/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 155 ms] (4) QTRS (5) DependencyPairsProof [EQUIVALENT, 66 ms] (6) QDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) QDP (9) QDPOrderProof [EQUIVALENT, 56 ms] (10) QDP (11) DependencyGraphProof [EQUIVALENT, 0 ms] (12) QDP (13) QDPOrderProof [EQUIVALENT, 88 ms] (14) QDP (15) PisEmptyProof [EQUIVALENT, 0 ms] (16) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0(0(1(0(2(x1))))) -> 0(0(1(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(0(2(1(2(x1))))) 0(0(1(0(2(x1))))) -> 0(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(1(1(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(1(2(0(2(x1))))) 0(0(1(0(2(x1))))) -> 0(1(2(2(0(x1))))) 0(0(1(0(2(x1))))) -> 0(1(2(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) -> 0(2(1(2(2(x1))))) 0(0(1(0(2(x1))))) -> 0(2(2(1(0(x1))))) 0(0(1(0(2(x1))))) -> 0(2(2(1(2(x1))))) 0(0(1(0(2(x1))))) -> 1(0(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 1(0(2(0(2(x1))))) 0(0(1(0(2(x1))))) -> 1(0(2(2(0(x1))))) 0(0(1(0(2(x1))))) -> 1(0(2(2(2(x1))))) 0(0(1(0(2(x1))))) -> 1(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 1(2(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 1(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) -> 1(2(2(0(2(x1))))) 0(0(1(0(2(x1))))) -> 1(2(2(2(0(x1))))) 0(0(1(0(2(x1))))) -> 2(1(0(2(2(x1))))) 0(0(1(0(2(x1))))) -> 2(2(1(0(2(x1))))) 0(0(1(0(2(x1))))) -> 2(2(2(1(0(x1))))) 0(1(2(0(2(x1))))) -> 0(1(0(2(2(x1))))) 0(1(2(0(2(x1))))) -> 0(1(1(2(2(x1))))) 0(1(2(0(2(x1))))) -> 0(1(2(2(2(x1))))) 0(1(2(0(2(x1))))) -> 0(2(1(0(2(x1))))) 0(1(2(0(2(x1))))) -> 0(2(1(2(2(x1))))) 0(1(2(0(2(x1))))) -> 0(2(2(1(0(x1))))) 0(1(2(0(2(x1))))) -> 0(2(2(1(2(x1))))) 0(1(2(0(2(x1))))) -> 1(0(2(2(2(x1))))) 0(1(2(0(2(x1))))) -> 1(2(0(2(2(x1))))) 0(1(2(0(2(x1))))) -> 1(2(2(0(2(x1))))) 0(1(2(0(2(x1))))) -> 1(2(2(2(0(x1))))) 1(0(1(0(2(x1))))) -> 0(1(2(2(2(x1))))) 1(0(1(0(2(x1))))) -> 0(2(1(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(0(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(1(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(2(0(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(2(1(2(x1))))) 1(0(1(0(2(x1))))) -> 1(0(2(2(0(x1))))) 1(0(1(0(2(x1))))) -> 1(0(2(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(1(0(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(2(0(2(2(x1))))) 1(0(1(0(2(x1))))) -> 1(2(1(0(2(x1))))) 1(0(1(0(2(x1))))) -> 1(2(2(0(2(x1))))) 1(0(1(0(2(x1))))) -> 1(2(2(2(0(x1))))) 1(0(1(0(2(x1))))) -> 2(0(1(2(2(x1))))) 1(0(1(0(2(x1))))) -> 2(0(2(1(2(x1))))) 1(0(1(0(2(x1))))) -> 2(1(0(2(2(x1))))) 1(0(1(0(2(x1))))) -> 2(1(2(0(2(x1))))) 1(0(1(0(2(x1))))) -> 2(1(2(2(0(x1))))) 