/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) DependencyPairsProof [EQUIVALENT, 273 ms] (4) QDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) QDP (7) QDPOrderProof [EQUIVALENT, 7944 ms] (8) QDP (9) TransformationProof [EQUIVALENT, 105 ms] (10) QDP (11) DependencyGraphProof [EQUIVALENT, 0 ms] (12) QDP (13) QDPOrderProof [EQUIVALENT, 8083 ms] (14) QDP (15) QDPOrderProof [EQUIVALENT, 5802 ms] (16) QDP (17) QDPOrderProof [EQUIVALENT, 8261 ms] (18) QDP (19) QDPOrderProof [EQUIVALENT, 7389 ms] (20) QDP (21) QDPOrderProof [EQUIVALENT, 5811 ms] (22) QDP (23) QDPOrderProof [EQUIVALENT, 6323 ms] (24) QDP (25) UsableRulesProof [EQUIVALENT, 0 ms] (26) QDP (27) QDPSizeChangeProof [EQUIVALENT, 0 ms] (28) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0(1(0(2(x1)))) -> 2(0(3(1(0(x1))))) 0(1(0(2(x1)))) -> 2(0(0(3(1(2(x1)))))) 0(1(0(2(x1)))) -> 2(0(3(1(0(4(x1)))))) 0(1(0(2(x1)))) -> 2(2(0(3(1(0(x1)))))) 0(1(0(2(x1)))) -> 2(3(1(0(0(2(x1)))))) 0(1(0(2(x1)))) -> 2(3(1(0(3(0(x1)))))) 0(1(0(2(x1)))) -> 4(1(0(3(0(2(x1)))))) 0(1(0(2(x1)))) -> 4(1(0(4(0(2(x1)))))) 0(1(4(2(x1)))) -> 2(3(1(0(4(x1))))) 0(1(4(2(x1)))) -> 2(4(0(3(1(x1))))) 0(1(4(2(x1)))) -> 3(2(1(0(4(x1))))) 0(1(4(2(x1)))) -> 3(2(1(4(0(x1))))) 0(1(4(2(x1)))) -> 4(0(3(1(2(x1))))) 0(1(4(2(x1)))) -> 4(1(0(3(2(x1))))) 0(1(4(2(x1)))) -> 4(1(0(4(2(x1))))) 0(1(4(2(x1)))) -> 4(1(0(5(2(x1))))) 0(1(4(2(x1)))) -> 2(0(3(1(0(4(x1)))))) 0(1(4(2(x1)))) -> 2(0(3(1(4(4(x1)))))) 0(1(4(2(x1)))) -> 2(3(1(4(0(4(x1)))))) 0(1(4(2(x1)))) -> 2(4(3(0(4(1(x1)))))) 0(1(4(2(x1)))) -> 2(4(3(1(0(3(x1)))))) 0(1(4(2(x1)))) -> 3(2(1(0(4(1(x1)))))) 0(1(4(2(x1)))) -> 3(2(2(1(4(0(x1)))))) 0(1(4(2(x1)))) -> 3(2(3(1(0(4(x1)))))) 0(1(4(2(x1)))) -> 3(2(3(1(4(0(x1)))))) 0(1(4(2(x1)))) -> 4(0(3(1(3(2(x1)))))) 0(1(4(2(x1)))) -> 4(0(3(1(4(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(0(4(3(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(0(4(5(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(0(5(3(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(1(0(5(2(x1)))))) 0(1(4(2(x1)))) -> 4(1(3(0(5(2(x1)))))) 0(1(4(2(x1)))) -> 4(3(0(3(1(2(x1)))))) 0(1(4(2(x1)))) -> 4(4(0(3(1(2(x1)))))) 0(0(1(0(2(x1))))) -> 1(0(0(2(0(4(x1)))))) 0(0(1(0(2(x1))))) -> 1(0(4(0(0(2(x1)))))) 0(0(1(0(2(x1))))) -> 2(1(0(3(0(0(x1)))))) 0(0(1(4(2(x1))))) -> 0(0(3(1(2(4(x1)))))) 0(0(1(4(2(x1))))) -> 0(2(3(1(0(4(x1)))))) 0(0(1(4(2(x1))))) -> 0(2(4(0(3(1(x1)))))) 0(0(1(4(2(x1))))) -> 0(3(1(0(2(4(x1)))))) 0(0(1(4(2(x1))))) -> 1(0(3(4(0(2(x1)))))) 0(0(1(4(2(x1))))) -> 2(0(0(3(1(4(x1)))))) 0(0(1(4(2(x1))))) -> 2(1(0(4(0(0(x1)))))) 0(1(2(0(2(x1))))) -> 2(0(1(0(4(2(x1)))))) 0(1(2(4(2(x1))))) -> 2(3(1(0(2(4(x1)))))) 0(1(2(4(2(x1))))) -> 4(1(0(2(2(4(x1)))))) 0(1(3(4(2(x1))))) -> 2(3(1(4(4(0(x1)))))) 0(1(3(4(2(x1))))) -> 2(4(3(0(4(1(x1)))))) 0(1(3(4(2(x1))))) -> 3(2(1(0(4(0(x1)))))) 0(1(3(4(2(x1))))) -> 4(0(3(3(1(2(x1)))))) 0(1(3(4(2(x1))))) -> 4(1(4(0(3(2(x1)))))) 0(1(4(0(2(x1))))) -> 2(0(3(1(0(4(x1)))))) 0(1(5(0(2(x1))))) -> 0(2(3(1(0(5(x1)))))) 0(1(5(0(2(x1))))) -> 3(0(5(1(0(2(x1)))))) 0(1(5(0(2(x1))))) -> 5(1(3(0(0(2(x1)))))) 0(1(5(4(2(x1))))) -> 0(4(4(1(2(5(x1)))))) 0(1(5(4(2(x1))))) -> 1(0(4(5(1(2(x1)))))) 0(1(5(4(2(x1))))) -> 2(0(4(4(5(1(x1)))))) 0(1(5(4(2(x1))))) -> 4(0(2(3(1(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(1(0(2(5(2(x1)))))) 0(1(5(4(2(x1))))) -> 4(1(0(5(2(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(2(1(3(0(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(3(1(0(2(5(x1)))))) 0(1(5(4(2(x1))))) -> 4(4(0(5(1(2(x1)))))) 0(1(5(4(2(x1))))) -> 4(4(2(1(0(5(x1)))))) 0(1(5(4(2(x1))))) -> 5(0(4(5(2(1(x1)))))) 0(1(5(4(2(x1))))) -> 5(1(2(0(4(3(x1)))))) 0(1(5(4(2(x1))))) -> 5(3(1(0(4(2(x1)))))) 0(2(1(4(2(x1))))) -> 0(4(4(1(2(2(x1)))))) 0(2(1(4(2(x1))))) -> 3(2(2(1(4(0(x1)))))) 0(2(1(4(2(x1))))) -> 4(1(0(3(2(2(x1)))))) 5(0(1(4(2(x1))))) -> 2(0(4(3(5(1(x1)))))) 5(0(1(4(2(x1))))) -> 2(4(0(3(1(5(x1)))))) 5(0(1(4(2(x1))))) -> 4(1(0(5(3(2(x1)))))) 5(0(1(4(2(x1))))) -> 5(2(1(1(0(4(x1)))))) 5(0(2(0(2(x1))))) -> 5(0(3(0(2(2(x1)))))) 5(0(2(0(2(x1))))) -> 5(0(4(0(2(2(x1)))))) 5(0(2(4(2(x1))))) -> 5(4(0(3(2(2(x1)))))) 5(1(5(0(2(x1))))) -> 5(2(1(4(5(0(x1)))))) 5(1(5(4(2(x1))))) -> 5(2(1(0(4(5(x1)))))) 5(4(1(4(2(x1))))) -> 3(2(1(4(4(5(x1)))))) 5(4(1(4(2(x1))))) -> 4(4(3(5(1(2(x1)))))) 5(4(2(0(2(x1))))) -> 3(0(5(2(2(4(x1)))))) 5(4(2(0(2(x1))))) -> 4(0(5(3(2(2(x1)))))) 5(4(2(0(2(x1))))) -> 5(2(2(2(4(0(x1)))))) 5(4(2(0(2(x1))))) -> 5(3(2(2(4(0(x1)))))) 5(4(2(0(2(x1))))) -> 5(4(2(2(4(0(x1)))))) 5(4(2(4(2(x1))))) -> 0(4(4(5(2(2(x1)))))) 5(4(2(4(2(x1))))) -> 5(4(4(3(2(2(x1)))))) 5(4(5(4(2(x1))))) -> 4(5(0(4(5(2(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(x1)))) -> 0(1(3(0(2(x1))))) 2(0(1(0(x1)))) -> 2(1(3(0(0(2(x1)))))) 2(0(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(0(1(0(x1)))) -> 0(1(3(0(2(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 0(3(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(3(0(1(4(x1)))))) 2(0(1(0(x1)))) -> 2(0(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(x1))))) 2(4(1(0(x1)))) -> 1(3(0(4(2(x1))))) 2(4(1(0(x1)))) -> 4(0(1(2(3(x1))))) 2(4(1(0(x1)))) -> 0(4(1(2(3(x1))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(x1))))) 2(4(1(0(x1)))) -> 2(3(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(4(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(5(0(1(4(x1))))) 2(4(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(4(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(0(4(1(3(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 3(0(1(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(1(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 2(3(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(4(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(1(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(3(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(3(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(4(x1)))))) 2(0(1(0(0(x1))))) -> 4(0(2(0(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 2(0(0(4(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 0(0(3(0(1(2(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(0(1(3(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 1(3(0(4(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(0(1(3(0(x1)))))) 2(4(1(0(0(x1))))) -> 2(0(4(3(0(1(x1)))))) 2(4(1(0(0(x1))))) -> 4(1(3(0(0(2(x1)))))) 2(4(1(0(0(x1))))) -> 0(0(4(0(1(2(x1)))))) 2(0(2(1(0(x1))))) -> 2(4(0(1(0(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(0(1(3(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(2(0(1(4(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(4(1(3(2(x1)))))) 2(4(3(1(0(x1))))) -> 1(4(0(3(4(2(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(0(1(2(3(x1)))))) 2(4(3(1(0(x1))))) -> 2(1(3(3(0(4(x1)))))) 2(4(3(1(0(x1))))) -> 2(3(0(4(1(4(x1)))))) 2(0(4(1(0(x1))))) -> 4(0(1(3(0(2(x1)))))) 2(0(5(1(0(x1))))) -> 5(0(1(3(2(0(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(1(5(0(3(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(0(3(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(1(4(4(0(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(4(0(1(x1)))))) 2(4(5(1(0(x1))))) -> 1(5(4(4(0(2(x1)))))) 2(4(5(1(0(x1))))) -> 5(1(3(2(0(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(5(2(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(3(1(2(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(0(1(3(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(0(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(1(2(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 1(2(5(4(0(5(x1)))))) 2(4(5(1(0(x1))))) -> 3(4(0(2(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 2(4(0(1(3(5(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(1(4(4(0(x1)))))) 2(4(1(2(0(x1))))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(3(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 1(5(3(4(0(2(x1)))))) 2(4(1(0(5(x1))))) -> 5(1(3(0(4(2(x1)))))) 2(4(1(0(5(x1))))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 4(0(1(1(2(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(3(0(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(4(0(5(x1)))))) 2(4(2(0(5(x1))))) -> 2(2(3(0(4(5(x1)))))) 2(0(5(1(5(x1))))) -> 0(5(4(1(2(5(x1)))))) 2(4(5(1(5(x1))))) -> 5(4(0(1(2(5(x1)))))) 2(4(1(4(5(x1))))) -> 5(4(4(1(2(3(x1)))))) 2(4(1(4(5(x1))))) -> 2(1(5(3(4(4(x1)))))) 2(0(2(4(5(x1))))) -> 4(2(2(5(0(3(x1)))))) 2(0(2(4(5(x1))))) -> 