43.35/11.86 YES 43.98/12.00 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 43.98/12.00 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 43.98/12.00 43.98/12.00 43.98/12.00 Termination w.r.t. Q of the given QTRS could be proven: 43.98/12.00 43.98/12.00 (0) QTRS 43.98/12.00 (1) QTRS Reverse [EQUIVALENT, 0 ms] 43.98/12.00 (2) QTRS 43.98/12.00 (3) QTRSRRRProof [EQUIVALENT, 331 ms] 43.98/12.00 (4) QTRS 43.98/12.00 (5) DependencyPairsProof [EQUIVALENT, 204 ms] 43.98/12.00 (6) QDP 43.98/12.00 (7) DependencyGraphProof [EQUIVALENT, 8 ms] 43.98/12.00 (8) QDP 43.98/12.00 (9) QDPOrderProof [EQUIVALENT, 341 ms] 43.98/12.00 (10) QDP 43.98/12.00 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 43.98/12.00 (12) QDP 43.98/12.00 (13) QDPOrderProof [EQUIVALENT, 56 ms] 43.98/12.00 (14) QDP 43.98/12.00 (15) PisEmptyProof [EQUIVALENT, 0 ms] 43.98/12.00 (16) YES 43.98/12.00 43.98/12.00 43.98/12.00 ---------------------------------------- 43.98/12.00 43.98/12.00 (0) 43.98/12.00 Obligation: 43.98/12.00 Q restricted rewrite system: 43.98/12.00 The TRS R consists of the following rules: 43.98/12.00 43.98/12.00 0(0(1(0(0(2(x1)))))) -> 0(0(2(3(4(4(x1)))))) 43.98/12.00 3(2(4(4(1(3(x1)))))) -> 5(5(3(2(1(3(x1)))))) 43.98/12.00 5(0(1(3(2(0(x1)))))) -> 5(5(1(0(1(x1))))) 43.98/12.00 1(5(1(0(2(1(1(x1))))))) -> 1(3(2(1(1(0(4(x1))))))) 43.98/12.00 0(1(0(0(5(5(5(0(x1)))))))) -> 0(2(2(1(0(4(0(0(x1)))))))) 43.98/12.00 1(2(5(1(2(0(1(3(x1)))))))) -> 1(4(2(5(4(3(5(3(x1)))))))) 43.98/12.00 4(5(0(4(2(3(2(0(x1)))))))) -> 1(1(4(2(3(2(0(x1))))))) 43.98/12.00 0(2(4(0(1(5(4(2(3(x1))))))))) -> 2(2(0(3(4(0(0(3(3(x1))))))))) 43.98/12.00 1(5(1(3(0(5(2(0(0(x1))))))))) -> 1(1(3(2(5(0(2(2(5(x1))))))))) 43.98/12.00 5(1(0(2(4(1(4(2(1(0(x1)))))))))) -> 1(3(5(1(2(4(0(3(2(3(x1)))))))))) 43.98/12.00 0(1(2(5(0(2(0(5(0(0(1(x1))))))))))) -> 0(1(4(1(2(0(3(1(2(1(x1)))))))))) 43.98/12.00 1(3(3(2(3(1(1(5(1(4(1(2(0(x1))))))))))))) -> 1(3(5(5(4(4(4(5(5(3(2(4(3(x1))))))))))))) 43.98/12.00 2(1(4(1(5(2(4(4(2(5(0(1(0(x1))))))))))))) -> 0(2(1(0(1(4(4(3(1(0(3(0(x1)))))))))))) 43.98/12.00 2(2(1(0(0(0(5(2(2(4(1(1(1(x1))))))))))))) -> 1(1(4(0(2(5(2(0(2(2(4(2(5(x1))))))))))))) 43.98/12.00 0(3(5(5(5(3(1(0(0(5(1(4(2(1(1(x1))))))))))))))) -> 0(3(5(2(1(0(1(2(1(3(2(4(5(2(1(x1))))))))))))))) 43.98/12.00 2(0(0(4(2(5(5(1(4(3(2(3(0(1(5(x1))))))))))))))) -> 2(1(0(1(3(2(4(5(2(4(4(5(2(4(x1)))))))))))))) 43.98/12.00 2(2(5(1(0(2(3(3(5(4(5(5(3(1(2(3(x1)))))))))))))))) -> 2(4(2(1(1(1(3(5(1(0(4(3(5(3(3(x1))))))))))))))) 43.98/12.00 3(1(2(0(2(4(1(4(4(4(2(3(3(2(4(0(0(x1))))))))))))))))) -> 2(5(1(2(1(4(4(0(2(1(5(4(1(3(2(0(0(x1))))))))))))))))) 43.98/12.00 0(5(0(0(4(2(0(0(3(4(4(0(5(0(4(1(2(0(x1)))))))))))))))))) -> 0(3(5(5(3(0(4(2(2(3(2(5(1(2(4(0(2(4(x1)))))))))))))))))) 43.98/12.00 5(5(5(2(3(4(0(4(2(2(2(4(1(4(5(5(5(4(x1)))))))))))))))))) -> 3(0(1(1(2(3(3(2(2(5(0(2(1(4(3(4(4(x1))))))))))))))))) 43.98/12.00 4(1(5(4(3(2(5(4(5(2(0(2(2(2(4(1(0(2(1(0(x1)))))))))))))))))))) -> 4(5(5(2(3(2(0(4(2(0(1(4(0(4(3(1(3(2(0(0(x1)))))))))))))))))))) 43.98/12.00 2(0(4(1(2(2(3(5(3(5(3(4(4(1(5(3(0(5(1(0(2(x1))))))))))))))))))))) -> 2(2(2(4(4(2(0(0(1(4(5(3(3(3(5(3(0(4(0(3(2(x1))))))))))))))))))))) 43.98/12.00 3(2(2(4(2(5(0(4(2(3(4(0(3(0(0(4(0(5(3(5(0(x1))))))))))))))))))))) -> 1(1(4(1(5(4(1(3(2(1(5(5(1(3(5(2(0(4(4(x1))))))))))))))))))) 43.98/12.00 5(1(0(1(1(5(2(1(5(5(4(3(5(2(5(2(4(3(1(5(3(x1))))))))))))))))))))) -> 1(1(0(3(4(5(4(3(2(2(5(4(0(1(4(3(0(2(5(0(3(x1))))))))))))))))))))) 43.98/12.00 5(1(0(4(5(2(3(0(2(3(2(1(4(3(1(3(3(1(3(4(2(x1))))))))))))))))))))) -> 0(1(1(3(5(0(4(5(2(2(3(0(4(4(3(1(0(0(1(3(x1)))))))))))))))))))) 43.98/12.00 43.98/12.00 Q is empty. 43.98/12.00 43.98/12.00 ---------------------------------------- 43.98/12.00 43.98/12.00 (1) QTRS Reverse (EQUIVALENT) 43.98/12.00 We applied the QTRS Reverse Processor [REVERSE]. 43.98/12.00 ---------------------------------------- 43.98/12.00 43.98/12.00 (2) 43.98/12.00 Obligation: 43.98/12.00 Q restricted rewrite system: 43.98/12.00 The TRS R consists of the following rules: 43.98/12.00 43.98/12.00 2(0(0(1(0(0(x1)))))) -> 4(4(3(2(0(0(x1)))))) 43.98/12.00 3(1(4(4(2(3(x1)))))) -> 3(1(2(3(5(5(x1)))))) 43.98/12.00 0(2(3(1(0(5(x1)))))) -> 1(0(1(5(5(x1))))) 43.98/12.00 1(1(2(0(1(5(1(x1))))))) -> 4(0(1(1(2(3(1(x1))))))) 43.98/12.00 0(5(5(5(0(0(1(0(x1)))))))) -> 0(0(4(0(1(2(2(0(x1)))))))) 43.98/12.00 3(1(0(2(1(5(2(1(x1)))))))) -> 3(5(3(4(5(2(4(1(x1)))))))) 43.98/12.00 0(2(3(2(4(0(5(4(x1)))))))) -> 0(2(3(2(4(1(1(x1))))))) 43.98/12.00 3(2(4(5(1(0(4(2(0(x1))))))))) -> 3(3(0(0(4(3(0(2(2(x1))))))))) 43.98/12.00 0(0(2(5(0(3(1(5(1(x1))))))))) -> 5(2(2(0(5(2(3(1(1(x1))))))))) 43.98/12.00 0(1(2(4(1(4(2(0(1(5(x1)))))))))) -> 3(2(3(0(4(2(1(5(3(1(x1)))))))))) 43.98/12.00 1(0(0(5(0(2(0(5(2(1(0(x1))))))))))) -> 1(2(1(3(0(2(1(4(1(0(x1)))))))))) 43.98/12.00 0(2(1(4(1(5(1(1(3(2(3(3(1(x1))))))))))))) -> 3(4(2(3(5(5(4(4(4(5(5(3(1(x1))))))))))))) 43.98/12.00 0(1(0(5(2(4(4(2(5(1(4(1(2(x1))))))))))))) -> 0(3(0(1(3(4(4(1(0(1(2(0(x1)))))))))))) 43.98/12.00 1(1(1(4(2(2(5(0(0(0(1(2(2(x1))))))))))))) -> 5(2(4(2(2(0(2(5(2(0(4(1(1(x1))))))))))))) 43.98/12.00 1(1(2(4(1(5(0(0(1(3(5(5(5(3(0(x1))))))))))))))) -> 