1(0(1(0(2(x1))))) -> 2(2(0(1(2(x1))))) 1(0(1(0(2(x1))))) -> 2(2(1(0(2(x1))))) 1(0(1(0(2(x1))))) -> 2(2(1(2(0(x1))))) 1(0(1(0(2(x1))))) -> 2(2(2(1(0(x1))))) 1(0(2(0(2(x1))))) -> 1(0(2(2(2(x1))))) 1(0(2(0(2(x1))))) -> 1(2(0(2(2(x1))))) 1(0(2(0(2(x1))))) -> 1(2(2(0(2(x1))))) 1(0(2(0(2(x1))))) -> 1(2(2(2(0(x1))))) 1(0(2(0(2(x1))))) -> 2(1(0(2(2(x1))))) 1(0(2(0(2(x1))))) -> 2(2(1(0(2(x1))))) 1(1(2(0(2(x1))))) -> 0(1(2(2(2(x1))))) 1(1(2(0(2(x1))))) -> 0(2(1(2(2(x1))))) 1(1(2(0(2(x1))))) -> 0(2(2(1(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(0(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(1(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(2(0(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(2(1(2(x1))))) 1(1(2(0(2(x1))))) -> 1(0(2(2(0(x1))))) 1(1(2(0(2(x1))))) -> 1(0(2(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(1(0(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(2(0(2(2(x1))))) 1(1(2(0(2(x1))))) -> 1(2(1(0(2(x1))))) 1(1(2(0(2(x1))))) -> 1(2(2(0(2(x1))))) 1(1(2(0(2(x1))))) -> 1(2(2(2(0(x1))))) 1(1(2(0(2(x1))))) -> 2(0(1(2(2(x1))))) 1(1(2(0(2(x1))))) -> 2(1(0(2(2(x1))))) 1(1(2(0(2(x1))))) -> 2(1(2(0(2(x1))))) 1(1(2(0(2(x1))))) -> 2(2(0(1(2(x1))))) 1(1(2(0(2(x1))))) -> 2(2(1(0(2(x1))))) 1(1(2(0(2(x1))))) -> 2(2(2(1(0(x1))))) 1(2(2(0(2(x1))))) -> 1(0(2(2(2(x1))))) 2(0(1(0(2(x1))))) -> 2(0(1(2(2(x1))))) 2(0(1(0(2(x1))))) -> 2(0(2(1(2(x1))))) 2(0(1(0(2(x1))))) -> 2(1(0(2(2(x1))))) 2(0(1(0(2(x1))))) -> 2(1(2(0(2(x1))))) 2(0(1(0(2(x1))))) -> 2(1(2(2(0(x1))))) 2(0(1(0(2(x1))))) -> 2(2(0(1(2(x1))))) 2(0(1(0(2(x1))))) -> 2(2(1(0(2(x1))))) 2(0(1(0(2(x1))))) -> 2(2(1(2(0(x1))))) 2(0(1(0(2(x1))))) -> 2(2(2(1(0(x1))))) 2(1(1(0(2(x1))))) -> 2(0(1(0(2(x1))))) 2(1(1(0(2(x1))))) -> 2(0(2(1(2(x1))))) 2(1(1(0(2(x1))))) -> 2(1(2(0(2(x1))))) 2(1(1(0(2(x1))))) -> 2(2(1(0(2(x1))))) 2(1(2(0(2(x1))))) -> 2(0(1(2(2(x1))))) 2(1(2(0(2(x1))))) -> 2(1(0(2(2(x1))))) 2(1(2(0(2(x1))))) -> 2(2(1(0(2(x1))))) 2(1(2(0(2(x1))))) -> 2(2(2(1(0(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: 2(0(1(0(0(x1))))) -> 2(2(1(0(0(x1))))) 2(0(1(0(0(x1))))) -> 2(1(2(0(0(x1))))) 2(0(1(0(0(x1))))) -> 2(2(0(1(0(x1))))) 2(0(1(0(0(x1))))) -> 2(2(1(1(0(x1))))) 2(0(1(0(0(x1))))) -> 2(0(2(1(0(x1))))) 2(0(1(0(0(x1))))) -> 0(2(2(1(0(x1))))) 2(0(1(0(0(x1))))) -> 2(2(2(1(0(x1))))) 2(0(1(0(0(x1))))) -> 2(0(1(2(0(x1))))) 2(0(1(0(0(x1))))) -> 2(2(1(2(0(x1))))) 2(0(1(0(0(x1))))) -> 0(1(2(2(0(x1))))) 2(0(1(0(0(x1))))) -> 2(1(2(2(0(x1))))) 2(0(1(0(0(x1))))) -> 2(2(0(0(1(x1))))) 2(0(1(0(0(x1))))) -> 2(0(2(0(1(x1))))) 2(0(1(0(0(x1))))) -> 0(2(2(0(1(x1))))) 2(0(1(0(0(x1))))) -> 2(2(2(0(1(x1))))) 2(0(1(0(0(x1))))) -> 2(2(0(1(1(x1))))) 2(0(1(0(0(x1))))) -> 