YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) DependencyPairsProof [EQUIVALENT, 2004 ms] (4) QDP (5) DependencyGraphProof [EQUIVALENT, 26 ms] (6) QDP (7) QDPOrderProof [EQUIVALENT, 786 ms] (8) QDP (9) DependencyGraphProof [EQUIVALENT, 0 ms] (10) QDP (11) QDPOrderProof [EQUIVALENT, 2141 ms] (12) QDP (13) QDPOrderProof [EQUIVALENT, 2464 ms] (14) QDP (15) DependencyGraphProof [EQUIVALENT, 0 ms] (16) QDP (17) UsableRulesProof [EQUIVALENT, 0 ms] (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0(0(0(0(x1)))) -> 0(0(0(1(0(2(x1)))))) 0(0(0(0(x1)))) -> 0(0(0(2(2(0(x1)))))) 0(0(0(0(x1)))) -> 0(1(0(2(0(0(x1)))))) 0(0(3(0(x1)))) -> 0(0(2(0(3(x1))))) 0(0(3(0(x1)))) -> 0(0(1(3(4(0(x1)))))) 0(0(3(0(x1)))) -> 0(2(0(3(2(0(x1)))))) 0(0(3(0(x1)))) -> 0(3(0(4(0(4(x1)))))) 0(0(3(0(x1)))) -> 0(4(3(0(2(0(x1)))))) 0(0(3(0(x1)))) -> 0(4(3(0(4(0(x1)))))) 0(0(3(0(x1)))) -> 1(0(2(0(0(3(x1)))))) 0(0(3(0(x1)))) -> 2(0(0(2(0(3(x1)))))) 0(0(3(0(x1)))) -> 3(0(0(2(2(0(x1)))))) 0(5(0(0(x1)))) -> 0(2(0(0(5(x1))))) 0(5(0(0(x1)))) -> 0(0(3(5(1(0(x1)))))) 0(5(0(0(x1)))) -> 0(0(5(1(0(4(x1)))))) 0(5(0(0(x1)))) -> 0(2(0(4(5(0(x1)))))) 0(5(0(0(x1)))) -> 0(2(5(0(4(0(x1)))))) 0(5(0(0(x1)))) -> 0(4(0(0(5(2(x1)))))) 0(5(0(0(x1)))) -> 0(4(0(1(5(0(x1)))))) 0(5(0(0(x1)))) -> 0(5(1(0(1(0(x1)))))) 0(5(0(0(x1)))) -> 2(0(0(2(0(5(x1)))))) 0(5(0(0(x1)))) -> 2(0(5(0(2(0(x1)))))) 0(5(0(0(x1)))) -> 5(0(1(0(2(0(x1)))))) 0(5(0(0(x1)))) -> 5(0(1(0(4(0(x1)))))) 0(5(3(0(x1)))) -> 0(0(3(2(5(x1))))) 0(5(3(0(x1)))) -> 0(0(3(5(1(x1))))) 0(5(3(0(x1)))) -> 0(0(3(5(2(x1))))) 0(5(3(0(x1)))) -> 0(0(5(2(3(x1))))) 0(5(3(0(x1)))) -> 0(2(0(3(5(x1))))) 0(5(3(0(x1)))) -> 0(3(4(0(5(x1))))) 0(5(3(0(x1)))) -> 5(3(0(1(0(x1))))) 0(5(3(0(x1)))) -> 0(0(1(3(5(4(x1)))))) 0(5(3(0(x1)))) -> 0(0(5(2(2(3(x1)))))) 0(5(3(0(x1)))) -> 0(0(5(2(4(3(x1)))))) 0(5(3(0(x1)))) -> 0(1(5(3(4(0(x1)))))) 0(5(3(0(x1)))) -> 0(2(0(3(1(5(x1)))))) 0(5(3(0(x1)))) -> 0(3(5(2(0(4(x1)))))) 0(5(3(0(x1)))) -> 0(5(2(0(4(3(x1)))))) 0(5(3(0(x1)))) -> 0(5(2(2(0(3(x1)))))) 0(5(3(0(x1)))) -> 0(5(3(4(4(0(x1)))))) 0(5(3(0(x1)))) -> 0(5(3(5(2(0(x1)))))) 0(5(3(0(x1)))) -> 1(0(3(5(4(0(x1)))))) 0(5(3(0(x1)))) -> 1(3(5(4(0(0(x1)))))) 0(5(3(0(x1)))) -> 2(3(0(5(1(0(x1)))))) 0(5(3(0(x1)))) -> 2(5(1(0(0(3(x1)))))) 0(5(3(0(x1)))) -> 3(0(0(2(5(2(x1)))))) 0(5(3(0(x1)))) -> 3(0(3(5(1(0(x1)))))) 0(5(3(0(x1)))) -> 3(0(4(0(5(1(x1)))))) 0(5(3(0(x1)))) -> 3(0(4(0(5(2(x1)))))) 0(5(3(0(x1)))) -> 3(0(5(1(0(1(x1)))))) 0(5(3(0(x1)))) -> 3(0(5(1(2(0(x1)))))) 0(5(3(0(x1)))) -> 3(2(0(1(5(0(x1)))))) 0(5(3(0(x1)))) -> 3(4(0(5(2(0(x1)))))) 0(5(3(0(x1)))) -> 3(4(5(0(0(2(x1)))))) 0(5(3(0(x1)))) -> 3(5(1(0(0(2(x1)))))) 0(5(3(0(x1)))) -> 5(0(1(3(2(0(x1)))))) 0(5(3(0(x1)))) -> 5(2(2(3(0(0(x1)))))) 0(5(3(0(x1)))) -> 5(3(0(1(1(0(x1)))))) 3(0(0(0(x1)))) -> 0(0(3(2(0(1(x1)))))) 3(0(0(0(x1)))) -> 0(2(1(0(3(0(x1)))))) 3(0(0(0(x1)))) -> 0(3(0(2(2(0(x1)))))) 3(0(0(0(x1)))) -> 0(3(0(2(4(0(x1)))))) 3(0(0(0(x1)))) -> 3(0(2(0(2(0(x1)))))) 3(0(3(0(x1)))) -> 0(0(3(2(3(4(x1)))))) 3(0(3(0(x1)))) -> 0(0(3(4(3(4(x1)))))) 3(0(3(0(x1)))) -> 1(0(2(0(3(3(x1)))))) 3(0(3(0(x1)))) -> 3(0(1(0(2(3(x1)))))) 3(0(3(0(x1)))) -> 3(0(2(0(3(1(x1)))))) 3(0(3(0(x1)))) -> 3(0(3(1(0(1(x1)))))) 3(0(3(0(x1)))) -> 3(0(3(5(2(0(x1)))))) 3(0(3(0(x1)))) -> 3(0(4(0(2(3(x1)))))) 3(0(3(0(x1)))) -> 3(1(0(3(2(0(x1)))))) 3(0(3(0(x1)))) -> 3(4(0(1(0(3(x1)))))) 3(3(0(0(x1)))) -> 0(3(3(4(0(x1))))) 3(3(0(0(x1)))) -> 3(0(2(0(3(x1))))) 3(3(0(0(x1)))) -> 3(0(3(2(0(x1))))) 3(3(0(0(x1)))) -> 0(1(0(1(3(3(x1)))))) 3(3(0(0(x1)))) -> 0(1(3(4(0(3(x1)))))) 3(3(0(0(x1)))) -> 0(2(2(3(3(0(x1)))))) 3(3(0(0(x1)))) -> 0(2(4(0(3(3(x1)))))) 3(3(0(0(x1)))) -> 2(0(3(3(2(0(x1)))))) 3(3(0(0(x1)))) -> 3(0(1(0(4(3(x1)))))) 3(3(0(0(x1)))) -> 3(0(2(2(0(3(x1)))))) 3(3(0(0(x1)))) -> 3(0(3(2(0(2(x1)))))) 3(3(0(0(x1)))) -> 3(0(3(4(1(0(x1)))))) 3(3(0(0(x1)))) -> 3(1(0(1(0(3(x1)))))) 3(3(0(0(x1)))) -> 3(3(0(4(0(2(x1)))))) 3(3(0(0(x1)))) -> 3(3(0(4(0(4(x1)))))) 3(3(0(0(x1)))) -> 3(3(0(4(1(0(x1)))))) 3(3(0(0(x1)))) -> 3(4(0(3(4(0(x1)))))) 3(3(3(0(x1)))) -> 3(3(1(0(3(x1))))) 3(3(3(0(x1)))) -> 3(3(3(2(0(x1))))) 3(3(3(0(x1)))) -> 0(3(1(3(2(3(x1)))))) 3(3(3(0(x1)))) -> 3(2(0(3(2(3(x1)))))) 3(3(3(0(x1)))) -> 3(3(1(2(0(3(x1)))))) 3(3(3(0(x1)))) -> 3(3(2(1(0(3(x1)))))) 3(3(3(0(x1)))) -> 3(3(5(1(0(3(x1)))))) 3(5(0(0(x1)))) -> 0(0(3(5(1(x1))))) 3(5(0(0(x1)))) -> 0(0(3(5(2(x1))))) 3(5(0(0(x1)))) -> 0(0(3(5(4(x1))))) 3(5(0(0(x1)))) -> 0(3(5(2(0(x1))))) 3(5(0(0(x1)))) -> 0(3(5(4(0(x1))))) 3(5(0(0(x1)))) -> 3(0(0(2(5(x1))))) 3(5(0(0(x1)))) -> 3(0(5(1(0(x1))))) 3(5(0(0(x1)))) -> 0(0(1(5(1(3(x1)))))) 3(5(0(0(x1)))) -> 0(0(1(5(2(3(x1)))))) 3(5(0(0(x1)))) -> 0(0(3(4(5(2(x1)))))) 3(5(0(0(x1)))) -> 0(1(3(5(2(0(x1)))))) 3(5(0(0(x1)))) -> 0(2(0(3(1(5(x1)))))) 3(5(0(0(x1)))) -> 0(3(1(0(4(5(x1)))))) 3(5(0(0(x1)))) -> 0(3(2(0(5(1(x1)))))) 3(5(0(0(x1)))) -> 0(3(5(2(2(0(x1)))))) 3(5(0(0(x1)))) -> 2(5(3(0(1(0(x1)))))) 3(5(0(0(x1)))) -> 3(0(0(2(4(5(x1)))))) 3(5(0(0(x1)))) -> 3(0(4(0(5(1(x1)))))) 3(5(0(0(x1)))) -> 3(0(4(1(0(5(x1)))))) 3(5(0(0(x1)))) -> 3(1(0(4(0(5(x1)))))) 3(5(0(0(x1)))) -> 3(2(0(5(2(0(x1)))))) 3(5(0(0(x1)))) -> 3(2(5(0(2(0(x1)))))) 3(5(0(0(x1)))) -> 3(5(0(5(1(0(x1)))))) 3(5(0(0(x1)))) -> 3(5(1(0(4(0(x1)))))) 3(5(0(0(x1)))) -> 3(5(4(0(4(0(x1)))))) 3(5(0(0(x1)))) -> 5(3(1(0(4(0(x1)))))) 3(5(3(0(x1)))) -> 0(3(5(2(3(x1))))) 3(5(3(0(x1)))) -> 3(0(3(2(5(x1))))) 3(5(3(0(x1)))) -> 3(0(3(5(4(x1))))) 3(5(3(0(x1)))) -> 3(3(0(5(2(x1))))) 3(5(3(0(x1)))) -> 3(3(4(0(5(x1))))) 3(5(3(0(x1)))) -> 3(5(1(0(3(x1))))) 3(5(3(0(x1)))) -> 0(2(3(3(4(5(x1)))))) 3(5(3(0(x1)))) -> 0(2(3(3(5(2(x1)))))) 3(5(3(0(x1)))) -> 0(3(1(3(5(4(x1)))))) 3(5(3(0(x1)))) -> 0(3(3(5(1(2(x1)))))) 3(5(3(0(x1)))) -> 0(3(3(5(2(1(x1)))))) 3(5(3(0(x1)))) -> 0(3(4(5(4(3(x1)))))) 3(5(3(0(x1)))) -> 0(4(3(3(5(4(x1)))))) 3(5(3(0(x1)))) -> 2(3(5(2(0(3(x1)))))) 3(5(3(0(x1)))) -> 3(0(3(2(2(5(x1)))))) 3(5(3(0(x1)))) -> 3(0(3(2(5(1(x1)))))) 3(5(3(0(x1)))) -> 3(0(3(4(2(5(x1)))))) 3(5(3(0(x1)))) -> 3(0(3(5(5(2(x1)))))) 3(5(3(0(x1)))) -> 3(0(5(1(3(1(x1)))))) 3(5(3(0(x1)))) -> 3(0(5(2(2(3(x1)))))) 3(5(3(0(x1)))) -> 3(1(0(3(5(1(x1)))))) 3(5(3(0(x1)))) -> 3(3(4(0(5(2(x1)))))) 3(5(3(0(x1)))) -> 3(3(4(1(0(5(x1)))))) 3(5(3(0(x1)))) -> 3(3(4(2(0(5(x1)))))) 3(5(3(0(x1)))) -> 3(5(3(4(4(0(x1)))))) 3(5(3(0(x1)))) -> 3(5(3(5(2(0(x1)))))) 3(5(3(0(x1)))) -> 3(5(3(5(4(0(x1)))))) 3(5(3(0(x1)))) -> 5(2(2(0(3(3(x1)))))) 3(5(3(0(x1)))) -> 5(3(0(1(3(1(x1)))))) 3(5(3(0(x1)))) -> 5(3(1(3(1(0(x1)))))) 3(5(3(0(x1)))) -> 5(3(3(4(0(2(x1)))))) 3(5(3(0(x1)))) -> 5(3(3(4(0(4(x1)))))) 3(5(3(0(x1)))) -> 5(3(3(4(1(0(x1)))))) 0(0(3(0(0(x1))))) -> 0(3(0(2(0(0(x1)))))) 0(0(3(0(5(x1))))) -> 0(0(2(3(0(5(x1)))))) 0(0(5(3(0(x1))))) -> 0(3(0(5(2(0(x1)))))) 0(5(3(0(5(x1))))) -> 3(0(5(1(0(5(x1)))))) 0(5(3(0(5(x1))))) -> 3(4(0(0(5(5(x1)))))) 0(5(3(3(0(x1))))) -> 5(3(0(2(0(3(x1)))))) 3(0(0(3(0(x1))))) -> 0(3(3(4(0(0(x1)))))) 3(0(3(3(0(x1))))) -> 3(3(2(0(0(3(x1)))))) 3(0(5(3(0(x1))))) -> 3(3(0(5(4(0(x1)))))) 3(1(5(3(0(x1))))) -> 3(0(5(2(3(1(x1)))))) 3(2(5(0(0(x1))))) -> 0(3(2(0(5(4(x1)))))) 3(2(5(0(0(x1))))) -> 0(5(2(2(0(3(x1)))))) 3(2(5(3(0(x1))))) -> 3(2(3(2(5(0(x1)))))) 3(2(5(3(0(x1))))) -> 3(5(2(0(3(1(x1)))))) 3(2(5(3(0(x1))))) -> 3(5(2(0(3(4(x1)))))) 3(3(0(0(5(x1))))) -> 3(5(0(3(4(0(x1)))))) 3(3(3(0(5(x1))))) -> 3(0(3(5(1(3(x1)))))) 3(4(5(0(0(x1))))) -> 0(1(3(4(0(5(x1)))))) 3(4(5(0(0(x1))))) -> 3(0(4(0(1(5(x1)))))) 3(4(5(0(0(x1))))) -> 3(4(0(5(1(0(x1)))))) 3(5(3(0(0(x1))))) -> 0(3(5(3(0(2(x1)))))) 3(5(3(0(0(x1))))) -> 3(5(3(2(0(0(x1)))))) 3(5(3(0(5(x1))))) -> 3(5(4(0(3(5(x1)))))) 3(5(3(3(0(x1))))) -> 3(5(1(0(3(3(x1)))))) 3(5(3(3(0(x1))))) -> 3(5(3(3(4(0(x1)))))) 3(5(5(3(0(x1))))) -> 0(5(3(5(1(3(x1)))))) 3(5(5(3(0(x1))))) -> 3(5(3(5(4(0(x1)))))) 5(0(3(3(0(x1))))) -> 0(3(5(3(0(1(x1)))))) 5(0(3(3(0(x1))))) -> 0(3(5(3(2(0(x1)))))) 5(0(3(3(0(x1))))) -> 3(3(0(2(0(5(x1)))))) 5(0(3(3(0(x1))))) -> 3(5(3(0(2(0(x1)))))) 5(0(5(3(0(x1))))) -> 5(3(4(5(0(0(x1)))))) 5(3(0(3(0(x1))))) -> 3(0(3(5(1(0(x1)))))) 5(3(0(3(0(x1))))) -> 3(5(1(3(0(0(x1)))))) 5(3(3(3(0(x1))))) -> 0(5(2(3(3(3(x1)))))) 5(3(3(3(0(x1))))) -> 5(3(0(2(3(3(x1)))))) 5(3(5(3(0(x1))))) -> 5(3(0(3(2(5(x1)))))) 5(5(3(0(5(x1))))) -> 5(3(0(2(5(5(x1)))))) Q is empty. ---------------------------------------- (1) QTRS Reverse (EQUIVALENT) We applied the QTRS Reverse Processor [REVERSE]. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0(0(0(0(x1)))) -> 2(0(1(0(0(0(x1)))))) 0(0(0(0(x1)))) -> 0(2(2(0(0(0(x1)))))) 0(0(0(0(x1)))) -> 0(0(2(0(1(0(x1)))))) 0(3(0(0(x1)))) -> 3(0(2(0(0(x1))))) 0(3(0(0(x1)))) -> 0(4(3(1(0(0(x1)))))) 0(3(0(0(x1)))) -> 0(2(3(0(2(0(x1)))))) 0(3(0(0(x1)))) -> 4(0(4(0(3(0(x1)))))) 0(3(0(0(x1)))) -> 0(2(0(3(4(0(x1)))))) 0(3(0(0(x1)))) -> 0(4(0(3(4(0(x1)))))) 0(3(0(0(x1)))) -> 3(0(0(2(0(1(x1)))))) 0(3(0(0(x1)))) -> 3(0(2(0(0(2(x1)))))) 0(3(0(0(x1)))) -> 0(2(2(0(0(3(x1)))))) 0(0(5(0(x1)))) -> 5(0(0(2(0(x1))))) 0(0(5(0(x1)))) -> 0(1(5(3(0(0(x1)))))) 0(0(5(0(x1)))) -> 4(0(1(5(0(0(x1)))))) 0(0(5(0(x1)))) -> 0(5(4(0(2(0(x1)))))) 0(0(5(0(x1)))) -> 0(4(0(5(2(0(x1)))))) 0(0(5(0(x1)))) -> 2(5(0(0(4(0(x1)))))) 0(0(5(0(x1)))) -> 0(5(1(0(4(0(x1)))))) 0(0(5(0(x1)))) -> 0(1(0(1(5(0(x1)))))) 0(0(5(0(x1)))) -> 5(0(2(0(0(2(x1)))))) 0(0(5(0(x1)))) -> 0(2(0(5(0(2(x1)))))) 0(0(5(0(x1)))) -> 0(2(0(1(0(5(x1)))))) 0(0(5(0(x1)))) -> 0(4(0(1(0(5(x1)))))) 0(3(5(0(x1)))) -> 5(2(3(0(0(x1))))) 0(3(5(0(x1)))) -> 1(5(3(0(0(x1))))) 0(3(5(0(x1)))) -> 2(5(3(0(0(x1))))) 0(3(5(0(x1)))) -> 3(2(5(0(0(x1))))) 0(3(5(0(x1)))) -> 5(3(0(2(0(x1))))) 0(3(5(0(x1)))) -> 5(0(4(3(0(x1))))) 0(3(5(0(x1)))) -> 0(1(0(3(5(x1))))) 0(3(5(0(x1)))) -> 4(5(3(1(0(0(x1)))))) 0(3(5(0(x1)))) -> 3(2(2(5(0(0(x1)))))) 0(3(5(0(x1)))) -> 3(4(2(5(0(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(3(5(1(0(x1)))))) 0(3(5(0(x1)))) -> 5(1(3(0(2(0(x1)))))) 0(3(5(0(x1)))) -> 4(0(2(5(3(0(x1)))))) 0(3(5(0(x1)))) -> 3(4(0(2(5(0(x1)))))) 0(3(5(0(x1)))) -> 3(0(2(2(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(4(3(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(2(5(3(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(5(3(0(1(x1)))))) 0(3(5(0(x1)))) -> 0(0(4(5(3(1(x1)))))) 0(3(5(0(x1)))) -> 0(1(5(0(3(2(x1)))))) 0(3(5(0(x1)))) -> 3(0(0(1(5(2(x1)))))) 0(3(5(0(x1)))) -> 2(5(2(0(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(1(5(3(0(3(x1)))))) 0(3(5(0(x1)))) -> 1(5(0(4(0(3(x1)))))) 0(3(5(0(x1)))) -> 2(5(0(4(0(3(x1)))))) 0(3(5(0(x1)))) -> 1(0(1(5(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(1(5(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(5(1(0(2(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(5(0(4(3(x1)))))) 0(3(5(0(x1)))) -> 2(0(0(5(4(3(x1)))))) 0(3(5(0(x1)))) -> 2(0(0(1(5(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(3(1(0(5(x1)))))) 0(3(5(0(x1)))) -> 0(0(3(2(2(5(x1)))))) 0(3(5(0(x1)))) -> 0(1(1(0(3(5(x1)))))) 0(0(0(3(x1)))) -> 1(0(2(3(0(0(x1)))))) 0(0(0(3(x1)))) -> 0(3(0(1(2(0(x1)))))) 0(0(0(3(x1)))) -> 0(2(2(0(3(0(x1)))))) 0(0(0(3(x1)))) -> 0(4(2(0(3(0(x1)))))) 0(0(0(3(x1)))) -> 0(2(0(2(0(3(x1)))))) 0(3(0(3(x1)))) -> 4(3(2(3(0(0(x1)))))) 0(3(0(3(x1)))) -> 4(3(4(3(0(0(x1)))))) 0(3(0(3(x1)))) -> 3(3(0(2(0(1(x1)))))) 0(3(0(3(x1)))) -> 3(2(0(1(0(3(x1)))))) 0(3(0(3(x1)))) -> 1(3(0(2(0(3(x1)))))) 0(3(0(3(x1)))) -> 1(0(1(3(0(3(x1)))))) 0(3(0(3(x1)))) -> 0(2(5(3(0(3(x1)))))) 0(3(0(3(x1)))) -> 3(2(0(4(0(3(x1)))))) 0(3(0(3(x1)))) -> 0(2(3(0(1(3(x1)))))) 0(3(0(3(x1)))) -> 3(0(1(0(4(3(x1)))))) 0(0(3(3(x1)))) -> 0(4(3(3(0(x1))))) 0(0(3(3(x1)))) -> 3(0(2(0(3(x1))))) 0(0(3(3(x1)))) -> 0(2(3(0(3(x1))))) 0(0(3(3(x1)))) -> 3(3(1(0(1(0(x1)))))) 0(0(3(3(x1)))) -> 3(0(4(3(1(0(x1)))))) 0(0(3(3(x1)))) -> 0(3(3(2(2(0(x1)))))) 0(0(3(3(x1)))) -> 3(3(0(4(2(0(x1)))))) 0(0(3(3(x1)))) -> 0(2(3(3(0(2(x1)))))) 0(0(3(3(x1)))) -> 3(4(0(1(0(3(x1)))))) 0(0(3(3(x1)))) -> 3(0(2(2(0(3(x1)))))) 0(0(3(3(x1)))) -> 2(0(2(3(0(3(x1)))))) 0(0(3(3(x1)))) -> 0(1(4(3(0(3(x1)))))) 0(0(3(3(x1)))) -> 3(0(1(0(1(3(x1)))))) 0(0(3(3(x1)))) -> 2(0(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 4(0(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 0(1(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 0(4(3(0(4(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(3(3(x1))))) 0(3(3(3(x1)))) -> 0(2(3(3(3(x1))))) 0(3(3(3(x1)))) -> 3(2(3(1(3(0(x1)))))) 0(3(3(3(x1)))) -> 3(2(3(0(2(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(2(1(3(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(2(3(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(5(3(3(x1)))))) 0(0(5(3(x1)))) -> 1(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 2(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 4(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 0(2(5(3(0(x1))))) 0(0(5(3(x1)))) -> 0(4(5(3(0(x1))))) 0(0(5(3(x1)))) -> 5(2(0(0(3(x1))))) 0(0(5(3(x1)))) -> 0(1(5(0(3(x1))))) 0(0(5(3(x1)))) -> 3(1(5(1(0(0(x1)))))) 0(0(5(3(x1)))) -> 3(2(5(1(0(0(x1)))))) 0(0(5(3(x1)))) -> 2(5(4(3(0(0(x1)))))) 0(0(5(3(x1)))) -> 0(2(5(3(1(0(x1)))))) 0(0(5(3(x1)))) -> 5(1(3(0(2(0(x1)))))) 0(0(5(3(x1)))) -> 5(4(0(1(3(0(x1)))))) 0(0(5(3(x1)))) -> 1(5(0(2(3(0(x1)))))) 0(0(5(3(x1)))) -> 0(2(2(5(3(0(x1)))))) 0(0(5(3(x1)))) -> 0(1(0(3(5(2(x1)))))) 0(0(5(3(x1)))) -> 5(4(2(0(0(3(x1)))))) 0(0(5(3(x1)))) -> 1(5(0(4(0(3(x1)))))) 0(0(5(3(x1)))) -> 5(0(1(4(0(3(x1)))))) 0(0(5(3(x1)))) -> 5(0(4(0(1(3(x1)))))) 0(0(5(3(x1)))) -> 0(2(5(0(2(3(x1)))))) 0(0(5(3(x1)))) -> 0(2(0(5(2(3(x1)))))) 0(0(5(3(x1)))) -> 0(1(5(0(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(1(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(4(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(1(3(5(x1)))))) 0(3(5(3(x1)))) -> 3(2(5(3(0(x1))))) 0(3(5(3(x1)))) -> 5(2(3(0(3(x1))))) 0(3(5(3(x1)))) -> 4(5(3(0(3(x1))))) 0(3(5(3(x1)))) -> 2(5(0(3(3(x1))))) 0(3(5(3(x1)))) -> 5(0(4(3(3(x1))))) 0(3(5(3(x1)))) -> 3(0(1(5(3(x1))))) 0(3(5(3(x1)))) -> 5(4(3(3(2(0(x1)))))) 0(3(5(3(x1)))) -> 2(5(3(3(2(0(x1)))))) 0(3(5(3(x1)))) -> 4(5(3(1(3(0(x1)))))) 0(3(5(3(x1)))) -> 2(1(5(3(3(0(x1)))))) 0(3(5(3(x1)))) -> 1(2(5(3(3(0(x1)))))) 0(3(5(3(x1)))) -> 3(4(5(4(3(0(x1)))))) 0(3(5(3(x1)))) -> 4(5(3(3(4(0(x1)))))) 0(3(5(3(x1)))) -> 3(0(2(5(3(2(x1)))))) 0(3(5(3(x1)))) -> 5(2(2(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(5(2(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 5(2(4(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 2(5(5(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(3(1(5(0(3(x1)))))) 0(3(5(3(x1)))) -> 3(2(2(5(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(5(3(0(1(3(x1)))))) 0(3(5(3(x1)))) -> 2(5(0(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 5(0(1(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 5(0(2(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 0(4(4(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 0(2(5(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 0(4(5(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 3(3(0(2(2(5(x1)))))) 0(3(5(3(x1)))) -> 1(3(1(0(3(5(x1)))))) 0(3(5(3(x1)))) -> 0(1(3(1(3(5(x1)))))) 0(3(5(3(x1)))) -> 2(0(4(3(3(5(x1)))))) 0(3(5(3(x1)))) -> 4(0(4(3(3(5(x1)))))) 0(3(5(3(x1)))) -> 0(1(4(3(3(5(x1)))))) 0(0(3(0(0(x1))))) -> 0(0(2(0(3(0(x1)))))) 5(0(3(0(0(x1))))) -> 5(0(3(2(0(0(x1)))))) 0(3(5(0(0(x1))))) -> 0(2(5(0(3(0(x1)))))) 5(0(3(5(0(x1))))) -> 5(0(1(5(0(3(x1)))))) 5(0(3(5(0(x1))))) -> 