2(2(3(5(0(4(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(2(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(3(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(4(5(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(5(4(4(0(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(3(4(4(5(x1)))))) 2(4(5(4(5(x1))))) -> 2(5(4(0(5(4(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: 2^1(0(1(0(x1)))) -> 2^1(x1) 2^1(0(1(0(x1)))) -> 2^1(1(3(0(0(2(x1)))))) 2^1(0(1(0(x1)))) -> 2^1(2(x1)) 2^1(0(1(0(x1)))) -> 2^1(0(0(1(3(2(x1)))))) 2^1(0(1(0(x1)))) -> 2^1(0(3(0(1(4(x1)))))) 2^1(0(1(0(x1)))) -> 2^1(0(4(0(1(4(x1)))))) 2^1(4(1(0(x1)))) -> 2^1(x1) 2^1(4(1(0(x1)))) -> 2^1(3(x1)) 2^1(4(1(0(x1)))) -> 2^1(1(3(0(4(x1))))) 2^1(4(1(0(x1)))) -> 2^1(3(0(1(4(x1))))) 2^1(4(1(0(x1)))) -> 2^1(4(0(1(4(x1))))) 2^1(4(1(0(x1)))) -> 2^1(5(0(1(4(x1))))) 2^1(4(1(0(x1)))) -> 2^1(2(3(x1))) 2^1(4(1(0(x1)))) -> 2^1(3(1(3(0(4(x1)))))) 2^1(4(1(0(x1)))) -> 2^1(4(1(3(0(4(x1)))))) 2^1(4(1(0(x1)))) -> 2^1(3(4(0(1(4(x1)))))) 2^1(4(1(0(x1)))) -> 2^1(5(4(0(1(4(x1)))))) 2^1(4(1(0(x1)))) -> 2^1(3(5(0(1(4(x1)))))) 2^1(4(1(0(x1)))) -> 2^1(5(0(1(1(4(x1)))))) 2^1(4(1(0(x1)))) -> 2^1(5(0(3(1(4(x1)))))) 2^1(4(1(0(x1)))) -> 2^1(1(3(0(3(4(x1)))))) 2^1(4(1(0(x1)))) -> 2^1(1(3(0(4(4(x1)))))) 2^1(0(1(0(0(x1))))) -> 2^1(0(0(1(x1)))) 2^1(0(1(0(0(x1))))) -> 2^1(0(0(4(0(1(x1)))))) 2^1(0(1(0(0(x1))))) -> 2^1(x1) 2^1(4(1(0(0(x1))))) -> 2^1(1(3(0(0(x1))))) 2^1(4(1(0(0(x1))))) -> 2^1(0(x1)) 2^1(4(1(0(0(x1))))) -> 2^1(0(1(3(0(x1))))) 2^1(4(1(0(0(x1))))) -> 2^1(0(4(3(0(1(x1)))))) 2^1(4(1(0(0(x1))))) -> 2^1(x1) 2^1(0(2(1(0(x1))))) -> 2^1(4(0(1(0(2(x1)))))) 2^1(0(2(1(0(x1))))) -> 2^1(x1) 2^1(4(2(1(0(x1))))) -> 2^1(0(1(3(2(x1))))) 2^1(4(2(1(0(x1))))) -> 2^1(x1) 2^1(4(2(1(0(x1))))) -> 2^1(2(0(1(4(x1))))) 2^1(4(2(1(0(x1))))) -> 2^1(0(1(4(x1)))) 2^1(4(3(1(0(x1))))) -> 2^1(x1) 2^1(4(3(1(0(x1))))) -> 2^1(3(x1)) 2^1(4(3(1(0(x1))))) -> 2^1(1(3(3(0(4(x1)))))) 2^1(4(3(1(0(x1))))) -> 2^1(3(0(4(1(4(x1)))))) 2^1(0(4(1(0(x1))))) -> 2^1(x1) 2^1(0(5(1(0(x1))))) -> 2^1(0(x1)) 2^1(0(5(1(0(x1))))) -> 2^1(0(1(5(0(3(x1)))))) 2^1(0(5(1(0(x1))))) -> 2^1(0(0(3(1(5(x1)))))) 2^1(4(5(1(0(x1))))) -> 2^1(1(4(4(0(x1))))) 2^1(4(5(1(0(x1))))) -> 2^1(1(5(4(0(1(x1)))))) 2^1(4(5(1(0(x1))))) -> 2^1(x1) 2^1(4(5(1(0(x1))))) -> 2^1(0(4(x1))) 2^1(4(5(1(0(x1))))) -> 2^1(5(2(0(1(4(x1)))))) 2^1(4(5(1(0(x1))))) -> 2^1(0(1(4(x1)))) 2^1(4(5(1(0(x1))))) -> 2^1(5(0(1(4(x1))))) 2^1(4(5(1(0(x1))))) -> 2^1(4(x1)) 2^1(4(5(1(0(x1))))) -> 2^1(0(1(3(4(x1))))) 2^1(4(5(1(0(x1))))) -> 2^1(1(5(0(4(4(x1)))))) 2^1(4(5(1(0(x1))))) -> 2^1(4(4(x1))) 2^1(4(5(1(0(x1))))) -> 2^1(5(4(0(5(x1))))) 2^1(4(5(1(0(x1))))) -> 2^1(1(5(x1))) 2^1(4(5(1(0(x1))))) -> 2^1(4(0(1(3(5(x1)))))) 2^1(4(1(2(0(x1))))) -> 2^1(2(1(4(4(0(x1)))))) 2^1(4(1(2(0(x1))))) -> 2^1(1(4(4(0(x1))))) 2^1(4(1(2(0(x1))))) -> 2^1(2(3(x1))) 2^1(4(1(2(0(x1))))) -> 2^1(3(x1)) 2^1(4(1(2(0(x1))))) -> 2^1(2(3(0(1(4(x1)))))) 2^1(4(1(2(0(x1))))) -> 2^1(3(0(1(4(x1))))) 2^1(4(1(0(5(x1))))) -> 2^1(x1) 2^1(4(1(0(5(x1))))) -> 2^1(3(5(0(1(4(x1)))))) 2^1(4(1(0(5(x1))))) -> 2^1(5(x1)) 2^1(0(2(0(5(x1))))) -> 2^1(2(0(3(0(5(x1)))))) 2^1(0(2(0(5(x1))))) -> 2^1(0(3(0(5(x1))))) 2^1(0(2(0(5(x1))))) -> 2^1(2(0(4(0(5(x1)))))) 2^1(0(2(0(5(x1))))) -> 2^1(0(4(0(5(x1))))) 2^1(4(2(0(5(x1))))) -> 2^1(2(3(0(4(5(x1)))))) 2^1(4(2(0(5(x1))))) -> 2^1(3(0(4(5(x1))))) 2^1(0(5(1(5(x1))))) -> 2^1(5(x1)) 2^1(4(5(1(5(x1))))) -> 2^1(5(x1)) 2^1(4(1(4(5(x1))))) -> 2^1(3(x1)) 2^1(4(1(4(5(x1))))) -> 2^1(1(5(3(4(4(x1)))))) 2^1(0(2(4(5(x1))))) -> 2^1(2(5(0(3(x1))))) 2^1(0(2(4(5(x1))))) -> 2^1(5(0(3(x1)))) 2^1(0(2(4(5(x1))))) -> 2^1(2(3(5(0(4(x1)))))) 2^1(0(2(4(5(x1))))) -> 2^1(3(5(0(4(x1))))) 2^1(0(2(4(5(x1))))) -> 2^1(2(2(5(x1)))) 2^1(0(2(4(5(x1))))) -> 2^1(2(5(x1))) 2^1(0(2(4(5(x1))))) -> 2^1(5(x1)) 2^1(0(2(4(5(x1))))) -> 2^1(2(3(5(x1)))) 2^1(0(2(4(5(x1))))) -> 2^1(3(5(x1))) 2^1(0(2(4(5(x1))))) -> 2^1(2(4(5(x1)))) 2^1(4(2(4(5(x1))))) -> 2^1(2(5(4(4(0(x1)))))) 2^1(4(2(4(5(x1))))) -> 2^1(5(4(4(0(x1))))) 2^1(4(2(4(5(x1))))) -> 2^1(2(3(4(4(5(x1)))))) 2^1(4(2(4(5(x1))))) -> 2^1(3(4(4(5(x1))))) 2^1(4(5(4(5(x1))))) -> 2^1(5(4(0(5(4(x1)))))) The TRS R consists of the following rules: 2(0(1(0(x1)))) -> 0(1(3(0(2(x1))))) 2(0(1(0(x1)))) -> 2(1(3(0(0(2(x1)))))) 2(0(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(0(1(0(x1)))) -> 0(1(3(0(2(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 0(3(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(3(0(1(4(x1)))))) 2(0(1(0(x1)))) -> 2(0(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(x1))))) 2(4(1(0(x1)))) -> 1(3(0(4(2(x1))))) 2(4(1(0(x1)))) -> 4(0(1(2(3(x1))))) 2(4(1(0(x1)))) -> 0(4(1(2(3(x1))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(x1))))) 2(4(1(0(x1)))) -> 2(3(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(4(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(5(0(1(4(x1))))) 2(4(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(4(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(0(4(1(3(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 3(0(1(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(1(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 2(3(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(4(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(1(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(3(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(3(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(4(x1)))))) 2(0(1(0(0(x1))))) -> 4(0(2(0(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 2(0(0(4(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 0(0(3(0(1(2(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(0(1(3(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 1(3(0(4(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(0(1(3(0(x1)))))) 2(4(1(0(0(x1))))) -> 2(0(4(3(0(1(x1)))))) 2(4(1(0(0(x1))))) -> 4(1(3(0(0(2(x1)))))) 2(4(1(0(0(x1))))) -> 0(0(4(0(1(2(x1)))))) 2(0(2(1(0(x1))))) -> 2(4(0(1(0(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(0(1(3(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(2(0(1(4(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(4(1(3(2(x1)))))) 2(4(3(1(0(x1))))) -> 1(4(0(3(4(2(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(0(1(2(3(x1)))))) 2(4(3(1(0(x1))))) -> 2(1(3(3(0(4(x1)))))) 2(4(3(1(0(x1))))) -> 2(3(0(4(1(4(x1)))))) 2(0(4(1(0(x1))))) -> 4(0(1(3(0(2(x1)))))) 2(0(5(1(0(x1))))) -> 5(0(1(3(2(0(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(1(5(0(3(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(0(3(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(1(4(4(0(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(4(0(1(x1)))))) 2(4(5(1(0(x1))))) -> 1(5(4(4(0(2(x1)))))) 2(4(5(1(0(x1))))) -> 5(1(3(2(0(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(5(2(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(3(1(2(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(0(1(3(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(0(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(1(2(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 1(2(5(4(0(5(x1)))))) 2(4(5(1(0(x1))))) -> 3(4(0(2(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 2(4(0(1(3(5(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(1(4(4(0(x1)))))) 2(4(1(2(0(x1))))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(3(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 1(5(3(4(0(2(x1)))))) 2(4(1(0(5(x1))))) -> 5(1(3(0(4(2(x1)))))) 2(4(1(0(5(x1))))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 4(0(1(1(2(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(3(0(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(4(0(5(x1)))))) 2(4(2(0(5(x1))))) -> 2(2(3(0(4(5(x1)))))) 2(0(5(1(5(x1))))) -> 0(5(4(1(2(5(x1)))))) 2(4(5(1(5(x1))))) -> 5(4(0(1(2(5(x1)))))) 2(4(1(4(5(x1))))) -> 5(4(4(1(2(3(x1)))))) 2(4(1(4(5(x1))))) -> 2(1(5(3(4(4(x1)))))) 2(0(2(4(5(x1))))) -> 4(2(2(5(0(3(x1)))))) 2(0(2(4(5(x1))))) -> 2(2(3(5(0(4(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(2(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(3(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(4(5(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(5(4(4(0(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(3(4(4(5(x1)))))) 2(4(5(4(5(x1))))) -> 2(5(4(0(5(4(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 76 less nodes. ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(0(1(0(x1)))) -> 2^1(2(x1)) 2^1(0(1(0(x1)))) -> 2^1(x1) 2^1(4(1(0(x1)))) -> 2^1(x1) 2^1(0(1(0(0(x1))))) -> 2^1(x1) 2^1(4(1(0(0(x1))))) -> 2^1(0(x1)) 2^1(0(2(1(0(x1))))) -> 2^1(x1) 2^1(4(1(0(0(x1))))) -> 2^1(x1) 2^1(4(2(1(0(x1))))) -> 2^1(x1) 2^1(4(3(1(0(x1))))) -> 2^1(x1) 2^1(0(4(1(0(x1))))) -> 2^1(x1) 2^1(0(5(1(0(x1))))) -> 2^1(0(x1)) 2^1(0(2(4(5(x1))))) -> 2^1(2(4(5(x1)))) 2^1(4(5(1(0(x1))))) -> 2^1(x1) 2^1(4(5(1(0(x1))))) -> 2^1(0(4(x1))) 2^1(4(5(1(0(x1))))) -> 2^1(4(x1)) 2^1(4(1(0(5(x1))))) -> 2^1(x1) The TRS R consists of the following rules: 2(0(1(0(x1)))) -> 0(1(3(0(2(x1))))) 2(0(1(0(x1)))) -> 2(1(3(0(0(2(x1)))))) 2(0(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(0(1(0(x1)))) -> 0(1(3(0(2(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 0(3(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(3(0(1(4(x1)))))) 2(0(1(0(x1)))) -> 2(0(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(x1))))) 2(4(1(0(x1)))) -> 1(3(0(4(2(x1))))) 2(4(1(0(x1)))) -> 4(0(1(2(3(x1))))) 2(4(1(0(x1)))) -> 0(4(1(2(3(x1))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(x1))))) 2(4(1(0(x1)))) -> 2(3(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(4(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(5(0(1(4(x1))))) 2(4(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(4(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(0(4(1(3(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 3(0(1(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(1(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 2(3(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(4(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(1(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(3(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(3(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(4(x1)))))) 2(0(1(0(0(x1))))) -> 4(0(2(0(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 2(0(0(4(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 0(0(3(0(1(2(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(0(1(3(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 1(3(0(4(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(0(1(3(0(x1)))))) 2(4(1(0(0(x1))))) -> 2(0(4(3(0(1(x1)))))) 2(4(1(0(0(x1))))) -> 4(1(3(0(0(2(x1)))))) 2(4(1(0(0(x1))))) -> 0(0(4(0(1(2(x1)))))) 2(0(2(1(0(x1))))) -> 2(4(0(1(0(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(0(1(3(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(2(0(1(4(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(4(1(3(2(x1)))))) 2(4(3(1(0(x1))))) -> 1(4(0(3(4(2(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(0(1(2(3(x1)))))) 2(4(3(1(0(x1))))) -> 2(1(3(3(0(4(x1)))))) 2(4(3(1(0(x1))))) -> 2(3(0(4(1(4(x1)))))) 2(0(4(1(0(x1))))) -> 4(0(1(3(0(2(x1)))))) 2(0(5(1(0(x1))))) -> 5(0(1(3(2(0(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(1(5(0(3(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(0(3(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(1(4(4(0(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(4(0(1(x1)))))) 2(4(5(1(0(x1))))) -> 1(5(4(4(0(2(x1)))))) 2(4(5(1(0(x1))))) -> 5(1(3(2(0(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(5(2(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(3(1(2(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(0(1(3(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(0(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(1(2(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 1(2(5(4(0(5(x1)))))) 2(4(5(1(0(x1))))) -> 3(4(0(2(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 2(4(0(1(3(5(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(1(4(4(0(x1)))))) 2(4(1(2(0(x1))))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(3(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 1(5(3(4(0(2(x1)))))) 2(4(1(0(5(x1))))) -> 5(1(3(0(4(2(x1)))))) 2(4(1(0(5(x1))))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 4(0(1(1(2(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(3(0(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(4(0(5(x1)))))) 2(4(2(0(5(x1))))) -> 2(2(3(0(4(5(x1)))))) 2(0(5(1(5(x1))))) -> 0(5(4(1(2(5(x1)))))) 2(4(5(1(5(x1))))) -> 5(4(0(1(2(5(x1)))))) 2(4(1(4(5(x1))))) -> 5(4(4(1(2(3(x1)))))) 2(4(1(4(5(x1))))) -> 2(1(5(3(4(4(x1)))))) 2(0(2(4(5(x1))))) -> 4(2(2(5(0(3(x1)))))) 2(0(2(4(5(x1))))) -> 2(2(3(5(0(4(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(2(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(3(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(4(5(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(5(4(4(0(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(3(4(4(5(x1)))))) 2(4(5(4(5(x1))))) -> 2(5(4(0(5(4(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. 2^1(4(1(0(5(x1))))) -> 2^1(x1) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(2^1(x_1)) = [[1A]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(0(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [-I, -I, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(1(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 1A], [0A, 0A, -I], [-I, -I, 0A]] * x_1 >>> <<< POL(2(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [-I, -I, 0A]] * x_1 >>> <<< POL(4(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, 0A], [0A, 0A, 0A], [0A, -I, 0A]] * x_1 >>> <<< POL(3(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, 0A], [0A, -I, 0A], [-I, -I, -I]] * x_1 >>> <<< POL(5(x_1)) = [[1A], [0A], [-I]] + [[-I, 0A, 0A], [0A, 0A, 0A], [-I, -I, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 2(0(1(0(x1)))) -> 0(1(3(0(2(x1))))) 2(0(1(0(x1)))) -> 2(1(3(0(0(2(x1)))))) 2(0(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(0(1(0(x1)))) -> 0(1(3(0(2(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 0(3(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(3(0(1(4(x1)))))) 2(0(1(0(x1)))) -> 2(0(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(x1))))) 2(4(1(0(x1)))) -> 1(3(0(4(2(x1))))) 2(4(1(0(x1)))) -> 4(0(1(2(3(x1))))) 2(4(1(0(x1)))) -> 0(4(1(2(3(x1))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(x1))))) 2(4(1(0(x1)))) -> 2(3(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(4(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(5(0(1(4(x1))))) 2(4(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(4(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(0(4(1(3(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 3(0(1(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(1(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 2(3(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(4(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(1(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(3(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(3(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(4(x1)))))) 2(0(1(0(0(x1))))) -> 4(0(2(0(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 2(0(0(4(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 0(0(3(0(1(2(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(0(1(3(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 1(3(0(4(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(0(1(3(0(x1)))))) 2(4(1(0(0(x1))))) -> 2(0(4(3(0(1(x1)))))) 2(4(1(0(0(x1))))) -> 4(1(3(0(0(2(x1)))))) 2(4(1(0(0(x1))))) -> 0(0(4(0(1(2(x1)))))) 2(0(2(1(0(x1))))) -> 2(4(0(1(0(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(0(1(3(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(2(0(1(4(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(4(1(3(2(x1)))))) 2(4(3(1(0(x1))))) -> 1(4(0(3(4(2(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(0(1(2(3(x1)))))) 2(4(3(1(0(x1))))) -> 2(1(3(3(0(4(x1)))))) 2(4(3(1(0(x1))))) -> 2(3(0(4(1(4(x1)))))) 2(0(4(1(0(x1))))) -> 4(0(1(3(0(2(x1)))))) 2(0(5(1(0(x1))))) -> 5(0(1(3(2(0(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(1(5(0(3(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(0(3(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(1(4(4(0(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(4(0(1(x1)))))) 2(4(5(1(0(x1))))) -> 1(5(4(4(0(2(x1)))))) 2(4(5(1(0(x1))))) -> 5(1(3(2(0(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(5(2(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(3(1(2(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(0(1(3(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(0(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(1(2(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 1(2(5(4(0(5(x1)))))) 2(4(5(1(0(x1))))) -> 3(4(0(2(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 