2(2(0(2(1(x1))))) 2(0(1(0(0(x1))))) -> 2(0(1(2(1(x1))))) 2(0(1(0(0(x1))))) -> 2(0(2(2(1(x1))))) 2(0(1(0(0(x1))))) -> 0(2(2(2(1(x1))))) 2(0(1(0(0(x1))))) -> 2(2(0(1(2(x1))))) 2(0(1(0(0(x1))))) -> 2(0(1(2(2(x1))))) 2(0(1(0(0(x1))))) -> 0(1(2(2(2(x1))))) 2(0(2(1(0(x1))))) -> 2(2(0(1(0(x1))))) 2(0(2(1(0(x1))))) -> 2(2(1(1(0(x1))))) 2(0(2(1(0(x1))))) -> 2(2(2(1(0(x1))))) 2(0(2(1(0(x1))))) -> 2(0(1(2(0(x1))))) 2(0(2(1(0(x1))))) -> 2(2(1(2(0(x1))))) 2(0(2(1(0(x1))))) -> 0(1(2(2(0(x1))))) 2(0(2(1(0(x1))))) -> 2(1(2(2(0(x1))))) 2(0(2(1(0(x1))))) -> 2(2(2(0(1(x1))))) 2(0(2(1(0(x1))))) -> 2(2(0(2(1(x1))))) 2(0(2(1(0(x1))))) -> 2(0(2(2(1(x1))))) 2(0(2(1(0(x1))))) -> 0(2(2(2(1(x1))))) 2(0(1(0(1(x1))))) -> 2(2(2(1(0(x1))))) 2(0(1(0(1(x1))))) -> 2(2(1(2(0(x1))))) 2(0(1(0(1(x1))))) -> 2(2(0(0(1(x1))))) 2(0(1(0(1(x1))))) -> 2(2(1(0(1(x1))))) 2(0(1(0(1(x1))))) -> 2(0(2(0(1(x1))))) 2(0(1(0(1(x1))))) -> 2(1(2(0(1(x1))))) 2(0(1(0(1(x1))))) -> 0(2(2(0(1(x1))))) 2(0(1(0(1(x1))))) -> 2(2(2(0(1(x1))))) 2(0(1(0(1(x1))))) -> 2(2(0(1(1(x1))))) 2(0(1(0(1(x1))))) -> 2(2(0(2(1(x1))))) 2(0(1(0(1(x1))))) -> 2(0(1(2(1(x1))))) 2(0(1(0(1(x1))))) -> 2(0(2(2(1(x1))))) 2(0(1(0(1(x1))))) -> 0(2(2(2(1(x1))))) 2(0(1(0(1(x1))))) -> 2(2(1(0(2(x1))))) 2(0(1(0(1(x1))))) -> 2(1(2(0(2(x1))))) 2(0(1(0(1(x1))))) -> 2(2(0(1(2(x1))))) 2(0(1(0(1(x1))))) -> 2(0(2(1(2(x1))))) 2(0(1(0(1(x1))))) -> 0(2(2(1(2(x1))))) 2(0(1(0(1(x1))))) -> 2(1(0(2(2(x1))))) 2(0(1(0(1(x1))))) -> 2(0(1(2(2(x1))))) 2(0(1(0(1(x1))))) -> 0(2(1(2(2(x1))))) 2(0(1(0(1(x1))))) -> 0(1(2(2(2(x1))))) 2(0(2(0(1(x1))))) -> 2(2(2(0(1(x1))))) 2(0(2(0(1(x1))))) -> 2(2(0(2(1(x1))))) 2(0(2(0(1(x1))))) -> 2(0(2(2(1(x1))))) 2(0(2(0(1(x1))))) -> 0(2(2(2(1(x1))))) 2(0(2(0(1(x1))))) -> 2(2(0(1(2(x1))))) 2(0(2(0(1(x1))))) -> 2(0(1(2(2(x1))))) 2(0(2(1(1(x1))))) -> 2(2(2(1(0(x1))))) 2(0(2(1(1(x1))))) -> 2(2(1(2(0(x1))))) 2(0(2(1(1(x1))))) -> 2(1(2(2(0(x1))))) 2(0(2(1(1(x1))))) -> 2(2(0(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(2(1(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(0(2(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(1(2(0(1(x1))))) 2(0(2(1(1(x1))))) -> 0(2(2(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(2(2(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(2(0(1(1(x1))))) 2(0(2(1(1(x1))))) -> 2(2(0(2(1(x1))))) 2(0(2(1(1(x1))))) -> 2(0(1(2(1(x1))))) 2(0(2(1(1(x1))))) -> 2(0(2(2(1(x1))))) 2(0(2(1(1(x1))))) -> 0(2(2(2(1(x1))))) 2(0(2(1(1(x1))))) -> 2(2(1(0(2(x1))))) 2(0(2(1(1(x1))))) -> 2(2(0(1(2(x1))))) 2(0(2(1(1(x1))))) -> 2(0(2(1(2(x1))))) 2(0(2(1(1(x1))))) -> 2(1(0(2(2(x1))))) 2(0(2(1(1(x1))))) -> 2(0(1(2(2(x1))))) 2(0(2(1(1(x1))))) -> 0(1(2(2(2(x1))))) 2(0(2(2(1(x1))))) -> 