5(5(0(0(4(3(x1)))))) 0(3(3(5(0(x1))))) -> 3(0(2(0(3(5(x1)))))) 0(3(0(0(3(x1))))) -> 0(0(4(3(3(0(x1)))))) 0(3(3(0(3(x1))))) -> 3(0(0(2(3(3(x1)))))) 0(3(5(0(3(x1))))) -> 0(4(5(0(3(3(x1)))))) 0(3(5(1(3(x1))))) -> 1(3(2(5(0(3(x1)))))) 0(0(5(2(3(x1))))) -> 4(5(0(2(3(0(x1)))))) 0(0(5(2(3(x1))))) -> 3(0(2(2(5(0(x1)))))) 0(3(5(2(3(x1))))) -> 0(5(2(3(2(3(x1)))))) 0(3(5(2(3(x1))))) -> 1(3(0(2(5(3(x1)))))) 0(3(5(2(3(x1))))) -> 4(3(0(2(5(3(x1)))))) 5(0(0(3(3(x1))))) -> 0(4(3(0(5(3(x1)))))) 5(0(3(3(3(x1))))) -> 3(1(5(3(0(3(x1)))))) 0(0(5(4(3(x1))))) -> 5(0(4(3(1(0(x1)))))) 0(0(5(4(3(x1))))) -> 5(1(0(4(0(3(x1)))))) 0(0(5(4(3(x1))))) -> 0(1(5(0(4(3(x1)))))) 0(0(3(5(3(x1))))) -> 2(0(3(5(3(0(x1)))))) 0(0(3(5(3(x1))))) -> 0(0(2(3(5(3(x1)))))) 5(0(3(5(3(x1))))) -> 5(3(0(4(5(3(x1)))))) 0(3(3(5(3(x1))))) -> 3(3(0(1(5(3(x1)))))) 0(3(3(5(3(x1))))) -> 0(4(3(3(5(3(x1)))))) 0(3(5(5(3(x1))))) -> 3(1(5(3(5(0(x1)))))) 0(3(5(5(3(x1))))) -> 0(4(5(3(5(3(x1)))))) 0(3(3(0(5(x1))))) -> 1(0(3(5(3(0(x1)))))) 0(3(3(0(5(x1))))) -> 0(2(3(5(3(0(x1)))))) 0(3(3(0(5(x1))))) -> 5(0(2(0(3(3(x1)))))) 0(3(3(0(5(x1))))) -> 0(2(0(3(5(3(x1)))))) 0(3(5(0(5(x1))))) -> 0(0(5(4(3(5(x1)))))) 0(3(0(3(5(x1))))) -> 0(1(5(3(0(3(x1)))))) 0(3(0(3(5(x1))))) -> 0(0(3(1(5(3(x1)))))) 0(3(3(3(5(x1))))) -> 3(3(3(2(5(0(x1)))))) 0(3(3(3(5(x1))))) -> 3(3(2(0(3(5(x1)))))) 0(3(5(3(5(x1))))) -> 5(2(3(0(3(5(x1)))))) 5(0(3(5(5(x1))))) -> 5(5(2(0(3(5(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(0(0(x1)))) -> 0^1(1(0(0(0(x1))))) 0^1(0(0(0(x1)))) -> 0^1(2(2(0(0(0(x1)))))) 0^1(0(0(0(x1)))) -> 0^1(0(2(0(1(0(x1)))))) 0^1(0(0(0(x1)))) -> 0^1(2(0(1(0(x1))))) 0^1(0(0(0(x1)))) -> 0^1(1(0(x1))) 0^1(3(0(0(x1)))) -> 0^1(2(0(0(x1)))) 0^1(3(0(0(x1)))) -> 0^1(4(3(1(0(0(x1)))))) 0^1(3(0(0(x1)))) -> 0^1(2(3(0(2(0(x1)))))) 0^1(3(0(0(x1)))) -> 0^1(2(0(x1))) 0^1(3(0(0(x1)))) -> 0^1(4(0(3(0(x1))))) 0^1(3(0(0(x1)))) -> 0^1(3(0(x1))) 0^1(3(0(0(x1)))) -> 0^1(2(0(3(4(0(x1)))))) 0^1(3(0(0(x1)))) -> 0^1(3(4(0(x1)))) 0^1(3(0(0(x1)))) -> 0^1(4(0(3(4(0(x1)))))) 0^1(3(0(0(x1)))) -> 0^1(0(2(0(1(x1))))) 0^1(3(0(0(x1)))) -> 0^1(2(0(1(x1)))) 0^1(3(0(0(x1)))) -> 0^1(1(x1)) 0^1(3(0(0(x1)))) -> 0^1(2(0(0(2(x1))))) 0^1(3(0(0(x1)))) -> 0^1(0(2(x1))) 0^1(3(0(0(x1)))) -> 0^1(2(x1)) 0^1(3(0(0(x1)))) -> 0^1(2(2(0(0(3(x1)))))) 0^1(3(0(0(x1)))) -> 0^1(0(3(x1))) 0^1(3(0(0(x1)))) -> 0^1(3(x1)) 0^1(0(5(0(x1)))) -> 5^1(0(0(2(0(x1))))) 0^1(0(5(0(x1)))) -> 0^1(0(2(0(x1)))) 0^1(0(5(0(x1)))) -> 0^1(2(0(x1))) 0^1(0(5(0(x1)))) -> 0^1(1(5(3(0(0(x1)))))) 0^1(0(5(0(x1)))) -> 5^1(3(0(0(x1)))) 0^1(0(5(0(x1)))) -> 0^1(0(x1)) 0^1(0(5(0(x1)))) -> 0^1(1(5(0(0(x1))))) 0^1(0(5(0(x1)))) -> 5^1(0(0(x1))) 0^1(0(5(0(x1)))) -> 0^1(5(4(0(2(0(x1)))))) 0^1(0(5(0(x1)))) -> 5^1(4(0(2(0(x1))))) 0^1(0(5(0(x1)))) -> 0^1(4(0(5(2(0(x1)))))) 0^1(0(5(0(x1)))) -> 0^1(5(2(0(x1)))) 0^1(0(5(0(x1)))) -> 5^1(2(0(x1))) 0^1(0(5(0(x1)))) -> 5^1(0(0(4(0(x1))))) 0^1(0(5(0(x1)))) -> 0^1(0(4(0(x1)))) 0^1(0(5(0(x1)))) -> 0^1(4(0(x1))) 0^1(0(5(0(x1)))) -> 0^1(5(1(0(4(0(x1)))))) 0^1(0(5(0(x1)))) -> 5^1(1(0(4(0(x1))))) 0^1(0(5(0(x1)))) -> 0^1(1(0(1(5(0(x1)))))) 0^1(0(5(0(x1)))) -> 0^1(1(5(0(x1)))) 0^1(0(5(0(x1)))) -> 5^1(0(2(0(0(2(x1)))))) 0^1(0(5(0(x1)))) -> 0^1(2(0(0(2(x1))))) 0^1(0(5(0(x1)))) -> 0^1(0(2(x1))) 0^1(0(5(0(x1)))) -> 0^1(2(x1)) 0^1(0(5(0(x1)))) -> 0^1(2(0(5(0(2(x1)))))) 0^1(0(5(0(x1)))) -> 0^1(5(0(2(x1)))) 0^1(0(5(0(x1)))) -> 5^1(0(2(x1))) 0^1(0(5(0(x1)))) -> 0^1(2(0(1(0(5(x1)))))) 0^1(0(5(0(x1)))) -> 0^1(1(0(5(x1)))) 0^1(0(5(0(x1)))) -> 0^1(5(x1)) 0^1(0(5(0(x1)))) -> 5^1(x1) 0^1(0(5(0(x1)))) -> 0^1(4(0(1(0(5(x1)))))) 0^1(3(5(0(x1)))) -> 5^1(2(3(0(0(x1))))) 0^1(3(5(0(x1)))) -> 0^1(0(x1)) 0^1(3(5(0(x1)))) -> 5^1(3(0(0(x1)))) 0^1(3(5(0(x1)))) -> 5^1(0(0(x1))) 0^1(3(5(0(x1)))) -> 5^1(3(0(2(0(x1))))) 0^1(3(5(0(x1)))) -> 0^1(2(0(x1))) 0^1(3(5(0(x1)))) -> 5^1(0(4(3(0(x1))))) 0^1(3(5(0(x1)))) -> 0^1(4(3(0(x1)))) 0^1(3(5(0(x1)))) -> 0^1(1(0(3(5(x1))))) 0^1(3(5(0(x1)))) -> 0^1(3(5(x1))) 0^1(3(5(0(x1)))) -> 5^1(x1) 0^1(3(5(0(x1)))) -> 5^1(3(1(0(0(x1))))) 0^1(3(5(0(x1)))) -> 0^1(4(3(5(1(0(x1)))))) 0^1(3(5(0(x1)))) -> 5^1(1(0(x1))) 0^1(3(5(0(x1)))) -> 5^1(1(3(0(2(0(x1)))))) 0^1(3(5(0(x1)))) -> 0^1(2(5(3(0(x1))))) 0^1(3(5(0(x1)))) -> 5^1(3(0(x1))) 0^1(3(5(0(x1)))) -> 0^1(2(5(0(x1)))) 0^1(3(5(0(x1)))) -> 0^1(2(2(5(0(x1))))) 0^1(3(5(0(x1)))) -> 0^1(4(4(3(5(0(x1)))))) 0^1(3(5(0(x1)))) -> 0^1(2(5(3(5(0(x1)))))) 0^1(3(5(0(x1)))) -> 5^1(3(5(0(x1)))) 0^1(3(5(0(x1)))) -> 0^1(4(5(3(0(1(x1)))))) 0^1(3(5(0(x1)))) -> 5^1(3(0(1(x1)))) 0^1(3(5(0(x1)))) -> 0^1(1(x1)) 0^1(3(5(0(x1)))) -> 0^1(0(4(5(3(1(x1)))))) 0^1(3(5(0(x1)))) -> 0^1(4(5(3(1(x1))))) 0^1(3(5(0(x1)))) -> 5^1(3(1(x1))) 0^1(3(5(0(x1)))) -> 0^1(1(5(0(3(2(x1)))))) 0^1(3(5(0(x1)))) -> 5^1(0(3(2(x1)))) 0^1(3(5(0(x1)))) -> 0^1(3(2(x1))) 0^1(3(5(0(x1)))) -> 0^1(0(1(5(2(x1))))) 0^1(3(5(0(x1)))) -> 0^1(1(5(2(x1)))) 0^1(3(5(0(x1)))) -> 5^1(2(x1)) 0^1(3(5(0(x1)))) -> 5^1(2(0(0(3(x1))))) 0^1(3(5(0(x1)))) -> 0^1(0(3(x1))) 0^1(3(5(0(x1)))) -> 0^1(3(x1)) 0^1(3(5(0(x1)))) -> 0^1(1(5(3(0(3(x1)))))) 0^1(3(5(0(x1)))) -> 5^1(3(0(3(x1)))) 0^1(3(5(0(x1)))) -> 5^1(0(4(0(3(x1))))) 0^1(3(5(0(x1)))) -> 0^1(4(0(3(x1)))) 0^1(3(5(0(x1)))) -> 0^1(1(5(0(3(x1))))) 0^1(3(5(0(x1)))) -> 5^1(0(3(x1))) 0^1(3(5(0(x1)))) -> 0^1(2(1(5(0(3(x1)))))) 0^1(3(5(0(x1)))) -> 0^1(5(1(0(2(3(x1)))))) 0^1(3(5(0(x1)))) -> 5^1(1(0(2(3(x1))))) 0^1(3(5(0(x1)))) -> 0^1(2(3(x1))) 0^1(3(5(0(x1)))) -> 0^1(2(5(0(4(3(x1)))))) 0^1(3(5(0(x1)))) -> 5^1(0(4(3(x1)))) 0^1(3(5(0(x1)))) -> 0^1(4(3(x1))) 0^1(3(5(0(x1)))) -> 0^1(0(5(4(3(x1))))) 0^1(3(5(0(x1)))) -> 0^1(5(4(3(x1)))) 0^1(3(5(0(x1)))) -> 5^1(4(3(x1))) 0^1(3(5(0(x1)))) -> 0^1(0(1(5(3(x1))))) 0^1(3(5(0(x1)))) -> 0^1(1(5(3(x1)))) 0^1(3(5(0(x1)))) -> 5^1(3(x1)) 0^1(3(5(0(x1)))) -> 0^1(2(3(1(0(5(x1)))))) 0^1(3(5(0(x1)))) -> 0^1(5(x1)) 0^1(3(5(0(x1)))) -> 0^1(0(3(2(2(5(x1)))))) 0^1(3(5(0(x1)))) -> 0^1(3(2(2(5(x1))))) 0^1(3(5(0(x1)))) -> 0^1(1(1(0(3(5(x1)))))) 0^1(0(0(3(x1)))) -> 0^1(2(3(0(0(x1))))) 0^1(0(0(3(x1)))) -> 0^1(0(x1)) 0^1(0(0(3(x1)))) -> 0^1(x1) 0^1(0(0(3(x1)))) -> 0^1(3(0(1(2(0(x1)))))) 0^1(0(0(3(x1)))) -> 0^1(1(2(0(x1)))) 0^1(0(0(3(x1)))) -> 0^1(2(2(0(3(0(x1)))))) 0^1(0(0(3(x1)))) -> 0^1(3(0(x1))) 0^1(0(0(3(x1)))) -> 0^1(4(2(0(3(0(x1)))))) 0^1(0(0(3(x1)))) -> 0^1(2(0(2(0(3(x1)))))) 0^1(0(0(3(x1)))) -> 0^1(2(0(3(x1)))) 0^1(3(0(3(x1)))) -> 0^1(0(x1)) 0^1(3(0(3(x1)))) -> 0^1(x1) 0^1(3(0(3(x1)))) -> 0^1(2(0(1(x1)))) 0^1(3(0(3(x1)))) -> 0^1(1(x1)) 0^1(3(0(3(x1)))) -> 0^1(1(0(3(x1)))) 0^1(3(0(3(x1)))) -> 0^1(2(0(3(x1)))) 0^1(3(0(3(x1)))) -> 0^1(1(3(0(3(x1))))) 0^1(3(0(3(x1)))) -> 0^1(2(5(3(0(3(x1)))))) 0^1(3(0(3(x1)))) -> 5^1(3(0(3(x1)))) 0^1(3(0(3(x1)))) -> 0^1(4(0(3(x1)))) 0^1(3(0(3(x1)))) -> 0^1(2(3(0(1(3(x1)))))) 0^1(3(0(3(x1)))) -> 0^1(1(3(x1))) 0^1(3(0(3(x1)))) -> 0^1(1(0(4(3(x1))))) 0^1(3(0(3(x1)))) -> 0^1(4(3(x1))) 0^1(0(3(3(x1)))) -> 0^1(4(3(3(0(x1))))) 0^1(0(3(3(x1)))) -> 0^1(x1) 0^1(0(3(3(x1)))) -> 0^1(2(0(3(x1)))) 0^1(0(3(3(x1)))) -> 0^1(3(x1)) 0^1(0(3(3(x1)))) -> 0^1(2(3(0(3(x1))))) 0^1(0(3(3(x1)))) -> 0^1(1(0(x1))) 0^1(0(3(3(x1)))) -> 0^1(4(3(1(0(x1))))) 0^1(0(3(3(x1)))) -> 0^1(3(3(2(2(0(x1)))))) 0^1(0(3(3(x1)))) -> 0^1(4(2(0(x1)))) 0^1(0(3(3(x1)))) -> 0^1(2(3(3(0(2(x1)))))) 0^1(0(3(3(x1)))) -> 0^1(2(x1)) 0^1(0(3(3(x1)))) -> 0^1(1(0(3(x1)))) 0^1(0(3(3(x1)))) -> 0^1(2(2(0(3(x1))))) 0^1(0(3(3(x1)))) -> 0^1(1(4(3(0(3(x1)))))) 0^1(0(3(3(x1)))) -> 0^1(1(0(1(3(x1))))) 0^1(0(3(3(x1)))) -> 0^1(1(3(x1))) 0^1(0(3(3(x1)))) -> 0^1(4(0(3(3(x1))))) 0^1(0(3(3(x1)))) -> 0^1(1(4(0(3(3(x1)))))) 0^1(0(3(3(x1)))) -> 0^1(4(3(0(4(3(x1)))))) 0^1(0(3(3(x1)))) -> 0^1(4(3(x1))) 0^1(3(3(3(x1)))) -> 0^1(1(3(3(x1)))) 0^1(3(3(3(x1)))) -> 0^1(2(3(3(3(x1))))) 0^1(3(3(3(x1)))) -> 0^1(x1) 0^1(3(3(3(x1)))) -> 0^1(2(3(x1))) 0^1(3(3(3(x1)))) -> 0^1(2(1(3(3(x1))))) 0^1(3(3(3(x1)))) -> 0^1(1(2(3(3(x1))))) 0^1(3(3(3(x1)))) -> 0^1(1(5(3(3(x1))))) 0^1(3(3(3(x1)))) -> 5^1(3(3(x1))) 0^1(0(5(3(x1)))) -> 5^1(3(0(0(x1)))) 0^1(0(5(3(x1)))) -> 0^1(0(x1)) 0^1(0(5(3(x1)))) -> 0^1(x1) 0^1(0(5(3(x1)))) -> 0^1(2(5(3(0(x1))))) 0^1(0(5(3(x1)))) -> 5^1(3(0(x1))) 0^1(0(5(3(x1)))) -> 0^1(4(5(3(0(x1))))) 0^1(0(5(3(x1)))) -> 5^1(2(0(0(3(x1))))) 0^1(0(5(3(x1)))) -> 0^1(0(3(x1))) 0^1(0(5(3(x1)))) -> 0^1(3(x1)) 0^1(0(5(3(x1)))) -> 0^1(1(5(0(3(x1))))) 0^1(0(5(3(x1)))) -> 5^1(0(3(x1))) 0^1(0(5(3(x1)))) -> 5^1(1(0(0(x1)))) 0^1(0(5(3(x1)))) -> 5^1(4(3(0(0(x1))))) 0^1(0(5(3(x1)))) -> 0^1(2(5(3(1(0(x1)))))) 0^1(0(5(3(x1)))) -> 5^1(3(1(0(x1)))) 0^1(0(5(3(x1)))) -> 5^1(1(3(0(2(0(x1)))))) 0^1(0(5(3(x1)))) -> 0^1(2(0(x1))) 0^1(0(5(3(x1)))) -> 5^1(4(0(1(3(0(x1)))))) 0^1(0(5(3(x1)))) -> 0^1(1(3(0(x1)))) 0^1(0(5(3(x1)))) -> 5^1(0(2(3(0(x1))))) 0^1(0(5(3(x1)))) -> 0^1(2(3(0(x1)))) 0^1(0(5(3(x1)))) -> 0^1(2(2(5(3(0(x1)))))) 0^1(0(5(3(x1)))) -> 0^1(1(0(3(5(2(x1)))))) 0^1(0(5(3(x1)))) -> 0^1(3(5(2(x1)))) 0^1(0(5(3(x1)))) -> 5^1(2(x1)) 0^1(0(5(3(x1)))) -> 5^1(4(2(0(0(3(x1)))))) 0^1(0(5(3(x1)))) -> 5^1(0(4(0(3(x1))))) 0^1(0(5(3(x1)))) -> 0^1(4(0(3(x1)))) 0^1(0(5(3(x1)))) -> 5^1(0(1(4(0(3(x1)))))) 0^1(0(5(3(x1)))) -> 0^1(1(4(0(3(x1))))) 0^1(0(5(3(x1)))) -> 5^1(0(4(0(1(3(x1)))))) 0^1(0(5(3(x1)))) -> 0^1(4(0(1(3(x1))))) 0^1(0(5(3(x1)))) -> 0^1(1(3(x1))) 0^1(0(5(3(x1)))) -> 0^1(2(5(0(2(3(x1)))))) 0^1(0(5(3(x1)))) -> 5^1(0(2(3(x1)))) 0^1(0(5(3(x1)))) -> 0^1(2(3(x1))) 0^1(0(5(3(x1)))) -> 0^1(2(0(5(2(3(x1)))))) 0^1(0(5(3(x1)))) -> 0^1(5(2(3(x1)))) 0^1(0(5(3(x1)))) -> 5^1(2(3(x1))) 0^1(0(5(3(x1)))) -> 0^1(1(5(0(5(3(x1)))))) 0^1(0(5(3(x1)))) -> 5^1(0(5(3(x1)))) 0^1(0(5(3(x1)))) -> 0^1(4(0(1(5(3(x1)))))) 0^1(0(5(3(x1)))) -> 0^1(1(5(3(x1)))) 0^1(0(5(3(x1)))) -> 0^1(4(0(4(5(3(x1)))))) 0^1(0(5(3(x1)))) -> 0^1(4(5(3(x1)))) 0^1(0(5(3(x1)))) -> 0^1(4(0(1(3(5(x1)))))) 0^1(0(5(3(x1)))) -> 0^1(1(3(5(x1)))) 0^1(0(5(3(x1)))) -> 5^1(x1) 0^1(3(5(3(x1)))) -> 5^1(3(0(x1))) 0^1(3(5(3(x1)))) -> 0^1(x1) 0^1(3(5(3(x1)))) -> 5^1(2(3(0(3(x1))))) 0^1(3(5(3(x1)))) -> 0^1(3(x1)) 0^1(3(5(3(x1)))) -> 5^1(3(0(3(x1)))) 0^1(3(5(3(x1)))) -> 5^1(0(3(3(x1)))) 0^1(3(5(3(x1)))) -> 0^1(3(3(x1))) 0^1(3(5(3(x1)))) -> 5^1(0(4(3(3(x1))))) 0^1(3(5(3(x1)))) -> 0^1(4(3(3(x1)))) 0^1(3(5(3(x1)))) -> 0^1(1(5(3(x1)))) 0^1(3(5(3(x1)))) -> 5^1(4(3(3(2(0(x1)))))) 0^1(3(5(3(x1)))) -> 5^1(3(3(2(0(x1))))) 0^1(3(5(3(x1)))) -> 5^1(3(1(3(0(x1))))) 0^1(3(5(3(x1)))) -> 5^1(3(3(0(x1)))) 0^1(3(5(3(x1)))) -> 5^1(4(3(0(x1)))) 0^1(3(5(3(x1)))) -> 5^1(3(3(4(0(x1))))) 0^1(3(5(3(x1)))) -> 0^1(2(5(3(2(x1))))) 0^1(3(5(3(x1)))) -> 5^1(3(2(x1))) 0^1(3(5(3(x1)))) -> 5^1(2(2(3(0(3(x1)))))) 0^1(3(5(3(x1)))) -> 5^1(2(4(3(0(3(x1)))))) 0^1(3(5(3(x1)))) -> 5^1(5(3(0(3(x1))))) 0^1(3(5(3(x1)))) -> 5^1(0(3(x1))) 0^1(3(5(3(x1)))) -> 5^1(3(0(1(3(x1))))) 0^1(3(5(3(x1)))) -> 0^1(1(3(x1))) 0^1(3(5(3(x1)))) -> 5^1(0(1(4(3(3(x1)))))) 0^1(3(5(3(x1)))) -> 0^1(1(4(3(3(x1))))) 0^1(3(5(3(x1)))) -> 5^1(0(2(4(3(3(x1)))))) 0^1(3(5(3(x1)))) -> 0^1(2(4(3(3(x1))))) 0^1(3(5(3(x1)))) -> 0^1(4(4(3(5(3(x1)))))) 0^1(3(5(3(x1)))) -> 0^1(2(5(3(5(3(x1)))))) 0^1(3(5(3(x1)))) -> 5^1(3(5(3(x1)))) 0^1(3(5(3(x1)))) -> 0^1(4(5(3(5(3(x1)))))) 0^1(3(5(3(x1)))) -> 0^1(2(2(5(x1)))) 0^1(3(5(3(x1)))) -> 5^1(x1) 0^1(3(5(3(x1)))) -> 0^1(3(5(x1))) 0^1(3(5(3(x1)))) -> 0^1(1(3(1(3(5(x1)))))) 0^1(3(5(3(x1)))) -> 0^1(4(3(3(5(x1))))) 0^1(3(5(3(x1)))) -> 0^1(1(4(3(3(5(x1)))))) 0^1(0(3(0(0(x1))))) -> 0^1(0(2(0(3(0(x1)))))) 0^1(0(3(0(0(x1))))) -> 0^1(2(0(3(0(x1))))) 0^1(0(3(0(0(x1))))) -> 0^1(3(0(x1))) 5^1(0(3(0(0(x1))))) -> 5^1(0(3(2(0(0(x1)))))) 5^1(0(3(0(0(x1))))) -> 0^1(3(2(0(0(x1))))) 0^1(3(5(0(0(x1))))) -> 0^1(2(5(0(3(0(x1)))))) 0^1(3(5(0(0(x1))))) -> 5^1(0(3(0(x1)))) 0^1(3(5(0(0(x1))))) -> 0^1(3(0(x1))) 5^1(0(3(5(0(x1))))) -> 5^1(0(1(5(0(3(x1)))))) 5^1(0(3(5(0(x1))))) -> 0^1(1(5(0(3(x1))))) 5^1(0(3(5(0(x1))))) -> 5^1(0(3(x1))) 5^1(0(3(5(0(x1))))) -> 0^1(3(x1)) 5^1(0(3(5(0(x1))))) -> 5^1(5(0(0(4(3(x1)))))) 5^1(0(3(5(0(x1))))) -> 5^1(0(0(4(3(x1))))) 5^1(0(3(5(0(x1))))) -> 0^1(0(4(3(x1)))) 5^1(0(3(5(0(x1))))) -> 0^1(4(3(x1))) 0^1(3(3(5(0(x1))))) -> 0^1(2(0(3(5(x1))))) 0^1(3(3(5(0(x1))))) -> 0^1(3(5(x1))) 0^1(3(3(5(0(x1))))) -> 5^1(x1) 0^1(3(0(0(3(x1))))) -> 0^1(0(4(3(3(0(x1)))))) 0^1(3(0(0(3(x1))))) -> 0^1(4(3(3(0(x1))))) 0^1(3(0(0(3(x1))))) -> 0^1(x1) 0^1(3(3(0(3(x1))))) -> 0^1(0(2(3(3(x1))))) 0^1(3(3(0(3(x1))))) -> 0^1(2(3(3(x1)))) 0^1(3(5(0(3(x1))))) -> 0^1(4(5(0(3(3(x1)))))) 0^1(3(5(0(3(x1))))) -> 5^1(0(3(3(x1)))) 0^1(3(5(0(3(x1))))) -> 0^1(3(3(x1))) 0^1(3(5(1(3(x1))))) -> 5^1(0(3(x1))) 0^1(3(5(1(3(x1))))) -> 0^1(3(x1)) 0^1(0(5(2(3(x1))))) -> 5^1(0(2(3(0(x1))))) 0^1(0(5(2(3(x1))))) -> 0^1(2(3(0(x1)))) 0^1(0(5(2(3(x1))))) -> 0^1(x1) 0^1(0(5(2(3(x1))))) -> 0^1(2(2(5(0(x1))))) 0^1(0(5(2(3(x1))))) -> 5^1(0(x1)) 0^1(3(5(2(3(x1))))) -> 0^1(5(2(3(2(3(x1)))))) 0^1(3(5(2(3(x1))))) -> 5^1(2(3(2(3(x1))))) 0^1(3(5(2(3(x1))))) -> 0^1(2(5(3(x1)))) 0^1(3(5(2(3(x1))))) -> 5^1(3(x1)) 5^1(0(0(3(3(x1))))) -> 0^1(4(3(0(5(3(x1)))))) 5^1(0(0(3(3(x1))))) -> 0^1(5(3(x1))) 5^1(0(0(3(3(x1))))) -> 5^1(3(x1)) 5^1(0(3(3(3(x1))))) -> 5^1(3(0(3(x1)))) 5^1(0(3(3(3(x1))))) -> 0^1(3(x1)) 0^1(0(5(4(3(x1))))) -> 5^1(0(4(3(1(0(x1)))))) 0^1(0(5(4(3(x1))))) -> 0^1(4(3(1(0(x1))))) 0^1(0(5(4(3(x1))))) -> 0^1(x1) 0^1(0(5(4(3(x1))))) -> 5^1(1(0(4(0(3(x1)))))) 0^1(0(5(4(3(x1))))) -> 0^1(4(0(3(x1)))) 0^1(0(5(4(3(x1))))) -> 0^1(3(x1)) 0^1(0(5(4(3(x1))))) -> 0^1(1(5(0(4(3(x1)))))) 0^1(0(5(4(3(x1))))) -> 5^1(0(4(3(x1)))) 0^1(0(5(4(3(x1))))) -> 0^1(4(3(x1))) 0^1(0(3(5(3(x1))))) -> 0^1(3(5(3(0(x1))))) 0^1(0(3(5(3(x1))))) -> 5^1(3(0(x1))) 0^1(0(3(5(3(x1))))) -> 0^1(x1) 0^1(0(3(5(3(x1))))) -> 0^1(0(2(3(5(3(x1)))))) 0^1(0(3(5(3(x1))))) -> 0^1(2(3(5(3(x1))))) 5^1(0(3(5(3(x1))))) -> 5^1(3(0(4(5(3(x1)))))) 5^1(0(3(5(3(x1))))) -> 0^1(4(5(3(x1)))) 0^1(3(3(5(3(x1))))) -> 0^1(1(5(3(x1)))) 0^1(3(3(5(3(x1))))) -> 0^1(4(3(3(5(3(x1)))))) 0^1(3(5(5(3(x1))))) -> 5^1(3(5(0(x1)))) 0^1(3(5(5(3(x1))))) -> 5^1(0(x1)) 0^1(3(5(5(3(x1))))) -> 0^1(x1) 0^1(3(5(5(3(x1))))) -> 0^1(4(5(3(5(3(x1)))))) 0^1(3(5(5(3(x1))))) -> 5^1(3(5(3(x1)))) 0^1(3(3(0(5(x1))))) -> 0^1(3(5(3(0(x1))))) 0^1(3(3(0(5(x1))))) -> 5^1(3(0(x1))) 0^1(3(3(0(5(x1))))) -> 0^1(x1) 0^1(3(3(0(5(x1))))) -> 0^1(2(3(5(3(0(x1)))))) 0^1(3(3(0(5(x1))))) -> 5^1(0(2(0(3(3(x1)))))) 0^1(3(3(0(5(x1))))) -> 0^1(2(0(3(3(x1))))) 0^1(3(3(0(5(x1))))) -> 0^1(3(3(x1))) 0^1(3(3(0(5(x1))))) -> 0^1(2(0(3(5(3(x1)))))) 0^1(3(3(0(5(x1))))) -> 0^1(3(5(3(x1)))) 0^1(3(3(0(5(x1))))) -> 5^1(3(x1)) 0^1(3(5(0(5(x1))))) -> 0^1(0(5(4(3(5(x1)))))) 0^1(3(5(0(5(x1))))) -> 0^1(5(4(3(5(x1))))) 0^1(3(5(0(5(x1))))) -> 5^1(4(3(5(x1)))) 0^1(3(0(3(5(x1))))) -> 0^1(1(5(3(0(3(x1)))))) 0^1(3(0(3(5(x1))))) -> 5^1(3(0(3(x1)))) 0^1(3(0(3(5(x1))))) -> 0^1(3(x1)) 0^1(3(0(3(5(x1))))) -> 0^1(0(3(1(5(3(x1)))))) 0^1(3(0(3(5(x1))))) -> 0^1(3(1(5(3(x1))))) 0^1(3(0(3(5(x1))))) -> 5^1(3(x1)) 0^1(3(3(3(5(x1))))) -> 5^1(0(x1)) 0^1(3(3(3(5(x1))))) -> 0^1(x1) 0^1(3(3(3(5(x1))))) -> 0^1(3(5(x1))) 0^1(3(5(3(5(x1))))) -> 5^1(2(3(0(3(5(x1)))))) 0^1(3(5(3(5(x1))))) -> 0^1(3(5(x1))) 5^1(0(3(5(5(x1))))) -> 5^1(5(2(0(3(5(x1)))))) 5^1(0(3(5(5(x1))))) -> 5^1(2(0(3(5(x1))))) 5^1(0(3(5(5(x1))))) -> 0^1(3(5(x1))) The TRS R consists of the following rules: 0(0(0(0(x1)))) -> 2(0(1(0(0(0(x1)))))) 0(0(0(0(x1)))) -> 0(2(2(0(0(0(x1)))))) 0(0(0(0(x1)))) -> 0(0(2(0(1(0(x1)))))) 0(3(0(0(x1)))) -> 3(0(2(0(0(x1))))) 0(3(0(0(x1)))) -> 0(4(3(1(0(0(x1)))))) 0(3(0(0(x1)))) -> 0(2(3(0(2(0(x1)))))) 0(3(0(0(x1)))) -> 4(0(4(0(3(0(x1)))))) 0(3(0(0(x1)))) -> 