1(2(5(4(2(3(1(2(1(0(1(2(5(3(0(x1))))))))))))))) 43.98/12.00 5(1(0(3(2(3(4(1(5(5(2(4(0(0(2(x1))))))))))))))) -> 4(2(5(4(4(2(5(4(2(3(1(0(1(2(x1)))))))))))))) 43.98/12.00 3(2(1(3(5(5(4(5(3(3(2(0(1(5(2(2(x1)))))))))))))))) -> 3(3(5(3(4(0(1(5(3(1(1(1(2(4(2(x1))))))))))))))) 43.98/12.00 0(0(4(2(3(3(2(4(4(4(1(4(2(0(2(1(3(x1))))))))))))))))) -> 0(0(2(3(1(4(5(1(2(0(4(4(1(2(1(5(2(x1))))))))))))))))) 43.98/12.00 0(2(1(4(0(5(0(4(4(3(0(0(2(4(0(0(5(0(x1)))))))))))))))))) -> 4(2(0(4(2(1(5(2(3(2(2(4(0(3(5(5(3(0(x1)))))))))))))))))) 43.98/12.00 4(5(5(5(4(1(4(2(2(2(4(0(4(3(2(5(5(5(x1)))))))))))))))))) -> 4(4(3(4(1(2(0(5(2(2(3(3(2(1(1(0(3(x1))))))))))))))))) 43.98/12.00 0(1(2(0(1(4(2(2(2(0(2(5(4(5(2(3(4(5(1(4(x1)))))))))))))))))))) -> 0(0(2(3(1(3(4(0(4(1(0(2(4(0(2(3(2(5(5(4(x1)))))))))))))))))))) 43.98/12.00 2(0(1(5(0(3(5(1(4(4(3(5(3(5(3(2(2(1(4(0(2(x1))))))))))))))))))))) -> 2(3(0(4(0(3(5(3(3(3(5(4(1(0(0(2(4(4(2(2(2(x1))))))))))))))))))))) 43.98/12.00 0(5(3(5(0(4(0(0(3(0(4(3(2(4(0(5(2(4(2(2(3(x1))))))))))))))))))))) -> 4(4(0(2(5(3(1(5(5(1(2(3(1(4(5(1(4(1(1(x1))))))))))))))))))) 43.98/12.00 3(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 3(0(5(2(0(3(4(1(0(4(5(2(2(3(4(5(4(3(0(1(1(x1))))))))))))))))))))) 43.98/12.00 2(4(3(1(3(3(1(3(4(1(2(3(2(0(3(2(5(4(0(1(5(x1))))))))))))))))))))) -> 3(1(0(0(1(3(4(4(0(3(2(2(5(4(0(5(3(1(1(0(x1)))))))))))))))))))) 43.98/12.00 43.98/12.00 Q is empty. 43.98/12.00 43.98/12.00 ---------------------------------------- 43.98/12.00 43.98/12.00 (3) QTRSRRRProof (EQUIVALENT) 43.98/12.00 Used ordering: 43.98/12.00 Polynomial interpretation [POLO]: 43.98/12.00 43.98/12.00 POL(0(x_1)) = 1 + x_1 43.98/12.00 POL(1(x_1)) = 1 + x_1 43.98/12.00 POL(2(x_1)) = 1 + x_1 43.98/12.00 POL(3(x_1)) = 1 + x_1 43.98/12.00 POL(4(x_1)) = 1 + x_1 43.98/12.00 POL(5(x_1)) = 1 + x_1 43.98/12.00 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 43.98/12.00 43.98/12.00 0(2(3(1(0(5(x1)))))) -> 1(0(1(5(5(x1))))) 43.98/12.00 0(2(3(2(4(0(5(4(x1)))))))) -> 0(2(3(2(4(1(1(x1))))))) 43.98/12.00 1(0(0(5(0(2(0(5(2(1(0(x1))))))))))) -> 1(2(1(3(0(2(1(4(1(0(x1)))))))))) 43.98/12.00 0(1(0(5(2(4(4(2(5(1(4(1(2(x1))))))))))))) -> 0(3(0(1(3(4(4(1(0(1(2(0(x1)))))))))))) 43.98/12.00 5(1(0(3(2(3(4(1(5(5(2(4(0(0(2(x1))))))))))))))) -> 4(2(5(4(4(2(5(4(2(3(1(0(1(2(x1)))))))))))))) 43.98/12.00 3(2(1(3(5(5(4(5(3(3(2(0(1(5(2(2(x1)))))))))))))))) -> 3(3(5(3(4(0(1(5(3(1(1(1(2(4(2(x1))))))))))))))) 43.98/12.00 4(5(5(5(4(1(4(2(2(2(4(0(4(3(2(5(5(5(x1)))))))))))))))))) -> 4(4(3(4(1(2(0(5(2(2(3(3(2(1(1(0(3(x1))))))))))))))))) 43.98/12.00 0(5(3(5(0(4(0(0(3(0(4(3(2(4(0(5(2(4(2(2(3(x1))))))))))))))))))))) -> 4(4(0(2(5(3(1(5(5(1(2(3(1(4(5(1(4(1(1(x1))))))))))))))))))) 43.98/12.00 2(4(3(1(3(3(1(3(4(1(2(3(2(0(3(2(5(4(0(1(5(x1))))))))))))))))))))) -> 3(1(0(0(1(3(4(4(0(3(2(2(5(4(0(5(3(1(1(0(x1)))))))))))))))))))) 43.98/12.00 43.98/12.00 43.98/12.00 43.98/12.00 43.98/12.00 ---------------------------------------- 43.98/12.00 43.98/12.00 (4) 43.98/12.00 Obligation: 43.98/12.00 Q restricted rewrite system: 43.98/12.00 The TRS R consists of the following rules: 43.98/12.00 43.98/12.00 2(0(0(1(0(0(x1)))))) -> 4(4(3(2(0(0(x1)))))) 43.98/12.00 3(1(4(4(2(3(x1)))))) -> 3(1(2(3(5(5(x1)))))) 43.98/12.00 1(1(2(0(1(5(1(x1))))))) -> 4(0(1(1(2(3(1(x1))))))) 43.98/12.00 0(5(5(5(0(0(1(0(x1)))))))) -> 0(0(4(0(1(2(2(0(x1)))))))) 43.98/12.00 3(1(0(2(1(5(2(1(x1)))))))) -> 3(5(3(4(5(2(4(1(x1)))))))) 43.98/12.00 3(2(4(5(1(0(4(2(0(x1))))))))) -> 3(3(0(0(4(3(0(2(2(x1))))))))) 43.98/12.00 0(0(2(5(0(3(1(5(1(x1))))))))) -> 5(2(2(0(5(2(3(1(1(x1))))))))) 43.98/12.00 0(1(2(4(1(4(2(0(1(5(x1)))))))))) -> 3(2(3(0(4(2(1(5(3(1(x1)))))))))) 43.98/12.00 0(2(1(4(1(5(1(1(3(2(3(3(1(x1))))))))))))) -> 3(4(2(3(5(5(4(4(4(5(5(3(1(x1))))))))))))) 43.98/12.00 1(1(1(4(2(2(5(0(0(0(1(2(2(x1))))))))))))) -> 5(2(4(2(2(0(2(5(2(0(4(1(1(x1))))))))))))) 43.98/12.00 1(1(2(4(1(5(0(0(1(3(5(5(5(3(0(x1))))))))))))))) -> 1(2(5(4(2(3(1(2(1(0(1(2(5(3(0(x1))))))))))))))) 43.98/12.00 0(0(4(2(3(3(2(4(4(4(1(4(2(0(2(1(3(x1))))))))))))))))) -> 0(0(2(3(1(4(5(1(2(0(4(4(1(2(1(5(2(x1))))))))))))))))) 43.98/12.00 0(2(1(4(0(5(0(4(4(3(0(0(2(4(0(0(5(0(x1)))))))))))))))))) -> 4(2(0(4(2(1(5(2(3(2(2(4(0(3(5(5(3(0(x1)))))))))))))))))) 43.98/12.00 0(1(2(0(1(4(2(2(2(0(2(5(4(5(2(3(4(5(1(4(x1)))))))))))))))))))) -> 0(0(2(3(1(3(4(0(4(1(0(2(4(0(2(3(2(5(5(4(x1)))))))))))))))))))) 43.98/12.00 2(0(1(5(0(3(5(1(4(4(3(5(3(5(3(2(2(1(4(0(2(x1))))))))))))))))))))) -> 2(3(0(4(0(3(5(3(3(3(5(4(1(0(0(2(4(4(2(2(2(x1))))))))))))))))))))) 43.98/12.00 3(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 3(0(5(2(0(3(4(1(0(4(5(2(2(3(4(5(4(3(0(1(1(x1))))))))))))))))))))) 43.98/12.00 43.98/12.00 Q is empty. 