2(2(2(0(1(x1))))) 2(0(1(0(2(x1))))) -> 2(2(1(0(2(x1))))) 2(0(1(0(2(x1))))) -> 2(1(2(0(2(x1))))) 2(0(1(0(2(x1))))) -> 2(2(0(1(2(x1))))) 2(0(1(0(2(x1))))) -> 2(0(2(1(2(x1))))) 2(0(1(0(2(x1))))) -> 0(2(2(1(2(x1))))) 2(0(1(0(2(x1))))) -> 2(1(0(2(2(x1))))) 2(0(1(0(2(x1))))) -> 2(0(1(2(2(x1))))) 2(0(1(0(2(x1))))) -> 0(2(1(2(2(x1))))) 2(0(1(0(2(x1))))) -> 0(1(2(2(2(x1))))) 2(0(1(1(2(x1))))) -> 2(0(1(0(2(x1))))) 2(0(1(1(2(x1))))) -> 2(1(2(0(2(x1))))) 2(0(1(1(2(x1))))) -> 2(0(2(1(2(x1))))) 2(0(1(1(2(x1))))) -> 2(0(1(2(2(x1))))) 2(0(2(1(2(x1))))) -> 2(2(1(0(2(x1))))) 2(0(2(1(2(x1))))) -> 2(2(0(1(2(x1))))) 2(0(2(1(2(x1))))) -> 2(0(1(2(2(x1))))) 2(0(2(1(2(x1))))) -> 0(1(2(2(2(x1))))) Q is empty. ---------------------------------------- (3) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 + x_1 POL(1(x_1)) = 1 + x_1 POL(2(x_1)) = x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 2(0(1(0(0(x1))))) -> 2(2(1(0(0(x1))))) 2(0(1(0(0(x1))))) -> 2(1(2(0(0(x1))))) 2(0(1(0(0(x1))))) -> 2(2(0(1(0(x1))))) 2(0(1(0(0(x1))))) -> 2(2(1(1(0(x1))))) 2(0(1(0(0(x1))))) -> 2(0(2(1(0(x1))))) 2(0(1(0(0(x1))))) -> 0(2(2(1(0(x1))))) 2(0(1(0(0(x1))))) -> 2(2(2(1(0(x1))))) 2(0(1(0(0(x1))))) -> 2(0(1(2(0(x1))))) 2(0(1(0(0(x1))))) -> 2(2(1(2(0(x1))))) 2(0(1(0(0(x1))))) -> 0(1(2(2(0(x1))))) 2(0(1(0(0(x1))))) -> 2(1(2(2(0(x1))))) 2(0(1(0(0(x1))))) -> 2(2(0(0(1(x1))))) 2(0(1(0(0(x1))))) -> 2(0(2(0(1(x1))))) 2(0(1(0(0(x1))))) -> 0(2(2(0(1(x1))))) 2(0(1(0(0(x1))))) -> 2(2(2(0(1(x1))))) 2(0(1(0(0(x1))))) -> 2(2(0(1(1(x1))))) 2(0(1(0(0(x1))))) -> 2(2(0(2(1(x1))))) 2(0(1(0(0(x1))))) -> 2(0(1(2(1(x1))))) 2(0(1(0(0(x1))))) -> 2(0(2(2(1(x1))))) 2(0(1(0(0(x1))))) -> 0(2(2(2(1(x1))))) 2(0(1(0(0(x1))))) -> 2(2(0(1(2(x1))))) 2(0(1(0(0(x1))))) -> 2(0(1(2(2(x1))))) 2(0(1(0(0(x1))))) -> 0(1(2(2(2(x1))))) 2(0(2(1(0(x1))))) -> 2(2(2(1(0(x1))))) 2(0(2(1(0(x1))))) -> 2(2(1(2(0(x1))))) 2(0(2(1(0(x1))))) -> 2(1(2(2(0(x1))))) 2(0(2(1(0(x1))))) -> 2(2(2(0(1(x1))))) 2(0(2(1(0(x1))))) -> 2(2(0(2(1(x1))))) 2(0(2(1(0(x1))))) -> 2(0(2(2(1(x1))))) 2(0(2(1(0(x1))))) -> 0(2(2(2(1(x1))))) 2(0(1(0(1(x1))))) -> 2(2(2(1(0(x1))))) 2(0(1(0(1(x1))))) -> 2(2(1(2(0(x1))))) 2(0(1(0(1(x1))))) -> 2(2(0(0(1(x1))))) 2(0(1(0(1(x1))))) -> 2(2(1(0(1(x1))))) 2(0(1(0(1(x1))))) -> 2(0(2(0(1(x1))))) 2(0(1(0(1(x1))))) -> 2(1(2(0(1(x1))))) 2(0(1(0(1(x1))))) -> 0(2(2(0(1(x1))))) 2(0(1(0(1(x1))))) -> 2(2(2(0(1(x1))))) 2(0(1(0(1(x1))))) -> 2(2(0(1(1(x1))))) 2(0(1(0(1(x1))))) -> 2(2(0(2(1(x1))))) 2(0(1(0(1(x1))))) -> 2(0(1(2(1(x1))))) 2(0(1(0(1(x1))))) -> 2(0(2(2(1(x1))))) 2(0(1(0(1(x1))))) -> 0(2(2(2(1(x1))))) 2(0(1(0(1(x1))))) -> 2(2(1(0(2(x1))))) 2(0(1(0(1(x1))))) -> 2(1(2(0(2(x1))))) 2(0(1(0(1(x1))))) -> 2(2(0(1(2(x1))))) 