2(4(0(1(3(5(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(1(4(4(0(x1)))))) 2(4(1(2(0(x1))))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(3(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 1(5(3(4(0(2(x1)))))) 2(4(1(0(5(x1))))) -> 5(1(3(0(4(2(x1)))))) 2(4(1(0(5(x1))))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 4(0(1(1(2(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(3(0(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(4(0(5(x1)))))) 2(4(2(0(5(x1))))) -> 2(2(3(0(4(5(x1)))))) 2(0(5(1(5(x1))))) -> 0(5(4(1(2(5(x1)))))) 2(4(5(1(5(x1))))) -> 5(4(0(1(2(5(x1)))))) 2(4(1(4(5(x1))))) -> 5(4(4(1(2(3(x1)))))) 2(4(1(4(5(x1))))) -> 2(1(5(3(4(4(x1)))))) 2(0(2(4(5(x1))))) -> 4(2(2(5(0(3(x1)))))) 2(0(2(4(5(x1))))) -> 2(2(3(5(0(4(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(2(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(3(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(4(5(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(5(4(4(0(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(3(4(4(5(x1)))))) 2(4(5(4(5(x1))))) -> 2(5(4(0(5(4(x1)))))) ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(0(1(0(x1)))) -> 2^1(2(x1)) 2^1(0(1(0(x1)))) -> 2^1(x1) 2^1(4(1(0(x1)))) -> 2^1(x1) 2^1(0(1(0(0(x1))))) -> 2^1(x1) 2^1(4(1(0(0(x1))))) -> 2^1(0(x1)) 2^1(0(2(1(0(x1))))) -> 2^1(x1) 2^1(4(1(0(0(x1))))) -> 2^1(x1) 2^1(4(2(1(0(x1))))) -> 2^1(x1) 2^1(4(3(1(0(x1))))) -> 2^1(x1) 2^1(0(4(1(0(x1))))) -> 2^1(x1) 2^1(0(5(1(0(x1))))) -> 2^1(0(x1)) 2^1(0(2(4(5(x1))))) -> 2^1(2(4(5(x1)))) 2^1(4(5(1(0(x1))))) -> 2^1(x1) 2^1(4(5(1(0(x1))))) -> 2^1(0(4(x1))) 2^1(4(5(1(0(x1))))) -> 2^1(4(x1)) The TRS R consists of the following rules: 2(0(1(0(x1)))) -> 0(1(3(0(2(x1))))) 2(0(1(0(x1)))) -> 2(1(3(0(0(2(x1)))))) 2(0(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(0(1(0(x1)))) -> 0(1(3(0(2(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 0(3(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(3(0(1(4(x1)))))) 2(0(1(0(x1)))) -> 2(0(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(x1))))) 2(4(1(0(x1)))) -> 1(3(0(4(2(x1))))) 2(4(1(0(x1)))) -> 4(0(1(2(3(x1))))) 2(4(1(0(x1)))) -> 0(4(1(2(3(x1))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(x1))))) 2(4(1(0(x1)))) -> 2(3(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(4(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(5(0(1(4(x1))))) 2(4(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(4(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(0(4(1(3(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 3(0(1(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(1(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 2(3(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(4(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(1(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(3(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(3(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(4(x1)))))) 2(0(1(0(0(x1))))) -> 4(0(2(0(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 2(0(0(4(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 0(0(3(0(1(2(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(0(1(3(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 1(3(0(4(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(0(1(3(0(x1)))))) 2(4(1(0(0(x1))))) -> 2(0(4(3(0(1(x1)))))) 2(4(1(0(0(x1))))) -> 4(1(3(0(0(2(x1)))))) 2(4(1(0(0(x1))))) -> 0(0(4(0(1(2(x1)))))) 2(0(2(1(0(x1))))) -> 2(4(0(1(0(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(0(1(3(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(2(0(1(4(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(4(1(3(2(x1)))))) 2(4(3(1(0(x1))))) -> 1(4(0(3(4(2(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(0(1(2(3(x1)))))) 2(4(3(1(0(x1))))) -> 2(1(3(3(0(4(x1)))))) 2(4(3(1(0(x1))))) -> 2(3(0(4(1(4(x1)))))) 2(0(4(1(0(x1))))) -> 4(0(1(3(0(2(x1)))))) 2(0(5(1(0(x1))))) -> 5(0(1(3(2(0(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(1(5(0(3(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(0(3(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(1(4(4(0(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(4(0(1(x1)))))) 2(4(5(1(0(x1))))) -> 1(5(4(4(0(2(x1)))))) 2(4(5(1(0(x1))))) -> 5(1(3(2(0(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(5(2(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(3(1(2(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(0(1(3(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(0(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(1(2(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 1(2(5(4(0(5(x1)))))) 2(4(5(1(0(x1))))) -> 3(4(0(2(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 2(4(0(1(3(5(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(1(4(4(0(x1)))))) 2(4(1(2(0(x1))))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(3(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 1(5(3(4(0(2(x1)))))) 2(4(1(0(5(x1))))) -> 5(1(3(0(4(2(x1)))))) 2(4(1(0(5(x1))))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 4(0(1(1(2(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(3(0(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(4(0(5(x1)))))) 2(4(2(0(5(x1))))) -> 2(2(3(0(4(5(x1)))))) 2(0(5(1(5(x1))))) -> 0(5(4(1(2(5(x1)))))) 2(4(5(1(5(x1))))) -> 5(4(0(1(2(5(x1)))))) 2(4(1(4(5(x1))))) -> 5(4(4(1(2(3(x1)))))) 2(4(1(4(5(x1))))) -> 2(1(5(3(4(4(x1)))))) 2(0(2(4(5(x1))))) -> 4(2(2(5(0(3(x1)))))) 2(0(2(4(5(x1))))) -> 2(2(3(5(0(4(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(2(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(3(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(4(5(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(5(4(4(0(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(3(4(4(5(x1)))))) 2(4(5(4(5(x1))))) -> 2(5(4(0(5(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule 2^1(0(2(4(5(x1))))) -> 2^1(2(4(5(x1)))) at position [0] we obtained the following new rules [LPAR04]: (2^1(0(2(4(5(1(0(x0))))))) -> 2^1(5(2(1(4(4(0(x0))))))),2^1(0(2(4(5(1(0(x0))))))) -> 2^1(5(2(1(4(4(0(x0)))))))) (2^1(0(2(4(5(1(0(x0))))))) -> 2^1(2(1(5(4(0(1(x0))))))),2^1(0(2(4(5(1(0(x0))))))) -> 2^1(2(1(5(4(0(1(x0)))))))) (2^1(0(2(4(5(1(0(x0))))))) -> 2^1(1(5(4(4(0(2(x0))))))),2^1(0(2(4(5(1(0(x0))))))) -> 2^1(1(5(4(4(0(2(x0)))))))) (2^1(0(2(4(5(1(0(x0))))))) -> 2^1(5(1(3(2(0(4(x0))))))),2^1(0(2(4(5(1(0(x0))))))) -> 2^1(5(1(3(2(0(4(x0)))))))) (2^1(0(2(4(5(1(0(x0))))))) -> 2^1(2(5(2(0(1(4(x0))))))),2^1(0(2(4(5(1(0(x0))))))) -> 2^1(2(5(2(0(1(4(x0)))))))) (2^1(0(2(4(5(1(0(x0))))))) -> 2^1(5(2(5(0(1(4(x0))))))),2^1(0(2(4(5(1(0(x0))))))) -> 2^1(5(2(5(0(1(4(x0)))))))) (2^1(0(2(4(5(1(0(x0))))))) -> 2^1(5(0(3(1(2(4(x0))))))),2^1(0(2(4(5(1(0(x0))))))) -> 2^1(5(0(3(1(2(4(x0)))))))) (2^1(0(2(4(5(1(0(x0))))))) -> 2^1(5(2(0(1(3(4(x0))))))),2^1(0(2(4(5(1(0(x0))))))) -> 2^1(5(2(0(1(3(4(x0)))))))) (2^1(0(2(4(5(1(0(x0))))))) -> 2^1(2(1(5(0(4(4(x0))))))),2^1(0(2(4(5(1(0(x0))))))) -> 2^1(2(1(5(0(4(4(x0)))))))) (2^1(0(2(4(5(1(0(x0))))))) -> 2^1(5(0(1(2(4(4(x0))))))),2^1(0(2(4(5(1(0(x0))))))) -> 2^1(5(0(1(2(4(4(x0)))))))) (2^1(0(2(4(5(1(0(x0))))))) -> 2^1(1(2(5(4(0(5(x0))))))),2^1(0(2(4(5(1(0(x0))))))) -> 2^1(1(2(5(4(0(5(x0)))))))) (2^1(0(2(4(5(1(0(x0))))))) -> 2^1(3(4(0(2(1(5(x0))))))),2^1(0(2(4(5(1(0(x0))))))) -> 2^1(3(4(0(2(1(5(x0)))))))) (2^1(0(2(4(5(1(0(x0))))))) -> 2^1(2(4(0(1(3(5(x0))))))),2^1(0(2(4(5(1(0(x0))))))) -> 2^1(2(4(0(1(3(5(x0)))))))) (2^1(0(2(4(5(1(5(x0))))))) -> 2^1(5(4(0(1(2(5(x0))))))),2^1(0(2(4(5(1(5(x0))))))) -> 2^1(5(4(0(1(2(5(x0)))))))) (2^1(0(2(4(5(4(5(x0))))))) -> 2^1(2(5(4(0(5(4(x0))))))),2^1(0(2(4(5(4(5(x0))))))) -> 2^1(2(5(4(0(5(4(x0)))))))) ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(0(1(0(x1)))) -> 2^1(2(x1)) 2^1(0(1(0(x1)))) -> 2^1(x1) 2^1(4(1(0(x1)))) -> 2^1(x1) 2^1(0(1(0(0(x1))))) -> 2^1(x1) 2^1(4(1(0(0(x1))))) -> 2^1(0(x1)) 2^1(0(2(1(0(x1))))) -> 2^1(x1) 2^1(4(1(0(0(x1))))) -> 2^1(x1) 2^1(4(2(1(0(x1))))) -> 2^1(x1) 2^1(4(3(1(0(x1))))) -> 2^1(x1) 2^1(0(4(1(0(x1))))) -> 2^1(x1) 2^1(0(5(1(0(x1))))) -> 2^1(0(x1)) 2^1(4(5(1(0(x1))))) -> 2^1(x1) 2^1(4(5(1(0(x1))))) -> 2^1(0(4(x1))) 2^1(4(5(1(0(x1))))) -> 2^1(4(x1)) 2^1(0(2(4(5(1(0(x0))))))) -> 2^1(5(2(1(4(4(0(x0))))))) 2^1(0(2(4(5(1(0(x0))))))) -> 2^1(2(1(5(4(0(1(x0))))))) 2^1(0(2(4(5(1(0(x0))))))) -> 2^1(1(5(4(4(0(2(x0))))))) 2^1(0(2(4(5(1(0(x0))))))) -> 2^1(5(1(3(2(0(4(x0))))))) 2^1(0(2(4(5(1(0(x0))))))) -> 2^1(2(5(2(0(1(4(x0))))))) 2^1(0(2(4(5(1(0(x0))))))) -> 