0(2(0(3(4(0(x1)))))) 0(3(0(0(x1)))) -> 0(4(0(3(4(0(x1)))))) 0(3(0(0(x1)))) -> 3(0(0(2(0(1(x1)))))) 0(3(0(0(x1)))) -> 3(0(2(0(0(2(x1)))))) 0(3(0(0(x1)))) -> 0(2(2(0(0(3(x1)))))) 0(0(5(0(x1)))) -> 5(0(0(2(0(x1))))) 0(0(5(0(x1)))) -> 0(1(5(3(0(0(x1)))))) 0(0(5(0(x1)))) -> 4(0(1(5(0(0(x1)))))) 0(0(5(0(x1)))) -> 0(5(4(0(2(0(x1)))))) 0(0(5(0(x1)))) -> 0(4(0(5(2(0(x1)))))) 0(0(5(0(x1)))) -> 2(5(0(0(4(0(x1)))))) 0(0(5(0(x1)))) -> 0(5(1(0(4(0(x1)))))) 0(0(5(0(x1)))) -> 0(1(0(1(5(0(x1)))))) 0(0(5(0(x1)))) -> 5(0(2(0(0(2(x1)))))) 0(0(5(0(x1)))) -> 0(2(0(5(0(2(x1)))))) 0(0(5(0(x1)))) -> 0(2(0(1(0(5(x1)))))) 0(0(5(0(x1)))) -> 0(4(0(1(0(5(x1)))))) 0(3(5(0(x1)))) -> 5(2(3(0(0(x1))))) 0(3(5(0(x1)))) -> 1(5(3(0(0(x1))))) 0(3(5(0(x1)))) -> 2(5(3(0(0(x1))))) 0(3(5(0(x1)))) -> 3(2(5(0(0(x1))))) 0(3(5(0(x1)))) -> 5(3(0(2(0(x1))))) 0(3(5(0(x1)))) -> 5(0(4(3(0(x1))))) 0(3(5(0(x1)))) -> 0(1(0(3(5(x1))))) 0(3(5(0(x1)))) -> 4(5(3(1(0(0(x1)))))) 0(3(5(0(x1)))) -> 3(2(2(5(0(0(x1)))))) 0(3(5(0(x1)))) -> 3(4(2(5(0(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(3(5(1(0(x1)))))) 0(3(5(0(x1)))) -> 5(1(3(0(2(0(x1)))))) 0(3(5(0(x1)))) -> 4(0(2(5(3(0(x1)))))) 0(3(5(0(x1)))) -> 3(4(0(2(5(0(x1)))))) 0(3(5(0(x1)))) -> 3(0(2(2(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(4(3(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(2(5(3(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(5(3(0(1(x1)))))) 0(3(5(0(x1)))) -> 0(0(4(5(3(1(x1)))))) 0(3(5(0(x1)))) -> 0(1(5(0(3(2(x1)))))) 0(3(5(0(x1)))) -> 3(0(0(1(5(2(x1)))))) 0(3(5(0(x1)))) -> 2(5(2(0(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(1(5(3(0(3(x1)))))) 0(3(5(0(x1)))) -> 1(5(0(4(0(3(x1)))))) 0(3(5(0(x1)))) -> 2(5(0(4(0(3(x1)))))) 0(3(5(0(x1)))) -> 1(0(1(5(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(1(5(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(5(1(0(2(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(5(0(4(3(x1)))))) 0(3(5(0(x1)))) -> 2(0(0(5(4(3(x1)))))) 0(3(5(0(x1)))) -> 2(0(0(1(5(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(3(1(0(5(x1)))))) 0(3(5(0(x1)))) -> 0(0(3(2(2(5(x1)))))) 0(3(5(0(x1)))) -> 0(1(1(0(3(5(x1)))))) 0(0(0(3(x1)))) -> 1(0(2(3(0(0(x1)))))) 0(0(0(3(x1)))) -> 0(3(0(1(2(0(x1)))))) 0(0(0(3(x1)))) -> 0(2(2(0(3(0(x1)))))) 0(0(0(3(x1)))) -> 0(4(2(0(3(0(x1)))))) 0(0(0(3(x1)))) -> 0(2(0(2(0(3(x1)))))) 0(3(0(3(x1)))) -> 4(3(2(3(0(0(x1)))))) 0(3(0(3(x1)))) -> 4(3(4(3(0(0(x1)))))) 0(3(0(3(x1)))) -> 3(3(0(2(0(1(x1)))))) 0(3(0(3(x1)))) -> 3(2(0(1(0(3(x1)))))) 0(3(0(3(x1)))) -> 1(3(0(2(0(3(x1)))))) 0(3(0(3(x1)))) -> 1(0(1(3(0(3(x1)))))) 0(3(0(3(x1)))) -> 0(2(5(3(0(3(x1)))))) 0(3(0(3(x1)))) -> 3(2(0(4(0(3(x1)))))) 0(3(0(3(x1)))) -> 0(2(3(0(1(3(x1)))))) 0(3(0(3(x1)))) -> 3(0(1(0(4(3(x1)))))) 0(0(3(3(x1)))) -> 0(4(3(3(0(x1))))) 0(0(3(3(x1)))) -> 3(0(2(0(3(x1))))) 0(0(3(3(x1)))) -> 0(2(3(0(3(x1))))) 0(0(3(3(x1)))) -> 3(3(1(0(1(0(x1)))))) 0(0(3(3(x1)))) -> 3(0(4(3(1(0(x1)))))) 0(0(3(3(x1)))) -> 0(3(3(2(2(0(x1)))))) 0(0(3(3(x1)))) -> 3(3(0(4(2(0(x1)))))) 0(0(3(3(x1)))) -> 0(2(3(3(0(2(x1)))))) 0(0(3(3(x1)))) -> 3(4(0(1(0(3(x1)))))) 0(0(3(3(x1)))) -> 3(0(2(2(0(3(x1)))))) 0(0(3(3(x1)))) -> 2(0(2(3(0(3(x1)))))) 0(0(3(3(x1)))) -> 0(1(4(3(0(3(x1)))))) 0(0(3(3(x1)))) -> 3(0(1(0(1(3(x1)))))) 0(0(3(3(x1)))) -> 2(0(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 4(0(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 0(1(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 0(4(3(0(4(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(3(3(x1))))) 0(3(3(3(x1)))) -> 0(2(3(3(3(x1))))) 0(3(3(3(x1)))) -> 3(2(3(1(3(0(x1)))))) 0(3(3(3(x1)))) -> 3(2(3(0(2(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(2(1(3(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(2(3(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(5(3(3(x1)))))) 0(0(5(3(x1)))) -> 1(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 2(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 4(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 0(2(5(3(0(x1))))) 0(0(5(3(x1)))) -> 0(4(5(3(0(x1))))) 0(0(5(3(x1)))) -> 5(2(0(0(3(x1))))) 0(0(5(3(x1)))) -> 0(1(5(0(3(x1))))) 0(0(5(3(x1)))) -> 3(1(5(1(0(0(x1)))))) 0(0(5(3(x1)))) -> 3(2(5(1(0(0(x1)))))) 0(0(5(3(x1)))) -> 2(5(4(3(0(0(x1)))))) 0(0(5(3(x1)))) -> 0(2(5(3(1(0(x1)))))) 0(0(5(3(x1)))) -> 5(1(3(0(2(0(x1)))))) 0(0(5(3(x1)))) -> 5(4(0(1(3(0(x1)))))) 0(0(5(3(x1)))) -> 1(5(0(2(3(0(x1)))))) 0(0(5(3(x1)))) -> 0(2(2(5(3(0(x1)))))) 0(0(5(3(x1)))) -> 0(1(0(3(5(2(x1)))))) 0(0(5(3(x1)))) -> 5(4(2(0(0(3(x1)))))) 0(0(5(3(x1)))) -> 1(5(0(4(0(3(x1)))))) 0(0(5(3(x1)))) -> 5(0(1(4(0(3(x1)))))) 0(0(5(3(x1)))) -> 5(0(4(0(1(3(x1)))))) 0(0(5(3(x1)))) -> 0(2(5(0(2(3(x1)))))) 0(0(5(3(x1)))) -> 0(2(0(5(2(3(x1)))))) 0(0(5(3(x1)))) -> 0(1(5(0(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(1(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(4(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(1(3(5(x1)))))) 0(3(5(3(x1)))) -> 3(2(5(3(0(x1))))) 0(3(5(3(x1)))) -> 5(2(3(0(3(x1))))) 0(3(5(3(x1)))) -> 4(5(3(0(3(x1))))) 0(3(5(3(x1)))) -> 2(5(0(3(3(x1))))) 0(3(5(3(x1)))) -> 5(0(4(3(3(x1))))) 0(3(5(3(x1)))) -> 3(0(1(5(3(x1))))) 0(3(5(3(x1)))) -> 5(4(3(3(2(0(x1)))))) 0(3(5(3(x1)))) -> 2(5(3(3(2(0(x1)))))) 0(3(5(3(x1)))) -> 4(5(3(1(3(0(x1)))))) 0(3(5(3(x1)))) -> 2(1(5(3(3(0(x1)))))) 0(3(5(3(x1)))) -> 1(2(5(3(3(0(x1)))))) 0(3(5(3(x1)))) -> 3(4(5(4(3(0(x1)))))) 0(3(5(3(x1)))) -> 4(5(3(3(4(0(x1)))))) 0(3(5(3(x1)))) -> 3(0(2(5(3(2(x1)))))) 0(3(5(3(x1)))) -> 5(2(2(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(5(2(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 5(2(4(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 2(5(5(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(3(1(5(0(3(x1)))))) 0(3(5(3(x1)))) -> 3(2(2(5(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(5(3(0(1(3(x1)))))) 0(3(5(3(x1)))) -> 2(5(0(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 5(0(1(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 5(0(2(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 0(4(4(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 0(2(5(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 0(4(5(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 3(3(0(2(2(5(x1)))))) 0(3(5(3(x1)))) -> 1(3(1(0(3(5(x1)))))) 0(3(5(3(x1)))) -> 0(1(3(1(3(5(x1)))))) 0(3(5(3(x1)))) -> 2(0(4(3(3(5(x1)))))) 0(3(5(3(x1)))) -> 4(0(4(3(3(5(x1)))))) 0(3(5(3(x1)))) -> 0(1(4(3(3(5(x1)))))) 0(0(3(0(0(x1))))) -> 0(0(2(0(3(0(x1)))))) 5(0(3(0(0(x1))))) -> 5(0(3(2(0(0(x1)))))) 0(3(5(0(0(x1))))) -> 0(2(5(0(3(0(x1)))))) 5(0(3(5(0(x1))))) -> 5(0(1(5(0(3(x1)))))) 5(0(3(5(0(x1))))) -> 5(5(0(0(4(3(x1)))))) 0(3(3(5(0(x1))))) -> 3(0(2(0(3(5(x1)))))) 0(3(0(0(3(x1))))) -> 0(0(4(3(3(0(x1)))))) 0(3(3(0(3(x1))))) -> 3(0(0(2(3(3(x1)))))) 0(3(5(0(3(x1))))) -> 0(4(5(0(3(3(x1)))))) 0(3(5(1(3(x1))))) -> 1(3(2(5(0(3(x1)))))) 0(0(5(2(3(x1))))) -> 4(5(0(2(3(0(x1)))))) 0(0(5(2(3(x1))))) -> 3(0(2(2(5(0(x1)))))) 0(3(5(2(3(x1))))) -> 0(5(2(3(2(3(x1)))))) 0(3(5(2(3(x1))))) -> 1(3(0(2(5(3(x1)))))) 0(3(5(2(3(x1))))) -> 4(3(0(2(5(3(x1)))))) 5(0(0(3(3(x1))))) -> 0(4(3(0(5(3(x1)))))) 5(0(3(3(3(x1))))) -> 3(1(5(3(0(3(x1)))))) 0(0(5(4(3(x1))))) -> 5(0(4(3(1(0(x1)))))) 0(0(5(4(3(x1))))) -> 5(1(0(4(0(3(x1)))))) 0(0(5(4(3(x1))))) -> 0(1(5(0(4(3(x1)))))) 0(0(3(5(3(x1))))) -> 2(0(3(5(3(0(x1)))))) 0(0(3(5(3(x1))))) -> 0(0(2(3(5(3(x1)))))) 5(0(3(5(3(x1))))) -> 5(3(0(4(5(3(x1)))))) 0(3(3(5(3(x1))))) -> 3(3(0(1(5(3(x1)))))) 0(3(3(5(3(x1))))) -> 0(4(3(3(5(3(x1)))))) 0(3(5(5(3(x1))))) -> 3(1(5(3(5(0(x1)))))) 0(3(5(5(3(x1))))) -> 0(4(5(3(5(3(x1)))))) 0(3(3(0(5(x1))))) -> 1(0(3(5(3(0(x1)))))) 0(3(3(0(5(x1))))) -> 0(2(3(5(3(0(x1)))))) 0(3(3(0(5(x1))))) -> 5(0(2(0(3(3(x1)))))) 0(3(3(0(5(x1))))) -> 0(2(0(3(5(3(x1)))))) 0(3(5(0(5(x1))))) -> 0(0(5(4(3(5(x1)))))) 0(3(0(3(5(x1))))) -> 0(1(5(3(0(3(x1)))))) 0(3(0(3(5(x1))))) -> 0(0(3(1(5(3(x1)))))) 0(3(3(3(5(x1))))) -> 3(3(3(2(5(0(x1)))))) 0(3(3(3(5(x1))))) -> 3(3(2(0(3(5(x1)))))) 0(3(5(3(5(x1))))) -> 5(2(3(0(3(5(x1)))))) 5(0(3(5(5(x1))))) -> 5(5(2(0(3(5(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 1 SCC with 278 less nodes. ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(3(0(0(x1)))) -> 0^1(3(0(x1))) 0^1(3(0(0(x1)))) -> 0^1(0(3(x1))) 0^1(3(0(0(x1)))) -> 0^1(3(x1)) 0^1(3(5(0(x1)))) -> 0^1(0(x1)) 0^1(0(5(0(x1)))) -> 0^1(0(x1)) 0^1(0(5(0(x1)))) -> 5^1(0(0(x1))) 5^1(0(3(5(0(x1))))) -> 5^1(0(3(x1))) 5^1(0(3(5(0(x1))))) -> 0^1(3(x1)) 0^1(3(5(0(x1)))) -> 5^1(0(0(x1))) 5^1(0(3(3(3(x1))))) -> 0^1(3(x1)) 0^1(3(5(0(x1)))) -> 0^1(3(5(x1))) 0^1(3(5(0(x1)))) -> 5^1(x1) 5^1(0(3(5(5(x1))))) -> 0^1(3(5(x1))) 0^1(3(5(0(x1)))) -> 0^1(0(3(x1))) 0^1(0(5(0(x1)))) -> 0^1(5(x1)) 0^1(0(5(0(x1)))) -> 5^1(x1) 0^1(3(5(0(x1)))) -> 0^1(3(x1)) 0^1(3(5(0(x1)))) -> 5^1(0(3(x1))) 0^1(3(5(0(x1)))) -> 0^1(0(5(4(3(x1))))) 0^1(0(5(4(3(x1))))) -> 0^1(x1) 0^1(3(5(0(x1)))) -> 0^1(5(x1)) 0^1(0(0(3(x1)))) -> 0^1(0(x1)) 0^1(0(0(3(x1)))) -> 0^1(x1) 0^1(0(0(3(x1)))) -> 0^1(3(0(x1))) 0^1(3(0(3(x1)))) -> 0^1(0(x1)) 0^1(3(0(3(x1)))) -> 0^1(x1) 0^1(0(3(3(x1)))) -> 0^1(x1) 0^1(0(3(3(x1)))) -> 0^1(3(x1)) 0^1(3(3(3(x1)))) -> 0^1(x1) 0^1(0(5(3(x1)))) -> 0^1(0(x1)) 0^1(0(5(3(x1)))) -> 0^1(x1) 0^1(0(5(3(x1)))) -> 0^1(0(3(x1))) 0^1(0(5(3(x1)))) -> 0^1(3(x1)) 0^1(3(5(3(x1)))) -> 0^1(x1) 0^1(0(5(3(x1)))) -> 5^1(0(3(x1))) 0^1(0(5(3(x1)))) -> 5^1(x1) 0^1(3(5(3(x1)))) -> 0^1(3(x1)) 0^1(3(5(3(x1)))) -> 5^1(0(3(3(x1)))) 0^1(3(5(3(x1)))) -> 0^1(3(3(x1))) 0^1(3(3(5(0(x1))))) -> 0^1(3(5(x1))) 0^1(3(5(3(x1)))) -> 5^1(0(3(x1))) 0^1(3(5(3(x1)))) -> 5^1(x1) 0^1(3(5(3(x1)))) -> 0^1(3(5(x1))) 0^1(3(5(0(0(x1))))) -> 5^1(0(3(0(x1)))) 0^1(3(5(0(0(x1))))) -> 0^1(3(0(x1))) 0^1(3(3(5(0(x1))))) -> 5^1(x1) 0^1(3(0(0(3(x1))))) -> 0^1(x1) 0^1(0(3(0(0(x1))))) -> 0^1(3(0(x1))) 0^1(3(5(0(3(x1))))) -> 5^1(0(3(3(x1)))) 0^1(3(5(0(3(x1))))) -> 0^1(3(3(x1))) 0^1(3(3(0(5(x1))))) -> 0^1(3(5(3(0(x1))))) 0^1(3(5(3(5(x1))))) -> 0^1(3(5(x1))) 0^1(3(5(1(3(x1))))) -> 5^1(0(3(x1))) 0^1(3(5(1(3(x1))))) -> 0^1(3(x1)) 0^1(3(5(5(3(x1))))) -> 5^1(0(x1)) 0^1(3(5(5(3(x1))))) -> 0^1(x1) 0^1(0(5(2(3(x1))))) -> 0^1(x1) 0^1(0(5(2(3(x1))))) -> 5^1(0(x1)) 0^1(0(5(4(3(x1))))) -> 0^1(3(x1)) 0^1(3(3(0(5(x1))))) -> 0^1(x1) 0^1(0(3(5(3(x1))))) -> 0^1(3(5(3(0(x1))))) 0^1(0(3(5(3(x1))))) -> 0^1(x1) 0^1(3(3(0(5(x1))))) -> 0^1(3(3(x1))) 0^1(3(3(0(5(x1))))) -> 0^1(3(5(3(x1)))) 0^1(3(3(3(5(x1))))) -> 5^1(0(x1)) 0^1(3(3(3(5(x1))))) -> 0^1(x1) 0^1(3(5(0(5(x1))))) -> 0^1(0(5(4(3(5(x1)))))) 0^1(3(0(3(5(x1))))) -> 0^1(3(x1)) 0^1(3(3(3(5(x1))))) -> 0^1(3(5(x1))) The TRS R consists of the following rules: 0(0(0(0(x1)))) -> 2(0(1(0(0(0(x1)))))) 0(0(0(0(x1)))) -> 0(2(2(0(0(0(x1)))))) 0(0(0(0(x1)))) -> 0(0(2(0(1(0(x1)))))) 0(3(0(0(x1)))) -> 3(0(2(0(0(x1))))) 0(3(0(0(x1)))) -> 0(4(3(1(0(0(x1)))))) 0(3(0(0(x1)))) -> 0(2(3(0(2(0(x1)))))) 0(3(0(0(x1)))) -> 4(0(4(0(3(0(x1)))))) 0(3(0(0(x1)))) -> 0(2(0(3(4(0(x1)))))) 0(3(0(0(x1)))) -> 0(4(0(3(4(0(x1)))))) 0(3(0(0(x1)))) -> 3(0(0(2(0(1(x1)))))) 0(3(0(0(x1)))) -> 3(0(2(0(0(2(x1)))))) 0(3(0(0(x1)))) -> 0(2(2(0(0(3(x1)))))) 0(0(5(0(x1)))) -> 5(0(0(2(0(x1))))) 0(0(5(0(x1)))) -> 0(1(5(3(0(0(x1)))))) 0(0(5(0(x1)))) -> 4(0(1(5(0(0(x1)))))) 0(0(5(0(x1)))) -> 0(5(4(0(2(0(x1)))))) 0(0(5(0(x1)))) -> 0(4(0(5(2(0(x1)))))) 0(0(5(0(x1)))) -> 2(5(0(0(4(0(x1)))))) 0(0(5(0(x1)))) -> 0(5(1(0(4(0(x1)))))) 0(0(5(0(x1)))) -> 0(1(0(1(5(0(x1)))))) 0(0(5(0(x1)))) -> 5(0(2(0(0(2(x1)))))) 0(0(5(0(x1)))) -> 0(2(0(5(0(2(x1)))))) 0(0(5(0(x1)))) -> 0(2(0(1(0(5(x1)))))) 0(0(5(0(x1)))) -> 0(4(0(1(0(5(x1)))))) 0(3(5(0(x1)))) -> 5(2(3(0(0(x1))))) 0(3(5(0(x1)))) -> 1(5(3(0(0(x1))))) 0(3(5(0(x1)))) -> 2(5(3(0(0(x1))))) 0(3(5(0(x1)))) -> 3(2(5(0(0(x1))))) 0(3(5(0(x1)))) -> 5(3(0(2(0(x1))))) 0(3(5(0(x1)))) -> 5(0(4(3(0(x1))))) 0(3(5(0(x1)))) -> 0(1(0(3(5(x1))))) 0(3(5(0(x1)))) -> 4(5(3(1(0(0(x1)))))) 0(3(5(0(x1)))) -> 3(2(2(5(0(0(x1)))))) 0(3(5(0(x1)))) -> 3(4(2(5(0(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(3(5(1(0(x1)))))) 0(3(5(0(x1)))) -> 5(1(3(0(2(0(x1)))))) 0(3(5(0(x1)))) -> 4(0(2(5(3(0(x1)))))) 0(3(5(0(x1)))) -> 3(4(0(2(5(0(x1)))))) 0(3(5(0(x1)))) -> 3(0(2(2(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(4(3(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(2(5(3(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(5(3(0(1(x1)))))) 0(3(5(0(x1)))) -> 0(0(4(5(3(1(x1)))))) 0(3(5(0(x1)))) -> 0(1(5(0(3(2(x1)))))) 0(3(5(0(x1)))) -> 3(0(0(1(5(2(x1)))))) 0(3(5(0(x1)))) -> 2(5(2(0(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(1(5(3(0(3(x1)))))) 0(3(5(0(x1)))) -> 1(5(0(4(0(3(x1)))))) 0(3(5(0(x1)))) -> 2(5(0(4(0(3(x1)))))) 0(3(5(0(x1)))) -> 1(0(1(5(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(1(5(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(5(1(0(2(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(5(0(4(3(x1)))))) 0(3(5(0(x1)))) -> 2(0(0(5(4(3(x1)))))) 0(3(5(0(x1)))) -> 2(0(0(1(5(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(3(1(0(5(x1)))))) 0(3(5(0(x1)))) -> 0(0(3(2(2(5(x1)))))) 0(3(5(0(x1)))) -> 0(1(1(0(3(5(x1)))))) 0(0(0(3(x1)))) -> 1(0(2(3(0(0(x1)))))) 0(0(0(3(x1)))) -> 0(3(0(1(2(0(x1)))))) 0(0(0(3(x1)))) -> 0(2(2(0(3(0(x1)))))) 0(0(0(3(x1)))) -> 0(4(2(0(3(0(x1)))))) 0(0(0(3(x1)))) -> 0(2(0(2(0(3(x1)))))) 0(3(0(3(x1)))) -> 4(3(2(3(0(0(x1)))))) 0(3(0(3(x1)))) -> 4(3(4(3(0(0(x1)))))) 0(3(0(3(x1)))) -> 3(3(0(2(0(1(x1)))))) 0(3(0(3(x1)))) -> 3(2(0(1(0(3(x1)))))) 0(3(0(3(x1)))) -> 1(3(0(2(0(3(x1)))))) 0(3(0(3(x1)))) -> 1(0(1(3(0(3(x1)))))) 0(3(0(3(x1)))) -> 0(2(5(3(0(3(x1)))))) 0(3(0(3(x1)))) -> 3(2(0(4(0(3(x1)))))) 0(3(0(3(x1)))) -> 0(2(3(0(1(3(x1)))))) 0(3(0(3(x1)))) -> 3(0(1(0(4(3(x1)))))) 0(0(3(3(x1)))) -> 0(4(3(3(0(x1))))) 0(0(3(3(x1)))) -> 3(0(2(0(3(x1))))) 0(0(3(3(x1)))) -> 0(2(3(0(3(x1))))) 0(0(3(3(x1)))) -> 3(3(1(0(1(0(x1)))))) 0(0(3(3(x1)))) -> 3(0(4(3(1(0(x1)))))) 0(0(3(3(x1)))) -> 0(3(3(2(2(0(x1)))))) 0(0(3(3(x1)))) -> 3(3(0(4(2(0(x1)))))) 0(0(3(3(x1)))) -> 0(2(3(3(0(2(x1)))))) 0(0(3(3(x1)))) -> 3(4(0(1(0(3(x1)))))) 0(0(3(3(x1)))) -> 3(0(2(2(0(3(x1)))))) 0(0(3(3(x1)))) -> 2(0(2(3(0(3(x1)))))) 0(0(3(3(x1)))) -> 0(1(4(3(0(3(x1)))))) 0(0(3(3(x1)))) -> 3(0(1(0(1(3(x1)))))) 0(0(3(3(x1)))) -> 2(0(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 4(0(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 0(1(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 0(4(3(0(4(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(3(3(x1))))) 0(3(3(3(x1)))) -> 0(2(3(3(3(x1))))) 0(3(3(3(x1)))) -> 3(2(3(1(3(0(x1)))))) 0(3(3(3(x1)))) -> 3(2(3(0(2(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(2(1(3(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(2(3(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(5(3(3(x1)))))) 0(0(5(3(x1)))) -> 1(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 2(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 4(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 0(2(5(3(0(x1))))) 0(0(5(3(x1)))) -> 0(4(5(3(0(x1))))) 0(0(5(3(x1)))) -> 5(2(0(0(3(x1))))) 0(0(5(3(x1)))) -> 0(1(5(0(3(x1))))) 0(0(5(3(x1)))) -> 3(1(5(1(0(0(x1)))))) 0(0(5(3(x1)))) -> 3(2(5(1(0(0(x1)))))) 0(0(5(3(x1)))) -> 2(5(4(3(0(0(x1)))))) 0(0(5(3(x1)))) -> 0(2(5(3(1(0(x1)))))) 0(0(5(3(x1)))) -> 5(1(3(0(2(0(x1)))))) 0(0(5(3(x1)))) -> 5(4(0(1(3(0(x1)))))) 0(0(5(3(x1)))) -> 1(5(0(2(3(0(x1)))))) 0(0(5(3(x1)))) -> 0(2(2(5(3(0(x1)))))) 0(0(5(3(x1)))) -> 