(0(5(2(1(3(x1))))))))) -> 3^1(5(2(2(2(3(x1)))))) 66.60/17.78 4^1(1(5(0(0(5(2(1(3(x1))))))))) -> 5^1(2(2(2(3(x1))))) 66.60/17.78 4^1(1(5(0(0(5(2(1(3(x1))))))))) -> 2^1(2(2(3(x1)))) 66.60/17.78 4^1(1(5(0(0(5(2(1(3(x1))))))))) -> 2^1(2(3(x1))) 66.60/17.78 4^1(1(5(0(0(5(2(1(3(x1))))))))) -> 2^1(3(x1)) 66.60/17.78 5^1(2(5(1(5(0(3(0(4(2(x1)))))))))) -> 3^1(3(2(5(1(3(3(5(0(2(x1)))))))))) 66.60/17.78 5^1(2(5(1(5(0(3(0(4(2(x1)))))))))) -> 3^1(2(5(1(3(3(5(0(2(x1))))))))) 66.60/17.78 5^1(2(5(1(5(0(3(0(4(2(x1)))))))))) -> 2^1(5(1(3(3(5(0(2(x1)))))))) 66.60/17.78 5^1(2(5(1(5(0(3(0(4(2(x1)))))))))) -> 5^1(1(3(3(5(0(2(x1))))))) 66.60/17.78 5^1(2(5(1(5(0(3(0(4(2(x1)))))))))) -> 1^1(3(3(5(0(2(x1)))))) 66.60/17.78 5^1(2(5(1(5(0(3(0(4(2(x1)))))))))) -> 3^1(3(5(0(2(x1))))) 66.60/17.78 5^1(2(5(1(5(0(3(0(4(2(x1)))))))))) -> 3^1(5(0(2(x1)))) 66.60/17.78 5^1(2(5(1(5(0(3(0(4(2(x1)))))))))) -> 5^1(0(2(x1))) 66.60/17.78 5^1(2(5(1(5(0(3(0(4(2(x1)))))))))) -> 0^1(2(x1)) 66.60/17.78 3^1(3(0(1(3(4(2(1(1(2(1(x1))))))))))) -> 5^1(2(4(4(3(5(3(5(5(1(1(x1))))))))))) 66.60/17.78 3^1(3(0(1(3(4(2(1(1(2(1(x1))))))))))) -> 2^1(4(4(3(5(3(5(5(1(1(x1)))))))))) 66.60/17.78 3^1(3(0(1(3(4(2(1(1(2(1(x1))))))))))) -> 4^1(4(3(5(3(5(5(1(1(x1))))))))) 66.60/17.78 3^1(3(0(1(3(4(2(1(1(2(1(x1))))))))))) -> 4^1(3(5(3(5(5(1(1(x1)))))))) 66.60/17.78 3^1(3(0(1(3(4(2(1(1(2(1(x1))))))))))) -> 3^1(5(3(5(5(1(1(x1))))))) 66.60/17.78 3^1(3(0(1(3(4(2(1(1(2(1(x1))))))))))) -> 5^1(3(5(5(1(1(x1)))))) 66.60/17.78 3^1(3(0(1(3(4(2(1(1(2(1(x1))))))))))) -> 3^1(5(5(1(1(x1))))) 66.60/17.78 3^1(3(0(1(3(4(2(1(1(2(1(x1))))))))))) -> 5^1(5(1(1(x1)))) 66.60/17.78 3^1(3(0(1(3(4(2(1(1(2(1(x1))))))))))) -> 5^1(1(1(x1))) 66.60/17.78 3^1(3(0(1(3(4(2(1(1(2(1(x1))))))))))) -> 1^1(1(x1)) 66.60/17.78 3^1(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 3^1(5(2(1(1(3(3(0(2(2(1(2(0(3(3(x1))))))))))))))) 66.60/17.78 3^1(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 5^1(2(1(1(3(3(0(2(2(1(2(0(3(3(x1)))))))))))))) 66.60/17.78 3^1(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 2^1(1(1(3(3(0(2(2(1(2(0(3(3(x1))))))))))))) 66.60/17.78 3^1(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 1^1(1(3(3(0(2(2(1(2(0(3(3(x1)))))))))))) 66.60/17.78 3^1(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 1^1(3(3(0(2(2(1(2(0(3(3(x1))))))))))) 66.60/17.78 3^1(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 3^1(3(0(2(2(1(2(0(3(3(x1)))))))))) 66.60/17.78 3^1(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 3^1(0(2(2(1(2(0(3(3(x1))))))))) 66.60/17.78 3^1(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 0^1(2(2(1(2(0(3(3(x1)))))))) 66.60/17.78 3^1(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 2^1(2(1(2(0(3(3(x1))))))) 66.60/17.78 3^1(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 2^1(1(2(0(3(3(x1)))))) 66.60/17.78 3^1(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 1^1(2(0(3(3(x1))))) 66.60/17.78 3^1(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 2^1(0(3(3(x1)))) 66.60/17.78 3^1(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 0^1(3(3(x1))) 66.60/17.78 3^1(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 3^1(5(1(2(5(3(5(4(5(3(1(1(4(2(2(3(x1)))))))))))))))) 66.60/17.78 3^1(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 5^1(1(2(5(3(5(4(5(3(1(1(4(2(2(3(x1))))))))))))))) 66.60/17.78 3^1(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 1^1(2(5(3(5(4(5(3(1(1(4(2(2(3(x1)))))))))))))) 66.60/17.78 3^1(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 2^1(5(3(5(4(5(3(1(1(4(2(2(3(x1))))))))))))) 66.60/17.78 3^1(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 5^1(3(5(4(5(3(1(1(4(2(2(3(x1)))))))))))) 66.60/17.78 3^1(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 3^1(5(4(5(3(1(1(4(2(2(3(x1))))))))))) 66.60/17.78 3^1(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 5^1(4(5(3(1(1(4(2(2(3(x1)))))))))) 66.60/17.78 3^1(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 4^1(5(3(1(1(4(2(2(3(x1))))))))) 66.60/17.78 3^1(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 5^1(3(1(1(4(2(2(3(x1)))))))) 66.60/17.78 3^1(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 3^1(1(1(4(2(2(3(x1))))))) 66.60/17.78 3^1(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 