2^1(5(2(5(0(1(4(x0))))))) 2^1(0(2(4(5(1(0(x0))))))) -> 2^1(5(0(3(1(2(4(x0))))))) 2^1(0(2(4(5(1(0(x0))))))) -> 2^1(5(2(0(1(3(4(x0))))))) 2^1(0(2(4(5(1(0(x0))))))) -> 2^1(2(1(5(0(4(4(x0))))))) 2^1(0(2(4(5(1(0(x0))))))) -> 2^1(5(0(1(2(4(4(x0))))))) 2^1(0(2(4(5(1(0(x0))))))) -> 2^1(1(2(5(4(0(5(x0))))))) 2^1(0(2(4(5(1(0(x0))))))) -> 2^1(3(4(0(2(1(5(x0))))))) 2^1(0(2(4(5(1(0(x0))))))) -> 2^1(2(4(0(1(3(5(x0))))))) 2^1(0(2(4(5(1(5(x0))))))) -> 2^1(5(4(0(1(2(5(x0))))))) 2^1(0(2(4(5(4(5(x0))))))) -> 2^1(2(5(4(0(5(4(x0))))))) The TRS R consists of the following rules: 2(0(1(0(x1)))) -> 0(1(3(0(2(x1))))) 2(0(1(0(x1)))) -> 2(1(3(0(0(2(x1)))))) 2(0(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(0(1(0(x1)))) -> 0(1(3(0(2(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 0(3(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(3(0(1(4(x1)))))) 2(0(1(0(x1)))) -> 2(0(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(x1))))) 2(4(1(0(x1)))) -> 1(3(0(4(2(x1))))) 2(4(1(0(x1)))) -> 4(0(1(2(3(x1))))) 2(4(1(0(x1)))) -> 0(4(1(2(3(x1))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(x1))))) 2(4(1(0(x1)))) -> 2(3(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(4(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(5(0(1(4(x1))))) 2(4(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(4(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(0(4(1(3(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 3(0(1(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(1(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 2(3(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(4(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(1(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(3(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(3(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(4(x1)))))) 2(0(1(0(0(x1))))) -> 4(0(2(0(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 2(0(0(4(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 0(0(3(0(1(2(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(0(1(3(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 1(3(0(4(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(0(1(3(0(x1)))))) 2(4(1(0(0(x1))))) -> 2(0(4(3(0(1(x1)))))) 2(4(1(0(0(x1))))) -> 4(1(3(0(0(2(x1)))))) 2(4(1(0(0(x1))))) -> 0(0(4(0(1(2(x1)))))) 2(0(2(1(0(x1))))) -> 2(4(0(1(0(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(0(1(3(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(2(0(1(4(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(4(1(3(2(x1)))))) 2(4(3(1(0(x1))))) -> 1(4(0(3(4(2(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(0(1(2(3(x1)))))) 2(4(3(1(0(x1))))) -> 2(1(3(3(0(4(x1)))))) 2(4(3(1(0(x1))))) -> 2(3(0(4(1(4(x1)))))) 2(0(4(1(0(x1))))) -> 4(0(1(3(0(2(x1)))))) 2(0(5(1(0(x1))))) -> 5(0(1(3(2(0(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(1(5(0(3(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(0(3(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(1(4(4(0(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(4(0(1(x1)))))) 2(4(5(1(0(x1))))) -> 1(5(4(4(0(2(x1)))))) 2(4(5(1(0(x1))))) -> 5(1(3(2(0(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(5(2(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(3(1(2(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(0(1(3(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(0(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(1(2(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 1(2(5(4(0(5(x1)))))) 2(4(5(1(0(x1))))) -> 3(4(0(2(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 2(4(0(1(3(5(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(1(4(4(0(x1)))))) 2(4(1(2(0(x1))))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(3(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 1(5(3(4(0(2(x1)))))) 2(4(1(0(5(x1))))) -> 5(1(3(0(4(2(x1)))))) 2(4(1(0(5(x1))))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 4(0(1(1(2(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(3(0(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(4(0(5(x1)))))) 2(4(2(0(5(x1))))) -> 2(2(3(0(4(5(x1)))))) 2(0(5(1(5(x1))))) -> 0(5(4(1(2(5(x1)))))) 2(4(5(1(5(x1))))) -> 5(4(0(1(2(5(x1)))))) 2(4(1(4(5(x1))))) -> 5(4(4(1(2(3(x1)))))) 2(4(1(4(5(x1))))) -> 2(1(5(3(4(4(x1)))))) 2(0(2(4(5(x1))))) -> 4(2(2(5(0(3(x1)))))) 2(0(2(4(5(x1))))) -> 2(2(3(5(0(4(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(2(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(3(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(4(5(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(5(4(4(0(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(3(4(4(5(x1)))))) 2(4(5(4(5(x1))))) -> 2(5(4(0(5(4(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 15 less nodes. ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(0(1(0(x1)))) -> 2^1(x1) 2^1(0(1(0(x1)))) -> 2^1(2(x1)) 2^1(4(1(0(x1)))) -> 2^1(x1) 2^1(0(1(0(0(x1))))) -> 2^1(x1) 2^1(4(1(0(0(x1))))) -> 2^1(0(x1)) 2^1(0(2(1(0(x1))))) -> 2^1(x1) 2^1(4(1(0(0(x1))))) -> 2^1(x1) 2^1(4(2(1(0(x1))))) -> 2^1(x1) 2^1(4(3(1(0(x1))))) -> 2^1(x1) 2^1(0(4(1(0(x1))))) -> 2^1(x1) 2^1(0(5(1(0(x1))))) -> 2^1(0(x1)) 2^1(4(5(1(0(x1))))) -> 2^1(x1) 2^1(4(5(1(0(x1))))) -> 2^1(0(4(x1))) 2^1(4(5(1(0(x1))))) -> 2^1(4(x1)) The TRS R consists of the following rules: 2(0(1(0(x1)))) -> 0(1(3(0(2(x1))))) 2(0(1(0(x1)))) -> 2(1(3(0(0(2(x1)))))) 2(0(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(0(1(0(x1)))) -> 0(1(3(0(2(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 0(3(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(3(0(1(4(x1)))))) 2(0(1(0(x1)))) -> 2(0(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(x1))))) 2(4(1(0(x1)))) -> 1(3(0(4(2(x1))))) 2(4(1(0(x1)))) -> 4(0(1(2(3(x1))))) 2(4(1(0(x1)))) -> 0(4(1(2(3(x1))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(x1))))) 2(4(1(0(x1)))) -> 2(3(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(4(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(5(0(1(4(x1))))) 2(4(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(4(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(0(4(1(3(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 3(0(1(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(1(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 2(3(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(4(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(1(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(3(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(3(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(4(x1)))))) 2(0(1(0(0(x1))))) -> 4(0(2(0(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 2(0(0(4(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 0(0(3(0(1(2(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(0(1(3(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 1(3(0(4(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(0(1(3(0(x1)))))) 2(4(1(0(0(x1))))) -> 2(0(4(3(0(1(x1)))))) 2(4(1(0(0(x1))))) -> 4(1(3(0(0(2(x1)))))) 2(4(1(0(0(x1))))) -> 0(0(4(0(1(2(x1)))))) 2(0(2(1(0(x1))))) -> 2(4(0(1(0(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(0(1(3(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(2(0(1(4(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(4(1(3(2(x1)))))) 2(4(3(1(0(x1))))) -> 1(4(0(3(4(2(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(0(1(2(3(x1)))))) 2(4(3(1(0(x1))))) -> 2(1(3(3(0(4(x1)))))) 2(4(3(1(0(x1))))) -> 2(3(0(4(1(4(x1)))))) 2(0(4(1(0(x1))))) -> 4(0(1(3(0(2(x1)))))) 2(0(5(1(0(x1))))) -> 5(0(1(3(2(0(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(1(5(0(3(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(0(3(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(1(4(4(0(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(4(0(1(x1)))))) 2(4(5(1(0(x1))))) -> 1(5(4(4(0(2(x1)))))) 2(4(5(1(0(x1))))) -> 5(1(3(2(0(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(5(2(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(3(1(2(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(0(1(3(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(0(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(1(2(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 1(2(5(4(0(5(x1)))))) 2(4(5(1(0(x1))))) -> 3(4(0(2(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 2(4(0(1(3(5(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(1(4(4(0(x1)))))) 