0(1(0(3(5(2(x1)))))) 0(0(5(3(x1)))) -> 5(4(2(0(0(3(x1)))))) 0(0(5(3(x1)))) -> 1(5(0(4(0(3(x1)))))) 0(0(5(3(x1)))) -> 5(0(1(4(0(3(x1)))))) 0(0(5(3(x1)))) -> 5(0(4(0(1(3(x1)))))) 0(0(5(3(x1)))) -> 0(2(5(0(2(3(x1)))))) 0(0(5(3(x1)))) -> 0(2(0(5(2(3(x1)))))) 0(0(5(3(x1)))) -> 0(1(5(0(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(1(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(4(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(1(3(5(x1)))))) 0(3(5(3(x1)))) -> 3(2(5(3(0(x1))))) 0(3(5(3(x1)))) -> 5(2(3(0(3(x1))))) 0(3(5(3(x1)))) -> 4(5(3(0(3(x1))))) 0(3(5(3(x1)))) -> 2(5(0(3(3(x1))))) 0(3(5(3(x1)))) -> 5(0(4(3(3(x1))))) 0(3(5(3(x1)))) -> 3(0(1(5(3(x1))))) 0(3(5(3(x1)))) -> 5(4(3(3(2(0(x1)))))) 0(3(5(3(x1)))) -> 2(5(3(3(2(0(x1)))))) 0(3(5(3(x1)))) -> 4(5(3(1(3(0(x1)))))) 0(3(5(3(x1)))) -> 2(1(5(3(3(0(x1)))))) 0(3(5(3(x1)))) -> 1(2(5(3(3(0(x1)))))) 0(3(5(3(x1)))) -> 3(4(5(4(3(0(x1)))))) 0(3(5(3(x1)))) -> 4(5(3(3(4(0(x1)))))) 0(3(5(3(x1)))) -> 3(0(2(5(3(2(x1)))))) 0(3(5(3(x1)))) -> 5(2(2(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(5(2(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 5(2(4(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 2(5(5(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(3(1(5(0(3(x1)))))) 0(3(5(3(x1)))) -> 3(2(2(5(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(5(3(0(1(3(x1)))))) 0(3(5(3(x1)))) -> 2(5(0(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 5(0(1(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 5(0(2(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 0(4(4(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 0(2(5(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 0(4(5(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 3(3(0(2(2(5(x1)))))) 0(3(5(3(x1)))) -> 1(3(1(0(3(5(x1)))))) 0(3(5(3(x1)))) -> 0(1(3(1(3(5(x1)))))) 0(3(5(3(x1)))) -> 2(0(4(3(3(5(x1)))))) 0(3(5(3(x1)))) -> 4(0(4(3(3(5(x1)))))) 0(3(5(3(x1)))) -> 0(1(4(3(3(5(x1)))))) 0(0(3(0(0(x1))))) -> 0(0(2(0(3(0(x1)))))) 5(0(3(0(0(x1))))) -> 5(0(3(2(0(0(x1)))))) 0(3(5(0(0(x1))))) -> 0(2(5(0(3(0(x1)))))) 5(0(3(5(0(x1))))) -> 5(0(1(5(0(3(x1)))))) 5(0(3(5(0(x1))))) -> 5(5(0(0(4(3(x1)))))) 0(3(3(5(0(x1))))) -> 3(0(2(0(3(5(x1)))))) 0(3(0(0(3(x1))))) -> 0(0(4(3(3(0(x1)))))) 0(3(3(0(3(x1))))) -> 3(0(0(2(3(3(x1)))))) 0(3(5(0(3(x1))))) -> 0(4(5(0(3(3(x1)))))) 0(3(5(1(3(x1))))) -> 1(3(2(5(0(3(x1)))))) 0(0(5(2(3(x1))))) -> 4(5(0(2(3(0(x1)))))) 0(0(5(2(3(x1))))) -> 3(0(2(2(5(0(x1)))))) 0(3(5(2(3(x1))))) -> 0(5(2(3(2(3(x1)))))) 0(3(5(2(3(x1))))) -> 1(3(0(2(5(3(x1)))))) 0(3(5(2(3(x1))))) -> 4(3(0(2(5(3(x1)))))) 5(0(0(3(3(x1))))) -> 0(4(3(0(5(3(x1)))))) 5(0(3(3(3(x1))))) -> 3(1(5(3(0(3(x1)))))) 0(0(5(4(3(x1))))) -> 5(0(4(3(1(0(x1)))))) 0(0(5(4(3(x1))))) -> 5(1(0(4(0(3(x1)))))) 0(0(5(4(3(x1))))) -> 0(1(5(0(4(3(x1)))))) 0(0(3(5(3(x1))))) -> 2(0(3(5(3(0(x1)))))) 0(0(3(5(3(x1))))) -> 0(0(2(3(5(3(x1)))))) 5(0(3(5(3(x1))))) -> 5(3(0(4(5(3(x1)))))) 0(3(3(5(3(x1))))) -> 3(3(0(1(5(3(x1)))))) 0(3(3(5(3(x1))))) -> 0(4(3(3(5(3(x1)))))) 0(3(5(5(3(x1))))) -> 3(1(5(3(5(0(x1)))))) 0(3(5(5(3(x1))))) -> 0(4(5(3(5(3(x1)))))) 0(3(3(0(5(x1))))) -> 1(0(3(5(3(0(x1)))))) 0(3(3(0(5(x1))))) -> 0(2(3(5(3(0(x1)))))) 0(3(3(0(5(x1))))) -> 5(0(2(0(3(3(x1)))))) 0(3(3(0(5(x1))))) -> 0(2(0(3(5(3(x1)))))) 0(3(5(0(5(x1))))) -> 0(0(5(4(3(5(x1)))))) 0(3(0(3(5(x1))))) -> 0(1(5(3(0(3(x1)))))) 0(3(0(3(5(x1))))) -> 0(0(3(1(5(3(x1)))))) 0(3(3(3(5(x1))))) -> 3(3(3(2(5(0(x1)))))) 0(3(3(3(5(x1))))) -> 3(3(2(0(3(5(x1)))))) 0(3(5(3(5(x1))))) -> 5(2(3(0(3(5(x1)))))) 5(0(3(5(5(x1))))) -> 5(5(2(0(3(5(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (7) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(3(0(0(x1)))) -> 0^1(3(0(x1))) 0^1(3(0(0(x1)))) -> 0^1(0(3(x1))) 0^1(3(0(0(x1)))) -> 0^1(3(x1)) 0^1(0(5(0(x1)))) -> 0^1(0(x1)) 0^1(0(5(0(x1)))) -> 5^1(0(0(x1))) 5^1(0(3(5(0(x1))))) -> 5^1(0(3(x1))) 5^1(0(3(5(0(x1))))) -> 0^1(3(x1)) 0^1(3(5(0(x1)))) -> 0^1(3(5(x1))) 0^1(3(5(0(x1)))) -> 5^1(x1) 0^1(0(5(0(x1)))) -> 0^1(5(x1)) 0^1(0(5(0(x1)))) -> 5^1(x1) 0^1(3(5(0(x1)))) -> 0^1(3(x1)) 0^1(3(5(0(x1)))) -> 5^1(0(3(x1))) 0^1(0(5(4(3(x1))))) -> 0^1(x1) 0^1(3(5(0(x1)))) -> 0^1(5(x1)) 0^1(0(0(3(x1)))) -> 0^1(0(x1)) 0^1(0(0(3(x1)))) -> 0^1(x1) 0^1(0(0(3(x1)))) -> 0^1(3(0(x1))) 0^1(3(0(3(x1)))) -> 0^1(x1) 0^1(0(3(3(x1)))) -> 0^1(x1) 0^1(0(3(3(x1)))) -> 0^1(3(x1)) 0^1(0(5(3(x1)))) -> 0^1(x1) 0^1(0(5(3(x1)))) -> 0^1(3(x1)) 0^1(0(5(3(x1)))) -> 5^1(0(3(x1))) 0^1(0(5(3(x1)))) -> 5^1(x1) 0^1(3(3(5(0(x1))))) -> 0^1(3(5(x1))) 0^1(3(5(3(x1)))) -> 5^1(x1) 0^1(3(5(0(0(x1))))) -> 5^1(0(3(0(x1)))) 0^1(3(5(0(0(x1))))) -> 0^1(3(0(x1))) 0^1(3(3(5(0(x1))))) -> 5^1(x1) 0^1(3(0(0(3(x1))))) -> 0^1(x1) 0^1(0(3(0(0(x1))))) -> 0^1(3(0(x1))) 0^1(3(5(0(3(x1))))) -> 5^1(0(3(3(x1)))) 0^1(3(5(0(3(x1))))) -> 0^1(3(3(x1))) 0^1(0(5(2(3(x1))))) -> 0^1(x1) 0^1(0(5(2(3(x1))))) -> 5^1(0(x1)) 0^1(0(5(4(3(x1))))) -> 0^1(3(x1)) 0^1(3(3(0(5(x1))))) -> 0^1(x1) 0^1(0(3(5(3(x1))))) -> 0^1(x1) 0^1(3(3(0(5(x1))))) -> 0^1(3(3(x1))) 0^1(3(3(0(5(x1))))) -> 0^1(3(5(3(x1)))) 0^1(3(0(3(5(x1))))) -> 0^1(3(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 + x_1 POL(0^1(x_1)) = 1 + x_1 POL(1(x_1)) = x_1 POL(2(x_1)) = x_1 POL(3(x_1)) = x_1 POL(4(x_1)) = x_1 POL(5(x_1)) = x_1 POL(5^1(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0(0(0(0(x1)))) -> 2(0(1(0(0(0(x1)))))) 0(0(0(0(x1)))) -> 0(2(2(0(0(0(x1)))))) 0(0(0(0(x1)))) -> 0(0(2(0(1(0(x1)))))) 0(3(0(0(x1)))) -> 3(0(2(0(0(x1))))) 0(3(0(0(x1)))) -> 0(4(3(1(0(0(x1)))))) 0(3(0(0(x1)))) -> 0(2(3(0(2(0(x1)))))) 0(3(0(0(x1)))) -> 4(0(4(0(3(0(x1)))))) 0(3(0(0(x1)))) -> 0(2(0(3(4(0(x1)))))) 0(3(0(0(x1)))) -> 0(4(0(3(4(0(x1)))))) 0(3(0(0(x1)))) -> 3(0(0(2(0(1(x1)))))) 0(3(0(0(x1)))) -> 3(0(2(0(0(2(x1)))))) 0(3(0(0(x1)))) -> 0(2(2(0(0(3(x1)))))) 0(0(5(0(x1)))) -> 5(0(0(2(0(x1))))) 0(0(5(0(x1)))) -> 0(1(5(3(0(0(x1)))))) 0(0(5(0(x1)))) -> 4(0(1(5(0(0(x1)))))) 0(0(5(0(x1)))) -> 0(5(4(0(2(0(x1)))))) 0(0(5(0(x1)))) -> 0(4(0(5(2(0(x1)))))) 0(0(5(0(x1)))) -> 2(5(0(0(4(0(x1)))))) 0(0(5(0(x1)))) -> 0(5(1(0(4(0(x1)))))) 0(0(5(0(x1)))) -> 0(1(0(1(5(0(x1)))))) 0(0(5(0(x1)))) -> 5(0(2(0(0(2(x1)))))) 0(0(5(0(x1)))) -> 0(2(0(5(0(2(x1)))))) 0(0(5(0(x1)))) -> 0(2(0(1(0(5(x1)))))) 0(0(5(0(x1)))) -> 0(4(0(1(0(5(x1)))))) 0(3(5(0(x1)))) -> 5(2(3(0(0(x1))))) 0(3(5(0(x1)))) -> 1(5(3(0(0(x1))))) 0(3(5(0(x1)))) -> 2(5(3(0(0(x1))))) 0(3(5(0(x1)))) -> 3(2(5(0(0(x1))))) 0(3(5(0(x1)))) -> 5(3(0(2(0(x1))))) 0(3(5(0(x1)))) -> 5(0(4(3(0(x1))))) 0(3(5(0(x1)))) -> 0(1(0(3(5(x1))))) 0(3(5(0(x1)))) -> 4(5(3(1(0(0(x1)))))) 0(3(5(0(x1)))) -> 3(2(2(5(0(0(x1)))))) 0(3(5(0(x1)))) -> 3(4(2(5(0(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(3(5(1(0(x1)))))) 0(3(5(0(x1)))) -> 5(1(3(0(2(0(x1)))))) 0(3(5(0(x1)))) -> 4(0(2(5(3(0(x1)))))) 0(3(5(0(x1)))) -> 3(4(0(2(5(0(x1)))))) 0(3(5(0(x1)))) -> 3(0(2(2(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(4(3(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(2(5(3(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(5(3(0(1(x1)))))) 0(3(5(0(x1)))) -> 0(0(4(5(3(1(x1)))))) 0(3(5(0(x1)))) -> 0(1(5(0(3(2(x1)))))) 0(3(5(0(x1)))) -> 3(0(0(1(5(2(x1)))))) 0(3(5(0(x1)))) -> 2(5(2(0(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(1(5(3(0(3(x1)))))) 0(3(5(0(x1)))) -> 1(5(0(4(0(3(x1)))))) 0(3(5(0(x1)))) -> 2(5(0(4(0(3(x1)))))) 0(3(5(0(x1)))) -> 1(0(1(5(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(1(5(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(5(1(0(2(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(5(0(4(3(x1)))))) 0(3(5(0(x1)))) -> 2(0(0(5(4(3(x1)))))) 0(3(5(0(x1)))) -> 2(0(0(1(5(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(3(1(0(5(x1)))))) 0(3(5(0(x1)))) -> 0(0(3(2(2(5(x1)))))) 0(3(5(0(x1)))) -> 0(1(1(0(3(5(x1)))))) 0(0(0(3(x1)))) -> 1(0(2(3(0(0(x1)))))) 0(0(0(3(x1)))) -> 0(3(0(1(2(0(x1)))))) 0(0(0(3(x1)))) -> 0(2(2(0(3(0(x1)))))) 0(0(0(3(x1)))) -> 0(4(2(0(3(0(x1)))))) 0(0(0(3(x1)))) -> 0(2(0(2(0(3(x1)))))) 0(3(0(3(x1)))) -> 4(3(2(3(0(0(x1)))))) 0(3(0(3(x1)))) -> 4(3(4(3(0(0(x1)))))) 0(3(0(3(x1)))) -> 3(3(0(2(0(1(x1)))))) 0(3(0(3(x1)))) -> 3(2(0(1(0(3(x1)))))) 0(3(0(3(x1)))) -> 1(3(0(2(0(3(x1)))))) 0(3(0(3(x1)))) -> 1(0(1(3(0(3(x1)))))) 0(3(0(3(x1)))) -> 0(2(5(3(0(3(x1)))))) 0(3(0(3(x1)))) -> 3(2(0(4(0(3(x1)))))) 0(3(0(3(x1)))) -> 0(2(3(0(1(3(x1)))))) 0(3(0(3(x1)))) -> 3(0(1(0(4(3(x1)))))) 0(0(3(3(x1)))) -> 0(4(3(3(0(x1))))) 0(0(3(3(x1)))) -> 3(0(2(0(3(x1))))) 0(0(3(3(x1)))) -> 0(2(3(0(3(x1))))) 0(0(3(3(x1)))) -> 3(3(1(0(1(0(x1)))))) 0(0(3(3(x1)))) -> 3(0(4(3(1(0(x1)))))) 0(0(3(3(x1)))) -> 0(3(3(2(2(0(x1)))))) 0(0(3(3(x1)))) -> 3(3(0(4(2(0(x1)))))) 0(0(3(3(x1)))) -> 0(2(3(3(0(2(x1)))))) 0(0(3(3(x1)))) -> 3(4(0(1(0(3(x1)))))) 0(0(3(3(x1)))) -> 3(0(2(2(0(3(x1)))))) 0(0(3(3(x1)))) -> 2(0(2(3(0(3(x1)))))) 0(0(3(3(x1)))) -> 0(1(4(3(0(3(x1)))))) 0(0(3(3(x1)))) -> 3(0(1(0(1(3(x1)))))) 0(0(3(3(x1)))) -> 2(0(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 4(0(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 0(1(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 0(4(3(0(4(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(3(3(x1))))) 0(3(3(3(x1)))) -> 0(2(3(3(3(x1))))) 0(3(3(3(x1)))) -> 3(2(3(1(3(0(x1)))))) 0(3(3(3(x1)))) -> 3(2(3(0(2(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(2(1(3(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(2(3(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(5(3(3(x1)))))) 0(0(5(3(x1)))) -> 1(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 2(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 4(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 0(2(5(3(0(x1))))) 0(0(5(3(x1)))) -> 0(4(5(3(0(x1))))) 0(0(5(3(x1)))) -> 5(2(0(0(3(x1))))) 0(0(5(3(x1)))) -> 0(1(5(0(3(x1))))) 0(0(5(3(x1)))) -> 3(1(5(1(0(0(x1)))))) 0(0(5(3(x1)))) -> 3(2(5(1(0(0(x1)))))) 0(0(5(3(x1)))) -> 2(5(4(3(0(0(x1)))))) 0(0(5(3(x1)))) -> 0(2(5(3(1(0(x1)))))) 0(0(5(3(x1)))) -> 5(1(3(0(2(0(x1)))))) 0(0(5(3(x1)))) -> 5(4(0(1(3(0(x1)))))) 0(0(5(3(x1)))) -> 1(5(0(2(3(0(x1)))))) 0(0(5(3(x1)))) -> 0(2(2(5(3(0(x1)))))) 0(0(5(3(x1)))) -> 0(1(0(3(5(2(x1)))))) 0(0(5(3(x1)))) -> 5(4(2(0(0(3(x1)))))) 0(0(5(3(x1)))) -> 1(5(0(4(0(3(x1)))))) 0(0(5(3(x1)))) -> 5(0(1(4(0(3(x1)))))) 0(0(5(3(x1)))) -> 5(0(4(0(1(3(x1)))))) 0(0(5(3(x1)))) -> 0(2(5(0(2(3(x1)))))) 0(0(5(3(x1)))) -> 0(2(0(5(2(3(x1)))))) 0(0(5(3(x1)))) -> 0(1(5(0(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(1(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(4(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(1(3(5(x1)))))) 0(3(5(3(x1)))) -> 3(2(5(3(0(x1))))) 0(3(5(3(x1)))) -> 5(2(3(0(3(x1))))) 0(3(5(3(x1)))) -> 4(5(3(0(3(x1))))) 0(3(5(3(x1)))) -> 2(5(0(3(3(x1))))) 0(3(5(3(x1)))) -> 5(0(4(3(3(x1))))) 0(3(5(3(x1)))) -> 3(0(1(5(3(x1))))) 0(3(5(3(x1)))) -> 5(4(3(3(2(0(x1)))))) 0(3(5(3(x1)))) -> 2(5(3(3(2(0(x1)))))) 0(3(5(3(x1)))) -> 4(5(3(1(3(0(x1)))))) 0(3(5(3(x1)))) -> 2(1(5(3(3(0(x1)))))) 0(3(5(3(x1)))) -> 1(2(5(3(3(0(x1)))))) 0(3(5(3(x1)))) -> 3(4(5(4(3(0(x1)))))) 0(3(5(3(x1)))) -> 4(5(3(3(4(0(x1)))))) 0(3(5(3(x1)))) -> 3(0(2(5(3(2(x1)))))) 0(3(5(3(x1)))) -> 5(2(2(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(5(2(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 5(2(4(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 2(5(5(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(3(1(5(0(3(x1)))))) 0(3(5(3(x1)))) -> 3(2(2(5(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(5(3(0(1(3(x1)))))) 0(3(5(3(x1)))) -> 2(5(0(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 5(0(1(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 5(0(2(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 0(4(4(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 0(2(5(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 0(4(5(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 3(3(0(2(2(5(x1)))))) 0(3(5(3(x1)))) -> 1(3(1(0(3(5(x1)))))) 0(3(5(3(x1)))) -> 0(1(3(1(3(5(x1)))))) 0(3(5(3(x1)))) -> 2(0(4(3(3(5(x1)))))) 0(3(5(3(x1)))) -> 4(0(4(3(3(5(x1)))))) 0(3(5(3(x1)))) -> 0(1(4(3(3(5(x1)))))) 0(0(3(0(0(x1))))) -> 0(0(2(0(3(0(x1)))))) 0(3(5(0(0(x1))))) -> 0(2(5(0(3(0(x1)))))) 0(3(3(5(0(x1))))) -> 3(0(2(0(3(5(x1)))))) 0(3(0(0(3(x1))))) -> 0(0(4(3(3(0(x1)))))) 0(3(3(0(3(x1))))) -> 3(0(0(2(3(3(x1)))))) 0(3(5(0(3(x1))))) -> 0(4(5(0(3(3(x1)))))) 0(3(5(1(3(x1))))) -> 1(3(2(5(0(3(x1)))))) 0(0(5(2(3(x1))))) -> 4(5(0(2(3(0(x1)))))) 0(0(5(2(3(x1))))) -> 3(0(2(2(5(0(x1)))))) 0(3(5(2(3(x1))))) -> 0(5(2(3(2(3(x1)))))) 0(3(5(2(3(x1))))) -> 1(3(0(2(5(3(x1)))))) 0(3(5(2(3(x1))))) -> 4(3(0(2(5(3(x1)))))) 0(0(5(4(3(x1))))) -> 5(0(4(3(1(0(x1)))))) 0(0(5(4(3(x1))))) -> 5(1(0(4(0(3(x1)))))) 0(0(5(4(3(x1))))) -> 0(1(5(0(4(3(x1)))))) 0(0(3(5(3(x1))))) -> 2(0(3(5(3(0(x1)))))) 0(0(3(5(3(x1))))) -> 0(0(2(3(5(3(x1)))))) 0(3(3(5(3(x1))))) -> 3(3(0(1(5(3(x1)))))) 0(3(3(5(3(x1))))) -> 0(4(3(3(5(3(x1)))))) 0(3(5(5(3(x1))))) -> 3(1(5(3(5(0(x1)))))) 0(3(5(5(3(x1))))) -> 0(4(5(3(5(3(x1)))))) 0(3(3(0(5(x1))))) -> 1(0(3(5(3(0(x1)))))) 0(3(3(0(5(x1))))) -> 0(2(3(5(3(0(x1)))))) 0(3(3(0(5(x1))))) -> 5(0(2(0(3(3(x1)))))) 0(3(3(0(5(x1))))) -> 0(2(0(3(5(3(x1)))))) 0(3(5(0(5(x1))))) -> 0(0(5(4(3(5(x1)))))) 0(3(0(3(5(x1))))) -> 0(1(5(3(0(3(x1)))))) 0(3(0(3(5(x1))))) -> 0(0(3(1(5(3(x1)))))) 0(3(3(3(5(x1))))) -> 3(3(3(2(5(0(x1)))))) 0(3(3(3(5(x1))))) -> 3(3(2(0(3(5(x1)))))) 0(3(5(3(5(x1))))) -> 5(2(3(0(3(5(x1)))))) 5(0(3(0(0(x1))))) -> 5(0(3(2(0(0(x1)))))) 5(0(3(5(0(x1))))) -> 5(0(1(5(0(3(x1)))))) 5(0(3(5(0(x1))))) -> 5(5(0(0(4(3(x1)))))) 5(0(0(3(3(x1))))) -> 0(4(3(0(5(3(x1)))))) 5(0(3(3(3(x1))))) -> 3(1(5(3(0(3(x1)))))) 5(0(3(5(3(x1))))) -> 5(3(0(4(5(3(x1)))))) 5(0(3(5(5(x1))))) -> 5(5(2(0(3(5(x1)))))) ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(3(5(0(x1)))) -> 0^1(0(x1)) 0^1(3(5(0(x1)))) -> 5^1(0(0(x1))) 5^1(0(3(3(3(x1))))) -> 0^1(3(x1)) 5^1(0(3(5(5(x1))))) -> 0^1(3(5(x1))) 0^1(3(5(0(x1)))) -> 0^1(0(3(x1))) 0^1(3(5(0(x1)))) -> 0^1(0(5(4(3(x1))))) 0^1(3(0(3(x1)))) -> 0^1(0(x1)) 0^1(3(3(3(x1)))) -> 0^1(x1) 0^1(0(5(3(x1)))) -> 0^1(0(x1)) 0^1(0(5(3(x1)))) -> 0^1(0(3(x1))) 0^1(3(5(3(x1)))) -> 0^1(x1) 0^1(3(5(3(x1)))) -> 0^1(3(x1)) 0^1(3(5(3(x1)))) -> 5^1(0(3(3(x1)))) 0^1(3(5(3(x1)))) -> 0^1(3(3(x1))) 0^1(3(5(3(x1)))) -> 5^1(0(3(x1))) 0^1(3(5(3(x1)))) -> 0^1(3(5(x1))) 0^1(3(3(0(5(x1))))) -> 0^1(3(5(3(0(x1))))) 0^1(3(5(3(5(x1))))) -> 0^1(3(5(x1))) 0^1(3(5(1(3(x1))))) -> 5^1(0(3(x1))) 0^1(3(5(1(3(x1))))) -> 0^1(3(x1)) 0^1(3(5(5(3(x1))))) -> 5^1(0(x1)) 0^1(3(5(5(3(x1))))) -> 0^1(x1) 0^1(0(3(5(3(x1))))) -> 0^1(3(5(3(0(x1))))) 0^1(3(3(3(5(x1))))) -> 5^1(0(x1)) 0^1(3(3(3(5(x1))))) -> 0^1(x1) 0^1(3(5(0(5(x1))))) -> 0^1(0(5(4(3(5(x1)))))) 0^1(3(3(3(5(x1))))) -> 0^1(3(5(x1))) The TRS R consists of the following rules: 0(0(0(0(x1)))) -> 2(0(1(0(0(0(x1)))))) 0(0(0(0(x1)))) -> 0(2(2(0(0(0(x1)))))) 0(0(0(0(x1)))) -> 0(0(2(0(1(0(x1)))))) 0(3(0(0(x1)))) -> 3(0(2(0(0(x1))))) 0(3(0(0(x1)))) -> 0(4(3(1(0(0(x1)))))) 0(3(0(0(x1)))) -> 0(2(3(0(2(0(x1)))))) 0(3(0(0(x1)))) -> 4(0(4(0(3(0(x1)))))) 0(3(0(0(x1)))) -> 0(2(0(3(4(0(x1)))))) 0(3(0(0(x1)))) -> 0(4(0(3(4(0(x1)))))) 0(3(0(0(x1)))) -> 3(0(0(2(0(1(x1)))))) 0(3(0(0(x1)))) -> 3(0(2(0(0(2(x1)))))) 0(3(0(0(x1)))) -> 0(2(2(0(0(3(x1)))))) 0(0(5(0(x1)))) -> 5(0(0(2(0(x1))))) 0(0(5(0(x1)))) -> 0(1(5(3(0(0(x1)))))) 0(0(5(0(x1)))) -> 4(0(1(5(0(0(x1)))))) 0(0(5(0(x1)))) -> 