1^1(1(4(2(2(3(x1)))))) 66.60/17.78 3^1(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 1^1(4(2(2(3(x1))))) 66.60/17.78 3^1(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 4^1(2(2(3(x1)))) 66.60/17.78 3^1(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 2^1(2(3(x1))) 66.60/17.78 3^1(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 2^1(3(x1)) 66.60/17.78 3^1(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 3^1(x1) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 3^1(4(0(5(0(0(1(0(4(0(4(3(4(5(0(2(3(1(x1)))))))))))))))))) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 4^1(0(5(0(0(1(0(4(0(4(3(4(5(0(2(3(1(x1))))))))))))))))) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 0^1(5(0(0(1(0(4(0(4(3(4(5(0(2(3(1(x1)))))))))))))))) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 5^1(0(0(1(0(4(0(4(3(4(5(0(2(3(1(x1))))))))))))))) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 0^1(0(1(0(4(0(4(3(4(5(0(2(3(1(x1)))))))))))))) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 0^1(1(0(4(0(4(3(4(5(0(2(3(1(x1))))))))))))) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 1^1(0(4(0(4(3(4(5(0(2(3(1(x1)))))))))))) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 0^1(4(0(4(3(4(5(0(2(3(1(x1))))))))))) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 4^1(0(4(3(4(5(0(2(3(1(x1)))))))))) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 0^1(4(3(4(5(0(2(3(1(x1))))))))) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 4^1(3(4(5(0(2(3(1(x1)))))))) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 3^1(4(5(0(2(3(1(x1))))))) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 4^1(5(0(2(3(1(x1)))))) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 5^1(0(2(3(1(x1))))) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 0^1(2(3(1(x1)))) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 2^1(3(1(x1))) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 3^1(1(x1)) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 1^1(x1) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 3^1(2(5(5(0(5(4(2(2(5(2(1(0(1(5(0(3(3(5(2(x1)))))))))))))))))))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 2^1(5(5(0(5(4(2(2(5(2(1(0(1(5(0(3(3(5(2(x1))))))))))))))))))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 5^1(5(0(5(4(2(2(5(2(1(0(1(5(0(3(3(5(2(x1)))))))))))))))))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 5^1(0(5(4(2(2(5(2(1(0(1(5(0(3(3(5(2(x1))))))))))))))))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 0^1(5(4(2(2(5(2(1(0(1(5(0(3(3(5(2(x1)))))))))))))))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 5^1(4(2(2(5(2(1(0(1(5(0(3(3(5(2(x1))))))))))))))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 4^1(2(2(5(2(1(0(1(5(0(3(3(5(2(x1)))))))))))))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 2^1(2(5(2(1(0(1(5(0(3(3(5(2(x1))))))))))))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 2^1(5(2(1(0(1(5(0(3(3(5(2(x1)))))))))))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 5^1(2(1(0(1(5(0(3(3(5(2(x1))))))))))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 2^1(1(0(1(5(0(3(3(5(2(x1)))))))))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 1^1(0(1(5(0(3(3(5(2(x1))))))))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 0^1(1(5(0(3(3(5(2(x1)))))))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 1^1(5(0(3(3(5(2(x1))))))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 5^1(0(3(3(5(2(x1)))))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 0^1(3(3(5(2(x1))))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 3^1(3(5(2(x1)))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 3^1(5(2(x1))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 5^1(2(x1)) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 2^1(x1) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 2^1(1(0(0(3(4(0(3(4(5(3(1(4(2(1(5(3(3(5(5(0(x1))))))))))))))))))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 1^1(0(0(3(4(0(3(4(5(3(1(4(2(1(5(3(3(5(5(0(x1)))))))))))))))))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 0^1(0(3(4(0(3(4(5(3(1(4(2(1(5(3(3(5(5(0(x1))))))))))))))))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 