43.98/12.00 43.98/12.00 ---------------------------------------- 43.98/12.00 43.98/12.00 (5) DependencyPairsProof (EQUIVALENT) 43.98/12.00 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 43.98/12.00 ---------------------------------------- 43.98/12.00 43.98/12.00 (6) 43.98/12.00 Obligation: 43.98/12.00 Q DP problem: 43.98/12.00 The TRS P consists of the following rules: 43.98/12.00 43.98/12.00 2^1(0(0(1(0(0(x1)))))) -> 3^1(2(0(0(x1)))) 43.98/12.00 2^1(0(0(1(0(0(x1)))))) -> 2^1(0(0(x1))) 43.98/12.00 3^1(1(4(4(2(3(x1)))))) -> 3^1(1(2(3(5(5(x1)))))) 43.98/12.00 3^1(1(4(4(2(3(x1)))))) -> 1^1(2(3(5(5(x1))))) 43.98/12.00 3^1(1(4(4(2(3(x1)))))) -> 2^1(3(5(5(x1)))) 43.98/12.00 3^1(1(4(4(2(3(x1)))))) -> 3^1(5(5(x1))) 43.98/12.00 1^1(1(2(0(1(5(1(x1))))))) -> 0^1(1(1(2(3(1(x1)))))) 43.98/12.00 1^1(1(2(0(1(5(1(x1))))))) -> 1^1(1(2(3(1(x1))))) 43.98/12.00 1^1(1(2(0(1(5(1(x1))))))) -> 1^1(2(3(1(x1)))) 43.98/12.00 1^1(1(2(0(1(5(1(x1))))))) -> 2^1(3(1(x1))) 43.98/12.00 1^1(1(2(0(1(5(1(x1))))))) -> 3^1(1(x1)) 43.98/12.00 0^1(5(5(5(0(0(1(0(x1)))))))) -> 0^1(0(4(0(1(2(2(0(x1)))))))) 43.98/12.00 0^1(5(5(5(0(0(1(0(x1)))))))) -> 0^1(4(0(1(2(2(0(x1))))))) 43.98/12.00 0^1(5(5(5(0(0(1(0(x1)))))))) -> 0^1(1(2(2(0(x1))))) 43.98/12.00 0^1(5(5(5(0(0(1(0(x1)))))))) -> 1^1(2(2(0(x1)))) 43.98/12.00 0^1(5(5(5(0(0(1(0(x1)))))))) -> 2^1(2(0(x1))) 43.98/12.00 0^1(5(5(5(0(0(1(0(x1)))))))) -> 2^1(0(x1)) 43.98/12.00 3^1(1(0(2(1(5(2(1(x1)))))))) -> 3^1(5(3(4(5(2(4(1(x1)))))))) 43.98/12.00 3^1(1(0(2(1(5(2(1(x1)))))))) -> 3^1(4(5(2(4(1(x1)))))) 43.98/12.00 3^1(1(0(2(1(5(2(1(x1)))))))) -> 2^1(4(1(x1))) 43.98/12.00 3^1(2(4(5(1(0(4(2(0(x1))))))))) -> 3^1(3(0(0(4(3(0(2(2(x1))))))))) 43.98/12.00 3^1(2(4(5(1(0(4(2(0(x1))))))))) -> 3^1(0(0(4(3(0(2(2(x1)))))))) 43.98/12.00 3^1(2(4(5(1(0(4(2(0(x1))))))))) -> 0^1(0(4(3(0(2(2(x1))))))) 43.98/12.00 3^1(2(4(5(1(0(4(2(0(x1))))))))) -> 0^1(4(3(0(2(2(x1)))))) 43.98/12.00 3^1(2(4(5(1(0(4(2(0(x1))))))))) -> 3^1(0(2(2(x1)))) 43.98/12.00 3^1(2(4(5(1(0(4(2(0(x1))))))))) -> 0^1(2(2(x1))) 43.98/12.00 3^1(2(4(5(1(0(4(2(0(x1))))))))) -> 2^1(2(x1)) 43.98/12.00 3^1(2(4(5(1(0(4(2(0(x1))))))))) -> 2^1(x1) 43.98/12.00 0^1(0(2(5(0(3(1(5(1(x1))))))))) -> 2^1(2(0(5(2(3(1(1(x1)))))))) 43.98/12.00 0^1(0(2(5(0(3(1(5(1(x1))))))))) -> 2^1(0(5(2(3(1(1(x1))))))) 43.98/12.00 0^1(0(2(5(0(3(1(5(1(x1))))))))) -> 0^1(5(2(3(1(1(x1)))))) 43.98/12.00 0^1(0(2(5(0(3(1(5(1(x1))))))))) -> 2^1(3(1(1(x1)))) 43.98/12.00 0^1(0(2(5(0(3(1(5(1(x1))))))))) -> 3^1(1(1(x1))) 43.98/12.00 0^1(0(2(5(0(3(1(5(1(x1))))))))) -> 1^1(1(x1)) 43.98/12.00 0^1(1(2(4(1(4(2(0(1(5(x1)))))))))) -> 3^1(2(3(0(4(2(1(5(3(1(x1)))))))))) 43.98/12.00 0^1(1(2(4(1(4(2(0(1(5(x1)))))))))) -> 2^1(3(0(4(2(1(5(3(1(x1))))))))) 43.98/12.00 0^1(1(2(4(1(4(2(0(1(5(x1)))))))))) -> 3^1(0(4(2(1(5(3(1(x1)))))))) 43.98/12.00 0^1(1(2(4(1(4(2(0(1(5(x1)))))))))) -> 0^1(4(2(1(5(3(1(x1))))))) 43.98/12.00 0^1(1(2(4(1(4(2(0(1(5(x1)))))))))) -> 2^1(1(5(3(1(x1))))) 43.98/12.00 0^1(1(2(4(1(4(2(0(1(5(x1)))))))))) -> 1^1(5(3(1(x1)))) 43.98/12.00 0^1(1(2(4(1(4(2(0(1(5(x1)))))))))) -> 3^1(1(x1)) 43.98/12.00 0^1(1(2(4(1(4(2(0(1(5(x1)))))))))) -> 1^1(x1) 43.98/12.00 0^1(2(1(4(1(5(1(1(3(2(3(3(1(x1))))))))))))) -> 3^1(4(2(3(5(5(4(4(4(5(5(3(1(x1))))))))))))) 43.98/12.00 0^1(2(1(4(1(5(1(1(3(2(3(3(1(x1))))))))))))) -> 2^1(3(5(5(4(4(4(5(5(3(1(x1))))))))))) 43.98/12.00 0^1(2(1(4(1(5(1(1(3(2(3(3(1(x1))))))))))))) -> 3^1(5(5(4(4(4(5(5(3(1(x1)))))))))) 43.98/12.00 1^1(1(1(4(2(2(5(0(0(0(1(2(2(x1))))))))))))) -> 2^1(4(2(2(0(2(5(2(0(4(1(1(x1)))))))))))) 43.98/12.00 1^1(1(1(4(2(2(5(0(0(0(1(2(2(x1))))))))))))) -> 2^1(2(0(2(5(2(0(4(1(1(x1)))))))))) 43.98/12.00 1^1(1(1(4(2(2(5(0(0(0(1(2(2(x1))))))))))))) -> 2^1(0(2(5(2(0(4(1(1(x1))))))))) 43.98/12.00 1^1(1(1(4(2(2(5(0(0(0(1(2(2(x1))))))))))))) -> 0^1(2(5(2(0(4(1(1(x1)))))))) 43.98/12.00 1^1(1(1(4(2(2(5(0(0(0(1(2(2(x1))))))))))))) -> 2^1(5(2(0(4(1(1(x1))))))) 43.98/12.00 1^1(1(1(4(2(2(5(0(0(0(1(2(2(x1))))))))))))) -> 2^1(0(4(1(1(x1))))) 43.98/12.00 1^1(1(1(4(2(2(5(0(0(0(1(2(2(x1))))))))))))) -> 0^1(4(1(1(x1)))) 43.98/12.00 1^1(1(1(4(2(2(5(0(0(0(1(2(2(x1))))))))))))) -> 1^1(1(x1)) 43.98/12.00 1^1(1(1(4(2(2(5(0(0(0(1(2(2(x1))))))))))))) -> 1^1(x1) 43.98/12.00 1^1(1(2(4(1(5(0(0(1(3(5(5(5(3(0(x1))))))))))))))) -> 1^1(2(5(4(2(3(1(2(1(0(1(2(5(3(0(x1))))))))))))))) 43.98/12.00 1^1(1(2(4(1(5(0(0(1(3(5(5(5(3(0(x1))))))))))))))) -> 2^1(5(4(2(3(1(2(1(0(1(2(5(3(0(x1)))))))))))))) 43.98/12.00 1^1(1(2(4(1(5(0(0(1(3(5(5(5(3(0(x1))))))))))))))) -> 2^1(3(1(2(1(0(1(2(5(3(0(x1))))))))))) 43.98/12.00 1^1(1(2(4(1(5(0(0(1(3(5(5(5(3(0(x1))))))))))))))) -> 3^1(1(2(1(0(1(2(5(3(0(x1)))))))))) 43.98/12.00 1^1(1(2(4(1(5(0(0(1(3(5(5(5(3(0(x1))))))))))))))) -> 1^1(2(1(0(1(2(5(3(0(x1))))))))) 43.98/12.00 1^1(1(2(4(1(5(0(0(1(3(5(5(5(3(0(x1))))))))))))))) -> 2^1(1(0(1(2(5(3(0(x1)))))))) 43.98/12.00 1^1(1(2(4(1(5(0(0(1(3(5(5(5(3(0(x1))))))))))))))) -> 1^1(0(1(2(5(3(0(x1))))))) 43.98/12.00 1^1(1(2(4(1(5(0(0(1(3(5(5(5(3(0(x1))))))))))))))) -> 0^1(1(2(5(3(0(x1)))))) 43.98/12.00 1^1(1(2(4(1(5(0(0(1(3(5(5(5(3(0(x1))))))))))))))) -> 1^1(2(5(3(0(x1))))) 43.98/12.00 1^1(1(2(4(1(5(0(0(1(3(5(5(5(3(0(x1))))))))))))))) -> 2^1(5(3(0(x1)))) 43.98/12.00 0^1(0(4(2(3(3(2(4(4(4(1(4(2(0(2(1(3(x1))))))))))))))))) -> 0^1(0(2(3(1(4(5(1(2(0(4(4(1(2(1(5(2(x1))))))))))))))))) 43.98/12.00 0^1(0(4(2(3(3(2(4(4(4(1(4(2(0(2(1(3(x1))))))))))))))))) -> 0^1(2(3(1(4(5(1(2(0(4(4(1(2(1(5(2(x1)))))))))))))))) 43.98/12.00 0^1(0(4(2(3(3(2(4(4(4(1(4(2(0(2(1(3(x1))))))))))))))))) -> 2^1(3(1(4(5(1(2(0(4(4(1(2(1(5(2(x1))))))))))))))) 43.98/12.00 0^1(0(4(2(3(3(2(4(4(4(1(4(2(0(2(1(3(x1))))))))))))))))) -> 