2(0(1(0(1(x1))))) -> 2(0(2(1(2(x1))))) 2(0(1(0(1(x1))))) -> 0(2(2(1(2(x1))))) 2(0(1(0(1(x1))))) -> 2(1(0(2(2(x1))))) 2(0(1(0(1(x1))))) -> 2(0(1(2(2(x1))))) 2(0(1(0(1(x1))))) -> 0(2(1(2(2(x1))))) 2(0(1(0(1(x1))))) -> 0(1(2(2(2(x1))))) 2(0(2(0(1(x1))))) -> 2(2(2(0(1(x1))))) 2(0(2(0(1(x1))))) -> 2(2(0(2(1(x1))))) 2(0(2(0(1(x1))))) -> 2(0(2(2(1(x1))))) 2(0(2(0(1(x1))))) -> 0(2(2(2(1(x1))))) 2(0(2(0(1(x1))))) -> 2(2(0(1(2(x1))))) 2(0(2(0(1(x1))))) -> 2(0(1(2(2(x1))))) 2(0(2(1(1(x1))))) -> 2(2(2(1(0(x1))))) 2(0(2(1(1(x1))))) -> 2(2(1(2(0(x1))))) 2(0(2(1(1(x1))))) -> 2(1(2(2(0(x1))))) 2(0(2(1(1(x1))))) -> 2(2(2(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(2(0(2(1(x1))))) 2(0(2(1(1(x1))))) -> 2(0(2(2(1(x1))))) 2(0(2(1(1(x1))))) -> 0(2(2(2(1(x1))))) 2(0(2(1(1(x1))))) -> 2(2(1(0(2(x1))))) 2(0(2(1(1(x1))))) -> 2(2(0(1(2(x1))))) 2(0(2(1(1(x1))))) -> 2(0(2(1(2(x1))))) 2(0(2(1(1(x1))))) -> 2(1(0(2(2(x1))))) 2(0(2(1(1(x1))))) -> 2(0(1(2(2(x1))))) 2(0(2(1(1(x1))))) -> 0(1(2(2(2(x1))))) 2(0(1(0(2(x1))))) -> 2(2(1(0(2(x1))))) 2(0(1(0(2(x1))))) -> 2(1(2(0(2(x1))))) 2(0(1(0(2(x1))))) -> 2(2(0(1(2(x1))))) 2(0(1(0(2(x1))))) -> 2(0(2(1(2(x1))))) 2(0(1(0(2(x1))))) -> 0(2(2(1(2(x1))))) 2(0(1(0(2(x1))))) -> 2(1(0(2(2(x1))))) 2(0(1(0(2(x1))))) -> 2(0(1(2(2(x1))))) 2(0(1(0(2(x1))))) -> 0(2(1(2(2(x1))))) 2(0(1(0(2(x1))))) -> 0(1(2(2(2(x1))))) 2(0(1(1(2(x1))))) -> 2(1(2(0(2(x1))))) 2(0(1(1(2(x1))))) -> 2(0(2(1(2(x1))))) 2(0(1(1(2(x1))))) -> 2(0(1(2(2(x1))))) ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 2(0(2(1(0(x1))))) -> 2(2(0(1(0(x1))))) 2(0(2(1(0(x1))))) -> 2(2(1(1(0(x1))))) 2(0(2(1(0(x1))))) -> 2(0(1(2(0(x1))))) 2(0(2(1(0(x1))))) -> 0(1(2(2(0(x1))))) 2(0(2(1(1(x1))))) -> 2(2(0(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(2(1(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(0(2(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(1(2(0(1(x1))))) 2(0(2(1(1(x1))))) -> 0(2(2(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(2(0(1(1(x1))))) 2(0(2(1(1(x1))))) -> 2(0(1(2(1(x1))))) 2(0(2(2(1(x1))))) -> 2(2(2(0(1(x1))))) 2(0(1(1(2(x1))))) -> 2(0(1(0(2(x1))))) 2(0(2(1(2(x1))))) -> 2(2(1(0(2(x1))))) 2(0(2(1(2(x1))))) -> 2(2(0(1(2(x1))))) 2(0(2(1(2(x1))))) -> 2(0(1(2(2(x1))))) 2(0(2(1(2(x1))))) -> 0(1(2(2(2(x1))))) Q is empty. ---------------------------------------- (5) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(0(2(1(0(x1))))) -> 2^1(2(0(1(0(x1))))) 2^1(0(2(1(0(x1))))) -> 2^1(0(1(0(x1)))) 2^1(0(2(1(0(x1))))) -> 2^1(2(1(1(0(x1))))) 2^1(0(2(1(0(x1))))) -> 2^1(1(1(0(x1)))) 2^1(0(2(1(0(x1))))) -> 2^1(0(1(2(0(x1))))) 2^1(0(2(1(0(x1))))) -> 2^1(0(x1)) 2^1(0(2(1(0(x1))))) -> 2^1(2(0(x1))) 2^1(0(2(1(1(x1))))) -> 2^1(2(0(0(1(x1))))) 