0^1(3(4(0(3(4(5(3(1(4(2(1(5(3(3(5(5(0(x1)))))))))))))))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 3^1(4(0(3(4(5(3(1(4(2(1(5(3(3(5(5(0(x1))))))))))))))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 4^1(0(3(4(5(3(1(4(2(1(5(3(3(5(5(0(x1)))))))))))))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 0^1(3(4(5(3(1(4(2(1(5(3(3(5(5(0(x1))))))))))))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 3^1(4(5(3(1(4(2(1(5(3(3(5(5(0(x1)))))))))))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 4^1(5(3(1(4(2(1(5(3(3(5(5(0(x1))))))))))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 5^1(3(1(4(2(1(5(3(3(5(5(0(x1)))))))))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 3^1(1(4(2(1(5(3(3(5(5(0(x1))))))))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 1^1(4(2(1(5(3(3(5(5(0(x1)))))))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 4^1(2(1(5(3(3(5(5(0(x1))))))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 2^1(1(5(3(3(5(5(0(x1)))))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 1^1(5(3(3(5(5(0(x1))))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 5^1(3(3(5(5(0(x1)))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 3^1(3(5(5(0(x1))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 3^1(5(5(0(x1)))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 5^1(5(0(x1))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 3^1(3(3(1(0(4(5(2(1(2(4(5(5(5(0(3(5(3(4(4(0(x1))))))))))))))))))))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 3^1(3(1(0(4(5(2(1(2(4(5(5(5(0(3(5(3(4(4(0(x1)))))))))))))))))))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 3^1(1(0(4(5(2(1(2(4(5(5(5(0(3(5(3(4(4(0(x1))))))))))))))))))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 1^1(0(4(5(2(1(2(4(5(5(5(0(3(5(3(4(4(0(x1)))))))))))))))))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 0^1(4(5(2(1(2(4(5(5(5(0(3(5(3(4(4(0(x1))))))))))))))))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 4^1(5(2(1(2(4(5(5(5(0(3(5(3(4(4(0(x1)))))))))))))))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 5^1(2(1(2(4(5(5(5(0(3(5(3(4(4(0(x1))))))))))))))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 2^1(1(2(4(5(5(5(0(3(5(3(4(4(0(x1)))))))))))))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 1^1(2(4(5(5(5(0(3(5(3(4(4(0(x1))))))))))))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 2^1(4(5(5(5(0(3(5(3(4(4(0(x1)))))))))))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 4^1(5(5(5(0(3(5(3(4(4(0(x1))))))))))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 5^1(5(5(0(3(5(3(4(4(0(x1)))))))))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 5^1(5(0(3(5(3(4(4(0(x1))))))))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 5^1(0(3(5(3(4(4(0(x1)))))))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 0^1(3(5(3(4(4(0(x1))))))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 3^1(5(3(4(4(0(x1)))))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 5^1(3(4(4(0(x1))))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 3^1(4(4(0(x1)))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 4^1(4(0(x1))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 4^1(0(x1)) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 0^1(x1) 66.60/17.78 66.60/17.78 The TRS R consists of the following rules: 66.60/17.78 66.60/17.78 0(1(2(1(2(2(x1)))))) -> 0(3(4(3(5(2(x1)))))) 66.60/17.78 4(0(1(0(3(0(x1)))))) -> 2(5(0(5(5(0(x1)))))) 66.60/17.78 2(1(3(0(3(3(5(x1))))))) -> 2(5(2(2(5(5(2(x1))))))) 66.60/17.78 3(3(0(0(0(5(2(x1))))))) -> 5(3(3(1(2(5(2(x1))))))) 66.60/17.78 3(3(3(0(1(1(0(x1))))))) -> 5(3(3(0(2(1(0(x1))))))) 66.60/17.78 4(1(5(0(0(5(2(1(3(x1))))))))) -> 4(5(0(3(5(2(2(2(3(x1))))))))) 66.60/17.78 5(2(5(1(5(0(3(0(4(2(x1)))))))))) -> 3(3(2(5(1(3(3(5(0(2(x1)))))))))) 66.60/17.78 3(3(0(1(3(4(2(1(1(2(1(x1))))))))))) -> 5(2(4(4(3(5(3(5(5(1(1(x1))))))))))) 66.60/17.78 3(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 3(5(2(1(1(3(3(0(2(2(1(2(0(3(3(x1))))))))))))))) 66.60/17.78 3(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 3(5(1(2(5(3(5(4(5(3(1(1(4(2(2(3(x1)))))))))))))))) 66.60/17.78 3(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 3(4(0(5(0(0(1(0(4(0(4(3(4(5(0(2(3(1(x1)))))))))))))))))) 66.60/17.78 3(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 3(2(5(5(0(5(4(2(2(5(2(1(0(1(5(0(3(3(5(2(x1)))))))))))))))))))) 66.60/17.78 0(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 2(1(0(0(3(4(0(3(4(5(3(1(4(2(1(5(3(3(5(5(0(x1))))))))))))))))))))) 66.60/17.78 1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 3(3(3(1(0(4(5(2(1(2(4(5(5(5(0(3(5(3(4(4(0(x1))))))))))))))))))))) 66.60/17.78 66.60/17.78 Q is empty. 