3^1(1(4(5(1(2(0(4(4(1(2(1(5(2(x1)))))))))))))) 43.98/12.00 0^1(0(4(2(3(3(2(4(4(4(1(4(2(0(2(1(3(x1))))))))))))))))) -> 1^1(4(5(1(2(0(4(4(1(2(1(5(2(x1))))))))))))) 43.98/12.00 0^1(0(4(2(3(3(2(4(4(4(1(4(2(0(2(1(3(x1))))))))))))))))) -> 1^1(2(0(4(4(1(2(1(5(2(x1)))))))))) 43.98/12.00 0^1(0(4(2(3(3(2(4(4(4(1(4(2(0(2(1(3(x1))))))))))))))))) -> 2^1(0(4(4(1(2(1(5(2(x1))))))))) 43.98/12.00 0^1(0(4(2(3(3(2(4(4(4(1(4(2(0(2(1(3(x1))))))))))))))))) -> 0^1(4(4(1(2(1(5(2(x1)))))))) 43.98/12.00 0^1(0(4(2(3(3(2(4(4(4(1(4(2(0(2(1(3(x1))))))))))))))))) -> 1^1(2(1(5(2(x1))))) 43.98/12.00 0^1(0(4(2(3(3(2(4(4(4(1(4(2(0(2(1(3(x1))))))))))))))))) -> 2^1(1(5(2(x1)))) 43.98/12.00 0^1(0(4(2(3(3(2(4(4(4(1(4(2(0(2(1(3(x1))))))))))))))))) -> 1^1(5(2(x1))) 43.98/12.00 0^1(0(4(2(3(3(2(4(4(4(1(4(2(0(2(1(3(x1))))))))))))))))) -> 2^1(x1) 43.98/12.00 0^1(2(1(4(0(5(0(4(4(3(0(0(2(4(0(0(5(0(x1)))))))))))))))))) -> 2^1(0(4(2(1(5(2(3(2(2(4(0(3(5(5(3(0(x1))))))))))))))))) 43.98/12.00 0^1(2(1(4(0(5(0(4(4(3(0(0(2(4(0(0(5(0(x1)))))))))))))))))) -> 0^1(4(2(1(5(2(3(2(2(4(0(3(5(5(3(0(x1)))))))))))))))) 43.98/12.00 0^1(2(1(4(0(5(0(4(4(3(0(0(2(4(0(0(5(0(x1)))))))))))))))))) -> 2^1(1(5(2(3(2(2(4(0(3(5(5(3(0(x1)))))))))))))) 43.98/12.00 0^1(2(1(4(0(5(0(4(4(3(0(0(2(4(0(0(5(0(x1)))))))))))))))))) -> 1^1(5(2(3(2(2(4(0(3(5(5(3(0(x1))))))))))))) 43.98/12.00 0^1(2(1(4(0(5(0(4(4(3(0(0(2(4(0(0(5(0(x1)))))))))))))))))) -> 2^1(3(2(2(4(0(3(5(5(3(0(x1))))))))))) 43.98/12.00 0^1(2(1(4(0(5(0(4(4(3(0(0(2(4(0(0(5(0(x1)))))))))))))))))) -> 3^1(2(2(4(0(3(5(5(3(0(x1)))))))))) 43.98/12.00 0^1(2(1(4(0(5(0(4(4(3(0(0(2(4(0(0(5(0(x1)))))))))))))))))) -> 2^1(2(4(0(3(5(5(3(0(x1))))))))) 43.98/12.00 0^1(2(1(4(0(5(0(4(4(3(0(0(2(4(0(0(5(0(x1)))))))))))))))))) -> 2^1(4(0(3(5(5(3(0(x1)))))))) 43.98/12.00 0^1(2(1(4(0(5(0(4(4(3(0(0(2(4(0(0(5(0(x1)))))))))))))))))) -> 0^1(3(5(5(3(0(x1)))))) 43.98/12.00 0^1(2(1(4(0(5(0(4(4(3(0(0(2(4(0(0(5(0(x1)))))))))))))))))) -> 3^1(5(5(3(0(x1))))) 43.98/12.00 0^1(2(1(4(0(5(0(4(4(3(0(0(2(4(0(0(5(0(x1)))))))))))))))))) -> 3^1(0(x1)) 43.98/12.00 0^1(1(2(0(1(4(2(2(2(0(2(5(4(5(2(3(4(5(1(4(x1)))))))))))))))))))) -> 0^1(0(2(3(1(3(4(0(4(1(0(2(4(0(2(3(2(5(5(4(x1)))))))))))))))))))) 43.98/12.00 0^1(1(2(0(1(4(2(2(2(0(2(5(4(5(2(3(4(5(1(4(x1)))))))))))))))))))) -> 0^1(2(3(1(3(4(0(4(1(0(2(4(0(2(3(2(5(5(4(x1))))))))))))))))))) 43.98/12.00 0^1(1(2(0(1(4(2(2(2(0(2(5(4(5(2(3(4(5(1(4(x1)))))))))))))))))))) -> 2^1(3(1(3(4(0(4(1(0(2(4(0(2(3(2(5(5(4(x1)))))))))))))))))) 43.98/12.00 0^1(1(2(0(1(4(2(2(2(0(2(5(4(5(2(3(4(5(1(4(x1)))))))))))))))))))) -> 3^1(1(3(4(0(4(1(0(2(4(0(2(3(2(5(5(4(x1))))))))))))))))) 43.98/12.00 0^1(1(2(0(1(4(2(2(2(0(2(5(4(5(2(3(4(5(1(4(x1)))))))))))))))))))) -> 1^1(3(4(0(4(1(0(2(4(0(2(3(2(5(5(4(x1)))))))))))))))) 43.98/12.00 0^1(1(2(0(1(4(2(2(2(0(2(5(4(5(2(3(4(5(1(4(x1)))))))))))))))))))) -> 3^1(4(0(4(1(0(2(4(0(2(3(2(5(5(4(x1))))))))))))))) 43.98/12.00 0^1(1(2(0(1(4(2(2(2(0(2(5(4(5(2(3(4(5(1(4(x1)))))))))))))))))))) -> 0^1(4(1(0(2(4(0(2(3(2(5(5(4(x1))))))))))))) 43.98/12.00 0^1(1(2(0(1(4(2(2(2(0(2(5(4(5(2(3(4(5(1(4(x1)))))))))))))))))))) -> 1^1(0(2(4(0(2(3(2(5(5(4(x1))))))))))) 43.98/12.00 0^1(1(2(0(1(4(2(2(2(0(2(5(4(5(2(3(4(5(1(4(x1)))))))))))))))))))) -> 0^1(2(4(0(2(3(2(5(5(4(x1)))))))))) 43.98/12.00 0^1(1(2(0(1(4(2(2(2(0(2(5(4(5(2(3(4(5(1(4(x1)))))))))))))))))))) -> 2^1(4(0(2(3(2(5(5(4(x1))))))))) 43.98/12.00 0^1(1(2(0(1(4(2(2(2(0(2(5(4(5(2(3(4(5(1(4(x1)))))))))))))))))))) -> 0^1(2(3(2(5(5(4(x1))))))) 43.98/12.00 0^1(1(2(0(1(4(2(2(2(0(2(5(4(5(2(3(4(5(1(4(x1)))))))))))))))))))) -> 2^1(3(2(5(5(4(x1)))))) 43.98/12.00 0^1(1(2(0(1(4(2(2(2(0(2(5(4(5(2(3(4(5(1(4(x1)))))))))))))))))))) -> 3^1(2(5(5(4(x1))))) 43.98/12.00 0^1(1(2(0(1(4(2(2(2(0(2(5(4(5(2(3(4(5(1(4(x1)))))))))))))))))))) -> 2^1(5(5(4(x1)))) 43.98/12.00 2^1(0(1(5(0(3(5(1(4(4(3(5(3(5(3(2(2(1(4(0(2(x1))))))))))))))))))))) -> 2^1(3(0(4(0(3(5(3(3(3(5(4(1(0(0(2(4(4(2(2(2(x1))))))))))))))))))))) 43.98/12.00 2^1(0(1(5(0(3(5(1(4(4(3(5(3(5(3(2(2(1(4(0(2(x1))))))))))))))))))))) -> 3^1(0(4(0(3(5(3(3(3(5(4(1(0(0(2(4(4(2(2(2(x1)))))))))))))))))))) 43.98/12.00 2^1(0(1(5(0(3(5(1(4(4(3(5(3(5(3(2(2(1(4(0(2(x1))))))))))))))))))))) -> 0^1(4(0(3(5(3(3(3(5(4(1(0(0(2(4(4(2(2(2(x1))))))))))))))))))) 43.98/12.00 2^1(0(1(5(0(3(5(1(4(4(3(5(3(5(3(2(2(1(4(0(2(x1))))))))))))))))))))) -> 0^1(3(5(3(3(3(5(4(1(0(0(2(4(4(2(2(2(x1))))))))))))))))) 43.98/12.00 2^1(0(1(5(0(3(5(1(4(4(3(5(3(5(3(2(2(1(4(0(2(x1))))))))))))))))))))) -> 3^1(5(3(3(3(5(4(1(0(0(2(4(4(2(2(2(x1)))))))))))))))) 43.98/12.00 2^1(0(1(5(0(3(5(1(4(4(3(5(3(5(3(2(2(1(4(0(2(x1))))))))))))))))))))) -> 3^1(3(3(5(4(1(0(0(2(4(4(2(2(2(x1)))))))))))))) 43.98/12.00 2^1(0(1(5(0(3(5(1(4(4(3(5(3(5(3(2(2(1(4(0(2(x1))))))))))))))))))))) -> 3^1(3(5(4(1(0(0(2(4(4(2(2(2(x1))))))))))))) 43.98/12.00 2^1(0(1(5(0(3(5(1(4(4(3(5(3(5(3(2(2(1(4(0(2(x1))))))))))))))))))))) -> 3^1(5(4(1(0(0(2(4(4(2(2(2(x1)))))))))))) 43.98/12.00 2^1(0(1(5(0(3(5(1(4(4(3(5(3(5(3(2(2(1(4(0(2(x1))))))))))))))))))))) -> 1^1(0(0(2(4(4(2(2(2(x1))))))))) 43.98/12.00 2^1(0(1(5(0(3(5(1(4(4(3(5(3(5(3(2(2(1(4(0(2(x1))))))))))))))))))))) -> 0^1(0(2(4(4(2(2(2(x1)))))))) 43.98/12.00 2^1(0(1(5(0(3(5(1(4(4(3(5(3(5(3(2(2(1(4(0(2(x1))))))))))))))))))))) -> 