0(5(4(0(2(0(x1)))))) 0(0(5(0(x1)))) -> 0(4(0(5(2(0(x1)))))) 0(0(5(0(x1)))) -> 2(5(0(0(4(0(x1)))))) 0(0(5(0(x1)))) -> 0(5(1(0(4(0(x1)))))) 0(0(5(0(x1)))) -> 0(1(0(1(5(0(x1)))))) 0(0(5(0(x1)))) -> 5(0(2(0(0(2(x1)))))) 0(0(5(0(x1)))) -> 0(2(0(5(0(2(x1)))))) 0(0(5(0(x1)))) -> 0(2(0(1(0(5(x1)))))) 0(0(5(0(x1)))) -> 0(4(0(1(0(5(x1)))))) 0(3(5(0(x1)))) -> 5(2(3(0(0(x1))))) 0(3(5(0(x1)))) -> 1(5(3(0(0(x1))))) 0(3(5(0(x1)))) -> 2(5(3(0(0(x1))))) 0(3(5(0(x1)))) -> 3(2(5(0(0(x1))))) 0(3(5(0(x1)))) -> 5(3(0(2(0(x1))))) 0(3(5(0(x1)))) -> 5(0(4(3(0(x1))))) 0(3(5(0(x1)))) -> 0(1(0(3(5(x1))))) 0(3(5(0(x1)))) -> 4(5(3(1(0(0(x1)))))) 0(3(5(0(x1)))) -> 3(2(2(5(0(0(x1)))))) 0(3(5(0(x1)))) -> 3(4(2(5(0(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(3(5(1(0(x1)))))) 0(3(5(0(x1)))) -> 5(1(3(0(2(0(x1)))))) 0(3(5(0(x1)))) -> 4(0(2(5(3(0(x1)))))) 0(3(5(0(x1)))) -> 3(4(0(2(5(0(x1)))))) 0(3(5(0(x1)))) -> 3(0(2(2(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(4(3(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(2(5(3(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(5(3(0(1(x1)))))) 0(3(5(0(x1)))) -> 0(0(4(5(3(1(x1)))))) 0(3(5(0(x1)))) -> 0(1(5(0(3(2(x1)))))) 0(3(5(0(x1)))) -> 3(0(0(1(5(2(x1)))))) 0(3(5(0(x1)))) -> 2(5(2(0(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(1(5(3(0(3(x1)))))) 0(3(5(0(x1)))) -> 1(5(0(4(0(3(x1)))))) 0(3(5(0(x1)))) -> 2(5(0(4(0(3(x1)))))) 0(3(5(0(x1)))) -> 1(0(1(5(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(1(5(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(5(1(0(2(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(5(0(4(3(x1)))))) 0(3(5(0(x1)))) -> 2(0(0(5(4(3(x1)))))) 0(3(5(0(x1)))) -> 2(0(0(1(5(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(3(1(0(5(x1)))))) 0(3(5(0(x1)))) -> 0(0(3(2(2(5(x1)))))) 0(3(5(0(x1)))) -> 0(1(1(0(3(5(x1)))))) 0(0(0(3(x1)))) -> 1(0(2(3(0(0(x1)))))) 0(0(0(3(x1)))) -> 0(3(0(1(2(0(x1)))))) 0(0(0(3(x1)))) -> 0(2(2(0(3(0(x1)))))) 0(0(0(3(x1)))) -> 0(4(2(0(3(0(x1)))))) 0(0(0(3(x1)))) -> 0(2(0(2(0(3(x1)))))) 0(3(0(3(x1)))) -> 4(3(2(3(0(0(x1)))))) 0(3(0(3(x1)))) -> 4(3(4(3(0(0(x1)))))) 0(3(0(3(x1)))) -> 3(3(0(2(0(1(x1)))))) 0(3(0(3(x1)))) -> 3(2(0(1(0(3(x1)))))) 0(3(0(3(x1)))) -> 1(3(0(2(0(3(x1)))))) 0(3(0(3(x1)))) -> 1(0(1(3(0(3(x1)))))) 0(3(0(3(x1)))) -> 0(2(5(3(0(3(x1)))))) 0(3(0(3(x1)))) -> 3(2(0(4(0(3(x1)))))) 0(3(0(3(x1)))) -> 0(2(3(0(1(3(x1)))))) 0(3(0(3(x1)))) -> 3(0(1(0(4(3(x1)))))) 0(0(3(3(x1)))) -> 0(4(3(3(0(x1))))) 0(0(3(3(x1)))) -> 3(0(2(0(3(x1))))) 0(0(3(3(x1)))) -> 0(2(3(0(3(x1))))) 0(0(3(3(x1)))) -> 3(3(1(0(1(0(x1)))))) 0(0(3(3(x1)))) -> 3(0(4(3(1(0(x1)))))) 0(0(3(3(x1)))) -> 0(3(3(2(2(0(x1)))))) 0(0(3(3(x1)))) -> 3(3(0(4(2(0(x1)))))) 0(0(3(3(x1)))) -> 0(2(3(3(0(2(x1)))))) 0(0(3(3(x1)))) -> 3(4(0(1(0(3(x1)))))) 0(0(3(3(x1)))) -> 3(0(2(2(0(3(x1)))))) 0(0(3(3(x1)))) -> 2(0(2(3(0(3(x1)))))) 0(0(3(3(x1)))) -> 0(1(4(3(0(3(x1)))))) 0(0(3(3(x1)))) -> 3(0(1(0(1(3(x1)))))) 0(0(3(3(x1)))) -> 2(0(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 4(0(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 0(1(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 0(4(3(0(4(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(3(3(x1))))) 0(3(3(3(x1)))) -> 0(2(3(3(3(x1))))) 0(3(3(3(x1)))) -> 3(2(3(1(3(0(x1)))))) 0(3(3(3(x1)))) -> 3(2(3(0(2(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(2(1(3(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(2(3(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(5(3(3(x1)))))) 0(0(5(3(x1)))) -> 1(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 2(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 4(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 0(2(5(3(0(x1))))) 0(0(5(3(x1)))) -> 0(4(5(3(0(x1))))) 0(0(5(3(x1)))) -> 5(2(0(0(3(x1))))) 0(0(5(3(x1)))) -> 0(1(5(0(3(x1))))) 0(0(5(3(x1)))) -> 3(1(5(1(0(0(x1)))))) 0(0(5(3(x1)))) -> 3(2(5(1(0(0(x1)))))) 0(0(5(3(x1)))) -> 2(5(4(3(0(0(x1)))))) 0(0(5(3(x1)))) -> 0(2(5(3(1(0(x1)))))) 0(0(5(3(x1)))) -> 5(1(3(0(2(0(x1)))))) 0(0(5(3(x1)))) -> 5(4(0(1(3(0(x1)))))) 0(0(5(3(x1)))) -> 1(5(0(2(3(0(x1)))))) 0(0(5(3(x1)))) -> 0(2(2(5(3(0(x1)))))) 0(0(5(3(x1)))) -> 0(1(0(3(5(2(x1)))))) 0(0(5(3(x1)))) -> 5(4(2(0(0(3(x1)))))) 0(0(5(3(x1)))) -> 1(5(0(4(0(3(x1)))))) 0(0(5(3(x1)))) -> 5(0(1(4(0(3(x1)))))) 0(0(5(3(x1)))) -> 5(0(4(0(1(3(x1)))))) 0(0(5(3(x1)))) -> 0(2(5(0(2(3(x1)))))) 0(0(5(3(x1)))) -> 0(2(0(5(2(3(x1)))))) 0(0(5(3(x1)))) -> 0(1(5(0(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(1(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(4(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(1(3(5(x1)))))) 0(3(5(3(x1)))) -> 3(2(5(3(0(x1))))) 0(3(5(3(x1)))) -> 5(2(3(0(3(x1))))) 0(3(5(3(x1)))) -> 4(5(3(0(3(x1))))) 0(3(5(3(x1)))) -> 2(5(0(3(3(x1))))) 0(3(5(3(x1)))) -> 5(0(4(3(3(x1))))) 0(3(5(3(x1)))) -> 3(0(1(5(3(x1))))) 0(3(5(3(x1)))) -> 5(4(3(3(2(0(x1)))))) 0(3(5(3(x1)))) -> 2(5(3(3(2(0(x1)))))) 0(3(5(3(x1)))) -> 4(5(3(1(3(0(x1)))))) 0(3(5(3(x1)))) -> 2(1(5(3(3(0(x1)))))) 0(3(5(3(x1)))) -> 1(2(5(3(3(0(x1)))))) 0(3(5(3(x1)))) -> 3(4(5(4(3(0(x1)))))) 0(3(5(3(x1)))) -> 4(5(3(3(4(0(x1)))))) 0(3(5(3(x1)))) -> 3(0(2(5(3(2(x1)))))) 0(3(5(3(x1)))) -> 5(2(2(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(5(2(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 5(2(4(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 2(5(5(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(3(1(5(0(3(x1)))))) 0(3(5(3(x1)))) -> 3(2(2(5(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(5(3(0(1(3(x1)))))) 0(3(5(3(x1)))) -> 2(5(0(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 5(0(1(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 5(0(2(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 0(4(4(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 0(2(5(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 0(4(5(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 3(3(0(2(2(5(x1)))))) 0(3(5(3(x1)))) -> 1(3(1(0(3(5(x1)))))) 0(3(5(3(x1)))) -> 0(1(3(1(3(5(x1)))))) 0(3(5(3(x1)))) -> 2(0(4(3(3(5(x1)))))) 0(3(5(3(x1)))) -> 4(0(4(3(3(5(x1)))))) 0(3(5(3(x1)))) -> 0(1(4(3(3(5(x1)))))) 0(0(3(0(0(x1))))) -> 0(0(2(0(3(0(x1)))))) 5(0(3(0(0(x1))))) -> 5(0(3(2(0(0(x1)))))) 0(3(5(0(0(x1))))) -> 0(2(5(0(3(0(x1)))))) 5(0(3(5(0(x1))))) -> 5(0(1(5(0(3(x1)))))) 5(0(3(5(0(x1))))) -> 5(5(0(0(4(3(x1)))))) 0(3(3(5(0(x1))))) -> 3(0(2(0(3(5(x1)))))) 0(3(0(0(3(x1))))) -> 0(0(4(3(3(0(x1)))))) 0(3(3(0(3(x1))))) -> 3(0(0(2(3(3(x1)))))) 0(3(5(0(3(x1))))) -> 0(4(5(0(3(3(x1)))))) 0(3(5(1(3(x1))))) -> 1(3(2(5(0(3(x1)))))) 0(0(5(2(3(x1))))) -> 4(5(0(2(3(0(x1)))))) 0(0(5(2(3(x1))))) -> 3(0(2(2(5(0(x1)))))) 0(3(5(2(3(x1))))) -> 0(5(2(3(2(3(x1)))))) 0(3(5(2(3(x1))))) -> 1(3(0(2(5(3(x1)))))) 0(3(5(2(3(x1))))) -> 4(3(0(2(5(3(x1)))))) 5(0(0(3(3(x1))))) -> 0(4(3(0(5(3(x1)))))) 5(0(3(3(3(x1))))) -> 3(1(5(3(0(3(x1)))))) 0(0(5(4(3(x1))))) -> 5(0(4(3(1(0(x1)))))) 0(0(5(4(3(x1))))) -> 5(1(0(4(0(3(x1)))))) 0(0(5(4(3(x1))))) -> 0(1(5(0(4(3(x1)))))) 0(0(3(5(3(x1))))) -> 2(0(3(5(3(0(x1)))))) 0(0(3(5(3(x1))))) -> 0(0(2(3(5(3(x1)))))) 5(0(3(5(3(x1))))) -> 5(3(0(4(5(3(x1)))))) 0(3(3(5(3(x1))))) -> 3(3(0(1(5(3(x1)))))) 0(3(3(5(3(x1))))) -> 0(4(3(3(5(3(x1)))))) 0(3(5(5(3(x1))))) -> 3(1(5(3(5(0(x1)))))) 0(3(5(5(3(x1))))) -> 0(4(5(3(5(3(x1)))))) 0(3(3(0(5(x1))))) -> 1(0(3(5(3(0(x1)))))) 0(3(3(0(5(x1))))) -> 0(2(3(5(3(0(x1)))))) 0(3(3(0(5(x1))))) -> 5(0(2(0(3(3(x1)))))) 0(3(3(0(5(x1))))) -> 0(2(0(3(5(3(x1)))))) 0(3(5(0(5(x1))))) -> 0(0(5(4(3(5(x1)))))) 0(3(0(3(5(x1))))) -> 0(1(5(3(0(3(x1)))))) 0(3(0(3(5(x1))))) -> 0(0(3(1(5(3(x1)))))) 0(3(3(3(5(x1))))) -> 3(3(3(2(5(0(x1)))))) 0(3(3(3(5(x1))))) -> 3(3(2(0(3(5(x1)))))) 0(3(5(3(5(x1))))) -> 5(2(3(0(3(5(x1)))))) 5(0(3(5(5(x1))))) -> 5(5(2(0(3(5(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(3(5(0(x1)))) -> 5^1(0(0(x1))) 5^1(0(3(3(3(x1))))) -> 0^1(3(x1)) 0^1(3(5(0(x1)))) -> 0^1(0(x1)) 0^1(3(5(0(x1)))) -> 0^1(0(3(x1))) 0^1(3(0(3(x1)))) -> 0^1(0(x1)) 0^1(3(3(3(x1)))) -> 0^1(x1) 0^1(0(5(3(x1)))) -> 0^1(0(x1)) 0^1(0(5(3(x1)))) -> 0^1(0(3(x1))) 0^1(3(5(3(x1)))) -> 0^1(x1) 0^1(3(5(3(x1)))) -> 0^1(3(x1)) 0^1(3(5(3(x1)))) -> 5^1(0(3(3(x1)))) 5^1(0(3(5(5(x1))))) -> 0^1(3(5(x1))) 0^1(3(5(3(x1)))) -> 0^1(3(3(x1))) 0^1(3(3(0(5(x1))))) -> 0^1(3(5(3(0(x1))))) 0^1(3(5(3(x1)))) -> 5^1(0(3(x1))) 0^1(3(5(3(x1)))) -> 0^1(3(5(x1))) 0^1(3(5(1(3(x1))))) -> 5^1(0(3(x1))) 0^1(3(5(1(3(x1))))) -> 0^1(3(x1)) 0^1(3(5(5(3(x1))))) -> 5^1(0(x1)) 0^1(3(5(5(3(x1))))) -> 0^1(x1) 0^1(0(3(5(3(x1))))) -> 0^1(3(5(3(0(x1))))) 0^1(3(5(3(5(x1))))) -> 0^1(3(5(x1))) 0^1(3(3(3(5(x1))))) -> 5^1(0(x1)) 0^1(3(3(3(5(x1))))) -> 0^1(x1) 0^1(3(3(3(5(x1))))) -> 0^1(3(5(x1))) The TRS R consists of the following rules: 0(0(0(0(x1)))) -> 2(0(1(0(0(0(x1)))))) 0(0(0(0(x1)))) -> 0(2(2(0(0(0(x1)))))) 0(0(0(0(x1)))) -> 0(0(2(0(1(0(x1)))))) 0(3(0(0(x1)))) -> 3(0(2(0(0(x1))))) 0(3(0(0(x1)))) -> 0(4(3(1(0(0(x1)))))) 0(3(0(0(x1)))) -> 0(2(3(0(2(0(x1)))))) 0(3(0(0(x1)))) -> 4(0(4(0(3(0(x1)))))) 0(3(0(0(x1)))) -> 0(2(0(3(4(0(x1)))))) 0(3(0(0(x1)))) -> 0(4(0(3(4(0(x1)))))) 0(3(0(0(x1)))) -> 3(0(0(2(0(1(x1)))))) 0(3(0(0(x1)))) -> 3(0(2(0(0(2(x1)))))) 0(3(0(0(x1)))) -> 0(2(2(0(0(3(x1)))))) 0(0(5(0(x1)))) -> 5(0(0(2(0(x1))))) 0(0(5(0(x1)))) -> 0(1(5(3(0(0(x1)))))) 0(0(5(0(x1)))) -> 4(0(1(5(0(0(x1)))))) 0(0(5(0(x1)))) -> 0(5(4(0(2(0(x1)))))) 0(0(5(0(x1)))) -> 0(4(0(5(2(0(x1)))))) 0(0(5(0(x1)))) -> 2(5(0(0(4(0(x1)))))) 0(0(5(0(x1)))) -> 0(5(1(0(4(0(x1)))))) 0(0(5(0(x1)))) -> 0(1(0(1(5(0(x1)))))) 0(0(5(0(x1)))) -> 5(0(2(0(0(2(x1)))))) 0(0(5(0(x1)))) -> 0(2(0(5(0(2(x1)))))) 0(0(5(0(x1)))) -> 0(2(0(1(0(5(x1)))))) 0(0(5(0(x1)))) -> 0(4(0(1(0(5(x1)))))) 0(3(5(0(x1)))) -> 5(2(3(0(0(x1))))) 0(3(5(0(x1)))) -> 1(5(3(0(0(x1))))) 0(3(5(0(x1)))) -> 2(5(3(0(0(x1))))) 0(3(5(0(x1)))) -> 3(2(5(0(0(x1))))) 0(3(5(0(x1)))) -> 5(3(0(2(0(x1))))) 0(3(5(0(x1)))) -> 5(0(4(3(0(x1))))) 0(3(5(0(x1)))) -> 0(1(0(3(5(x1))))) 0(3(5(0(x1)))) -> 4(5(3(1(0(0(x1)))))) 0(3(5(0(x1)))) -> 3(2(2(5(0(0(x1)))))) 0(3(5(0(x1)))) -> 3(4(2(5(0(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(3(5(1(0(x1)))))) 0(3(5(0(x1)))) -> 5(1(3(0(2(0(x1)))))) 0(3(5(0(x1)))) -> 4(0(2(5(3(0(x1)))))) 0(3(5(0(x1)))) -> 3(4(0(2(5(0(x1)))))) 0(3(5(0(x1)))) -> 3(0(2(2(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(4(3(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(2(5(3(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(5(3(0(1(x1)))))) 0(3(5(0(x1)))) -> 0(0(4(5(3(1(x1)))))) 0(3(5(0(x1)))) -> 0(1(5(0(3(2(x1)))))) 0(3(5(0(x1)))) -> 3(0(0(1(5(2(x1)))))) 0(3(5(0(x1)))) -> 2(5(2(0(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(1(5(3(0(3(x1)))))) 0(3(5(0(x1)))) -> 1(5(0(4(0(3(x1)))))) 0(3(5(0(x1)))) -> 2(5(0(4(0(3(x1)))))) 0(3(5(0(x1)))) -> 1(0(1(5(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(1(5(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(5(1(0(2(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(5(0(4(3(x1)))))) 0(3(5(0(x1)))) -> 2(0(0(5(4(3(x1)))))) 0(3(5(0(x1)))) -> 2(0(0(1(5(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(3(1(0(5(x1)))))) 0(3(5(0(x1)))) -> 0(0(3(2(2(5(x1)))))) 0(3(5(0(x1)))) -> 0(1(1(0(3(5(x1)))))) 0(0(0(3(x1)))) -> 1(0(2(3(0(0(x1)))))) 0(0(0(3(x1)))) -> 0(3(0(1(2(0(x1)))))) 0(0(0(3(x1)))) -> 0(2(2(0(3(0(x1)))))) 0(0(0(3(x1)))) -> 0(4(2(0(3(0(x1)))))) 0(0(0(3(x1)))) -> 0(2(0(2(0(3(x1)))))) 0(3(0(3(x1)))) -> 4(3(2(3(0(0(x1)))))) 0(3(0(3(x1)))) -> 4(3(4(3(0(0(x1)))))) 0(3(0(3(x1)))) -> 3(3(0(2(0(1(x1)))))) 0(3(0(3(x1)))) -> 3(2(0(1(0(3(x1)))))) 0(3(0(3(x1)))) -> 1(3(0(2(0(3(x1)))))) 0(3(0(3(x1)))) -> 1(0(1(3(0(3(x1)))))) 0(3(0(3(x1)))) -> 0(2(5(3(0(3(x1)))))) 0(3(0(3(x1)))) -> 3(2(0(4(0(3(x1)))))) 0(3(0(3(x1)))) -> 0(2(3(0(1(3(x1)))))) 0(3(0(3(x1)))) -> 3(0(1(0(4(3(x1)))))) 0(0(3(3(x1)))) -> 0(4(3(3(0(x1))))) 0(0(3(3(x1)))) -> 3(0(2(0(3(x1))))) 0(0(3(3(x1)))) -> 0(2(3(0(3(x1))))) 0(0(3(3(x1)))) -> 3(3(1(0(1(0(x1)))))) 0(0(3(3(x1)))) -> 3(0(4(3(1(0(x1)))))) 0(0(3(3(x1)))) -> 0(3(3(2(2(0(x1)))))) 0(0(3(3(x1)))) -> 3(3(0(4(2(0(x1)))))) 0(0(3(3(x1)))) -> 0(2(3(3(0(2(x1)))))) 0(0(3(3(x1)))) -> 3(4(0(1(0(3(x1)))))) 0(0(3(3(x1)))) -> 3(0(2(2(0(3(x1)))))) 0(0(3(3(x1)))) -> 2(0(2(3(0(3(x1)))))) 0(0(3(3(x1)))) -> 0(1(4(3(0(3(x1)))))) 0(0(3(3(x1)))) -> 3(0(1(0(1(3(x1)))))) 0(0(3(3(x1)))) -> 2(0(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 4(0(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 0(1(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 0(4(3(0(4(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(3(3(x1))))) 0(3(3(3(x1)))) -> 0(2(3(3(3(x1))))) 0(3(3(3(x1)))) -> 3(2(3(1(3(0(x1)))))) 0(3(3(3(x1)))) -> 3(2(3(0(2(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(2(1(3(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(2(3(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(5(3(3(x1)))))) 0(0(5(3(x1)))) -> 1(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 2(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 4(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 0(2(5(3(0(x1))))) 0(0(5(3(x1)))) -> 0(4(5(3(0(x1))))) 0(0(5(3(x1)))) -> 5(2(0(0(3(x1))))) 0(0(5(3(x1)))) -> 0(1(5(0(3(x1))))) 0(0(5(3(x1)))) -> 3(1(5(1(0(0(x1)))))) 0(0(5(3(x1)))) -> 3(2(5(1(0(0(x1)))))) 0(0(5(3(x1)))) -> 2(5(4(3(0(0(x1)))))) 0(0(5(3(x1)))) -> 0(2(5(3(1(0(x1)))))) 0(0(5(3(x1)))) -> 5(1(3(0(2(0(x1)))))) 0(0(5(3(x1)))) -> 5(4(0(1(3(0(x1)))))) 0(0(5(3(x1)))) -> 1(5(0(2(3(0(x1)))))) 0(0(5(3(x1)))) -> 0(2(2(5(3(0(x1)))))) 0(0(5(3(x1)))) -> 0(1(0(3(5(2(x1)))))) 0(0(5(3(x1)))) -> 5(4(2(0(0(3(x1)))))) 0(0(5(3(x1)))) -> 1(5(0(4(0(3(x1)))))) 0(0(5(3(x1)))) -> 5(0(1(4(0(3(x1)))))) 0(0(5(3(x1)))) -> 5(0(4(0(1(3(x1)))))) 0(0(5(3(x1)))) -> 0(2(5(0(2(3(x1)))))) 0(0(5(3(x1)))) -> 0(2(0(5(2(3(x1)))))) 0(0(5(3(x1)))) -> 0(1(5(0(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(1(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(4(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(1(3(5(x1)))))) 0(3(5(3(x1)))) -> 3(2(5(3(0(x1))))) 0(3(5(3(x1)))) -> 5(2(3(0(3(x1))))) 0(3(5(3(x1)))) -> 4(5(3(0(3(x1))))) 0(3(5(3(x1)))) -> 2(5(0(3(3(x1))))) 0(3(5(3(x1)))) -> 5(0(4(3(3(x1))))) 0(3(5(3(x1)))) -> 3(0(1(5(3(x1))))) 0(3(5(3(x1)))) -> 5(4(3(3(2(0(x1)))))) 0(3(5(3(x1)))) -> 2(5(3(3(2(0(x1)))))) 0(3(5(3(x1)))) -> 4(5(3(1(3(0(x1)))))) 0(3(5(3(x1)))) -> 2(1(5(3(3(0(x1)))))) 0(3(5(3(x1)))) -> 1(2(5(3(3(0(x1)))))) 0(3(5(3(x1)))) -> 3(4(5(4(3(0(x1)))))) 0(3(5(3(x1)))) -> 4(5(3(3(4(0(x1)))))) 0(3(5(3(x1)))) -> 3(0(2(5(3(2(x1)))))) 0(3(5(3(x1)))) -> 5(2(2(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(5(2(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 5(2(4(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 2(5(5(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(3(1(5(0(3(x1)))))) 0(3(5(3(x1)))) -> 3(2(2(5(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(5(3(0(1(3(x1)))))) 0(3(5(3(x1)))) -> 2(5(0(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 5(0(1(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 5(0(2(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 0(4(4(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 0(2(5(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 0(4(5(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 3(3(0(2(2(5(x1)))))) 0(3(5(3(x1)))) -> 1(3(1(0(3(5(x1)))))) 0(3(5(3(x1)))) -> 0(1(3(1(3(5(x1)))))) 