66.60/17.78 We have to consider all minimal (P,Q,R)-chains. 66.60/17.78 ---------------------------------------- 66.60/17.78 66.60/17.78 (5) DependencyGraphProof (EQUIVALENT) 66.60/17.78 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 109 less nodes. 66.60/17.78 ---------------------------------------- 66.60/17.78 66.60/17.78 (6) 66.60/17.78 Obligation: 66.60/17.78 Q DP problem: 66.60/17.78 The TRS P consists of the following rules: 66.60/17.78 66.60/17.78 0^1(1(2(1(2(2(x1)))))) -> 3^1(4(3(5(2(x1))))) 66.60/17.78 3^1(3(0(0(0(5(2(x1))))))) -> 3^1(3(1(2(5(2(x1)))))) 66.60/17.78 3^1(3(0(0(0(5(2(x1))))))) -> 3^1(1(2(5(2(x1))))) 66.60/17.78 3^1(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 3^1(5(1(2(5(3(5(4(5(3(1(1(4(2(2(3(x1)))))))))))))))) 66.60/17.78 3^1(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 3^1(5(4(5(3(1(1(4(2(2(3(x1))))))))))) 66.60/17.78 3^1(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 3^1(1(1(4(2(2(3(x1))))))) 66.60/17.78 3^1(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 3^1(x1) 66.60/17.78 3^1(3(3(0(1(1(0(x1))))))) -> 3^1(3(0(2(1(0(x1)))))) 66.60/17.78 3^1(3(3(0(1(1(0(x1))))))) -> 3^1(0(2(1(0(x1))))) 66.60/17.78 3^1(3(3(0(1(1(0(x1))))))) -> 2^1(1(0(x1))) 66.60/17.78 2^1(1(3(0(3(3(5(x1))))))) -> 5^1(2(x1)) 66.60/17.78 5^1(2(5(1(5(0(3(0(4(2(x1)))))))))) -> 3^1(3(2(5(1(3(3(5(0(2(x1)))))))))) 66.60/17.78 5^1(2(5(1(5(0(3(0(4(2(x1)))))))))) -> 3^1(2(5(1(3(3(5(0(2(x1))))))))) 66.60/17.78 5^1(2(5(1(5(0(3(0(4(2(x1)))))))))) -> 3^1(3(5(0(2(x1))))) 66.60/17.78 5^1(2(5(1(5(0(3(0(4(2(x1)))))))))) -> 3^1(5(0(2(x1)))) 66.60/17.78 2^1(1(3(0(3(3(5(x1))))))) -> 2^1(x1) 66.60/17.78 3^1(3(0(1(3(4(2(1(1(2(1(x1))))))))))) -> 3^1(5(3(5(5(1(1(x1))))))) 66.60/17.78 3^1(3(0(1(3(4(2(1(1(2(1(x1))))))))))) -> 3^1(5(5(1(1(x1))))) 66.60/17.78 3^1(3(0(1(3(4(2(1(1(2(1(x1))))))))))) -> 1^1(1(x1)) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 3^1(3(3(1(0(4(5(2(1(2(4(5(5(5(0(3(5(3(4(4(0(x1))))))))))))))))))))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 3^1(3(1(0(4(5(2(1(2(4(5(5(5(0(3(5(3(4(4(0(x1)))))))))))))))))))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 3^1(1(0(4(5(2(1(2(4(5(5(5(0(3(5(3(4(4(0(x1))))))))))))))))))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 2^1(1(2(4(5(5(5(0(3(5(3(4(4(0(x1)))))))))))))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 3^1(5(3(4(4(0(x1)))))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 3^1(4(4(0(x1)))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 0^1(x1) 66.60/17.78 0^1(1(2(1(2(2(x1)))))) -> 3^1(5(2(x1))) 66.60/17.78 0^1(1(2(1(2(2(x1)))))) -> 5^1(2(x1)) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 2^1(1(0(0(3(4(0(3(4(5(3(1(4(2(1(5(3(3(5(5(0(x1))))))))))))))))))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 3^1(4(0(3(4(5(3(1(4(2(1(5(3(3(5(5(0(x1))))))))))))))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 3^1(4(5(3(1(4(2(1(5(3(3(5(5(0(x1)))))))))))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 3^1(1(4(2(1(5(3(3(5(5(0(x1))))))))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 2^1(1(5(3(3(5(5(0(x1)))))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 3^1(3(5(5(0(x1))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 3^1(5(5(0(x1)))) 66.60/17.78 3^1(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 3^1(5(2(1(1(3(3(0(2(2(1(2(0(3(3(x1))))))))))))))) 66.60/17.78 3^1(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 2^1(1(1(3(3(0(2(2(1(2(0(3(3(x1))))))))))))) 66.60/17.78 3^1(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 3^1(3(0(2(2(1(2(0(3(3(x1)))))))))) 66.60/17.78 3^1(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 3^1(0(2(2(1(2(0(3(3(x1))))))))) 66.60/17.78 3^1(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 