0^1(2(4(4(2(2(2(x1))))))) 43.98/12.00 2^1(0(1(5(0(3(5(1(4(4(3(5(3(5(3(2(2(1(4(0(2(x1))))))))))))))))))))) -> 2^1(4(4(2(2(2(x1)))))) 43.98/12.00 2^1(0(1(5(0(3(5(1(4(4(3(5(3(5(3(2(2(1(4(0(2(x1))))))))))))))))))))) -> 2^1(2(2(x1))) 43.98/12.00 2^1(0(1(5(0(3(5(1(4(4(3(5(3(5(3(2(2(1(4(0(2(x1))))))))))))))))))))) -> 2^1(2(x1)) 43.98/12.00 3^1(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 3^1(0(5(2(0(3(4(1(0(4(5(2(2(3(4(5(4(3(0(1(1(x1))))))))))))))))))))) 43.98/12.00 3^1(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 0^1(5(2(0(3(4(1(0(4(5(2(2(3(4(5(4(3(0(1(1(x1)))))))))))))))))))) 43.98/12.00 3^1(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 2^1(0(3(4(1(0(4(5(2(2(3(4(5(4(3(0(1(1(x1)))))))))))))))))) 43.98/12.00 3^1(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 0^1(3(4(1(0(4(5(2(2(3(4(5(4(3(0(1(1(x1))))))))))))))))) 43.98/12.00 3^1(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 3^1(4(1(0(4(5(2(2(3(4(5(4(3(0(1(1(x1)))))))))))))))) 43.98/12.00 3^1(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 1^1(0(4(5(2(2(3(4(5(4(3(0(1(1(x1)))))))))))))) 43.98/12.00 3^1(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 0^1(4(5(2(2(3(4(5(4(3(0(1(1(x1))))))))))))) 43.98/12.00 3^1(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 2^1(2(3(4(5(4(3(0(1(1(x1)))))))))) 43.98/12.00 3^1(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 2^1(3(4(5(4(3(0(1(1(x1))))))))) 43.98/12.00 3^1(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 3^1(4(5(4(3(0(1(1(x1)))))))) 43.98/12.00 3^1(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 3^1(0(1(1(x1)))) 43.98/12.00 3^1(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 0^1(1(1(x1))) 43.98/12.00 3^1(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 1^1(1(x1)) 43.98/12.00 3^1(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 1^1(x1) 43.98/12.00 43.98/12.00 The TRS R consists of the following rules: 43.98/12.00 43.98/12.00 2(0(0(1(0(0(x1)))))) -> 4(4(3(2(0(0(x1)))))) 43.98/12.00 3(1(4(4(2(3(x1)))))) -> 3(1(2(3(5(5(x1)))))) 43.98/12.00 1(1(2(0(1(5(1(x1))))))) -> 4(0(1(1(2(3(1(x1))))))) 43.98/12.00 0(5(5(5(0(0(1(0(x1)))))))) -> 0(0(4(0(1(2(2(0(x1)))))))) 43.98/12.00 3(1(0(2(1(5(2(1(x1)))))))) -> 3(5(3(4(5(2(4(1(x1)))))))) 43.98/12.00 3(2(4(5(1(0(4(2(0(x1))))))))) -> 3(3(0(0(4(3(0(2(2(x1))))))))) 43.98/12.00 0(0(2(5(0(3(1(5(1(x1))))))))) -> 5(2(2(0(5(2(3(1(1(x1))))))))) 43.98/12.00 0(1(2(4(1(4(2(0(1(5(x1)))))))))) -> 3(2(3(0(4(2(1(5(3(1(x1)))))))))) 43.98/12.00 0(2(1(4(1(5(1(1(3(2(3(3(1(x1))))))))))))) -> 3(4(2(3(5(5(4(4(4(5(5(3(1(x1))))))))))))) 43.98/12.00 1(1(1(4(2(2(5(0(0(0(1(2(2(x1))))))))))))) -> 5(2(4(2(2(0(2(5(2(0(4(1(1(x1))))))))))))) 43.98/12.00 1(1(2(4(1(5(0(0(1(3(5(5(5(3(0(x1))))))))))))))) -> 1(2(5(4(2(3(1(2(1(0(1(2(5(3(0(x1))))))))))))))) 43.98/12.00 0(0(4(2(3(3(2(4(4(4(1(4(2(0(2(1(3(x1))))))))))))))))) -> 0(0(2(3(1(4(5(1(2(0(4(4(1(2(1(5(2(x1))))))))))))))))) 43.98/12.00 0(2(1(4(0(5(0(4(4(3(0(0(2(4(0(0(5(0(x1)))))))))))))))))) -> 4(2(0(4(2(1(5(2(3(2(2(4(0(3(5(5(3(0(x1)))))))))))))))))) 43.98/12.00 0(1(2(0(1(4(2(2(2(0(2(5(4(5(2(3(4(5(1(4(x1)))))))))))))))))))) -> 0(0(2(3(1(3(4(0(4(1(0(2(4(0(2(3(2(5(5(4(x1)))))))))))))))))))) 43.98/12.00 2(0(1(5(0(3(5(1(4(4(3(5(3(5(3(2(2(1(4(0(2(x1))))))))))))))))))))) -> 2(3(0(4(0(3(5(3(3(3(5(4(1(0(0(2(4(4(2(2(2(x1))))))))))))))))))))) 43.98/12.00 3(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 3(0(5(2(0(3(4(1(0(4(5(2(2(3(4(5(4(3(0(1(1(x1))))))))))))))))))))) 43.98/12.00 43.98/12.00 Q is empty. 43.98/12.00 We have to consider all minimal (P,Q,R)-chains. 43.98/12.00 ---------------------------------------- 43.98/12.00 43.98/12.00 (7) DependencyGraphProof (EQUIVALENT) 43.98/12.00 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 92 less nodes. 43.98/12.00 ---------------------------------------- 43.98/12.00 43.98/12.00 (8) 43.98/12.00 Obligation: 43.98/12.00 Q DP problem: 43.98/12.00 The TRS P consists of the following rules: 43.98/12.00 43.98/12.00 3^1(2(4(5(1(0(4(2(0(x1))))))))) -> 3^1(3(0(0(4(3(0(2(2(x1))))))))) 43.98/12.00 3^1(2(4(5(1(0(4(2(0(x1))))))))) -> 3^1(0(0(4(3(0(2(2(x1)))))))) 43.98/12.00 3^1(2(4(5(1(0(4(2(0(x1))))))))) -> 0^1(0(4(3(0(2(2(x1))))))) 43.98/12.00 0^1(0(4(2(3(3(2(4(4(4(1(4(2(0(2(1(3(x1))))))))))))))))) -> 2^1(x1) 43.98/12.00 2^1(0(0(1(0(0(x1)))))) -> 3^1(2(0(0(x1)))) 43.98/12.00 3^1(2(4(5(1(0(4(2(0(x1))))))))) -> 3^1(0(2(2(x1)))) 43.98/12.00 3^1(2(4(5(1(0(4(2(0(x1))))))))) -> 0^1(2(2(x1))) 43.98/12.00 0^1(5(5(5(0(0(1(0(x1)))))))) -> 0^1(0(4(0(1(2(2(0(x1)))))))) 43.98/12.00 0^1(5(5(5(0(0(1(0(x1)))))))) -> 0^1(1(2(2(0(x1))))) 43.98/12.00 0^1(5(5(5(0(0(1(0(x1)))))))) -> 1^1(2(2(0(x1)))) 43.98/12.00 1^1(1(2(0(1(5(1(x1))))))) -> 0^1(1(1(2(3(1(x1)))))) 43.98/12.00 0^1(5(5(5(0(0(1(0(x1)))))))) -> 2^1(2(0(x1))) 43.98/12.00 2^1(0(0(1(0(0(x1)))))) -> 2^1(0(0(x1))) 43.98/12.00 2^1(0(1(5(0(3(5(1(4(4(3(5(3(5(3(2(2(1(4(0(2(x1))))))))))))))))))))) -> 2^1(2(2(x1))) 43.98/12.00 2^1(0(1(5(0(3(5(1(4(4(3(5(3(5(3(2(2(1(4(0(2(x1))))))))))))))))))))) -> 2^1(2(x1)) 43.98/12.00 0^1(5(5(5(0(0(1(0(x1)))))))) -> 2^1(0(x1)) 43.98/12.00 0^1(0(2(5(0(3(1(5(1(x1))))))))) -> 2^1(2(0(5(2(3(1(1(x1)))))))) 