0(3(5(3(x1)))) -> 2(0(4(3(3(5(x1)))))) 0(3(5(3(x1)))) -> 4(0(4(3(3(5(x1)))))) 0(3(5(3(x1)))) -> 0(1(4(3(3(5(x1)))))) 0(0(3(0(0(x1))))) -> 0(0(2(0(3(0(x1)))))) 5(0(3(0(0(x1))))) -> 5(0(3(2(0(0(x1)))))) 0(3(5(0(0(x1))))) -> 0(2(5(0(3(0(x1)))))) 5(0(3(5(0(x1))))) -> 5(0(1(5(0(3(x1)))))) 5(0(3(5(0(x1))))) -> 5(5(0(0(4(3(x1)))))) 0(3(3(5(0(x1))))) -> 3(0(2(0(3(5(x1)))))) 0(3(0(0(3(x1))))) -> 0(0(4(3(3(0(x1)))))) 0(3(3(0(3(x1))))) -> 3(0(0(2(3(3(x1)))))) 0(3(5(0(3(x1))))) -> 0(4(5(0(3(3(x1)))))) 0(3(5(1(3(x1))))) -> 1(3(2(5(0(3(x1)))))) 0(0(5(2(3(x1))))) -> 4(5(0(2(3(0(x1)))))) 0(0(5(2(3(x1))))) -> 3(0(2(2(5(0(x1)))))) 0(3(5(2(3(x1))))) -> 0(5(2(3(2(3(x1)))))) 0(3(5(2(3(x1))))) -> 1(3(0(2(5(3(x1)))))) 0(3(5(2(3(x1))))) -> 4(3(0(2(5(3(x1)))))) 5(0(0(3(3(x1))))) -> 0(4(3(0(5(3(x1)))))) 5(0(3(3(3(x1))))) -> 3(1(5(3(0(3(x1)))))) 0(0(5(4(3(x1))))) -> 5(0(4(3(1(0(x1)))))) 0(0(5(4(3(x1))))) -> 5(1(0(4(0(3(x1)))))) 0(0(5(4(3(x1))))) -> 0(1(5(0(4(3(x1)))))) 0(0(3(5(3(x1))))) -> 2(0(3(5(3(0(x1)))))) 0(0(3(5(3(x1))))) -> 0(0(2(3(5(3(x1)))))) 5(0(3(5(3(x1))))) -> 5(3(0(4(5(3(x1)))))) 0(3(3(5(3(x1))))) -> 3(3(0(1(5(3(x1)))))) 0(3(3(5(3(x1))))) -> 0(4(3(3(5(3(x1)))))) 0(3(5(5(3(x1))))) -> 3(1(5(3(5(0(x1)))))) 0(3(5(5(3(x1))))) -> 0(4(5(3(5(3(x1)))))) 0(3(3(0(5(x1))))) -> 1(0(3(5(3(0(x1)))))) 0(3(3(0(5(x1))))) -> 0(2(3(5(3(0(x1)))))) 0(3(3(0(5(x1))))) -> 5(0(2(0(3(3(x1)))))) 0(3(3(0(5(x1))))) -> 0(2(0(3(5(3(x1)))))) 0(3(5(0(5(x1))))) -> 0(0(5(4(3(5(x1)))))) 0(3(0(3(5(x1))))) -> 0(1(5(3(0(3(x1)))))) 0(3(0(3(5(x1))))) -> 0(0(3(1(5(3(x1)))))) 0(3(3(3(5(x1))))) -> 3(3(3(2(5(0(x1)))))) 0(3(3(3(5(x1))))) -> 3(3(2(0(3(5(x1)))))) 0(3(5(3(5(x1))))) -> 5(2(3(0(3(5(x1)))))) 5(0(3(5(5(x1))))) -> 5(5(2(0(3(5(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(3(5(0(x1)))) -> 5^1(0(0(x1))) 0^1(3(5(0(x1)))) -> 0^1(0(x1)) 0^1(3(5(0(x1)))) -> 0^1(0(3(x1))) 0^1(0(5(3(x1)))) -> 0^1(0(x1)) 0^1(0(5(3(x1)))) -> 0^1(0(3(x1))) 0^1(3(5(3(x1)))) -> 0^1(x1) 0^1(3(5(3(x1)))) -> 0^1(3(x1)) 0^1(3(5(3(x1)))) -> 5^1(0(3(3(x1)))) 5^1(0(3(5(5(x1))))) -> 0^1(3(5(x1))) 0^1(3(5(3(x1)))) -> 0^1(3(3(x1))) 0^1(3(3(0(5(x1))))) -> 0^1(3(5(3(0(x1))))) 0^1(3(5(3(x1)))) -> 5^1(0(3(x1))) 0^1(3(5(5(3(x1))))) -> 5^1(0(x1)) 0^1(3(5(5(3(x1))))) -> 0^1(x1) 0^1(0(3(5(3(x1))))) -> 0^1(3(5(3(0(x1))))) 0^1(3(5(3(5(x1))))) -> 0^1(3(5(x1))) 0^1(3(3(3(5(x1))))) -> 5^1(0(x1)) 0^1(3(3(3(5(x1))))) -> 0^1(x1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( 0^1_1(x_1) ) = 2x_1 POL( 5^1_1(x_1) ) = max{0, x_1 - 2} POL( 0_1(x_1) ) = 2x_1 + 2 POL( 2_1(x_1) ) = max{0, -2} POL( 1_1(x_1) ) = max{0, x_1 - 1} POL( 3_1(x_1) ) = x_1 POL( 4_1(x_1) ) = max{0, -2} POL( 5_1(x_1) ) = x_1 + 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0(0(0(0(x1)))) -> 2(0(1(0(0(0(x1)))))) 0(0(0(0(x1)))) -> 0(2(2(0(0(0(x1)))))) 0(0(0(0(x1)))) -> 0(0(2(0(1(0(x1)))))) 0(3(0(0(x1)))) -> 3(0(2(0(0(x1))))) 0(3(0(0(x1)))) -> 0(4(3(1(0(0(x1)))))) 0(3(0(0(x1)))) -> 0(2(3(0(2(0(x1)))))) 0(3(0(0(x1)))) -> 4(0(4(0(3(0(x1)))))) 0(3(0(0(x1)))) -> 0(2(0(3(4(0(x1)))))) 0(3(0(0(x1)))) -> 0(4(0(3(4(0(x1)))))) 0(3(0(0(x1)))) -> 3(0(0(2(0(1(x1)))))) 0(3(0(0(x1)))) -> 3(0(2(0(0(2(x1)))))) 0(3(0(0(x1)))) -> 0(2(2(0(0(3(x1)))))) 0(0(5(0(x1)))) -> 5(0(0(2(0(x1))))) 0(0(5(0(x1)))) -> 0(1(5(3(0(0(x1)))))) 0(0(5(0(x1)))) -> 4(0(1(5(0(0(x1)))))) 0(0(5(0(x1)))) -> 0(5(4(0(2(0(x1)))))) 0(0(5(0(x1)))) -> 0(4(0(5(2(0(x1)))))) 0(0(5(0(x1)))) -> 2(5(0(0(4(0(x1)))))) 0(0(5(0(x1)))) -> 0(5(1(0(4(0(x1)))))) 0(0(5(0(x1)))) -> 0(1(0(1(5(0(x1)))))) 0(0(5(0(x1)))) -> 5(0(2(0(0(2(x1)))))) 0(0(5(0(x1)))) -> 0(2(0(5(0(2(x1)))))) 0(0(5(0(x1)))) -> 0(2(0(1(0(5(x1)))))) 0(0(5(0(x1)))) -> 0(4(0(1(0(5(x1)))))) 0(3(5(0(x1)))) -> 5(2(3(0(0(x1))))) 0(3(5(0(x1)))) -> 1(5(3(0(0(x1))))) 0(3(5(0(x1)))) -> 2(5(3(0(0(x1))))) 0(3(5(0(x1)))) -> 3(2(5(0(0(x1))))) 0(3(5(0(x1)))) -> 5(3(0(2(0(x1))))) 0(3(5(0(x1)))) -> 5(0(4(3(0(x1))))) 0(3(5(0(x1)))) -> 0(1(0(3(5(x1))))) 0(3(5(0(x1)))) -> 4(5(3(1(0(0(x1)))))) 0(3(5(0(x1)))) -> 3(2(2(5(0(0(x1)))))) 0(3(5(0(x1)))) -> 3(4(2(5(0(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(3(5(1(0(x1)))))) 0(3(5(0(x1)))) -> 5(1(3(0(2(0(x1)))))) 0(3(5(0(x1)))) -> 4(0(2(5(3(0(x1)))))) 0(3(5(0(x1)))) -> 3(4(0(2(5(0(x1)))))) 0(3(5(0(x1)))) -> 3(0(2(2(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(4(3(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(2(5(3(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(5(3(0(1(x1)))))) 0(3(5(0(x1)))) -> 0(0(4(5(3(1(x1)))))) 0(3(5(0(x1)))) -> 0(1(5(0(3(2(x1)))))) 0(3(5(0(x1)))) -> 3(0(0(1(5(2(x1)))))) 0(3(5(0(x1)))) -> 2(5(2(0(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(1(5(3(0(3(x1)))))) 0(3(5(0(x1)))) -> 1(5(0(4(0(3(x1)))))) 0(3(5(0(x1)))) -> 2(5(0(4(0(3(x1)))))) 0(3(5(0(x1)))) -> 1(0(1(5(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(1(5(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(5(1(0(2(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(5(0(4(3(x1)))))) 0(3(5(0(x1)))) -> 2(0(0(5(4(3(x1)))))) 0(3(5(0(x1)))) -> 2(0(0(1(5(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(3(1(0(5(x1)))))) 0(3(5(0(x1)))) -> 0(0(3(2(2(5(x1)))))) 0(3(5(0(x1)))) -> 0(1(1(0(3(5(x1)))))) 0(0(0(3(x1)))) -> 1(0(2(3(0(0(x1)))))) 0(0(0(3(x1)))) -> 0(3(0(1(2(0(x1)))))) 0(0(0(3(x1)))) -> 0(2(2(0(3(0(x1)))))) 0(0(0(3(x1)))) -> 0(4(2(0(3(0(x1)))))) 0(0(0(3(x1)))) -> 0(2(0(2(0(3(x1)))))) 0(3(0(3(x1)))) -> 4(3(2(3(0(0(x1)))))) 0(3(0(3(x1)))) -> 4(3(4(3(0(0(x1)))))) 0(3(0(3(x1)))) -> 3(3(0(2(0(1(x1)))))) 0(3(0(3(x1)))) -> 3(2(0(1(0(3(x1)))))) 0(3(0(3(x1)))) -> 1(3(0(2(0(3(x1)))))) 0(3(0(3(x1)))) -> 1(0(1(3(0(3(x1)))))) 0(3(0(3(x1)))) -> 0(2(5(3(0(3(x1)))))) 0(3(0(3(x1)))) -> 3(2(0(4(0(3(x1)))))) 0(3(0(3(x1)))) -> 0(2(3(0(1(3(x1)))))) 0(3(0(3(x1)))) -> 3(0(1(0(4(3(x1)))))) 0(0(3(3(x1)))) -> 0(4(3(3(0(x1))))) 0(0(3(3(x1)))) -> 3(0(2(0(3(x1))))) 0(0(3(3(x1)))) -> 0(2(3(0(3(x1))))) 0(0(3(3(x1)))) -> 3(3(1(0(1(0(x1)))))) 0(0(3(3(x1)))) -> 3(0(4(3(1(0(x1)))))) 0(0(3(3(x1)))) -> 0(3(3(2(2(0(x1)))))) 0(0(3(3(x1)))) -> 3(3(0(4(2(0(x1)))))) 0(0(3(3(x1)))) -> 0(2(3(3(0(2(x1)))))) 0(0(3(3(x1)))) -> 3(4(0(1(0(3(x1)))))) 0(0(3(3(x1)))) -> 3(0(2(2(0(3(x1)))))) 0(0(3(3(x1)))) -> 2(0(2(3(0(3(x1)))))) 0(0(3(3(x1)))) -> 0(1(4(3(0(3(x1)))))) 0(0(3(3(x1)))) -> 3(0(1(0(1(3(x1)))))) 0(0(3(3(x1)))) -> 2(0(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 4(0(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 0(1(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 0(4(3(0(4(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(3(3(x1))))) 0(3(3(3(x1)))) -> 0(2(3(3(3(x1))))) 0(3(3(3(x1)))) -> 3(2(3(1(3(0(x1)))))) 0(3(3(3(x1)))) -> 3(2(3(0(2(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(2(1(3(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(2(3(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(5(3(3(x1)))))) 0(0(5(3(x1)))) -> 1(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 2(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 4(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 0(2(5(3(0(x1))))) 0(0(5(3(x1)))) -> 0(4(5(3(0(x1))))) 0(0(5(3(x1)))) -> 5(2(0(0(3(x1))))) 0(0(5(3(x1)))) -> 0(1(5(0(3(x1))))) 0(0(5(3(x1)))) -> 3(1(5(1(0(0(x1)))))) 0(0(5(3(x1)))) -> 3(2(5(1(0(0(x1)))))) 0(0(5(3(x1)))) -> 2(5(4(3(0(0(x1)))))) 0(0(5(3(x1)))) -> 0(2(5(3(1(0(x1)))))) 0(0(5(3(x1)))) -> 5(1(3(0(2(0(x1)))))) 0(0(5(3(x1)))) -> 5(4(0(1(3(0(x1)))))) 0(0(5(3(x1)))) -> 1(5(0(2(3(0(x1)))))) 0(0(5(3(x1)))) -> 0(2(2(5(3(0(x1)))))) 0(0(5(3(x1)))) -> 0(1(0(3(5(2(x1)))))) 0(0(5(3(x1)))) -> 5(4(2(0(0(3(x1)))))) 0(0(5(3(x1)))) -> 1(5(0(4(0(3(x1)))))) 0(0(5(3(x1)))) -> 5(0(1(4(0(3(x1)))))) 0(0(5(3(x1)))) -> 5(0(4(0(1(3(x1)))))) 0(0(5(3(x1)))) -> 0(2(5(0(2(3(x1)))))) 0(0(5(3(x1)))) -> 0(2(0(5(2(3(x1)))))) 0(0(5(3(x1)))) -> 0(1(5(0(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(1(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(4(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(1(3(5(x1)))))) 0(3(5(3(x1)))) -> 3(2(5(3(0(x1))))) 0(3(5(3(x1)))) -> 5(2(3(0(3(x1))))) 0(3(5(3(x1)))) -> 4(5(3(0(3(x1))))) 0(3(5(3(x1)))) -> 2(5(0(3(3(x1))))) 0(3(5(3(x1)))) -> 5(0(4(3(3(x1))))) 0(3(5(3(x1)))) -> 3(0(1(5(3(x1))))) 0(3(5(3(x1)))) -> 5(4(3(3(2(0(x1)))))) 0(3(5(3(x1)))) -> 2(5(3(3(2(0(x1)))))) 0(3(5(3(x1)))) -> 4(5(3(1(3(0(x1)))))) 0(3(5(3(x1)))) -> 2(1(5(3(3(0(x1)))))) 0(3(5(3(x1)))) -> 1(2(5(3(3(0(x1)))))) 0(3(5(3(x1)))) -> 3(4(5(4(3(0(x1)))))) 0(3(5(3(x1)))) -> 4(5(3(3(4(0(x1)))))) 0(3(5(3(x1)))) -> 3(0(2(5(3(2(x1)))))) 0(3(5(3(x1)))) -> 5(2(2(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(5(2(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 5(2(4(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 2(5(5(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(3(1(5(0(3(x1)))))) 0(3(5(3(x1)))) -> 3(2(2(5(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(5(3(0(1(3(x1)))))) 0(3(5(3(x1)))) -> 2(5(0(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 5(0(1(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 5(0(2(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 0(4(4(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 0(2(5(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 0(4(5(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 3(3(0(2(2(5(x1)))))) 0(3(5(3(x1)))) -> 1(3(1(0(3(5(x1)))))) 0(3(5(3(x1)))) -> 0(1(3(1(3(5(x1)))))) 0(3(5(3(x1)))) -> 2(0(4(3(3(5(x1)))))) 0(3(5(3(x1)))) -> 4(0(4(3(3(5(x1)))))) 0(3(5(3(x1)))) -> 0(1(4(3(3(5(x1)))))) 0(0(3(0(0(x1))))) -> 0(0(2(0(3(0(x1)))))) 0(3(5(0(0(x1))))) -> 0(2(5(0(3(0(x1)))))) 0(3(3(5(0(x1))))) -> 3(0(2(0(3(5(x1)))))) 0(3(0(0(3(x1))))) -> 0(0(4(3(3(0(x1)))))) 0(3(3(0(3(x1))))) -> 3(0(0(2(3(3(x1)))))) 0(3(5(0(3(x1))))) -> 0(4(5(0(3(3(x1)))))) 0(3(5(1(3(x1))))) -> 1(3(2(5(0(3(x1)))))) 0(0(5(2(3(x1))))) -> 4(5(0(2(3(0(x1)))))) 0(0(5(2(3(x1))))) -> 3(0(2(2(5(0(x1)))))) 0(3(5(2(3(x1))))) -> 0(5(2(3(2(3(x1)))))) 0(3(5(2(3(x1))))) -> 1(3(0(2(5(3(x1)))))) 0(3(5(2(3(x1))))) -> 4(3(0(2(5(3(x1)))))) 0(0(5(4(3(x1))))) -> 5(0(4(3(1(0(x1)))))) 0(0(5(4(3(x1))))) -> 5(1(0(4(0(3(x1)))))) 0(0(5(4(3(x1))))) -> 0(1(5(0(4(3(x1)))))) 0(0(3(5(3(x1))))) -> 2(0(3(5(3(0(x1)))))) 0(0(3(5(3(x1))))) -> 0(0(2(3(5(3(x1)))))) 0(3(3(5(3(x1))))) -> 3(3(0(1(5(3(x1)))))) 0(3(3(5(3(x1))))) -> 0(4(3(3(5(3(x1)))))) 0(3(5(5(3(x1))))) -> 3(1(5(3(5(0(x1)))))) 0(3(5(5(3(x1))))) -> 0(4(5(3(5(3(x1)))))) 0(3(3(0(5(x1))))) -> 1(0(3(5(3(0(x1)))))) 0(3(3(0(5(x1))))) -> 0(2(3(5(3(0(x1)))))) 0(3(3(0(5(x1))))) -> 5(0(2(0(3(3(x1)))))) 0(3(3(0(5(x1))))) -> 0(2(0(3(5(3(x1)))))) 0(3(5(0(5(x1))))) -> 0(0(5(4(3(5(x1)))))) 0(3(0(3(5(x1))))) -> 0(1(5(3(0(3(x1)))))) 0(3(0(3(5(x1))))) -> 0(0(3(1(5(3(x1)))))) 0(3(3(3(5(x1))))) -> 3(3(3(2(5(0(x1)))))) 0(3(3(3(5(x1))))) -> 3(3(2(0(3(5(x1)))))) 0(3(5(3(5(x1))))) -> 5(2(3(0(3(5(x1)))))) 5(0(3(0(0(x1))))) -> 5(0(3(2(0(0(x1)))))) 5(0(3(5(0(x1))))) -> 5(0(1(5(0(3(x1)))))) 5(0(3(5(0(x1))))) -> 5(5(0(0(4(3(x1)))))) 5(0(0(3(3(x1))))) -> 0(4(3(0(5(3(x1)))))) 5(0(3(3(3(x1))))) -> 3(1(5(3(0(3(x1)))))) 5(0(3(5(3(x1))))) -> 5(3(0(4(5(3(x1)))))) 5(0(3(5(5(x1))))) -> 5(5(2(0(3(5(x1)))))) ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: 5^1(0(3(3(3(x1))))) -> 0^1(3(x1)) 0^1(3(0(3(x1)))) -> 0^1(0(x1)) 0^1(3(3(3(x1)))) -> 0^1(x1) 0^1(3(5(3(x1)))) -> 0^1(3(5(x1))) 0^1(3(5(1(3(x1))))) -> 5^1(0(3(x1))) 0^1(3(5(1(3(x1))))) -> 0^1(3(x1)) 0^1(3(3(3(5(x1))))) -> 0^1(3(5(x1))) The TRS R consists of the following rules: 0(0(0(0(x1)))) -> 2(0(1(0(0(0(x1)))))) 0(0(0(0(x1)))) -> 0(2(2(0(0(0(x1)))))) 0(0(0(0(x1)))) -> 0(0(2(0(1(0(x1)))))) 0(3(0(0(x1)))) -> 3(0(2(0(0(x1))))) 0(3(0(0(x1)))) -> 0(4(3(1(0(0(x1)))))) 0(3(0(0(x1)))) -> 0(2(3(0(2(0(x1)))))) 0(3(0(0(x1)))) -> 4(0(4(0(3(0(x1)))))) 0(3(0(0(x1)))) -> 0(2(0(3(4(0(x1)))))) 0(3(0(0(x1)))) -> 0(4(0(3(4(0(x1)))))) 0(3(0(0(x1)))) -> 3(0(0(2(0(1(x1)))))) 0(3(0(0(x1)))) -> 3(0(2(0(0(2(x1)))))) 0(3(0(0(x1)))) -> 0(2(2(0(0(3(x1)))))) 0(0(5(0(x1)))) -> 5(0(0(2(0(x1))))) 0(0(5(0(x1)))) -> 0(1(5(3(0(0(x1)))))) 0(0(5(0(x1)))) -> 4(0(1(5(0(0(x1)))))) 0(0(5(0(x1)))) -> 0(5(4(0(2(0(x1)))))) 0(0(5(0(x1)))) -> 0(4(0(5(2(0(x1)))))) 0(0(5(0(x1)))) -> 2(5(0(0(4(0(x1)))))) 0(0(5(0(x1)))) -> 0(5(1(0(4(0(x1)))))) 0(0(5(0(x1)))) -> 0(1(0(1(5(0(x1)))))) 0(0(5(0(x1)))) -> 5(0(2(0(0(2(x1)))))) 0(0(5(0(x1)))) -> 0(2(0(5(0(2(x1)))))) 0(0(5(0(x1)))) -> 0(2(0(1(0(5(x1)))))) 0(0(5(0(x1)))) -> 0(4(0(1(0(5(x1)))))) 0(3(5(0(x1)))) -> 5(2(3(0(0(x1))))) 0(3(5(0(x1)))) -> 1(5(3(0(0(x1))))) 0(3(5(0(x1)))) -> 2(5(3(0(0(x1))))) 0(3(5(0(x1)))) -> 3(2(5(0(0(x1))))) 0(3(5(0(x1)))) -> 5(3(0(2(0(x1))))) 0(3(5(0(x1)))) -> 5(0(4(3(0(x1))))) 0(3(5(0(x1)))) -> 0(1(0(3(5(x1))))) 0(3(5(0(x1)))) -> 4(5(3(1(0(0(x1)))))) 0(3(5(0(x1)))) -> 3(2(2(5(0(0(x1)))))) 0(3(5(0(x1)))) -> 3(4(2(5(0(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(3(5(1(0(x1)))))) 0(3(5(0(x1)))) -> 5(1(3(0(2(0(x1)))))) 0(3(5(0(x1)))) -> 4(0(2(5(3(0(x1)))))) 0(3(5(0(x1)))) -> 3(4(0(2(5(0(x1)))))) 0(3(5(0(x1)))) -> 3(0(2(2(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(4(3(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(2(5(3(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(5(3(0(1(x1)))))) 0(3(5(0(x1)))) -> 0(0(4(5(3(1(x1)))))) 0(3(5(0(x1)))) -> 0(1(5(0(3(2(x1)))))) 0(3(5(0(x1)))) -> 3(0(0(1(5(2(x1)))))) 0(3(5(0(x1)))) -> 2(5(2(0(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(1(5(3(0(3(x1)))))) 0(3(5(0(x1)))) -> 1(5(0(4(0(3(x1)))))) 0(3(5(0(x1)))) -> 2(5(0(4(0(3(x1)))))) 0(3(5(0(x1)))) -> 1(0(1(5(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(1(5(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(5(1(0(2(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(5(0(4(3(x1)))))) 0(3(5(0(x1)))) -> 2(0(0(5(4(3(x1)))))) 0(3(5(0(x1)))) -> 2(0(0(1(5(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(3(1(0(5(x1)))))) 0(3(5(0(x1)))) -> 0(0(3(2(2(5(x1)))))) 0(3(5(0(x1)))) -> 0(1(1(0(3(5(x1)))))) 0(0(0(3(x1)))) -> 1(0(2(3(0(0(x1)))))) 0(0(0(3(x1)))) -> 0(3(0(1(2(0(x1)))))) 0(0(0(3(x1)))) -> 0(2(2(0(3(0(x1)))))) 0(0(0(3(x1)))) -> 0(4(2(0(3(0(x1)))))) 0(0(0(3(x1)))) -> 0(2(0(2(0(3(x1)))))) 0(3(0(3(x1)))) -> 4(3(2(3(0(0(x1)))))) 0(3(0(3(x1)))) -> 4(3(4(3(0(0(x1)))))) 0(3(0(3(x1)))) -> 3(3(0(2(0(1(x1)))))) 0(3(0(3(x1)))) -> 3(2(0(1(0(3(x1)))))) 0(3(0(3(x1)))) -> 1(3(0(2(0(3(x1)))))) 0(3(0(3(x1)))) -> 1(0(1(3(0(3(x1)))))) 0(3(0(3(x1)))) -> 0(2(5(3(0(3(x1)))))) 0(3(0(3(x1)))) -> 3(2(0(4(0(3(x1)))))) 0(3(0(3(x1)))) -> 0(2(3(0(1(3(x1)))))) 0(3(0(3(x1)))) -> 3(0(1(0(4(3(x1)))))) 0(0(3(3(x1)))) -> 0(4(3(3(0(x1))))) 0(0(3(3(x1)))) -> 3(0(2(0(3(x1))))) 0(0(3(3(x1)))) -> 0(2(3(0(3(x1))))) 0(0(3(3(x1)))) -> 3(3(1(0(1(0(x1)))))) 0(0(3(3(x1)))) -> 3(0(4(3(1(0(x1)))))) 0(0(3(3(x1)))) -> 0(3(3(2(2(0(x1)))))) 0(0(3(3(x1)))) -> 3(3(0(4(2(0(x1)))))) 0(0(3(3(x1)))) -> 0(2(3(3(0(2(x1)))))) 0(0(3(3(x1)))) -> 3(4(0(1(0(3(x1)))))) 0(0(3(3(x1)))) -> 3(0(2(2(0(3(x1)))))) 0(0(3(3(x1)))) -> 2(0(2(3(0(3(x1)))))) 0(0(3(3(x1)))) -> 0(1(4(3(0(3(x1)))))) 0(0(3(3(x1)))) -> 3(0(1(0(1(3(x1)))))) 0(0(3(3(x1)))) -> 2(0(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 4(0(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 0(1(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 0(4(3(0(4(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(3(3(x1))))) 0(3(3(3(x1)))) -> 0(2(3(3(3(x1))))) 0(3(3(3(x1)))) -> 3(2(3(1(3(0(x1)))))) 