2^1(0(2(1(1(x1))))) -> 2^1(0(0(1(x1)))) 2^1(0(2(1(1(x1))))) -> 2^1(2(1(0(1(x1))))) 2^1(0(2(1(1(x1))))) -> 2^1(1(0(1(x1)))) 2^1(0(2(1(1(x1))))) -> 2^1(0(2(0(1(x1))))) 2^1(0(2(1(1(x1))))) -> 2^1(0(1(x1))) 2^1(0(2(1(1(x1))))) -> 2^1(1(2(0(1(x1))))) 2^1(0(2(1(1(x1))))) -> 2^1(2(0(1(x1)))) 2^1(0(2(1(1(x1))))) -> 2^1(2(0(1(1(x1))))) 2^1(0(2(1(1(x1))))) -> 2^1(0(1(1(x1)))) 2^1(0(2(1(1(x1))))) -> 2^1(0(1(2(1(x1))))) 2^1(0(2(1(1(x1))))) -> 2^1(1(x1)) 2^1(0(2(2(1(x1))))) -> 2^1(2(2(0(1(x1))))) 2^1(0(2(2(1(x1))))) -> 2^1(2(0(1(x1)))) 2^1(0(2(2(1(x1))))) -> 2^1(0(1(x1))) 2^1(0(1(1(2(x1))))) -> 2^1(0(1(0(2(x1))))) 2^1(0(2(1(2(x1))))) -> 2^1(2(1(0(2(x1))))) 2^1(0(2(1(2(x1))))) -> 2^1(1(0(2(x1)))) 2^1(0(2(1(2(x1))))) -> 2^1(2(0(1(2(x1))))) 2^1(0(2(1(2(x1))))) -> 2^1(0(1(2(x1)))) 2^1(0(2(1(2(x1))))) -> 2^1(0(1(2(2(x1))))) 2^1(0(2(1(2(x1))))) -> 2^1(2(x1)) 2^1(0(2(1(2(x1))))) -> 2^1(2(2(x1))) The TRS R consists of the following rules: 2(0(2(1(0(x1))))) -> 2(2(0(1(0(x1))))) 2(0(2(1(0(x1))))) -> 2(2(1(1(0(x1))))) 2(0(2(1(0(x1))))) -> 2(0(1(2(0(x1))))) 2(0(2(1(0(x1))))) -> 0(1(2(2(0(x1))))) 2(0(2(1(1(x1))))) -> 2(2(0(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(2(1(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(0(2(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(1(2(0(1(x1))))) 2(0(2(1(1(x1))))) -> 0(2(2(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(2(0(1(1(x1))))) 2(0(2(1(1(x1))))) -> 2(0(1(2(1(x1))))) 2(0(2(2(1(x1))))) -> 2(2(2(0(1(x1))))) 2(0(1(1(2(x1))))) -> 2(0(1(0(2(x1))))) 2(0(2(1(2(x1))))) -> 2(2(1(0(2(x1))))) 2(0(2(1(2(x1))))) -> 2(2(0(1(2(x1))))) 2(0(2(1(2(x1))))) -> 2(0(1(2(2(x1))))) 2(0(2(1(2(x1))))) -> 0(1(2(2(2(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 20 less nodes. ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(0(2(1(0(x1))))) -> 2^1(2(0(x1))) 2^1(0(2(1(0(x1))))) -> 2^1(0(x1)) 2^1(0(2(1(1(x1))))) -> 2^1(0(2(0(1(x1))))) 2^1(0(2(2(1(x1))))) -> 2^1(2(2(0(1(x1))))) 2^1(0(2(2(1(x1))))) -> 2^1(2(0(1(x1)))) 2^1(0(2(1(2(x1))))) -> 2^1(2(0(1(2(x1))))) 2^1(0(2(1(2(x1))))) -> 2^1(2(x1)) 2^1(0(2(1(2(x1))))) -> 2^1(2(2(x1))) 2^1(0(2(1(1(x1))))) -> 2^1(2(0(1(x1)))) 2^1(0(2(1(1(x1))))) -> 2^1(2(0(1(1(x1))))) The TRS R consists of the following rules: 2(0(2(1(0(x1))))) -> 2(2(0(1(0(x1))))) 2(0(2(1(0(x1))))) -> 2(2(1(1(0(x1))))) 2(0(2(1(0(x1))))) -> 2(0(1(2(0(x1))))) 2(0(2(1(0(x1))))) -> 0(1(2(2(0(x1))))) 2(0(2(1(1(x1))))) -> 2(2(0(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(2(1(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(0(2(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(1(2(0(1(x1))))) 2(0(2(1(1(x1))))) -> 0(2(2(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(2(0(1(1(x1))))) 