2^1(1(2(0(3(3(x1)))))) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 3^1(4(0(5(0(0(1(0(4(0(4(3(4(5(0(2(3(1(x1)))))))))))))))))) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 0^1(1(0(4(0(4(3(4(5(0(2(3(1(x1))))))))))))) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 3^1(4(5(0(2(3(1(x1))))))) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 3^1(1(x1)) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 1^1(x1) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 3^1(2(5(5(0(5(4(2(2(5(2(1(0(1(5(0(3(3(5(2(x1)))))))))))))))))))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 2^1(1(0(1(5(0(3(3(5(2(x1)))))))))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 0^1(1(5(0(3(3(5(2(x1)))))))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 3^1(3(5(2(x1)))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 3^1(5(2(x1))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 5^1(2(x1)) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 2^1(x1) 66.60/17.78 66.60/17.78 The TRS R consists of the following rules: 66.60/17.78 66.60/17.78 0(1(2(1(2(2(x1)))))) -> 0(3(4(3(5(2(x1)))))) 66.60/17.78 4(0(1(0(3(0(x1)))))) -> 2(5(0(5(5(0(x1)))))) 66.60/17.78 2(1(3(0(3(3(5(x1))))))) -> 2(5(2(2(5(5(2(x1))))))) 66.60/17.78 3(3(0(0(0(5(2(x1))))))) -> 5(3(3(1(2(5(2(x1))))))) 66.60/17.78 3(3(3(0(1(1(0(x1))))))) -> 5(3(3(0(2(1(0(x1))))))) 66.60/17.78 4(1(5(0(0(5(2(1(3(x1))))))))) -> 4(5(0(3(5(2(2(2(3(x1))))))))) 66.60/17.78 5(2(5(1(5(0(3(0(4(2(x1)))))))))) -> 3(3(2(5(1(3(3(5(0(2(x1)))))))))) 66.60/17.78 3(3(0(1(3(4(2(1(1(2(1(x1))))))))))) -> 5(2(4(4(3(5(3(5(5(1(1(x1))))))))))) 66.60/17.78 3(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 3(5(2(1(1(3(3(0(2(2(1(2(0(3(3(x1))))))))))))))) 66.60/17.78 3(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 3(5(1(2(5(3(5(4(5(3(1(1(4(2(2(3(x1)))))))))))))))) 66.60/17.78 3(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 3(4(0(5(0(0(1(0(4(0(4(3(4(5(0(2(3(1(x1)))))))))))))))))) 66.60/17.78 3(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 3(2(5(5(0(5(4(2(2(5(2(1(0(1(5(0(3(3(5(2(x1)))))))))))))))))))) 66.60/17.78 0(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 2(1(0(0(3(4(0(3(4(5(3(1(4(2(1(5(3(3(5(5(0(x1))))))))))))))))))))) 66.60/17.78 1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 3(3(3(1(0(4(5(2(1(2(4(5(5(5(0(3(5(3(4(4(0(x1))))))))))))))))))))) 66.60/17.78 66.60/17.78 Q is empty. 66.60/17.78 We have to consider all minimal (P,Q,R)-chains. 66.60/17.78 ---------------------------------------- 66.60/17.78 66.60/17.78 (7) QDPOrderProof (EQUIVALENT) 66.60/17.78 We use the reduction pair processor [LPAR04,JAR06]. 66.60/17.78 66.60/17.78 66.60/17.78 The following pairs can be oriented strictly and are deleted. 66.60/17.78 66.60/17.78 3^1(3(0(0(0(5(2(x1))))))) -> 3^1(3(1(2(5(2(x1)))))) 66.60/17.78 3^1(3(0(0(0(5(2(x1))))))) -> 3^1(1(2(5(2(x1))))) 66.60/17.78 3^1(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 3^1(5(4(5(3(1(1(4(2(2(3(x1))))))))))) 66.60/17.78 3^1(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 3^1(1(1(4(2(2(3(x1))))))) 66.60/17.78 3^1(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 3^1(x1) 66.60/17.78 3^1(3(3(0(1(1(0(x1))))))) -> 3^1(3(0(2(1(0(x1)))))) 66.60/17.78 3^1(3(3(0(1(1(0(x1))))))) -> 3^1(0(2(1(0(x1))))) 66.60/17.78 3^1(3(3(0(1(1(0(x1))))))) -> 2^1(1(0(x1))) 66.60/17.78 2^1(1(3(0(3(3(5(x1))))))) -> 5^1(2(x1)) 66.60/17.78 5^1(2(5(1(5(0(3(0(4(2(x1)))))))))) -> 3^1(2(5(1(3(3(5(0(2(x1))))))))) 66.60/17.78 5^1(2(5(1(5(0(3(0(4(2(x1)))))))))) -> 3^1(3(5(0(2(x1))))) 66.60/17.78 5^1(2(5(1(5(0(3(0(4(2(x1)))))))))) -> 3^1(5(0(2(x1)))) 66.60/17.78 2^1(1(3(0(3(3(5(x1))))))) -> 2^1(x1) 66.60/17.78 3^1(3(0(1(3(4(2(1(1(2(1(x1))))))))))) -> 3^1(5(3(5(5(1(1(x1))))))) 66.60/17.78 3^1(3(0(1(3(4(2(1(1(2(1(x1))))))))))) -> 3^1(5(5(1(1(x1))))) 66.60/17.78 3^1(3(0(1(3(4(2(1(1(2(1(x1))))))))))) -> 1^1(1(x1)) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 3^1(3(1(0(4(5(2(1(2(4(5(5(5(0(3(5(3(4(4(0(x1)))))))))))))))))))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 3^1(1(0(4(5(2(1(2(4(5(5(5(0(3(5(3(4(4(0(x1))))))))))))))))))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 2^1(1(2(4(5(5(5(0(3(5(3(4(4(0(x1)))))))))))))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 