0(3(3(3(x1)))) -> 3(2(3(0(2(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(2(1(3(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(2(3(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(5(3(3(x1)))))) 0(0(5(3(x1)))) -> 1(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 2(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 4(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 0(2(5(3(0(x1))))) 0(0(5(3(x1)))) -> 0(4(5(3(0(x1))))) 0(0(5(3(x1)))) -> 5(2(0(0(3(x1))))) 0(0(5(3(x1)))) -> 0(1(5(0(3(x1))))) 0(0(5(3(x1)))) -> 3(1(5(1(0(0(x1)))))) 0(0(5(3(x1)))) -> 3(2(5(1(0(0(x1)))))) 0(0(5(3(x1)))) -> 2(5(4(3(0(0(x1)))))) 0(0(5(3(x1)))) -> 0(2(5(3(1(0(x1)))))) 0(0(5(3(x1)))) -> 5(1(3(0(2(0(x1)))))) 0(0(5(3(x1)))) -> 5(4(0(1(3(0(x1)))))) 0(0(5(3(x1)))) -> 1(5(0(2(3(0(x1)))))) 0(0(5(3(x1)))) -> 0(2(2(5(3(0(x1)))))) 0(0(5(3(x1)))) -> 0(1(0(3(5(2(x1)))))) 0(0(5(3(x1)))) -> 5(4(2(0(0(3(x1)))))) 0(0(5(3(x1)))) -> 1(5(0(4(0(3(x1)))))) 0(0(5(3(x1)))) -> 5(0(1(4(0(3(x1)))))) 0(0(5(3(x1)))) -> 5(0(4(0(1(3(x1)))))) 0(0(5(3(x1)))) -> 0(2(5(0(2(3(x1)))))) 0(0(5(3(x1)))) -> 0(2(0(5(2(3(x1)))))) 0(0(5(3(x1)))) -> 0(1(5(0(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(1(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(4(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(1(3(5(x1)))))) 0(3(5(3(x1)))) -> 3(2(5(3(0(x1))))) 0(3(5(3(x1)))) -> 5(2(3(0(3(x1))))) 0(3(5(3(x1)))) -> 4(5(3(0(3(x1))))) 0(3(5(3(x1)))) -> 2(5(0(3(3(x1))))) 0(3(5(3(x1)))) -> 5(0(4(3(3(x1))))) 0(3(5(3(x1)))) -> 3(0(1(5(3(x1))))) 0(3(5(3(x1)))) -> 5(4(3(3(2(0(x1)))))) 0(3(5(3(x1)))) -> 2(5(3(3(2(0(x1)))))) 0(3(5(3(x1)))) -> 4(5(3(1(3(0(x1)))))) 0(3(5(3(x1)))) -> 2(1(5(3(3(0(x1)))))) 0(3(5(3(x1)))) -> 1(2(5(3(3(0(x1)))))) 0(3(5(3(x1)))) -> 3(4(5(4(3(0(x1)))))) 0(3(5(3(x1)))) -> 4(5(3(3(4(0(x1)))))) 0(3(5(3(x1)))) -> 3(0(2(5(3(2(x1)))))) 0(3(5(3(x1)))) -> 5(2(2(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(5(2(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 5(2(4(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 2(5(5(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(3(1(5(0(3(x1)))))) 0(3(5(3(x1)))) -> 3(2(2(5(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(5(3(0(1(3(x1)))))) 0(3(5(3(x1)))) -> 2(5(0(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 5(0(1(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 5(0(2(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 0(4(4(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 0(2(5(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 0(4(5(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 3(3(0(2(2(5(x1)))))) 0(3(5(3(x1)))) -> 1(3(1(0(3(5(x1)))))) 0(3(5(3(x1)))) -> 0(1(3(1(3(5(x1)))))) 0(3(5(3(x1)))) -> 2(0(4(3(3(5(x1)))))) 0(3(5(3(x1)))) -> 4(0(4(3(3(5(x1)))))) 0(3(5(3(x1)))) -> 0(1(4(3(3(5(x1)))))) 0(0(3(0(0(x1))))) -> 0(0(2(0(3(0(x1)))))) 5(0(3(0(0(x1))))) -> 5(0(3(2(0(0(x1)))))) 0(3(5(0(0(x1))))) -> 0(2(5(0(3(0(x1)))))) 5(0(3(5(0(x1))))) -> 5(0(1(5(0(3(x1)))))) 5(0(3(5(0(x1))))) -> 5(5(0(0(4(3(x1)))))) 0(3(3(5(0(x1))))) -> 3(0(2(0(3(5(x1)))))) 0(3(0(0(3(x1))))) -> 0(0(4(3(3(0(x1)))))) 0(3(3(0(3(x1))))) -> 3(0(0(2(3(3(x1)))))) 0(3(5(0(3(x1))))) -> 0(4(5(0(3(3(x1)))))) 0(3(5(1(3(x1))))) -> 1(3(2(5(0(3(x1)))))) 0(0(5(2(3(x1))))) -> 4(5(0(2(3(0(x1)))))) 0(0(5(2(3(x1))))) -> 3(0(2(2(5(0(x1)))))) 0(3(5(2(3(x1))))) -> 0(5(2(3(2(3(x1)))))) 0(3(5(2(3(x1))))) -> 1(3(0(2(5(3(x1)))))) 0(3(5(2(3(x1))))) -> 4(3(0(2(5(3(x1)))))) 5(0(0(3(3(x1))))) -> 0(4(3(0(5(3(x1)))))) 5(0(3(3(3(x1))))) -> 3(1(5(3(0(3(x1)))))) 0(0(5(4(3(x1))))) -> 5(0(4(3(1(0(x1)))))) 0(0(5(4(3(x1))))) -> 5(1(0(4(0(3(x1)))))) 0(0(5(4(3(x1))))) -> 0(1(5(0(4(3(x1)))))) 0(0(3(5(3(x1))))) -> 2(0(3(5(3(0(x1)))))) 0(0(3(5(3(x1))))) -> 0(0(2(3(5(3(x1)))))) 5(0(3(5(3(x1))))) -> 5(3(0(4(5(3(x1)))))) 0(3(3(5(3(x1))))) -> 3(3(0(1(5(3(x1)))))) 0(3(3(5(3(x1))))) -> 0(4(3(3(5(3(x1)))))) 0(3(5(5(3(x1))))) -> 3(1(5(3(5(0(x1)))))) 0(3(5(5(3(x1))))) -> 0(4(5(3(5(3(x1)))))) 0(3(3(0(5(x1))))) -> 1(0(3(5(3(0(x1)))))) 0(3(3(0(5(x1))))) -> 0(2(3(5(3(0(x1)))))) 0(3(3(0(5(x1))))) -> 5(0(2(0(3(3(x1)))))) 0(3(3(0(5(x1))))) -> 0(2(0(3(5(3(x1)))))) 0(3(5(0(5(x1))))) -> 0(0(5(4(3(5(x1)))))) 0(3(0(3(5(x1))))) -> 0(1(5(3(0(3(x1)))))) 0(3(0(3(5(x1))))) -> 0(0(3(1(5(3(x1)))))) 0(3(3(3(5(x1))))) -> 3(3(3(2(5(0(x1)))))) 0(3(3(3(5(x1))))) -> 3(3(2(0(3(5(x1)))))) 0(3(5(3(5(x1))))) -> 5(2(3(0(3(5(x1)))))) 5(0(3(5(5(x1))))) -> 5(5(2(0(3(5(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 5^1(0(3(3(3(x1))))) -> 0^1(3(x1)) 0^1(3(0(3(x1)))) -> 0^1(0(x1)) 0^1(3(3(3(x1)))) -> 0^1(x1) 0^1(3(5(3(x1)))) -> 0^1(3(5(x1))) 0^1(3(3(3(5(x1))))) -> 0^1(3(5(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( 0^1_1(x_1) ) = 2x_1 POL( 0_1(x_1) ) = 2x_1 + 2 POL( 2_1(x_1) ) = max{0, -2} POL( 5^1_1(x_1) ) = max{0, x_1 - 2} POL( 5_1(x_1) ) = x_1 + 1 POL( 3_1(x_1) ) = x_1 + 1 POL( 4_1(x_1) ) = max{0, -2} POL( 1_1(x_1) ) = max{0, x_1 - 2} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0(0(0(0(x1)))) -> 2(0(1(0(0(0(x1)))))) 0(0(0(0(x1)))) -> 0(2(2(0(0(0(x1)))))) 0(0(0(0(x1)))) -> 0(0(2(0(1(0(x1)))))) 0(3(0(0(x1)))) -> 3(0(2(0(0(x1))))) 0(3(0(0(x1)))) -> 0(4(3(1(0(0(x1)))))) 0(3(0(0(x1)))) -> 0(2(3(0(2(0(x1)))))) 0(3(0(0(x1)))) -> 4(0(4(0(3(0(x1)))))) 0(3(0(0(x1)))) -> 0(2(0(3(4(0(x1)))))) 0(3(0(0(x1)))) -> 0(4(0(3(4(0(x1)))))) 0(3(0(0(x1)))) -> 3(0(0(2(0(1(x1)))))) 0(3(0(0(x1)))) -> 3(0(2(0(0(2(x1)))))) 0(3(0(0(x1)))) -> 0(2(2(0(0(3(x1)))))) 0(0(5(0(x1)))) -> 5(0(0(2(0(x1))))) 0(0(5(0(x1)))) -> 0(1(5(3(0(0(x1)))))) 0(0(5(0(x1)))) -> 4(0(1(5(0(0(x1)))))) 0(0(5(0(x1)))) -> 0(5(4(0(2(0(x1)))))) 0(0(5(0(x1)))) -> 0(4(0(5(2(0(x1)))))) 0(0(5(0(x1)))) -> 2(5(0(0(4(0(x1)))))) 0(0(5(0(x1)))) -> 0(5(1(0(4(0(x1)))))) 0(0(5(0(x1)))) -> 0(1(0(1(5(0(x1)))))) 0(0(5(0(x1)))) -> 5(0(2(0(0(2(x1)))))) 0(0(5(0(x1)))) -> 0(2(0(5(0(2(x1)))))) 0(0(5(0(x1)))) -> 0(2(0(1(0(5(x1)))))) 0(0(5(0(x1)))) -> 0(4(0(1(0(5(x1)))))) 0(3(5(0(x1)))) -> 5(2(3(0(0(x1))))) 0(3(5(0(x1)))) -> 1(5(3(0(0(x1))))) 0(3(5(0(x1)))) -> 2(5(3(0(0(x1))))) 0(3(5(0(x1)))) -> 3(2(5(0(0(x1))))) 0(3(5(0(x1)))) -> 5(3(0(2(0(x1))))) 0(3(5(0(x1)))) -> 5(0(4(3(0(x1))))) 0(3(5(0(x1)))) -> 0(1(0(3(5(x1))))) 0(3(5(0(x1)))) -> 4(5(3(1(0(0(x1)))))) 0(3(5(0(x1)))) -> 3(2(2(5(0(0(x1)))))) 0(3(5(0(x1)))) -> 3(4(2(5(0(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(3(5(1(0(x1)))))) 0(3(5(0(x1)))) -> 5(1(3(0(2(0(x1)))))) 0(3(5(0(x1)))) -> 4(0(2(5(3(0(x1)))))) 0(3(5(0(x1)))) -> 3(4(0(2(5(0(x1)))))) 0(3(5(0(x1)))) -> 3(0(2(2(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(4(3(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(2(5(3(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(5(3(0(1(x1)))))) 0(3(5(0(x1)))) -> 0(0(4(5(3(1(x1)))))) 0(3(5(0(x1)))) -> 0(1(5(0(3(2(x1)))))) 0(3(5(0(x1)))) -> 3(0(0(1(5(2(x1)))))) 0(3(5(0(x1)))) -> 2(5(2(0(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(1(5(3(0(3(x1)))))) 0(3(5(0(x1)))) -> 1(5(0(4(0(3(x1)))))) 0(3(5(0(x1)))) -> 2(5(0(4(0(3(x1)))))) 0(3(5(0(x1)))) -> 1(0(1(5(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(1(5(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(5(1(0(2(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(5(0(4(3(x1)))))) 0(3(5(0(x1)))) -> 2(0(0(5(4(3(x1)))))) 0(3(5(0(x1)))) -> 2(0(0(1(5(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(3(1(0(5(x1)))))) 0(3(5(0(x1)))) -> 0(0(3(2(2(5(x1)))))) 0(3(5(0(x1)))) -> 0(1(1(0(3(5(x1)))))) 0(0(0(3(x1)))) -> 1(0(2(3(0(0(x1)))))) 0(0(0(3(x1)))) -> 0(3(0(1(2(0(x1)))))) 0(0(0(3(x1)))) -> 0(2(2(0(3(0(x1)))))) 0(0(0(3(x1)))) -> 0(4(2(0(3(0(x1)))))) 0(0(0(3(x1)))) -> 0(2(0(2(0(3(x1)))))) 0(3(0(3(x1)))) -> 4(3(2(3(0(0(x1)))))) 0(3(0(3(x1)))) -> 4(3(4(3(0(0(x1)))))) 0(3(0(3(x1)))) -> 3(3(0(2(0(1(x1)))))) 0(3(0(3(x1)))) -> 3(2(0(1(0(3(x1)))))) 0(3(0(3(x1)))) -> 1(3(0(2(0(3(x1)))))) 0(3(0(3(x1)))) -> 1(0(1(3(0(3(x1)))))) 0(3(0(3(x1)))) -> 0(2(5(3(0(3(x1)))))) 0(3(0(3(x1)))) -> 3(2(0(4(0(3(x1)))))) 0(3(0(3(x1)))) -> 0(2(3(0(1(3(x1)))))) 0(3(0(3(x1)))) -> 3(0(1(0(4(3(x1)))))) 0(0(3(3(x1)))) -> 0(4(3(3(0(x1))))) 0(0(3(3(x1)))) -> 3(0(2(0(3(x1))))) 0(0(3(3(x1)))) -> 0(2(3(0(3(x1))))) 0(0(3(3(x1)))) -> 3(3(1(0(1(0(x1)))))) 0(0(3(3(x1)))) -> 3(0(4(3(1(0(x1)))))) 0(0(3(3(x1)))) -> 0(3(3(2(2(0(x1)))))) 0(0(3(3(x1)))) -> 3(3(0(4(2(0(x1)))))) 0(0(3(3(x1)))) -> 0(2(3(3(0(2(x1)))))) 0(0(3(3(x1)))) -> 3(4(0(1(0(3(x1)))))) 0(0(3(3(x1)))) -> 3(0(2(2(0(3(x1)))))) 0(0(3(3(x1)))) -> 2(0(2(3(0(3(x1)))))) 0(0(3(3(x1)))) -> 0(1(4(3(0(3(x1)))))) 0(0(3(3(x1)))) -> 3(0(1(0(1(3(x1)))))) 0(0(3(3(x1)))) -> 2(0(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 4(0(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 0(1(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 0(4(3(0(4(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(3(3(x1))))) 0(3(3(3(x1)))) -> 0(2(3(3(3(x1))))) 0(3(3(3(x1)))) -> 3(2(3(1(3(0(x1)))))) 0(3(3(3(x1)))) -> 3(2(3(0(2(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(2(1(3(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(2(3(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(5(3(3(x1)))))) 0(0(5(3(x1)))) -> 1(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 2(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 4(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 0(2(5(3(0(x1))))) 0(0(5(3(x1)))) -> 0(4(5(3(0(x1))))) 0(0(5(3(x1)))) -> 5(2(0(0(3(x1))))) 0(0(5(3(x1)))) -> 0(1(5(0(3(x1))))) 0(0(5(3(x1)))) -> 3(1(5(1(0(0(x1)))))) 0(0(5(3(x1)))) -> 3(2(5(1(0(0(x1)))))) 0(0(5(3(x1)))) -> 2(5(4(3(0(0(x1)))))) 0(0(5(3(x1)))) -> 0(2(5(3(1(0(x1)))))) 0(0(5(3(x1)))) -> 5(1(3(0(2(0(x1)))))) 0(0(5(3(x1)))) -> 5(4(0(1(3(0(x1)))))) 0(0(5(3(x1)))) -> 1(5(0(2(3(0(x1)))))) 0(0(5(3(x1)))) -> 0(2(2(5(3(0(x1)))))) 0(0(5(3(x1)))) -> 0(1(0(3(5(2(x1)))))) 0(0(5(3(x1)))) -> 5(4(2(0(0(3(x1)))))) 0(0(5(3(x1)))) -> 1(5(0(4(0(3(x1)))))) 0(0(5(3(x1)))) -> 5(0(1(4(0(3(x1)))))) 0(0(5(3(x1)))) -> 5(0(4(0(1(3(x1)))))) 0(0(5(3(x1)))) -> 0(2(5(0(2(3(x1)))))) 0(0(5(3(x1)))) -> 0(2(0(5(2(3(x1)))))) 0(0(5(3(x1)))) -> 0(1(5(0(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(1(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(4(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(1(3(5(x1)))))) 0(3(5(3(x1)))) -> 3(2(5(3(0(x1))))) 0(3(5(3(x1)))) -> 5(2(3(0(3(x1))))) 0(3(5(3(x1)))) -> 4(5(3(0(3(x1))))) 0(3(5(3(x1)))) -> 2(5(0(3(3(x1))))) 0(3(5(3(x1)))) -> 5(0(4(3(3(x1))))) 0(3(5(3(x1)))) -> 3(0(1(5(3(x1))))) 0(3(5(3(x1)))) -> 5(4(3(3(2(0(x1)))))) 0(3(5(3(x1)))) -> 2(5(3(3(2(0(x1)))))) 0(3(5(3(x1)))) -> 4(5(3(1(3(0(x1)))))) 0(3(5(3(x1)))) -> 2(1(5(3(3(0(x1)))))) 0(3(5(3(x1)))) -> 1(2(5(3(3(0(x1)))))) 0(3(5(3(x1)))) -> 3(4(5(4(3(0(x1)))))) 0(3(5(3(x1)))) -> 4(5(3(3(4(0(x1)))))) 0(3(5(3(x1)))) -> 3(0(2(5(3(2(x1)))))) 0(3(5(3(x1)))) -> 5(2(2(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(5(2(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 5(2(4(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 2(5(5(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(3(1(5(0(3(x1)))))) 0(3(5(3(x1)))) -> 3(2(2(5(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(5(3(0(1(3(x1)))))) 0(3(5(3(x1)))) -> 2(5(0(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 5(0(1(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 5(0(2(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 0(4(4(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 0(2(5(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 0(4(5(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 3(3(0(2(2(5(x1)))))) 0(3(5(3(x1)))) -> 1(3(1(0(3(5(x1)))))) 0(3(5(3(x1)))) -> 0(1(3(1(3(5(x1)))))) 0(3(5(3(x1)))) -> 2(0(4(3(3(5(x1)))))) 0(3(5(3(x1)))) -> 4(0(4(3(3(5(x1)))))) 0(3(5(3(x1)))) -> 0(1(4(3(3(5(x1)))))) 0(0(3(0(0(x1))))) -> 0(0(2(0(3(0(x1)))))) 0(3(5(0(0(x1))))) -> 0(2(5(0(3(0(x1)))))) 0(3(3(5(0(x1))))) -> 3(0(2(0(3(5(x1)))))) 0(3(0(0(3(x1))))) -> 0(0(4(3(3(0(x1)))))) 0(3(3(0(3(x1))))) -> 3(0(0(2(3(3(x1)))))) 0(3(5(0(3(x1))))) -> 0(4(5(0(3(3(x1)))))) 0(3(5(1(3(x1))))) -> 1(3(2(5(0(3(x1)))))) 0(0(5(2(3(x1))))) -> 4(5(0(2(3(0(x1)))))) 0(0(5(2(3(x1))))) -> 3(0(2(2(5(0(x1)))))) 0(3(5(2(3(x1))))) -> 0(5(2(3(2(3(x1)))))) 0(3(5(2(3(x1))))) -> 1(3(0(2(5(3(x1)))))) 0(3(5(2(3(x1))))) -> 4(3(0(2(5(3(x1)))))) 0(0(5(4(3(x1))))) -> 5(0(4(3(1(0(x1)))))) 0(0(5(4(3(x1))))) -> 5(1(0(4(0(3(x1)))))) 0(0(5(4(3(x1))))) -> 0(1(5(0(4(3(x1)))))) 0(0(3(5(3(x1))))) -> 2(0(3(5(3(0(x1)))))) 0(0(3(5(3(x1))))) -> 0(0(2(3(5(3(x1)))))) 0(3(3(5(3(x1))))) -> 3(3(0(1(5(3(x1)))))) 0(3(3(5(3(x1))))) -> 0(4(3(3(5(3(x1)))))) 0(3(5(5(3(x1))))) -> 3(1(5(3(5(0(x1)))))) 0(3(5(5(3(x1))))) -> 0(4(5(3(5(3(x1)))))) 0(3(3(0(5(x1))))) -> 1(0(3(5(3(0(x1)))))) 0(3(3(0(5(x1))))) -> 0(2(3(5(3(0(x1)))))) 0(3(3(0(5(x1))))) -> 5(0(2(0(3(3(x1)))))) 0(3(3(0(5(x1))))) -> 0(2(0(3(5(3(x1)))))) 0(3(5(0(5(x1))))) -> 0(0(5(4(3(5(x1)))))) 0(3(0(3(5(x1))))) -> 0(1(5(3(0(3(x1)))))) 0(3(0(3(5(x1))))) -> 0(0(3(1(5(3(x1)))))) 0(3(3(3(5(x1))))) -> 3(3(3(2(5(0(x1)))))) 0(3(3(3(5(x1))))) -> 3(3(2(0(3(5(x1)))))) 0(3(5(3(5(x1))))) -> 5(2(3(0(3(5(x1)))))) 5(0(3(0(0(x1))))) -> 5(0(3(2(0(0(x1)))))) 5(0(3(5(0(x1))))) -> 5(0(1(5(0(3(x1)))))) 5(0(3(5(0(x1))))) -> 5(5(0(0(4(3(x1)))))) 5(0(0(3(3(x1))))) -> 0(4(3(0(5(3(x1)))))) 5(0(3(3(3(x1))))) -> 3(1(5(3(0(3(x1)))))) 5(0(3(5(3(x1))))) -> 5(3(0(4(5(3(x1)))))) 5(0(3(5(5(x1))))) -> 5(5(2(0(3(5(x1)))))) ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(3(5(1(3(x1))))) -> 5^1(0(3(x1))) 0^1(3(5(1(3(x1))))) -> 0^1(3(x1)) The TRS R consists of the following rules: 0(0(0(0(x1)))) -> 2(0(1(0(0(0(x1)))))) 0(0(0(0(x1)))) -> 0(2(2(0(0(0(x1)))))) 0(0(0(0(x1)))) -> 0(0(2(0(1(0(x1)))))) 0(3(0(0(x1)))) -> 3(0(2(0(0(x1))))) 0(3(0(0(x1)))) -> 0(4(3(1(0(0(x1)))))) 0(3(0(0(x1)))) -> 0(2(3(0(2(0(x1)))))) 0(3(0(0(x1)))) -> 4(0(4(0(3(0(x1)))))) 0(3(0(0(x1)))) -> 0(2(0(3(4(0(x1)))))) 0(3(0(0(x1)))) -> 0(4(0(3(4(0(x1)))))) 0(3(0(0(x1)))) -> 3(0(0(2(0(1(x1)))))) 0(3(0(0(x1)))) -> 3(0(2(0(0(2(x1)))))) 0(3(0(0(x1)))) -> 0(2(2(0(0(3(x1)))))) 0(0(5(0(x1)))) -> 5(0(0(2(0(x1))))) 0(0(5(0(x1)))) -> 0(1(5(3(0(0(x1)))))) 0(0(5(0(x1)))) -> 4(0(1(5(0(0(x1)))))) 0(0(5(0(x1)))) -> 0(5(4(0(2(0(x1)))))) 0(0(5(0(x1)))) -> 0(4(0(5(2(0(x1)))))) 0(0(5(0(x1)))) -> 2(5(0(0(4(0(x1)))))) 0(0(5(0(x1)))) -> 0(5(1(0(4(0(x1)))))) 0(0(5(0(x1)))) -> 0(1(0(1(5(0(x1)))))) 0(0(5(0(x1)))) -> 5(0(2(0(0(2(x1)))))) 0(0(5(0(x1)))) -> 0(2(0(5(0(2(x1)))))) 0(0(5(0(x1)))) -> 0(2(0(1(0(5(x1)))))) 0(0(5(0(x1)))) -> 0(4(0(1(0(5(x1)))))) 0(3(5(0(x1)))) -> 5(2(3(0(0(x1))))) 0(3(5(0(x1)))) -> 1(5(3(0(0(x1))))) 0(3(5(0(x1)))) -> 2(5(3(0(0(x1))))) 0(3(5(0(x1)))) -> 3(2(5(0(0(x1))))) 0(3(5(0(x1)))) -> 5(3(0(2(0(x1))))) 0(3(5(0(x1)))) -> 5(0(4(3(0(x1))))) 0(3(5(0(x1)))) -> 0(1(0(3(5(x1))))) 0(3(5(0(x1)))) -> 4(5(3(1(0(0(x1)))))) 0(3(5(0(x1)))) -> 3(2(2(5(0(0(x1)))))) 0(3(5(0(x1)))) -> 3(4(2(5(0(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(3(5(1(0(x1)))))) 0(3(5(0(x1)))) -> 5(1(3(0(2(0(x1)))))) 0(3(5(0(x1)))) -> 4(0(2(5(3(0(x1)))))) 0(3(5(0(x1)))) -> 3(4(0(2(5(0(x1)))))) 0(3(5(0(x1)))) -> 3(0(2(2(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(4(3(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(2(5(3(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(5(3(0(1(x1)))))) 0(3(5(0(x1)))) -> 0(0(4(5(3(1(x1)))))) 0(3(5(0(x1)))) -> 0(1(5(0(3(2(x1)))))) 