2(0(2(1(1(x1))))) -> 2(0(1(2(1(x1))))) 2(0(2(2(1(x1))))) -> 2(2(2(0(1(x1))))) 2(0(1(1(2(x1))))) -> 2(0(1(0(2(x1))))) 2(0(2(1(2(x1))))) -> 2(2(1(0(2(x1))))) 2(0(2(1(2(x1))))) -> 2(2(0(1(2(x1))))) 2(0(2(1(2(x1))))) -> 2(0(1(2(2(x1))))) 2(0(2(1(2(x1))))) -> 0(1(2(2(2(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 2^1(0(2(1(0(x1))))) -> 2^1(2(0(x1))) 2^1(0(2(1(0(x1))))) -> 2^1(0(x1)) 2^1(0(2(2(1(x1))))) -> 2^1(2(0(1(x1)))) 2^1(0(2(1(2(x1))))) -> 2^1(2(x1)) 2^1(0(2(1(2(x1))))) -> 2^1(2(2(x1))) 2^1(0(2(1(1(x1))))) -> 2^1(2(0(1(x1)))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 + x_1 POL(1(x_1)) = 1 + x_1 POL(2(x_1)) = 1 + x_1 POL(2^1(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 2(0(2(1(0(x1))))) -> 2(2(0(1(0(x1))))) 2(0(2(1(0(x1))))) -> 2(2(1(1(0(x1))))) 2(0(2(1(0(x1))))) -> 2(0(1(2(0(x1))))) 2(0(2(1(0(x1))))) -> 0(1(2(2(0(x1))))) 2(0(2(1(1(x1))))) -> 2(2(0(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(2(1(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(2(0(1(1(x1))))) 2(0(2(1(1(x1))))) -> 2(0(2(0(1(x1))))) 2(0(2(2(1(x1))))) -> 2(2(2(0(1(x1))))) 2(0(2(1(2(x1))))) -> 2(2(0(1(2(x1))))) 2(0(2(1(1(x1))))) -> 2(1(2(0(1(x1))))) 2(0(2(1(1(x1))))) -> 0(2(2(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(0(1(2(1(x1))))) 2(0(1(1(2(x1))))) -> 2(0(1(0(2(x1))))) 2(0(2(1(2(x1))))) -> 2(2(1(0(2(x1))))) 2(0(2(1(2(x1))))) -> 2(0(1(2(2(x1))))) 2(0(2(1(2(x1))))) -> 0(1(2(2(2(x1))))) ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(0(2(1(1(x1))))) -> 2^1(0(2(0(1(x1))))) 2^1(0(2(2(1(x1))))) -> 2^1(2(2(0(1(x1))))) 2^1(0(2(1(2(x1))))) -> 2^1(2(0(1(2(x1))))) 2^1(0(2(1(1(x1))))) -> 2^1(2(0(1(1(x1))))) The TRS R consists of the following rules: 2(0(2(1(0(x1))))) -> 2(2(0(1(0(x1))))) 2(0(2(1(0(x1))))) -> 2(2(1(1(0(x1))))) 2(0(2(1(0(x1))))) -> 2(0(1(2(0(x1))))) 2(0(2(1(0(x1))))) -> 0(1(2(2(0(x1))))) 2(0(2(1(1(x1))))) -> 2(2(0(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(2(1(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(0(2(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(1(2(0(1(x1))))) 2(0(2(1(1(x1))))) -> 0(2(2(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(2(0(1(1(x1))))) 2(0(2(1(1(x1))))) -> 2(0(1(2(1(x1))))) 2(0(2(2(1(x1))))) -> 2(2(2(0(1(x1))))) 2(0(1(1(2(x1))))) -> 2(0(1(0(2(x1))))) 2(0(2(1(2(x1))))) -> 2(2(1(0(2(x1))))) 2(0(2(1(2(x1))))) -> 2(2(0(1(2(x1))))) 2(0(2(1(2(x1))))) -> 2(0(1(2(2(x1))))) 2(0(2(1(2(x1))))) -> 0(1(2(2(2(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(0(2(1(2(x1))))) -> 2^1(2(0(1(2(x1))))) 2^1(0(2(2(1(x1))))) -> 2^1(2(2(0(1(x1))))) The TRS R consists of the following rules: 2(0(2(1(0(x1))))) -> 2(2(0(1(0(x1))))) 2(0(2(1(0(x1))))) -> 