43.98/12.00 0^1(0(2(5(0(3(1(5(1(x1))))))))) -> 2^1(0(5(2(3(1(1(x1))))))) 43.98/12.00 0^1(0(2(5(0(3(1(5(1(x1))))))))) -> 0^1(5(2(3(1(1(x1)))))) 43.98/12.00 0^1(0(2(5(0(3(1(5(1(x1))))))))) -> 2^1(3(1(1(x1)))) 43.98/12.00 0^1(0(2(5(0(3(1(5(1(x1))))))))) -> 3^1(1(1(x1))) 43.98/12.00 3^1(2(4(5(1(0(4(2(0(x1))))))))) -> 2^1(2(x1)) 43.98/12.00 3^1(2(4(5(1(0(4(2(0(x1))))))))) -> 2^1(x1) 43.98/12.00 3^1(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 3^1(0(1(1(x1)))) 43.98/12.00 3^1(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 0^1(1(1(x1))) 43.98/12.00 0^1(0(2(5(0(3(1(5(1(x1))))))))) -> 1^1(1(x1)) 43.98/12.00 1^1(1(2(0(1(5(1(x1))))))) -> 1^1(1(2(3(1(x1))))) 43.98/12.00 1^1(1(2(0(1(5(1(x1))))))) -> 1^1(2(3(1(x1)))) 43.98/12.00 1^1(1(2(0(1(5(1(x1))))))) -> 2^1(3(1(x1))) 43.98/12.00 1^1(1(2(0(1(5(1(x1))))))) -> 3^1(1(x1)) 43.98/12.00 3^1(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 1^1(1(x1)) 43.98/12.00 1^1(1(1(4(2(2(5(0(0(0(1(2(2(x1))))))))))))) -> 1^1(1(x1)) 43.98/12.00 1^1(1(1(4(2(2(5(0(0(0(1(2(2(x1))))))))))))) -> 1^1(x1) 43.98/12.00 3^1(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 1^1(x1) 43.98/12.00 0^1(1(2(4(1(4(2(0(1(5(x1)))))))))) -> 3^1(1(x1)) 43.98/12.00 0^1(1(2(4(1(4(2(0(1(5(x1)))))))))) -> 1^1(x1) 43.98/12.00 0^1(2(1(4(0(5(0(4(4(3(0(0(2(4(0(0(5(0(x1)))))))))))))))))) -> 3^1(0(x1)) 43.98/12.00 43.98/12.00 The TRS R consists of the following rules: 43.98/12.00 43.98/12.00 2(0(0(1(0(0(x1)))))) -> 4(4(3(2(0(0(x1)))))) 43.98/12.00 3(1(4(4(2(3(x1)))))) -> 3(1(2(3(5(5(x1)))))) 43.98/12.00 1(1(2(0(1(5(1(x1))))))) -> 4(0(1(1(2(3(1(x1))))))) 43.98/12.00 0(5(5(5(0(0(1(0(x1)))))))) -> 0(0(4(0(1(2(2(0(x1)))))))) 43.98/12.00 3(1(0(2(1(5(2(1(x1)))))))) -> 3(5(3(4(5(2(4(1(x1)))))))) 43.98/12.00 3(2(4(5(1(0(4(2(0(x1))))))))) -> 3(3(0(0(4(3(0(2(2(x1))))))))) 43.98/12.00 0(0(2(5(0(3(1(5(1(x1))))))))) -> 5(2(2(0(5(2(3(1(1(x1))))))))) 43.98/12.00 0(1(2(4(1(4(2(0(1(5(x1)))))))))) -> 3(2(3(0(4(2(1(5(3(1(x1)))))))))) 43.98/12.00 0(2(1(4(1(5(1(1(3(2(3(3(1(x1))))))))))))) -> 3(4(2(3(5(5(4(4(4(5(5(3(1(x1))))))))))))) 43.98/12.00 1(1(1(4(2(2(5(0(0(0(1(2(2(x1))))))))))))) -> 5(2(4(2(2(0(2(5(2(0(4(1(1(x1))))))))))))) 43.98/12.00 1(1(2(4(1(5(0(0(1(3(5(5(5(3(0(x1))))))))))))))) -> 1(2(5(4(2(3(1(2(1(0(1(2(5(3(0(x1))))))))))))))) 43.98/12.00 0(0(4(2(3(3(2(4(4(4(1(4(2(0(2(1(3(x1))))))))))))))))) -> 0(0(2(3(1(4(5(1(2(0(4(4(1(2(1(5(2(x1))))))))))))))))) 43.98/12.00 0(2(1(4(0(5(0(4(4(3(0(0(2(4(0(0(5(0(x1)))))))))))))))))) -> 4(2(0(4(2(1(5(2(3(2(2(4(0(3(5(5(3(0(x1)))))))))))))))))) 43.98/12.00 0(1(2(0(1(4(2(2(2(0(2(5(4(5(2(3(4(5(1(4(x1)))))))))))))))))))) -> 0(0(2(3(1(3(4(0(4(1(0(2(4(0(2(3(2(5(5(4(x1)))))))))))))))))))) 43.98/12.00 2(0(1(5(0(3(5(1(4(4(3(5(3(5(3(2(2(1(4(0(2(x1))))))))))))))))))))) -> 2(3(0(4(0(3(5(3(3(3(5(4(1(0(0(2(4(4(2(2(2(x1))))))))))))))))))))) 43.98/12.00 3(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 3(0(5(2(0(3(4(1(0(4(5(2(2(3(4(5(4(3(0(1(1(x1))))))))))))))))))))) 43.98/12.00 43.98/12.00 Q is empty. 43.98/12.00 We have to consider all minimal (P,Q,R)-chains. 43.98/12.00 ---------------------------------------- 43.98/12.00 43.98/12.00 (9) QDPOrderProof (EQUIVALENT) 43.98/12.00 We use the reduction pair processor [LPAR04,JAR06]. 43.98/12.00 43.98/12.00 43.98/12.00 The following pairs can be oriented strictly and are deleted. 43.98/12.00 43.98/12.00 3^1(2(4(5(1(0(4(2(0(x1))))))))) -> 3^1(0(0(4(3(0(2(2(x1)))))))) 43.98/12.00 3^1(2(4(5(1(0(4(2(0(x1))))))))) -> 0^1(0(4(3(0(2(2(x1))))))) 43.98/12.00 0^1(0(4(2(3(3(2(4(4(4(1(4(2(0(2(1(3(x1))))))))))))))))) -> 2^1(x1) 43.98/12.00 2^1(0(0(1(0(0(x1)))))) -> 3^1(2(0(0(x1)))) 43.98/12.00 3^1(2(4(5(1(0(4(2(0(x1))))))))) -> 3^1(0(2(2(x1)))) 43.98/12.00 3^1(2(4(5(1(0(4(2(0(x1))))))))) -> 0^1(2(2(x1))) 43.98/12.00 0^1(5(5(5(0(0(1(0(x1)))))))) -> 0^1(1(2(2(0(x1))))) 43.98/12.00 0^1(5(5(5(0(0(1(0(x1)))))))) -> 1^1(2(2(0(x1)))) 43.98/12.00 0^1(5(5(5(0(0(1(0(x1)))))))) -> 2^1(2(0(x1))) 43.98/12.00 2^1(0(0(1(0(0(x1)))))) -> 2^1(0(0(x1))) 43.98/12.00 2^1(0(1(5(0(3(5(1(4(4(3(5(3(5(3(2(2(1(4(0(2(x1))))))))))))))))))))) -> 2^1(2(2(x1))) 43.98/12.00 2^1(0(1(5(0(3(5(1(4(4(3(5(3(5(3(2(2(1(4(0(2(x1))))))))))))))))))))) -> 2^1(2(x1)) 43.98/12.00 0^1(5(5(5(0(0(1(0(x1)))))))) -> 2^1(0(x1)) 43.98/12.00 0^1(0(2(5(0(3(1(5(1(x1))))))))) -> 2^1(2(0(5(2(3(1(1(x1)))))))) 43.98/12.00 0^1(0(2(5(0(3(1(5(1(x1))))))))) -> 2^1(0(5(2(3(1(1(x1))))))) 43.98/12.00 0^1(0(2(5(0(3(1(5(1(x1))))))))) -> 0^1(5(2(3(1(1(x1)))))) 43.98/12.00 0^1(0(2(5(0(3(1(5(1(x1))))))))) -> 2^1(3(1(1(x1)))) 43.98/12.00 0^1(0(2(5(0(3(1(5(1(x1))))))))) -> 3^1(1(1(x1))) 43.98/12.00 3^1(2(4(5(1(0(4(2(0(x1))))))))) -> 2^1(2(x1)) 43.98/12.00 3^1(2(4(5(1(0(4(2(0(x1))))))))) -> 2^1(x1) 43.98/12.00 3^1(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 3^1(0(1(1(x1)))) 43.98/12.00 3^1(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 0^1(1(1(x1))) 43.98/12.00 0^1(0(2(5(0(3(1(5(1(x1))))))))) -> 1^1(1(x1)) 43.98/12.00 1^1(1(2(0(1(5(1(x1))))))) -> 1^1(1(2(3(1(x1))))) 43.98/12.00 1^1(1(2(0(1(5(1(x1))))))) -> 1^1(2(3(1(x1)))) 43.98/12.00 1^1(1(2(0(1(5(1(x1))))))) -> 2^1(3(1(x1))) 43.98/12.00 1^1(1(2(0(1(5(1(x1))))))) -> 3^1(1(x1)) 43.98/12.00 3^1(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 1^1(1(x1)) 43.98/12.00 1^1(1(1(4(2(2(5(0(0(0(1(2(2(x1))))))))))))) -> 1^1(1(x1)) 43.98/12.00 1^1(1(1(4(2(2(5(0(0(0(1(2(2(x1))))))))))))) -> 1^1(x1) 43.98/12.00 3^1(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 1^1(x1) 43.98/12.00 0^1(1(2(4(1(4(2(0(1(5(x1)))))))))) -> 3^1(1(x1)) 43.98/12.00 0^1(1(2(4(1(4(2(0(1(5(x1)))))))))) -> 1^1(x1) 43.98/12.00 0^1(2(1(4(0(5(0(4(4(3(0(0(2(4(0(0(5(0(x1)))))))))))))))))) -> 3^1(0(x1)) 43.98/12.00 The remaining pairs can at least be oriented weakly. 