3^1(5(3(4(4(0(x1)))))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 3^1(4(4(0(x1)))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 0^1(x1) 66.60/17.78 0^1(1(2(1(2(2(x1)))))) -> 3^1(5(2(x1))) 66.60/17.78 0^1(1(2(1(2(2(x1)))))) -> 5^1(2(x1)) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 3^1(4(0(3(4(5(3(1(4(2(1(5(3(3(5(5(0(x1))))))))))))))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 3^1(4(5(3(1(4(2(1(5(3(3(5(5(0(x1)))))))))))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 3^1(1(4(2(1(5(3(3(5(5(0(x1))))))))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 2^1(1(5(3(3(5(5(0(x1)))))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 3^1(3(5(5(0(x1))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 3^1(5(5(0(x1)))) 66.60/17.78 3^1(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 2^1(1(1(3(3(0(2(2(1(2(0(3(3(x1))))))))))))) 66.60/17.78 3^1(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 3^1(3(0(2(2(1(2(0(3(3(x1)))))))))) 66.60/17.78 3^1(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 3^1(0(2(2(1(2(0(3(3(x1))))))))) 66.60/17.78 3^1(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 2^1(1(2(0(3(3(x1)))))) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 0^1(1(0(4(0(4(3(4(5(0(2(3(1(x1))))))))))))) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 3^1(4(5(0(2(3(1(x1))))))) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 3^1(1(x1)) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 1^1(x1) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 2^1(1(0(1(5(0(3(3(5(2(x1)))))))))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 0^1(1(5(0(3(3(5(2(x1)))))))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 3^1(3(5(2(x1)))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 3^1(5(2(x1))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 5^1(2(x1)) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 2^1(x1) 66.60/17.78 The remaining pairs can at least be oriented weakly. 66.60/17.78 Used ordering: Polynomial interpretation [POLO]: 66.60/17.78 66.60/17.78 POL(0(x_1)) = 1 + x_1 66.60/17.78 POL(0^1(x_1)) = x_1 66.60/17.78 POL(1(x_1)) = 1 + x_1 66.60/17.78 POL(1^1(x_1)) = 1 + x_1 66.60/17.78 POL(2(x_1)) = 1 + x_1 66.60/17.78 POL(2^1(x_1)) = x_1 66.60/17.78 POL(3(x_1)) = 1 + x_1 66.60/17.78 POL(3^1(x_1)) = 1 + x_1 66.60/17.78 POL(4(x_1)) = 1 + x_1 66.60/17.78 POL(5(x_1)) = 1 + x_1 66.60/17.78 POL(5^1(x_1)) = 1 + x_1 66.60/17.78 66.60/17.78 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 66.60/17.78 66.60/17.78 2(1(3(0(3(3(5(x1))))))) -> 2(5(2(2(5(5(2(x1))))))) 66.60/17.78 5(2(5(1(5(0(3(0(4(2(x1)))))))))) -> 3(3(2(5(1(3(3(5(0(2(x1)))))))))) 66.60/17.78 3(3(0(0(0(5(2(x1))))))) -> 5(3(3(1(2(5(2(x1))))))) 66.60/17.78 3(3(3(0(1(1(0(x1))))))) -> 5(3(3(0(2(1(0(x1))))))) 66.60/17.78 3(3(0(1(3(4(2(1(1(2(1(x1))))))))))) -> 5(2(4(4(3(5(3(5(5(1(1(x1))))))))))) 66.60/17.78 3(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 3(5(2(1(1(3(3(0(2(2(1(2(0(3(3(x1))))))))))))))) 66.60/17.78 3(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 3(5(1(2(5(3(5(4(5(3(1(1(4(2(2(3(x1)))))))))))))))) 66.60/17.78 3(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 3(4(0(5(0(0(1(0(4(0(4(3(4(5(0(2(3(1(x1)))))))))))))))))) 66.60/17.78 3(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 3(2(5(5(0(5(4(2(2(5(2(1(0(1(5(0(3(3(5(2(x1)))))))))))))))))))) 66.60/17.78 4(0(1(0(3(0(x1)))))) -> 2(5(0(5(5(0(x1)))))) 66.60/17.78 4(1(5(0(0(5(2(1(3(x1))))))))) -> 4(5(0(3(5(2(2(2(3(x1))))))))) 66.60/17.78 1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 3(3(3(1(0(4(5(2(1(2(4(5(5(5(0(3(5(3(4(4(0(x1))))))))))))))))))))) 66.60/17.78 0(1(2(1(2(2(x1)))))) -> 0(3(4(3(5(2(x1)))))) 66.60/17.78 0(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 2(1(0(0(3(4(0(3(4(5(3(1(4(2(1(5(3(3(5(5(0(x1))))))))))))))))))))) 66.60/17.78 66.60/17.78 66.60/17.78 ---------------------------------------- 66.60/17.78 66.60/17.78 (8) 66.60/17.78 Obligation: 66.60/17.78 Q DP problem: 66.60/17.78 The TRS P consists of the following rules: 66.60/17.78 66.60/17.78 0^1(1(2(1(2(2(x1)))))) -> 3^1(4(3(5(2(x1))))) 66.60/17.78 3^1(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 