2(4(1(2(0(x1))))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(3(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 1(5(3(4(0(2(x1)))))) 2(4(1(0(5(x1))))) -> 5(1(3(0(4(2(x1)))))) 2(4(1(0(5(x1))))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 4(0(1(1(2(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(3(0(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(4(0(5(x1)))))) 2(4(2(0(5(x1))))) -> 2(2(3(0(4(5(x1)))))) 2(0(5(1(5(x1))))) -> 0(5(4(1(2(5(x1)))))) 2(4(5(1(5(x1))))) -> 5(4(0(1(2(5(x1)))))) 2(4(1(4(5(x1))))) -> 5(4(4(1(2(3(x1)))))) 2(4(1(4(5(x1))))) -> 2(1(5(3(4(4(x1)))))) 2(0(2(4(5(x1))))) -> 4(2(2(5(0(3(x1)))))) 2(0(2(4(5(x1))))) -> 2(2(3(5(0(4(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(2(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(3(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(4(5(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(5(4(4(0(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(3(4(4(5(x1)))))) 2(4(5(4(5(x1))))) -> 2(5(4(0(5(4(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(5(1(0(x1))))) -> 2^1(0(x1)) 2^1(4(5(1(0(x1))))) -> 2^1(x1) 2^1(4(5(1(0(x1))))) -> 2^1(0(4(x1))) 2^1(4(5(1(0(x1))))) -> 2^1(4(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(2^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(0(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(1(x_1)) = [[0A], [-I], [-I]] + [[-I, -I, -I], [0A, 0A, 0A], [0A, -I, 0A]] * x_1 >>> <<< POL(2(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, -I], [0A, 0A, -I], [-I, 0A, -I]] * x_1 >>> <<< POL(4(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, -I], [0A, 0A, 0A], [0A, -I, -I]] * x_1 >>> <<< POL(3(x_1)) = [[0A], [-I], [0A]] + [[-I, -I, -I], [0A, 0A, -I], [-I, -I, -I]] * x_1 >>> <<< POL(5(x_1)) = [[0A], [0A], [1A]] + [[0A, 0A, 0A], [-I, -I, -I], [-I, 0A, 1A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 2(0(1(0(x1)))) -> 0(1(3(0(2(x1))))) 2(0(1(0(x1)))) -> 2(1(3(0(0(2(x1)))))) 2(0(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(0(1(0(x1)))) -> 0(1(3(0(2(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 0(3(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(3(0(1(4(x1)))))) 2(0(1(0(x1)))) -> 2(0(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(x1))))) 2(4(1(0(x1)))) -> 1(3(0(4(2(x1))))) 2(4(1(0(x1)))) -> 4(0(1(2(3(x1))))) 2(4(1(0(x1)))) -> 0(4(1(2(3(x1))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(x1))))) 2(4(1(0(x1)))) -> 2(3(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(4(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(5(0(1(4(x1))))) 2(4(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(4(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(0(4(1(3(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 3(0(1(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(1(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 2(3(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(4(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(1(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(3(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(3(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(4(x1)))))) 2(0(1(0(0(x1))))) -> 4(0(2(0(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 2(0(0(4(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 0(0(3(0(1(2(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(0(1(3(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 1(3(0(4(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(0(1(3(0(x1)))))) 2(4(1(0(0(x1))))) -> 2(0(4(3(0(1(x1)))))) 2(4(1(0(0(x1))))) -> 4(1(3(0(0(2(x1)))))) 2(4(1(0(0(x1))))) -> 0(0(4(0(1(2(x1)))))) 2(0(2(1(0(x1))))) -> 2(4(0(1(0(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(0(1(3(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(2(0(1(4(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(4(1(3(2(x1)))))) 2(4(3(1(0(x1))))) -> 1(4(0(3(4(2(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(0(1(2(3(x1)))))) 2(4(3(1(0(x1))))) -> 2(1(3(3(0(4(x1)))))) 2(4(3(1(0(x1))))) -> 2(3(0(4(1(4(x1)))))) 2(0(4(1(0(x1))))) -> 4(0(1(3(0(2(x1)))))) 2(0(5(1(0(x1))))) -> 5(0(1(3(2(0(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(1(5(0(3(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(0(3(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(1(4(4(0(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(4(0(1(x1)))))) 2(4(5(1(0(x1))))) -> 1(5(4(4(0(2(x1)))))) 2(4(5(1(0(x1))))) -> 5(1(3(2(0(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(5(2(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(3(1(2(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(0(1(3(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(0(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(1(2(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 1(2(5(4(0(5(x1)))))) 2(4(5(1(0(x1))))) -> 3(4(0(2(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 2(4(0(1(3(5(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(1(4(4(0(x1)))))) 2(4(1(2(0(x1))))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(3(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 1(5(3(4(0(2(x1)))))) 2(4(1(0(5(x1))))) -> 5(1(3(0(4(2(x1)))))) 2(4(1(0(5(x1))))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 4(0(1(1(2(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(3(0(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(4(0(5(x1)))))) 2(4(2(0(5(x1))))) -> 2(2(3(0(4(5(x1)))))) 2(0(5(1(5(x1))))) -> 0(5(4(1(2(5(x1)))))) 2(4(5(1(5(x1))))) -> 5(4(0(1(2(5(x1)))))) 2(4(1(4(5(x1))))) -> 5(4(4(1(2(3(x1)))))) 2(4(1(4(5(x1))))) -> 2(1(5(3(4(4(x1)))))) 2(0(2(4(5(x1))))) -> 4(2(2(5(0(3(x1)))))) 2(0(2(4(5(x1))))) -> 2(2(3(5(0(4(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(2(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(3(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(4(5(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(5(4(4(0(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(3(4(4(5(x1)))))) 2(4(5(4(5(x1))))) -> 2(5(4(0(5(4(x1)))))) ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(0(1(0(x1)))) -> 2^1(x1) 2^1(0(1(0(x1)))) -> 2^1(2(x1)) 2^1(4(1(0(x1)))) -> 2^1(x1) 2^1(0(1(0(0(x1))))) -> 2^1(x1) 2^1(4(1(0(0(x1))))) -> 2^1(0(x1)) 2^1(0(2(1(0(x1))))) -> 2^1(x1) 2^1(4(1(0(0(x1))))) -> 2^1(x1) 2^1(4(2(1(0(x1))))) -> 2^1(x1) 2^1(4(3(1(0(x1))))) -> 2^1(x1) 2^1(0(4(1(0(x1))))) -> 2^1(x1) The TRS R consists of the following rules: 2(0(1(0(x1)))) -> 0(1(3(0(2(x1))))) 2(0(1(0(x1)))) -> 2(1(3(0(0(2(x1)))))) 2(0(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(0(1(0(x1)))) -> 0(1(3(0(2(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 0(3(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(3(0(1(4(x1)))))) 2(0(1(0(x1)))) -> 2(0(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(x1))))) 2(4(1(0(x1)))) -> 1(3(0(4(2(x1))))) 2(4(1(0(x1)))) -> 4(0(1(2(3(x1))))) 2(4(1(0(x1)))) -> 0(4(1(2(3(x1))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(x1))))) 2(4(1(0(x1)))) -> 2(3(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(4(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(5(0(1(4(x1))))) 2(4(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(4(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(0(4(1(3(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 3(0(1(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(1(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 2(3(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(4(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(1(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(3(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(3(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(4(x1)))))) 2(0(1(0(0(x1))))) -> 4(0(2(0(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 2(0(0(4(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 0(0(3(0(1(2(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(0(1(3(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 1(3(0(4(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(0(1(3(0(x1)))))) 2(4(1(0(0(x1))))) -> 2(0(4(3(0(1(x1)))))) 2(4(1(0(0(x1))))) -> 4(1(3(0(0(2(x1)))))) 2(4(1(0(0(x1))))) -> 0(0(4(0(1(2(x1)))))) 2(0(2(1(0(x1))))) -> 2(4(0(1(0(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(0(1(3(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(2(0(1(4(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(4(1(3(2(x1)))))) 2(4(3(1(0(x1))))) -> 1(4(0(3(4(2(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(0(1(2(3(x1)))))) 