43.98/12.00 Used ordering: Polynomial interpretation [POLO]: 43.98/12.00 43.98/12.00 POL(0(x_1)) = 1 + x_1 43.98/12.00 POL(0^1(x_1)) = 1 + x_1 43.98/12.00 POL(1(x_1)) = 1 + x_1 43.98/12.00 POL(1^1(x_1)) = x_1 43.98/12.00 POL(2(x_1)) = 1 + x_1 43.98/12.00 POL(2^1(x_1)) = x_1 43.98/12.00 POL(3(x_1)) = 1 + x_1 43.98/12.00 POL(3^1(x_1)) = x_1 43.98/12.00 POL(4(x_1)) = 1 + x_1 43.98/12.00 POL(5(x_1)) = 1 + x_1 43.98/12.00 43.98/12.00 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 43.98/12.00 43.98/12.00 2(0(0(1(0(0(x1)))))) -> 4(4(3(2(0(0(x1)))))) 43.98/12.00 2(0(1(5(0(3(5(1(4(4(3(5(3(5(3(2(2(1(4(0(2(x1))))))))))))))))))))) -> 2(3(0(4(0(3(5(3(3(3(5(4(1(0(0(2(4(4(2(2(2(x1))))))))))))))))))))) 43.98/12.00 0(5(5(5(0(0(1(0(x1)))))))) -> 0(0(4(0(1(2(2(0(x1)))))))) 43.98/12.00 0(0(2(5(0(3(1(5(1(x1))))))))) -> 5(2(2(0(5(2(3(1(1(x1))))))))) 43.98/12.00 0(1(2(4(1(4(2(0(1(5(x1)))))))))) -> 3(2(3(0(4(2(1(5(3(1(x1)))))))))) 43.98/12.00 0(2(1(4(1(5(1(1(3(2(3(3(1(x1))))))))))))) -> 3(4(2(3(5(5(4(4(4(5(5(3(1(x1))))))))))))) 43.98/12.00 0(0(4(2(3(3(2(4(4(4(1(4(2(0(2(1(3(x1))))))))))))))))) -> 0(0(2(3(1(4(5(1(2(0(4(4(1(2(1(5(2(x1))))))))))))))))) 43.98/12.00 0(2(1(4(0(5(0(4(4(3(0(0(2(4(0(0(5(0(x1)))))))))))))))))) -> 4(2(0(4(2(1(5(2(3(2(2(4(0(3(5(5(3(0(x1)))))))))))))))))) 43.98/12.00 0(1(2(0(1(4(2(2(2(0(2(5(4(5(2(3(4(5(1(4(x1)))))))))))))))))))) -> 0(0(2(3(1(3(4(0(4(1(0(2(4(0(2(3(2(5(5(4(x1)))))))))))))))))))) 43.98/12.00 3(1(4(4(2(3(x1)))))) -> 3(1(2(3(5(5(x1)))))) 43.98/12.00 3(1(0(2(1(5(2(1(x1)))))))) -> 3(5(3(4(5(2(4(1(x1)))))))) 43.98/12.00 3(2(4(5(1(0(4(2(0(x1))))))))) -> 3(3(0(0(4(3(0(2(2(x1))))))))) 43.98/12.00 3(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 3(0(5(2(0(3(4(1(0(4(5(2(2(3(4(5(4(3(0(1(1(x1))))))))))))))))))))) 43.98/12.00 1(1(2(0(1(5(1(x1))))))) -> 4(0(1(1(2(3(1(x1))))))) 43.98/12.00 1(1(1(4(2(2(5(0(0(0(1(2(2(x1))))))))))))) -> 5(2(4(2(2(0(2(5(2(0(4(1(1(x1))))))))))))) 43.98/12.00 1(1(2(4(1(5(0(0(1(3(5(5(5(3(0(x1))))))))))))))) -> 1(2(5(4(2(3(1(2(1(0(1(2(5(3(0(x1))))))))))))))) 43.98/12.00 43.98/12.00 43.98/12.00 ---------------------------------------- 43.98/12.00 43.98/12.00 (10) 43.98/12.00 Obligation: 43.98/12.00 Q DP problem: 43.98/12.00 The TRS P consists of the following rules: 43.98/12.00 43.98/12.00 3^1(2(4(5(1(0(4(2(0(x1))))))))) -> 3^1(3(0(0(4(3(0(2(2(x1))))))))) 43.98/12.00 0^1(5(5(5(0(0(1(0(x1)))))))) -> 0^1(0(4(0(1(2(2(0(x1)))))))) 43.98/12.00 1^1(1(2(0(1(5(1(x1))))))) -> 0^1(1(1(2(3(1(x1)))))) 43.98/12.00 43.98/12.00 The TRS R consists of the following rules: 43.98/12.00 43.98/12.00 2(0(0(1(0(0(x1)))))) -> 4(4(3(2(0(0(x1)))))) 43.98/12.00 3(1(4(4(2(3(x1)))))) -> 3(1(2(3(5(5(x1)))))) 43.98/12.00 1(1(2(0(1(5(1(x1))))))) -> 4(0(1(1(2(3(1(x1))))))) 43.98/12.00 0(5(5(5(0(0(1(0(x1)))))))) -> 0(0(4(0(1(2(2(0(x1)))))))) 43.98/12.00 3(1(0(2(1(5(2(1(x1)))))))) -> 3(5(3(4(5(2(4(1(x1)))))))) 43.98/12.00 3(2(4(5(1(0(4(2(0(x1))))))))) -> 3(3(0(0(4(3(0(2(2(x1))))))))) 43.98/12.00 0(0(2(5(0(3(1(5(1(x1))))))))) -> 5(2(2(0(5(2(3(1(1(x1))))))))) 43.98/12.00 0(1(2(4(1(4(2(0(1(5(x1)))))))))) -> 3(2(3(0(4(2(1(5(3(1(x1)))))))))) 43.98/12.00 0(2(1(4(1(5(1(1(3(2(3(3(1(x1))))))))))))) -> 3(4(2(3(5(5(4(4(4(5(5(3(1(x1))))))))))))) 43.98/12.00 1(1(1(4(2(2(5(0(0(0(1(2(2(x1))))))))))))) -> 5(2(4(2(2(0(2(5(2(0(4(1(1(x1))))))))))))) 43.98/12.00 1(1(2(4(1(5(0(0(1(3(5(5(5(3(0(x1))))))))))))))) -> 1(2(5(4(2(3(1(2(1(0(1(2(5(3(0(x1))))))))))))))) 43.98/12.00 0(0(4(2(3(3(2(4(4(4(1(4(2(0(2(1(3(x1))))))))))))))))) -> 0(0(2(3(1(4(5(1(2(0(4(4(1(2(1(5(2(x1))))))))))))))))) 43.98/12.00 0(2(1(4(0(5(0(4(4(3(0(0(2(4(0(0(5(0(x1)))))))))))))))))) -> 4(2(0(4(2(1(5(2(3(2(2(4(0(3(5(5(3(0(x1)))))))))))))))))) 43.98/12.00 0(1(2(0(1(4(2(2(2(0(2(5(4(5(2(3(4(5(1(4(x1)))))))))))))))))))) -> 0(0(2(3(1(3(4(0(4(1(0(2(4(0(2(3(2(5(5(4(x1)))))))))))))))))))) 43.98/12.00 2(0(1(5(0(3(5(1(4(4(3(5(3(5(3(2(2(1(4(0(2(x1))))))))))))))))))))) -> 2(3(0(4(0(3(5(3(3(3(5(4(1(0(0(2(4(4(2(2(2(x1))))))))))))))))))))) 43.98/12.00 3(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 3(0(5(2(0(3(4(1(0(4(5(2(2(3(4(5(4(3(0(1(1(x1))))))))))))))))))))) 43.98/12.00 43.98/12.00 Q is empty. 43.98/12.00 We have to consider all minimal (P,Q,R)-chains. 