2(2(1(1(0(x1))))) 2(0(2(1(0(x1))))) -> 2(0(1(2(0(x1))))) 2(0(2(1(0(x1))))) -> 0(1(2(2(0(x1))))) 2(0(2(1(1(x1))))) -> 2(2(0(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(2(1(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(0(2(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(1(2(0(1(x1))))) 2(0(2(1(1(x1))))) -> 0(2(2(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(2(0(1(1(x1))))) 2(0(2(1(1(x1))))) -> 2(0(1(2(1(x1))))) 2(0(2(2(1(x1))))) -> 2(2(2(0(1(x1))))) 2(0(1(1(2(x1))))) -> 2(0(1(0(2(x1))))) 2(0(2(1(2(x1))))) -> 2(2(1(0(2(x1))))) 2(0(2(1(2(x1))))) -> 2(2(0(1(2(x1))))) 2(0(2(1(2(x1))))) -> 2(0(1(2(2(x1))))) 2(0(2(1(2(x1))))) -> 0(1(2(2(2(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. 2^1(0(2(1(2(x1))))) -> 2^1(2(0(1(2(x1))))) 2^1(0(2(2(1(x1))))) -> 2^1(2(2(0(1(x1))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( 2^1_1(x_1) ) = max{0, 2x_1 - 2} POL( 2_1(x_1) ) = x_1 + 2 POL( 0_1(x_1) ) = max{0, 2x_1 - 2} POL( 1_1(x_1) ) = 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 2(0(2(1(0(x1))))) -> 2(2(0(1(0(x1))))) 2(0(2(1(0(x1))))) -> 2(2(1(1(0(x1))))) 2(0(2(1(0(x1))))) -> 2(0(1(2(0(x1))))) 2(0(2(1(0(x1))))) -> 0(1(2(2(0(x1))))) 2(0(2(1(1(x1))))) -> 2(2(0(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(2(1(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(2(0(1(1(x1))))) 2(0(2(1(1(x1))))) -> 2(0(2(0(1(x1))))) 2(0(2(2(1(x1))))) -> 2(2(2(0(1(x1))))) 2(0(2(1(2(x1))))) -> 2(2(0(1(2(x1))))) 2(0(2(1(1(x1))))) -> 2(1(2(0(1(x1))))) 2(0(2(1(1(x1))))) -> 0(2(2(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(0(1(2(1(x1))))) 2(0(1(1(2(x1))))) -> 2(0(1(0(2(x1))))) 2(0(2(1(2(x1))))) -> 2(2(1(0(2(x1))))) 2(0(2(1(2(x1))))) -> 2(0(1(2(2(x1))))) 2(0(2(1(2(x1))))) -> 0(1(2(2(2(x1))))) ---------------------------------------- (14) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: 2(0(2(1(0(x1))))) -> 2(2(0(1(0(x1))))) 2(0(2(1(0(x1))))) -> 2(2(1(1(0(x1))))) 2(0(2(1(0(x1))))) -> 2(0(1(2(0(x1))))) 2(0(2(1(0(x1))))) -> 0(1(2(2(0(x1))))) 2(0(2(1(1(x1))))) -> 2(2(0(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(2(1(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(0(2(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(1(2(0(1(x1))))) 2(0(2(1(1(x1))))) -> 0(2(2(0(1(x1))))) 2(0(2(1(1(x1))))) -> 2(2(0(1(1(x1))))) 2(0(2(1(1(x1))))) -> 2(0(1(2(1(x1))))) 2(0(2(2(1(x1))))) -> 2(2(2(0(1(x1))))) 2(0(1(1(2(x1))))) -> 2(0(1(0(2(x1))))) 2(0(2(1(2(x1))))) -> 2(2(1(0(2(x1))))) 2(0(2(1(2(x1))))) -> 2(2(0(1(2(x1))))) 2(0(2(1(2(x1))))) -> 2(0(1(2(2(x1))))) 2(0(2(1(2(x1))))) -> 0(1(2(2(2(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (15) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (16) YES