2(4(3(1(0(x1))))) -> 2(1(3(3(0(4(x1)))))) 2(4(3(1(0(x1))))) -> 2(3(0(4(1(4(x1)))))) 2(0(4(1(0(x1))))) -> 4(0(1(3(0(2(x1)))))) 2(0(5(1(0(x1))))) -> 5(0(1(3(2(0(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(1(5(0(3(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(0(3(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(1(4(4(0(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(4(0(1(x1)))))) 2(4(5(1(0(x1))))) -> 1(5(4(4(0(2(x1)))))) 2(4(5(1(0(x1))))) -> 5(1(3(2(0(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(5(2(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(3(1(2(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(0(1(3(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(0(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(1(2(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 1(2(5(4(0(5(x1)))))) 2(4(5(1(0(x1))))) -> 3(4(0(2(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 2(4(0(1(3(5(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(1(4(4(0(x1)))))) 2(4(1(2(0(x1))))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(3(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 1(5(3(4(0(2(x1)))))) 2(4(1(0(5(x1))))) -> 5(1(3(0(4(2(x1)))))) 2(4(1(0(5(x1))))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 4(0(1(1(2(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(3(0(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(4(0(5(x1)))))) 2(4(2(0(5(x1))))) -> 2(2(3(0(4(5(x1)))))) 2(0(5(1(5(x1))))) -> 0(5(4(1(2(5(x1)))))) 2(4(5(1(5(x1))))) -> 5(4(0(1(2(5(x1)))))) 2(4(1(4(5(x1))))) -> 5(4(4(1(2(3(x1)))))) 2(4(1(4(5(x1))))) -> 2(1(5(3(4(4(x1)))))) 2(0(2(4(5(x1))))) -> 4(2(2(5(0(3(x1)))))) 2(0(2(4(5(x1))))) -> 2(2(3(5(0(4(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(2(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(3(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(4(5(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(5(4(4(0(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(3(4(4(5(x1)))))) 2(4(5(4(5(x1))))) -> 2(5(4(0(5(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (15) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 2^1(4(3(1(0(x1))))) -> 2^1(x1) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(2^1(x_1)) = [[0A]] + [[0A, -I, 0A]] * x_1 >>> <<< POL(0(x_1)) = [[0A], [-I], [-I]] + [[0A, 0A, -I], [-I, 0A, 0A], [-I, -I, -I]] * x_1 >>> <<< POL(1(x_1)) = [[1A], [0A], [0A]] + [[0A, 0A, -I], [-I, 0A, -I], [0A, 0A, -I]] * x_1 >>> <<< POL(2(x_1)) = [[1A], [1A], [1A]] + [[-I, -I, -I], [0A, 0A, -I], [0A, -I, -I]] * x_1 >>> <<< POL(4(x_1)) = [[0A], [-I], [0A]] + [[-I, -I, 0A], [-I, -I, 0A], [-I, -I, -I]] * x_1 >>> <<< POL(3(x_1)) = [[0A], [-I], [-I]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, 1A]] * x_1 >>> <<< POL(5(x_1)) = [[0A], [-I], [0A]] + [[-I, -I, -I], [-I, -I, -I], [0A, -I, 0A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 2(0(1(0(x1)))) -> 0(1(3(0(2(x1))))) 2(0(1(0(x1)))) -> 2(1(3(0(0(2(x1)))))) 2(0(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(0(1(0(x1)))) -> 0(1(3(0(2(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 0(3(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(3(0(1(4(x1)))))) 2(0(1(0(x1)))) -> 2(0(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(x1))))) 2(4(1(0(x1)))) -> 1(3(0(4(2(x1))))) 2(4(1(0(x1)))) -> 4(0(1(2(3(x1))))) 2(4(1(0(x1)))) -> 0(4(1(2(3(x1))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(x1))))) 2(4(1(0(x1)))) -> 2(3(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(4(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(5(0(1(4(x1))))) 2(4(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(4(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(0(4(1(3(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 3(0(1(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(1(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 2(3(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(4(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(1(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(3(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(3(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(4(x1)))))) 2(0(1(0(0(x1))))) -> 4(0(2(0(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 2(0(0(4(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 0(0(3(0(1(2(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(0(1(3(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 1(3(0(4(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(0(1(3(0(x1)))))) 2(4(1(0(0(x1))))) -> 2(0(4(3(0(1(x1)))))) 2(4(1(0(0(x1))))) -> 4(1(3(0(0(2(x1)))))) 2(4(1(0(0(x1))))) -> 0(0(4(0(1(2(x1)))))) 2(0(2(1(0(x1))))) -> 2(4(0(1(0(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(0(1(3(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(2(0(1(4(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(4(1(3(2(x1)))))) 2(4(3(1(0(x1))))) -> 1(4(0(3(4(2(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(0(1(2(3(x1)))))) 2(4(3(1(0(x1))))) -> 2(1(3(3(0(4(x1)))))) 2(4(3(1(0(x1))))) -> 2(3(0(4(1(4(x1)))))) 2(0(4(1(0(x1))))) -> 4(0(1(3(0(2(x1)))))) 2(0(5(1(0(x1))))) -> 5(0(1(3(2(0(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(1(5(0(3(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(0(3(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(1(4(4(0(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(4(0(1(x1)))))) 2(4(5(1(0(x1))))) -> 1(5(4(4(0(2(x1)))))) 2(4(5(1(0(x1))))) -> 5(1(3(2(0(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(5(2(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(3(1(2(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(0(1(3(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(0(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(1(2(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 1(2(5(4(0(5(x1)))))) 2(4(5(1(0(x1))))) -> 3(4(0(2(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 2(4(0(1(3(5(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(1(4(4(0(x1)))))) 2(4(1(2(0(x1))))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(3(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 1(5(3(4(0(2(x1)))))) 2(4(1(0(5(x1))))) -> 5(1(3(0(4(2(x1)))))) 2(4(1(0(5(x1))))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 4(0(1(1(2(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(3(0(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(4(0(5(x1)))))) 2(4(2(0(5(x1))))) -> 2(2(3(0(4(5(x1)))))) 2(0(5(1(5(x1))))) -> 0(5(4(1(2(5(x1)))))) 2(4(5(1(5(x1))))) -> 5(4(0(1(2(5(x1)))))) 2(4(1(4(5(x1))))) -> 5(4(4(1(2(3(x1)))))) 2(4(1(4(5(x1))))) -> 2(1(5(3(4(4(x1)))))) 2(0(2(4(5(x1))))) -> 4(2(2(5(0(3(x1)))))) 2(0(2(4(5(x1))))) -> 2(2(3(5(0(4(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(2(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(3(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(4(5(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(5(4(4(0(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(3(4(4(5(x1)))))) 2(4(5(4(5(x1))))) -> 2(5(4(0(5(4(x1)))))) ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(0(1(0(x1)))) -> 2^1(x1) 2^1(0(1(0(x1)))) -> 2^1(2(x1)) 2^1(4(1(0(x1)))) -> 2^1(x1) 2^1(0(1(0(0(x1))))) -> 2^1(x1) 2^1(4(1(0(0(x1))))) -> 2^1(0(x1)) 2^1(0(2(1(0(x1))))) -> 2^1(x1) 2^1(4(1(0(0(x1))))) -> 2^1(x1) 2^1(4(2(1(0(x1))))) -> 2^1(x1) 2^1(0(4(1(0(x1))))) -> 2^1(x1) The TRS R consists of the following rules: 2(0(1(0(x1)))) -> 0(1(3(0(2(x1))))) 2(0(1(0(x1)))) -> 2(1(3(0(0(2(x1)))))) 2(0(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(0(1(0(x1)))) -> 0(1(3(0(2(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 0(3(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(3(0(1(4(x1)))))) 2(0(1(0(x1)))) -> 2(0(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(x1))))) 2(4(1(0(x1)))) -> 1(3(0(4(2(x1))))) 2(4(1(0(x1)))) -> 4(0(1(2(3(x1))))) 2(4(1(0(x1)))) -> 0(4(1(2(3(x1))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(x1))))) 2(4(1(0(x1)))) -> 2(3(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(4(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(5(0(1(4(x1))))) 2(4(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(4(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(0(4(1(3(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 3(0(1(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(1(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 