43.98/12.00 ---------------------------------------- 43.98/12.00 43.98/12.00 (11) DependencyGraphProof (EQUIVALENT) 43.98/12.00 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 43.98/12.00 ---------------------------------------- 43.98/12.00 43.98/12.00 (12) 43.98/12.00 Obligation: 43.98/12.00 Q DP problem: 43.98/12.00 The TRS P consists of the following rules: 43.98/12.00 43.98/12.00 3^1(2(4(5(1(0(4(2(0(x1))))))))) -> 3^1(3(0(0(4(3(0(2(2(x1))))))))) 43.98/12.00 43.98/12.00 The TRS R consists of the following rules: 43.98/12.00 43.98/12.00 2(0(0(1(0(0(x1)))))) -> 4(4(3(2(0(0(x1)))))) 43.98/12.00 3(1(4(4(2(3(x1)))))) -> 3(1(2(3(5(5(x1)))))) 43.98/12.00 1(1(2(0(1(5(1(x1))))))) -> 4(0(1(1(2(3(1(x1))))))) 43.98/12.00 0(5(5(5(0(0(1(0(x1)))))))) -> 0(0(4(0(1(2(2(0(x1)))))))) 43.98/12.00 3(1(0(2(1(5(2(1(x1)))))))) -> 3(5(3(4(5(2(4(1(x1)))))))) 43.98/12.00 3(2(4(5(1(0(4(2(0(x1))))))))) -> 3(3(0(0(4(3(0(2(2(x1))))))))) 43.98/12.00 0(0(2(5(0(3(1(5(1(x1))))))))) -> 5(2(2(0(5(2(3(1(1(x1))))))))) 43.98/12.00 0(1(2(4(1(4(2(0(1(5(x1)))))))))) -> 3(2(3(0(4(2(1(5(3(1(x1)))))))))) 43.98/12.00 0(2(1(4(1(5(1(1(3(2(3(3(1(x1))))))))))))) -> 3(4(2(3(5(5(4(4(4(5(5(3(1(x1))))))))))))) 43.98/12.00 1(1(1(4(2(2(5(0(0(0(1(2(2(x1))))))))))))) -> 5(2(4(2(2(0(2(5(2(0(4(1(1(x1))))))))))))) 43.98/12.00 1(1(2(4(1(5(0(0(1(3(5(5(5(3(0(x1))))))))))))))) -> 1(2(5(4(2(3(1(2(1(0(1(2(5(3(0(x1))))))))))))))) 43.98/12.00 0(0(4(2(3(3(2(4(4(4(1(4(2(0(2(1(3(x1))))))))))))))))) -> 0(0(2(3(1(4(5(1(2(0(4(4(1(2(1(5(2(x1))))))))))))))))) 43.98/12.00 0(2(1(4(0(5(0(4(4(3(0(0(2(4(0(0(5(0(x1)))))))))))))))))) -> 4(2(0(4(2(1(5(2(3(2(2(4(0(3(5(5(3(0(x1)))))))))))))))))) 43.98/12.00 0(1(2(0(1(4(2(2(2(0(2(5(4(5(2(3(4(5(1(4(x1)))))))))))))))))))) -> 0(0(2(3(1(3(4(0(4(1(0(2(4(0(2(3(2(5(5(4(x1)))))))))))))))))))) 43.98/12.00 2(0(1(5(0(3(5(1(4(4(3(5(3(5(3(2(2(1(4(0(2(x1))))))))))))))))))))) -> 2(3(0(4(0(3(5(3(3(3(5(4(1(0(0(2(4(4(2(2(2(x1))))))))))))))))))))) 43.98/12.00 3(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 3(0(5(2(0(3(4(1(0(4(5(2(2(3(4(5(4(3(0(1(1(x1))))))))))))))))))))) 43.98/12.00 43.98/12.00 Q is empty. 43.98/12.00 We have to consider all minimal (P,Q,R)-chains. 43.98/12.00 ---------------------------------------- 43.98/12.00 43.98/12.00 (13) QDPOrderProof (EQUIVALENT) 43.98/12.00 We use the reduction pair processor [LPAR04,JAR06]. 43.98/12.00 43.98/12.00 43.98/12.00 The following pairs can be oriented strictly and are deleted. 43.98/12.00 43.98/12.00 3^1(2(4(5(1(0(4(2(0(x1))))))))) -> 3^1(3(0(0(4(3(0(2(2(x1))))))))) 43.98/12.00 The remaining pairs can at least be oriented weakly. 43.98/12.00 Used ordering: Polynomial interpretation [POLO]: 43.98/12.00 43.98/12.00 POL(0(x_1)) = 1 43.98/12.00 POL(1(x_1)) = 1 + x_1 43.98/12.00 POL(2(x_1)) = 1 + x_1 43.98/12.00 POL(3(x_1)) = 0 43.98/12.00 POL(3^1(x_1)) = x_1 43.98/12.00 POL(4(x_1)) = 1 + x_1 43.98/12.00 POL(5(x_1)) = x_1 43.98/12.00 43.98/12.00 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 43.98/12.00 43.98/12.00 3(1(4(4(2(3(x1)))))) -> 3(1(2(3(5(5(x1)))))) 43.98/12.00 3(1(0(2(1(5(2(1(x1)))))))) -> 3(5(3(4(5(2(4(1(x1)))))))) 43.98/12.00 3(2(4(5(1(0(4(2(0(x1))))))))) -> 3(3(0(0(4(3(0(2(2(x1))))))))) 43.98/12.00 3(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 3(0(5(2(0(3(4(1(0(4(5(2(2(3(4(5(4(3(0(1(1(x1))))))))))))))))))))) 43.98/12.00 43.98/12.00 43.98/12.00 ---------------------------------------- 43.98/12.00 43.98/12.00 (14) 43.98/12.00 Obligation: 43.98/12.00 Q DP problem: 43.98/12.00 P is empty. 43.98/12.00 The TRS R consists of the following rules: 43.98/12.00 43.98/12.00 2(0(0(1(0(0(x1)))))) -> 4(4(3(2(0(0(x1)))))) 43.98/12.00 3(1(4(4(2(3(x1)))))) -> 3(1(2(3(5(5(x1)))))) 43.98/12.00 1(1(2(0(1(5(1(x1))))))) -> 4(0(1(1(2(3(1(x1))))))) 43.98/12.00 0(5(5(5(0(0(1(0(x1)))))))) -> 0(0(4(0(1(2(2(0(x1)))))))) 43.98/12.00 3(1(0(2(1(5(2(1(x1)))))))) -> 3(5(3(4(5(2(4(1(x1)))))))) 43.98/12.00 3(2(4(5(1(0(4(2(0(x1))))))))) -> 3(3(0(0(4(3(0(2(2(x1))))))))) 43.98/12.00 0(0(2(5(0(3(1(5(1(x1))))))))) -> 5(2(2(0(5(2(3(1(1(x1))))))))) 43.98/12.00 0(1(2(4(1(4(2(0(1(5(x1)))))))))) -> 3(2(3(0(4(2(1(5(3(1(x1)))))))))) 43.98/12.00 0(2(1(4(1(5(1(1(3(2(3(3(1(x1))))))))))))) -> 3(4(2(3(5(5(4(4(4(5(5(3(1(x1))))))))))))) 43.98/12.00 1(1(1(4(2(2(5(0(0(0(1(2(2(x1))))))))))))) -> 5(2(4(2(2(0(2(5(2(0(4(1(1(x1))))))))))))) 43.98/12.00 1(1(2(4(1(5(0(0(1(3(5(5(5(3(0(x1))))))))))))))) -> 1(2(5(4(2(3(1(2(1(0(1(2(5(3(0(x1))))))))))))))) 43.98/12.00 0(0(4(2(3(3(2(4(4(4(1(4(2(0(2(1(3(x1))))))))))))))))) -> 0(0(2(3(1(4(5(1(2(0(4(4(1(2(1(5(2(x1))))))))))))))))) 43.98/12.00 0(2(1(4(0(5(0(4(4(3(0(0(2(4(0(0(5(0(x1)))))))))))))))))) -> 4(2(0(4(2(1(5(2(3(2(2(4(0(3(5(5(3(0(x1)))))))))))))))))) 43.98/12.00 0(1(2(0(1(4(2(2(2(0(2(5(4(5(2(3(4(5(1(4(x1)))))))))))))))))))) -> 0(0(2(3(1(3(4(0(4(1(0(2(4(0(2(3(2(5(5(4(x1)))))))))))))))))))) 43.98/12.00 2(0(1(5(0(3(5(1(4(4(3(5(3(5(3(2(2(1(4(0(2(x1))))))))))))))))))))) -> 2(3(0(4(0(3(5(3(3(3(5(4(1(0(0(2(4(4(2(2(2(x1))))))))))))))))))))) 43.98/12.00 3(5(1(3(4(2(5(2(5(3(4(5(5(1(2(5(1(1(0(1(5(x1))))))))))))))))))))) -> 3(0(5(2(0(3(4(1(0(4(5(2(2(3(4(5(4(3(0(1(1(x1))))))))))))))))))))) 43.98/12.00 43.98/12.00 Q is empty. 43.98/12.00 We have to consider all minimal (P,Q,R)-chains. 43.98/12.00 ---------------------------------------- 43.98/12.00 43.98/12.00 (15) PisEmptyProof (EQUIVALENT) 43.98/12.00 The TRS P is empty. Hence, there is no (P,Q,R) chain. 43.98/12.00 ---------------------------------------- 43.98/12.00 43.98/12.00 (16) 43.98/12.00 YES 44.38/12.08 EOF