3^1(5(1(2(5(3(5(4(5(3(1(1(4(2(2(3(x1)))))))))))))))) 66.60/17.78 5^1(2(5(1(5(0(3(0(4(2(x1)))))))))) -> 3^1(3(2(5(1(3(3(5(0(2(x1)))))))))) 66.60/17.78 1^1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 3^1(3(3(1(0(4(5(2(1(2(4(5(5(5(0(3(5(3(4(4(0(x1))))))))))))))))))))) 66.60/17.78 0^1(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 2^1(1(0(0(3(4(0(3(4(5(3(1(4(2(1(5(3(3(5(5(0(x1))))))))))))))))))))) 66.60/17.78 3^1(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 3^1(5(2(1(1(3(3(0(2(2(1(2(0(3(3(x1))))))))))))))) 66.60/17.78 3^1(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 3^1(4(0(5(0(0(1(0(4(0(4(3(4(5(0(2(3(1(x1)))))))))))))))))) 66.60/17.78 3^1(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 3^1(2(5(5(0(5(4(2(2(5(2(1(0(1(5(0(3(3(5(2(x1)))))))))))))))))))) 66.60/17.78 66.60/17.78 The TRS R consists of the following rules: 66.60/17.78 66.60/17.78 0(1(2(1(2(2(x1)))))) -> 0(3(4(3(5(2(x1)))))) 66.60/17.78 4(0(1(0(3(0(x1)))))) -> 2(5(0(5(5(0(x1)))))) 66.60/17.78 2(1(3(0(3(3(5(x1))))))) -> 2(5(2(2(5(5(2(x1))))))) 66.60/17.78 3(3(0(0(0(5(2(x1))))))) -> 5(3(3(1(2(5(2(x1))))))) 66.60/17.78 3(3(3(0(1(1(0(x1))))))) -> 5(3(3(0(2(1(0(x1))))))) 66.60/17.78 4(1(5(0(0(5(2(1(3(x1))))))))) -> 4(5(0(3(5(2(2(2(3(x1))))))))) 66.60/17.78 5(2(5(1(5(0(3(0(4(2(x1)))))))))) -> 3(3(2(5(1(3(3(5(0(2(x1)))))))))) 66.60/17.78 3(3(0(1(3(4(2(1(1(2(1(x1))))))))))) -> 5(2(4(4(3(5(3(5(5(1(1(x1))))))))))) 66.60/17.78 3(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 3(5(2(1(1(3(3(0(2(2(1(2(0(3(3(x1))))))))))))))) 66.60/17.78 3(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 3(5(1(2(5(3(5(4(5(3(1(1(4(2(2(3(x1)))))))))))))))) 66.60/17.78 3(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 3(4(0(5(0(0(1(0(4(0(4(3(4(5(0(2(3(1(x1)))))))))))))))))) 66.60/17.78 3(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 3(2(5(5(0(5(4(2(2(5(2(1(0(1(5(0(3(3(5(2(x1)))))))))))))))))))) 66.60/17.78 0(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 2(1(0(0(3(4(0(3(4(5(3(1(4(2(1(5(3(3(5(5(0(x1))))))))))))))))))))) 66.60/17.78 1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 3(3(3(1(0(4(5(2(1(2(4(5(5(5(0(3(5(3(4(4(0(x1))))))))))))))))))))) 66.60/17.78 66.60/17.78 Q is empty. 66.60/17.78 We have to consider all minimal (P,Q,R)-chains. 66.60/17.78 ---------------------------------------- 66.60/17.78 66.60/17.78 (9) DependencyGraphProof (EQUIVALENT) 66.60/17.78 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 7 less nodes. 66.60/17.78 ---------------------------------------- 66.60/17.78 66.60/17.78 (10) 66.60/17.78 Obligation: 66.60/17.78 Q DP problem: 66.60/17.78 The TRS P consists of the following rules: 66.60/17.78 66.60/17.78 3^1(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 3^1(5(1(2(5(3(5(4(5(3(1(1(4(2(2(3(x1)))))))))))))))) 66.60/17.78 66.60/17.78 The TRS R consists of the following rules: 66.60/17.78 66.60/17.78 0(1(2(1(2(2(x1)))))) -> 0(3(4(3(5(2(x1)))))) 66.60/17.78 4(0(1(0(3(0(x1)))))) -> 2(5(0(5(5(0(x1)))))) 66.60/17.78 2(1(3(0(3(3(5(x1))))))) -> 2(5(2(2(5(5(2(x1))))))) 66.60/17.78 3(3(0(0(0(5(2(x1))))))) -> 5(3(3(1(2(5(2(x1))))))) 66.60/17.78 3(3(3(0(1(1(0(x1))))))) -> 5(3(3(0(2(1(0(x1))))))) 66.60/17.78 4(1(5(0(0(5(2(1(3(x1))))))))) -> 4(5(0(3(5(2(2(2(3(x1))))))))) 66.60/17.78 5(2(5(1(5(0(3(0(4(2(x1)))))))))) -> 3(3(2(5(1(3(3(5(0(2(x1)))))))))) 66.60/17.78 3(3(0(1(3(4(2(1(1(2(1(x1))))))))))) -> 5(2(4(4(3(5(3(5(5(1(1(x1))))))))))) 66.60/17.78 3(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 3(5(2(1(1(3(3(0(2(2(1(2(0(3(3(x1))))))))))))))) 66.60/17.78 3(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 3(5(1(2(5(3(5(4(5(3(1(1(4(2(2(3(x1)))))))))))))))) 66.60/17.78 3(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 3(4(0(5(0(0(1(0(4(0(4(3(4(5(0(2(3(1(x1)))))))))))))))))) 66.60/17.78 3(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 3(2(5(5(0(5(4(2(2(5(2(1(0(1(5(0(3(3(5(2(x1)))))))))))))))))))) 66.60/17.78 0(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 2(1(0(0(3(4(0(3(4(5(3(1(4(2(1(5(3(3(5(5(0(x1))))))))))))))))))))) 66.60/17.78 1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 3(3(3(1(0(4(5(2(1(2(4(5(5(5(0(3(5(3(4(4(0(x1))))))))))))))))))))) 66.60/17.78 66.60/17.78 Q is empty. 