0(3(5(0(x1)))) -> 3(0(0(1(5(2(x1)))))) 0(3(5(0(x1)))) -> 2(5(2(0(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(1(5(3(0(3(x1)))))) 0(3(5(0(x1)))) -> 1(5(0(4(0(3(x1)))))) 0(3(5(0(x1)))) -> 2(5(0(4(0(3(x1)))))) 0(3(5(0(x1)))) -> 1(0(1(5(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(1(5(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(5(1(0(2(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(5(0(4(3(x1)))))) 0(3(5(0(x1)))) -> 2(0(0(5(4(3(x1)))))) 0(3(5(0(x1)))) -> 2(0(0(1(5(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(3(1(0(5(x1)))))) 0(3(5(0(x1)))) -> 0(0(3(2(2(5(x1)))))) 0(3(5(0(x1)))) -> 0(1(1(0(3(5(x1)))))) 0(0(0(3(x1)))) -> 1(0(2(3(0(0(x1)))))) 0(0(0(3(x1)))) -> 0(3(0(1(2(0(x1)))))) 0(0(0(3(x1)))) -> 0(2(2(0(3(0(x1)))))) 0(0(0(3(x1)))) -> 0(4(2(0(3(0(x1)))))) 0(0(0(3(x1)))) -> 0(2(0(2(0(3(x1)))))) 0(3(0(3(x1)))) -> 4(3(2(3(0(0(x1)))))) 0(3(0(3(x1)))) -> 4(3(4(3(0(0(x1)))))) 0(3(0(3(x1)))) -> 3(3(0(2(0(1(x1)))))) 0(3(0(3(x1)))) -> 3(2(0(1(0(3(x1)))))) 0(3(0(3(x1)))) -> 1(3(0(2(0(3(x1)))))) 0(3(0(3(x1)))) -> 1(0(1(3(0(3(x1)))))) 0(3(0(3(x1)))) -> 0(2(5(3(0(3(x1)))))) 0(3(0(3(x1)))) -> 3(2(0(4(0(3(x1)))))) 0(3(0(3(x1)))) -> 0(2(3(0(1(3(x1)))))) 0(3(0(3(x1)))) -> 3(0(1(0(4(3(x1)))))) 0(0(3(3(x1)))) -> 0(4(3(3(0(x1))))) 0(0(3(3(x1)))) -> 3(0(2(0(3(x1))))) 0(0(3(3(x1)))) -> 0(2(3(0(3(x1))))) 0(0(3(3(x1)))) -> 3(3(1(0(1(0(x1)))))) 0(0(3(3(x1)))) -> 3(0(4(3(1(0(x1)))))) 0(0(3(3(x1)))) -> 0(3(3(2(2(0(x1)))))) 0(0(3(3(x1)))) -> 3(3(0(4(2(0(x1)))))) 0(0(3(3(x1)))) -> 0(2(3(3(0(2(x1)))))) 0(0(3(3(x1)))) -> 3(4(0(1(0(3(x1)))))) 0(0(3(3(x1)))) -> 3(0(2(2(0(3(x1)))))) 0(0(3(3(x1)))) -> 2(0(2(3(0(3(x1)))))) 0(0(3(3(x1)))) -> 0(1(4(3(0(3(x1)))))) 0(0(3(3(x1)))) -> 3(0(1(0(1(3(x1)))))) 0(0(3(3(x1)))) -> 2(0(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 4(0(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 0(1(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 0(4(3(0(4(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(3(3(x1))))) 0(3(3(3(x1)))) -> 0(2(3(3(3(x1))))) 0(3(3(3(x1)))) -> 3(2(3(1(3(0(x1)))))) 0(3(3(3(x1)))) -> 3(2(3(0(2(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(2(1(3(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(2(3(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(5(3(3(x1)))))) 0(0(5(3(x1)))) -> 1(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 2(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 4(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 0(2(5(3(0(x1))))) 0(0(5(3(x1)))) -> 0(4(5(3(0(x1))))) 0(0(5(3(x1)))) -> 5(2(0(0(3(x1))))) 0(0(5(3(x1)))) -> 0(1(5(0(3(x1))))) 0(0(5(3(x1)))) -> 3(1(5(1(0(0(x1)))))) 0(0(5(3(x1)))) -> 3(2(5(1(0(0(x1)))))) 0(0(5(3(x1)))) -> 2(5(4(3(0(0(x1)))))) 0(0(5(3(x1)))) -> 0(2(5(3(1(0(x1)))))) 0(0(5(3(x1)))) -> 5(1(3(0(2(0(x1)))))) 0(0(5(3(x1)))) -> 5(4(0(1(3(0(x1)))))) 0(0(5(3(x1)))) -> 1(5(0(2(3(0(x1)))))) 0(0(5(3(x1)))) -> 0(2(2(5(3(0(x1)))))) 0(0(5(3(x1)))) -> 0(1(0(3(5(2(x1)))))) 0(0(5(3(x1)))) -> 5(4(2(0(0(3(x1)))))) 0(0(5(3(x1)))) -> 1(5(0(4(0(3(x1)))))) 0(0(5(3(x1)))) -> 5(0(1(4(0(3(x1)))))) 0(0(5(3(x1)))) -> 5(0(4(0(1(3(x1)))))) 0(0(5(3(x1)))) -> 0(2(5(0(2(3(x1)))))) 0(0(5(3(x1)))) -> 0(2(0(5(2(3(x1)))))) 0(0(5(3(x1)))) -> 0(1(5(0(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(1(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(4(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(1(3(5(x1)))))) 0(3(5(3(x1)))) -> 3(2(5(3(0(x1))))) 0(3(5(3(x1)))) -> 5(2(3(0(3(x1))))) 0(3(5(3(x1)))) -> 4(5(3(0(3(x1))))) 0(3(5(3(x1)))) -> 2(5(0(3(3(x1))))) 0(3(5(3(x1)))) -> 5(0(4(3(3(x1))))) 0(3(5(3(x1)))) -> 3(0(1(5(3(x1))))) 0(3(5(3(x1)))) -> 5(4(3(3(2(0(x1)))))) 0(3(5(3(x1)))) -> 2(5(3(3(2(0(x1)))))) 0(3(5(3(x1)))) -> 4(5(3(1(3(0(x1)))))) 0(3(5(3(x1)))) -> 2(1(5(3(3(0(x1)))))) 0(3(5(3(x1)))) -> 1(2(5(3(3(0(x1)))))) 0(3(5(3(x1)))) -> 3(4(5(4(3(0(x1)))))) 0(3(5(3(x1)))) -> 4(5(3(3(4(0(x1)))))) 0(3(5(3(x1)))) -> 3(0(2(5(3(2(x1)))))) 0(3(5(3(x1)))) -> 5(2(2(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(5(2(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 5(2(4(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 2(5(5(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(3(1(5(0(3(x1)))))) 0(3(5(3(x1)))) -> 3(2(2(5(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(5(3(0(1(3(x1)))))) 0(3(5(3(x1)))) -> 2(5(0(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 5(0(1(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 5(0(2(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 0(4(4(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 0(2(5(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 0(4(5(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 3(3(0(2(2(5(x1)))))) 0(3(5(3(x1)))) -> 1(3(1(0(3(5(x1)))))) 0(3(5(3(x1)))) -> 0(1(3(1(3(5(x1)))))) 0(3(5(3(x1)))) -> 2(0(4(3(3(5(x1)))))) 0(3(5(3(x1)))) -> 4(0(4(3(3(5(x1)))))) 0(3(5(3(x1)))) -> 0(1(4(3(3(5(x1)))))) 0(0(3(0(0(x1))))) -> 0(0(2(0(3(0(x1)))))) 5(0(3(0(0(x1))))) -> 5(0(3(2(0(0(x1)))))) 0(3(5(0(0(x1))))) -> 0(2(5(0(3(0(x1)))))) 5(0(3(5(0(x1))))) -> 5(0(1(5(0(3(x1)))))) 5(0(3(5(0(x1))))) -> 5(5(0(0(4(3(x1)))))) 0(3(3(5(0(x1))))) -> 3(0(2(0(3(5(x1)))))) 0(3(0(0(3(x1))))) -> 0(0(4(3(3(0(x1)))))) 0(3(3(0(3(x1))))) -> 3(0(0(2(3(3(x1)))))) 0(3(5(0(3(x1))))) -> 0(4(5(0(3(3(x1)))))) 0(3(5(1(3(x1))))) -> 1(3(2(5(0(3(x1)))))) 0(0(5(2(3(x1))))) -> 4(5(0(2(3(0(x1)))))) 0(0(5(2(3(x1))))) -> 3(0(2(2(5(0(x1)))))) 0(3(5(2(3(x1))))) -> 0(5(2(3(2(3(x1)))))) 0(3(5(2(3(x1))))) -> 1(3(0(2(5(3(x1)))))) 0(3(5(2(3(x1))))) -> 4(3(0(2(5(3(x1)))))) 5(0(0(3(3(x1))))) -> 0(4(3(0(5(3(x1)))))) 5(0(3(3(3(x1))))) -> 3(1(5(3(0(3(x1)))))) 0(0(5(4(3(x1))))) -> 5(0(4(3(1(0(x1)))))) 0(0(5(4(3(x1))))) -> 5(1(0(4(0(3(x1)))))) 0(0(5(4(3(x1))))) -> 0(1(5(0(4(3(x1)))))) 0(0(3(5(3(x1))))) -> 2(0(3(5(3(0(x1)))))) 0(0(3(5(3(x1))))) -> 0(0(2(3(5(3(x1)))))) 5(0(3(5(3(x1))))) -> 5(3(0(4(5(3(x1)))))) 0(3(3(5(3(x1))))) -> 3(3(0(1(5(3(x1)))))) 0(3(3(5(3(x1))))) -> 0(4(3(3(5(3(x1)))))) 0(3(5(5(3(x1))))) -> 3(1(5(3(5(0(x1)))))) 0(3(5(5(3(x1))))) -> 0(4(5(3(5(3(x1)))))) 0(3(3(0(5(x1))))) -> 1(0(3(5(3(0(x1)))))) 0(3(3(0(5(x1))))) -> 0(2(3(5(3(0(x1)))))) 0(3(3(0(5(x1))))) -> 5(0(2(0(3(3(x1)))))) 0(3(3(0(5(x1))))) -> 0(2(0(3(5(3(x1)))))) 0(3(5(0(5(x1))))) -> 0(0(5(4(3(5(x1)))))) 0(3(0(3(5(x1))))) -> 0(1(5(3(0(3(x1)))))) 0(3(0(3(5(x1))))) -> 0(0(3(1(5(3(x1)))))) 0(3(3(3(5(x1))))) -> 3(3(3(2(5(0(x1)))))) 0(3(3(3(5(x1))))) -> 3(3(2(0(3(5(x1)))))) 0(3(5(3(5(x1))))) -> 5(2(3(0(3(5(x1)))))) 5(0(3(5(5(x1))))) -> 5(5(2(0(3(5(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (15) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(3(5(1(3(x1))))) -> 0^1(3(x1)) The TRS R consists of the following rules: 0(0(0(0(x1)))) -> 2(0(1(0(0(0(x1)))))) 0(0(0(0(x1)))) -> 0(2(2(0(0(0(x1)))))) 0(0(0(0(x1)))) -> 0(0(2(0(1(0(x1)))))) 0(3(0(0(x1)))) -> 3(0(2(0(0(x1))))) 0(3(0(0(x1)))) -> 0(4(3(1(0(0(x1)))))) 0(3(0(0(x1)))) -> 0(2(3(0(2(0(x1)))))) 0(3(0(0(x1)))) -> 4(0(4(0(3(0(x1)))))) 0(3(0(0(x1)))) -> 0(2(0(3(4(0(x1)))))) 0(3(0(0(x1)))) -> 0(4(0(3(4(0(x1)))))) 0(3(0(0(x1)))) -> 3(0(0(2(0(1(x1)))))) 0(3(0(0(x1)))) -> 3(0(2(0(0(2(x1)))))) 0(3(0(0(x1)))) -> 0(2(2(0(0(3(x1)))))) 0(0(5(0(x1)))) -> 5(0(0(2(0(x1))))) 0(0(5(0(x1)))) -> 0(1(5(3(0(0(x1)))))) 0(0(5(0(x1)))) -> 4(0(1(5(0(0(x1)))))) 0(0(5(0(x1)))) -> 0(5(4(0(2(0(x1)))))) 0(0(5(0(x1)))) -> 0(4(0(5(2(0(x1)))))) 0(0(5(0(x1)))) -> 2(5(0(0(4(0(x1)))))) 0(0(5(0(x1)))) -> 0(5(1(0(4(0(x1)))))) 0(0(5(0(x1)))) -> 0(1(0(1(5(0(x1)))))) 0(0(5(0(x1)))) -> 5(0(2(0(0(2(x1)))))) 0(0(5(0(x1)))) -> 0(2(0(5(0(2(x1)))))) 0(0(5(0(x1)))) -> 0(2(0(1(0(5(x1)))))) 0(0(5(0(x1)))) -> 0(4(0(1(0(5(x1)))))) 0(3(5(0(x1)))) -> 5(2(3(0(0(x1))))) 0(3(5(0(x1)))) -> 1(5(3(0(0(x1))))) 0(3(5(0(x1)))) -> 2(5(3(0(0(x1))))) 0(3(5(0(x1)))) -> 3(2(5(0(0(x1))))) 0(3(5(0(x1)))) -> 5(3(0(2(0(x1))))) 0(3(5(0(x1)))) -> 5(0(4(3(0(x1))))) 0(3(5(0(x1)))) -> 0(1(0(3(5(x1))))) 0(3(5(0(x1)))) -> 4(5(3(1(0(0(x1)))))) 0(3(5(0(x1)))) -> 3(2(2(5(0(0(x1)))))) 0(3(5(0(x1)))) -> 3(4(2(5(0(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(3(5(1(0(x1)))))) 0(3(5(0(x1)))) -> 5(1(3(0(2(0(x1)))))) 0(3(5(0(x1)))) -> 4(0(2(5(3(0(x1)))))) 0(3(5(0(x1)))) -> 3(4(0(2(5(0(x1)))))) 0(3(5(0(x1)))) -> 3(0(2(2(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(4(3(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(2(5(3(5(0(x1)))))) 0(3(5(0(x1)))) -> 0(4(5(3(0(1(x1)))))) 0(3(5(0(x1)))) -> 0(0(4(5(3(1(x1)))))) 0(3(5(0(x1)))) -> 0(1(5(0(3(2(x1)))))) 0(3(5(0(x1)))) -> 3(0(0(1(5(2(x1)))))) 0(3(5(0(x1)))) -> 2(5(2(0(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(1(5(3(0(3(x1)))))) 0(3(5(0(x1)))) -> 1(5(0(4(0(3(x1)))))) 0(3(5(0(x1)))) -> 2(5(0(4(0(3(x1)))))) 0(3(5(0(x1)))) -> 1(0(1(5(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(1(5(0(3(x1)))))) 0(3(5(0(x1)))) -> 0(5(1(0(2(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(5(0(4(3(x1)))))) 0(3(5(0(x1)))) -> 2(0(0(5(4(3(x1)))))) 0(3(5(0(x1)))) -> 2(0(0(1(5(3(x1)))))) 0(3(5(0(x1)))) -> 0(2(3(1(0(5(x1)))))) 0(3(5(0(x1)))) -> 0(0(3(2(2(5(x1)))))) 0(3(5(0(x1)))) -> 0(1(1(0(3(5(x1)))))) 0(0(0(3(x1)))) -> 1(0(2(3(0(0(x1)))))) 0(0(0(3(x1)))) -> 0(3(0(1(2(0(x1)))))) 0(0(0(3(x1)))) -> 0(2(2(0(3(0(x1)))))) 0(0(0(3(x1)))) -> 0(4(2(0(3(0(x1)))))) 0(0(0(3(x1)))) -> 0(2(0(2(0(3(x1)))))) 0(3(0(3(x1)))) -> 4(3(2(3(0(0(x1)))))) 0(3(0(3(x1)))) -> 4(3(4(3(0(0(x1)))))) 0(3(0(3(x1)))) -> 3(3(0(2(0(1(x1)))))) 0(3(0(3(x1)))) -> 3(2(0(1(0(3(x1)))))) 0(3(0(3(x1)))) -> 1(3(0(2(0(3(x1)))))) 0(3(0(3(x1)))) -> 1(0(1(3(0(3(x1)))))) 0(3(0(3(x1)))) -> 0(2(5(3(0(3(x1)))))) 0(3(0(3(x1)))) -> 3(2(0(4(0(3(x1)))))) 0(3(0(3(x1)))) -> 0(2(3(0(1(3(x1)))))) 0(3(0(3(x1)))) -> 3(0(1(0(4(3(x1)))))) 0(0(3(3(x1)))) -> 0(4(3(3(0(x1))))) 0(0(3(3(x1)))) -> 3(0(2(0(3(x1))))) 0(0(3(3(x1)))) -> 0(2(3(0(3(x1))))) 0(0(3(3(x1)))) -> 3(3(1(0(1(0(x1)))))) 0(0(3(3(x1)))) -> 3(0(4(3(1(0(x1)))))) 0(0(3(3(x1)))) -> 0(3(3(2(2(0(x1)))))) 0(0(3(3(x1)))) -> 3(3(0(4(2(0(x1)))))) 0(0(3(3(x1)))) -> 0(2(3(3(0(2(x1)))))) 0(0(3(3(x1)))) -> 3(4(0(1(0(3(x1)))))) 0(0(3(3(x1)))) -> 3(0(2(2(0(3(x1)))))) 0(0(3(3(x1)))) -> 2(0(2(3(0(3(x1)))))) 0(0(3(3(x1)))) -> 0(1(4(3(0(3(x1)))))) 0(0(3(3(x1)))) -> 3(0(1(0(1(3(x1)))))) 0(0(3(3(x1)))) -> 2(0(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 4(0(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 0(1(4(0(3(3(x1)))))) 0(0(3(3(x1)))) -> 0(4(3(0(4(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(3(3(x1))))) 0(3(3(3(x1)))) -> 0(2(3(3(3(x1))))) 0(3(3(3(x1)))) -> 3(2(3(1(3(0(x1)))))) 0(3(3(3(x1)))) -> 3(2(3(0(2(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(2(1(3(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(2(3(3(x1)))))) 0(3(3(3(x1)))) -> 3(0(1(5(3(3(x1)))))) 0(0(5(3(x1)))) -> 1(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 2(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 4(5(3(0(0(x1))))) 0(0(5(3(x1)))) -> 0(2(5(3(0(x1))))) 0(0(5(3(x1)))) -> 0(4(5(3(0(x1))))) 0(0(5(3(x1)))) -> 5(2(0(0(3(x1))))) 0(0(5(3(x1)))) -> 0(1(5(0(3(x1))))) 0(0(5(3(x1)))) -> 3(1(5(1(0(0(x1)))))) 0(0(5(3(x1)))) -> 3(2(5(1(0(0(x1)))))) 0(0(5(3(x1)))) -> 2(5(4(3(0(0(x1)))))) 0(0(5(3(x1)))) -> 0(2(5(3(1(0(x1)))))) 0(0(5(3(x1)))) -> 5(1(3(0(2(0(x1)))))) 0(0(5(3(x1)))) -> 5(4(0(1(3(0(x1)))))) 0(0(5(3(x1)))) -> 1(5(0(2(3(0(x1)))))) 0(0(5(3(x1)))) -> 0(2(2(5(3(0(x1)))))) 0(0(5(3(x1)))) -> 0(1(0(3(5(2(x1)))))) 0(0(5(3(x1)))) -> 5(4(2(0(0(3(x1)))))) 0(0(5(3(x1)))) -> 1(5(0(4(0(3(x1)))))) 0(0(5(3(x1)))) -> 5(0(1(4(0(3(x1)))))) 0(0(5(3(x1)))) -> 5(0(4(0(1(3(x1)))))) 0(0(5(3(x1)))) -> 0(2(5(0(2(3(x1)))))) 0(0(5(3(x1)))) -> 0(2(0(5(2(3(x1)))))) 0(0(5(3(x1)))) -> 0(1(5(0(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(1(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(4(5(3(x1)))))) 0(0(5(3(x1)))) -> 0(4(0(1(3(5(x1)))))) 0(3(5(3(x1)))) -> 3(2(5(3(0(x1))))) 0(3(5(3(x1)))) -> 5(2(3(0(3(x1))))) 0(3(5(3(x1)))) -> 4(5(3(0(3(x1))))) 0(3(5(3(x1)))) -> 2(5(0(3(3(x1))))) 0(3(5(3(x1)))) -> 5(0(4(3(3(x1))))) 0(3(5(3(x1)))) -> 3(0(1(5(3(x1))))) 0(3(5(3(x1)))) -> 5(4(3(3(2(0(x1)))))) 0(3(5(3(x1)))) -> 2(5(3(3(2(0(x1)))))) 0(3(5(3(x1)))) -> 4(5(3(1(3(0(x1)))))) 0(3(5(3(x1)))) -> 2(1(5(3(3(0(x1)))))) 0(3(5(3(x1)))) -> 1(2(5(3(3(0(x1)))))) 0(3(5(3(x1)))) -> 3(4(5(4(3(0(x1)))))) 0(3(5(3(x1)))) -> 4(5(3(3(4(0(x1)))))) 0(3(5(3(x1)))) -> 3(0(2(5(3(2(x1)))))) 0(3(5(3(x1)))) -> 5(2(2(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(5(2(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 5(2(4(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 2(5(5(3(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(3(1(5(0(3(x1)))))) 0(3(5(3(x1)))) -> 3(2(2(5(0(3(x1)))))) 0(3(5(3(x1)))) -> 1(5(3(0(1(3(x1)))))) 0(3(5(3(x1)))) -> 2(5(0(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 5(0(1(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 5(0(2(4(3(3(x1)))))) 0(3(5(3(x1)))) -> 0(4(4(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 0(2(5(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 0(4(5(3(5(3(x1)))))) 0(3(5(3(x1)))) -> 3(3(0(2(2(5(x1)))))) 0(3(5(3(x1)))) -> 1(3(1(0(3(5(x1)))))) 0(3(5(3(x1)))) -> 0(1(3(1(3(5(x1)))))) 0(3(5(3(x1)))) -> 2(0(4(3(3(5(x1)))))) 0(3(5(3(x1)))) -> 4(0(4(3(3(5(x1)))))) 0(3(5(3(x1)))) -> 0(1(4(3(3(5(x1)))))) 0(0(3(0(0(x1))))) -> 0(0(2(0(3(0(x1)))))) 5(0(3(0(0(x1))))) -> 5(0(3(2(0(0(x1)))))) 0(3(5(0(0(x1))))) -> 0(2(5(0(3(0(x1)))))) 5(0(3(5(0(x1))))) -> 5(0(1(5(0(3(x1)))))) 5(0(3(5(0(x1))))) -> 5(5(0(0(4(3(x1)))))) 0(3(3(5(0(x1))))) -> 3(0(2(0(3(5(x1)))))) 0(3(0(0(3(x1))))) -> 0(0(4(3(3(0(x1)))))) 0(3(3(0(3(x1))))) -> 3(0(0(2(3(3(x1)))))) 0(3(5(0(3(x1))))) -> 0(4(5(0(3(3(x1)))))) 0(3(5(1(3(x1))))) -> 1(3(2(5(0(3(x1)))))) 0(0(5(2(3(x1))))) -> 4(5(0(2(3(0(x1)))))) 0(0(5(2(3(x1))))) -> 3(0(2(2(5(0(x1)))))) 0(3(5(2(3(x1))))) -> 0(5(2(3(2(3(x1)))))) 0(3(5(2(3(x1))))) -> 1(3(0(2(5(3(x1)))))) 0(3(5(2(3(x1))))) -> 4(3(0(2(5(3(x1)))))) 5(0(0(3(3(x1))))) -> 0(4(3(0(5(3(x1)))))) 5(0(3(3(3(x1))))) -> 3(1(5(3(0(3(x1)))))) 0(0(5(4(3(x1))))) -> 5(0(4(3(1(0(x1)))))) 0(0(5(4(3(x1))))) -> 5(1(0(4(0(3(x1)))))) 0(0(5(4(3(x1))))) -> 0(1(5(0(4(3(x1)))))) 0(0(3(5(3(x1))))) -> 2(0(3(5(3(0(x1)))))) 0(0(3(5(3(x1))))) -> 0(0(2(3(5(3(x1)))))) 5(0(3(5(3(x1))))) -> 5(3(0(4(5(3(x1)))))) 0(3(3(5(3(x1))))) -> 3(3(0(1(5(3(x1)))))) 0(3(3(5(3(x1))))) -> 0(4(3(3(5(3(x1)))))) 0(3(5(5(3(x1))))) -> 3(1(5(3(5(0(x1)))))) 0(3(5(5(3(x1))))) -> 0(4(5(3(5(3(x1)))))) 0(3(3(0(5(x1))))) -> 1(0(3(5(3(0(x1)))))) 0(3(3(0(5(x1))))) -> 0(2(3(5(3(0(x1)))))) 0(3(3(0(5(x1))))) -> 5(0(2(0(3(3(x1)))))) 0(3(3(0(5(x1))))) -> 0(2(0(3(5(3(x1)))))) 0(3(5(0(5(x1))))) -> 0(0(5(4(3(5(x1)))))) 0(3(0(3(5(x1))))) -> 0(1(5(3(0(3(x1)))))) 0(3(0(3(5(x1))))) -> 0(0(3(1(5(3(x1)))))) 0(3(3(3(5(x1))))) -> 3(3(3(2(5(0(x1)))))) 0(3(3(3(5(x1))))) -> 3(3(2(0(3(5(x1)))))) 0(3(5(3(5(x1))))) -> 5(2(3(0(3(5(x1)))))) 5(0(3(5(5(x1))))) -> 5(5(2(0(3(5(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (17) 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. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(3(5(1(3(x1))))) -> 0^1(3(x1)) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) 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: *0^1(3(5(1(3(x1))))) -> 0^1(3(x1)) The graph contains the following edges 1 > 1 ---------------------------------------- (20) YES