66.60/17.78 We have to consider all minimal (P,Q,R)-chains. 66.60/17.78 ---------------------------------------- 66.60/17.78 66.60/17.78 (11) QDPOrderProof (EQUIVALENT) 66.60/17.78 We use the reduction pair processor [LPAR04,JAR06]. 66.60/17.78 66.60/17.78 66.60/17.78 The following pairs can be oriented strictly and are deleted. 66.60/17.78 66.60/17.78 3^1(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 3^1(5(1(2(5(3(5(4(5(3(1(1(4(2(2(3(x1)))))))))))))))) 66.60/17.78 The remaining pairs can at least be oriented weakly. 66.60/17.78 Used ordering: Polynomial interpretation [POLO]: 66.60/17.78 66.60/17.78 POL(0(x_1)) = 1 66.60/17.78 POL(1(x_1)) = 1 66.60/17.78 POL(2(x_1)) = 0 66.60/17.78 POL(3(x_1)) = x_1 66.60/17.78 POL(3^1(x_1)) = x_1 66.60/17.78 POL(4(x_1)) = 1 + x_1 66.60/17.78 POL(5(x_1)) = x_1 66.60/17.78 66.60/17.78 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 66.60/17.78 66.60/17.78 5(2(5(1(5(0(3(0(4(2(x1)))))))))) -> 3(3(2(5(1(3(3(5(0(2(x1)))))))))) 66.60/17.78 3(3(0(0(0(5(2(x1))))))) -> 5(3(3(1(2(5(2(x1))))))) 66.60/17.78 3(3(3(0(1(1(0(x1))))))) -> 5(3(3(0(2(1(0(x1))))))) 66.60/17.78 3(3(0(1(3(4(2(1(1(2(1(x1))))))))))) -> 5(2(4(4(3(5(3(5(5(1(1(x1))))))))))) 66.60/17.78 3(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 3(5(2(1(1(3(3(0(2(2(1(2(0(3(3(x1))))))))))))))) 66.60/17.78 3(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 3(5(1(2(5(3(5(4(5(3(1(1(4(2(2(3(x1)))))))))))))))) 66.60/17.78 3(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 3(4(0(5(0(0(1(0(4(0(4(3(4(5(0(2(3(1(x1)))))))))))))))))) 66.60/17.78 3(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 3(2(5(5(0(5(4(2(2(5(2(1(0(1(5(0(3(3(5(2(x1)))))))))))))))))))) 66.60/17.78 2(1(3(0(3(3(5(x1))))))) -> 2(5(2(2(5(5(2(x1))))))) 66.60/17.78 4(0(1(0(3(0(x1)))))) -> 2(5(0(5(5(0(x1)))))) 66.60/17.78 4(1(5(0(0(5(2(1(3(x1))))))))) -> 4(5(0(3(5(2(2(2(3(x1))))))))) 66.60/17.78 1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 3(3(3(1(0(4(5(2(1(2(4(5(5(5(0(3(5(3(4(4(0(x1))))))))))))))))))))) 66.60/17.78 0(1(2(1(2(2(x1)))))) -> 0(3(4(3(5(2(x1)))))) 66.60/17.78 0(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 2(1(0(0(3(4(0(3(4(5(3(1(4(2(1(5(3(3(5(5(0(x1))))))))))))))))))))) 66.60/17.78 66.60/17.78 66.60/17.78 ---------------------------------------- 66.60/17.78 66.60/17.78 (12) 66.60/17.78 Obligation: 66.60/17.78 Q DP problem: 66.60/17.78 P is empty. 66.60/17.78 The TRS R consists of the following rules: 66.60/17.78 66.60/17.78 0(1(2(1(2(2(x1)))))) -> 0(3(4(3(5(2(x1)))))) 66.60/17.78 4(0(1(0(3(0(x1)))))) -> 2(5(0(5(5(0(x1)))))) 66.60/17.78 2(1(3(0(3(3(5(x1))))))) -> 2(5(2(2(5(5(2(x1))))))) 66.60/17.78 3(3(0(0(0(5(2(x1))))))) -> 5(3(3(1(2(5(2(x1))))))) 66.60/17.78 3(3(3(0(1(1(0(x1))))))) -> 5(3(3(0(2(1(0(x1))))))) 66.60/17.78 4(1(5(0(0(5(2(1(3(x1))))))))) -> 4(5(0(3(5(2(2(2(3(x1))))))))) 66.60/17.78 5(2(5(1(5(0(3(0(4(2(x1)))))))))) -> 3(3(2(5(1(3(3(5(0(2(x1)))))))))) 66.60/17.78 3(3(0(1(3(4(2(1(1(2(1(x1))))))))))) -> 5(2(4(4(3(5(3(5(5(1(1(x1))))))))))) 66.60/17.78 3(3(5(3(2(1(2(3(3(1(0(0(2(3(3(x1))))))))))))))) -> 3(5(2(1(1(3(3(0(2(2(1(2(0(3(3(x1))))))))))))))) 66.60/17.78 3(3(3(4(0(5(4(4(3(2(1(4(2(0(0(2(x1)))))))))))))))) -> 3(5(1(2(5(3(5(4(5(3(1(1(4(2(2(3(x1)))))))))))))))) 66.60/17.78 3(5(4(1(4(3(0(5(5(0(2(5(1(4(3(1(5(3(x1)))))))))))))))))) -> 3(4(0(5(0(0(1(0(4(0(4(3(4(5(0(2(3(1(x1)))))))))))))))))) 66.60/17.78 3(2(1(4(2(1(1(0(4(0(1(2(2(3(1(3(5(1(4(5(x1)))))))))))))))))))) -> 3(2(5(5(0(5(4(2(2(5(2(1(0(1(5(0(3(3(5(2(x1)))))))))))))))))))) 66.60/17.78 0(4(2(4(5(0(4(4(4(0(2(1(1(5(3(1(5(1(1(5(0(x1))))))))))))))))))))) -> 2(1(0(0(3(4(0(3(4(5(3(1(4(2(1(5(3(3(5(5(0(x1))))))))))))))))))))) 66.60/17.78 1(1(2(2(3(0(3(1(2(3(3(5(4(2(0(1(3(0(0(2(3(x1))))))))))))))))))))) -> 3(3(3(1(0(4(5(2(1(2(4(5(5(5(0(3(5(3(4(4(0(x1))))))))))))))))))))) 66.60/17.78 66.60/17.78 Q is empty. 66.60/17.78 We have to consider all minimal (P,Q,R)-chains. 66.60/17.78 ---------------------------------------- 66.60/17.78 66.60/17.78 (13) PisEmptyProof (EQUIVALENT) 66.60/17.78 The TRS P is empty. Hence, there is no (P,Q,R) chain. 66.60/17.78 ---------------------------------------- 66.60/17.78 66.60/17.78 (14) 66.60/17.78 YES 66.69/17.84 EOF