2(3(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(4(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(1(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(3(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(3(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(4(x1)))))) 2(0(1(0(0(x1))))) -> 4(0(2(0(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 2(0(0(4(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 0(0(3(0(1(2(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(0(1(3(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 1(3(0(4(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(0(1(3(0(x1)))))) 2(4(1(0(0(x1))))) -> 2(0(4(3(0(1(x1)))))) 2(4(1(0(0(x1))))) -> 4(1(3(0(0(2(x1)))))) 2(4(1(0(0(x1))))) -> 0(0(4(0(1(2(x1)))))) 2(0(2(1(0(x1))))) -> 2(4(0(1(0(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(0(1(3(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(2(0(1(4(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(4(1(3(2(x1)))))) 2(4(3(1(0(x1))))) -> 1(4(0(3(4(2(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(0(1(2(3(x1)))))) 2(4(3(1(0(x1))))) -> 2(1(3(3(0(4(x1)))))) 2(4(3(1(0(x1))))) -> 2(3(0(4(1(4(x1)))))) 2(0(4(1(0(x1))))) -> 4(0(1(3(0(2(x1)))))) 2(0(5(1(0(x1))))) -> 5(0(1(3(2(0(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(1(5(0(3(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(0(3(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(1(4(4(0(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(4(0(1(x1)))))) 2(4(5(1(0(x1))))) -> 1(5(4(4(0(2(x1)))))) 2(4(5(1(0(x1))))) -> 5(1(3(2(0(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(5(2(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(3(1(2(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(0(1(3(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(0(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(1(2(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 1(2(5(4(0(5(x1)))))) 2(4(5(1(0(x1))))) -> 3(4(0(2(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 2(4(0(1(3(5(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(1(4(4(0(x1)))))) 2(4(1(2(0(x1))))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(3(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 1(5(3(4(0(2(x1)))))) 2(4(1(0(5(x1))))) -> 5(1(3(0(4(2(x1)))))) 2(4(1(0(5(x1))))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 4(0(1(1(2(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(3(0(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(4(0(5(x1)))))) 2(4(2(0(5(x1))))) -> 2(2(3(0(4(5(x1)))))) 2(0(5(1(5(x1))))) -> 0(5(4(1(2(5(x1)))))) 2(4(5(1(5(x1))))) -> 5(4(0(1(2(5(x1)))))) 2(4(1(4(5(x1))))) -> 5(4(4(1(2(3(x1)))))) 2(4(1(4(5(x1))))) -> 2(1(5(3(4(4(x1)))))) 2(0(2(4(5(x1))))) -> 4(2(2(5(0(3(x1)))))) 2(0(2(4(5(x1))))) -> 2(2(3(5(0(4(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(2(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(3(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(4(5(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(5(4(4(0(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(3(4(4(5(x1)))))) 2(4(5(4(5(x1))))) -> 2(5(4(0(5(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (17) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 2^1(0(1(0(0(x1))))) -> 2^1(x1) 2^1(0(4(1(0(x1))))) -> 2^1(x1) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(2^1(x_1)) = [[0A]] + [[0A, 0A, -I]] * x_1 >>> <<< POL(0(x_1)) = [[-I], [0A], [1A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [-I, -I, 0A]] * x_1 >>> <<< POL(1(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, -I], [1A, 0A, -I], [0A, 0A, -I]] * x_1 >>> <<< POL(2(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(4(x_1)) = [[0A], [0A], [1A]] + [[-I, -I, -I], [0A, -I, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(3(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [0A, 0A, -I], [0A, 0A, 0A]] * x_1 >>> <<< POL(5(x_1)) = [[0A], [-I], [-I]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 2(0(1(0(x1)))) -> 0(1(3(0(2(x1))))) 2(0(1(0(x1)))) -> 2(1(3(0(0(2(x1)))))) 2(0(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(0(1(0(x1)))) -> 0(1(3(0(2(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 0(3(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(3(0(1(4(x1)))))) 2(0(1(0(x1)))) -> 2(0(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(x1))))) 2(4(1(0(x1)))) -> 1(3(0(4(2(x1))))) 2(4(1(0(x1)))) -> 4(0(1(2(3(x1))))) 2(4(1(0(x1)))) -> 0(4(1(2(3(x1))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(x1))))) 2(4(1(0(x1)))) -> 2(3(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(4(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(5(0(1(4(x1))))) 2(4(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(4(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(0(4(1(3(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 3(0(1(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(1(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 2(3(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(4(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(1(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(3(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(3(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(4(x1)))))) 2(0(1(0(0(x1))))) -> 4(0(2(0(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 2(0(0(4(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 0(0(3(0(1(2(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(0(1(3(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 1(3(0(4(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(0(1(3(0(x1)))))) 2(4(1(0(0(x1))))) -> 2(0(4(3(0(1(x1)))))) 2(4(1(0(0(x1))))) -> 4(1(3(0(0(2(x1)))))) 2(4(1(0(0(x1))))) -> 0(0(4(0(1(2(x1)))))) 2(0(2(1(0(x1))))) -> 2(4(0(1(0(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(0(1(3(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(2(0(1(4(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(4(1(3(2(x1)))))) 2(4(3(1(0(x1))))) -> 1(4(0(3(4(2(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(0(1(2(3(x1)))))) 2(4(3(1(0(x1))))) -> 2(1(3(3(0(4(x1)))))) 2(4(3(1(0(x1))))) -> 2(3(0(4(1(4(x1)))))) 2(0(4(1(0(x1))))) -> 4(0(1(3(0(2(x1)))))) 2(0(5(1(0(x1))))) -> 5(0(1(3(2(0(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(1(5(0(3(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(0(3(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(1(4(4(0(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(4(0(1(x1)))))) 2(4(5(1(0(x1))))) -> 1(5(4(4(0(2(x1)))))) 2(4(5(1(0(x1))))) -> 5(1(3(2(0(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(5(2(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(3(1(2(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(0(1(3(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(0(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(1(2(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 1(2(5(4(0(5(x1)))))) 2(4(5(1(0(x1))))) -> 3(4(0(2(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 2(4(0(1(3(5(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(1(4(4(0(x1)))))) 2(4(1(2(0(x1))))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(3(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 1(5(3(4(0(2(x1)))))) 2(4(1(0(5(x1))))) -> 5(1(3(0(4(2(x1)))))) 2(4(1(0(5(x1))))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 4(0(1(1(2(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(3(0(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(4(0(5(x1)))))) 2(4(2(0(5(x1))))) -> 2(2(3(0(4(5(x1)))))) 2(0(5(1(5(x1))))) -> 0(5(4(1(2(5(x1)))))) 2(4(5(1(5(x1))))) -> 5(4(0(1(2(5(x1)))))) 2(4(1(4(5(x1))))) -> 5(4(4(1(2(3(x1)))))) 2(4(1(4(5(x1))))) -> 2(1(5(3(4(4(x1)))))) 2(0(2(4(5(x1))))) -> 4(2(2(5(0(3(x1)))))) 2(0(2(4(5(x1))))) -> 2(2(3(5(0(4(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(2(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(3(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(4(5(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(5(4(4(0(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(3(4(4(5(x1)))))) 2(4(5(4(5(x1))))) -> 2(5(4(0(5(4(x1)))))) ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(0(1(0(x1)))) -> 2^1(x1) 2^1(0(1(0(x1)))) -> 2^1(2(x1)) 2^1(4(1(0(x1)))) -> 2^1(x1) 2^1(4(1(0(0(x1))))) -> 2^1(0(x1)) 2^1(0(2(1(0(x1))))) -> 2^1(x1) 2^1(4(1(0(0(x1))))) -> 2^1(x1) 2^1(4(2(1(0(x1))))) -> 2^1(x1) The TRS R consists of the following rules: 2(0(1(0(x1)))) -> 0(1(3(0(2(x1))))) 2(0(1(0(x1)))) -> 2(1(3(0(0(2(x1)))))) 2(0(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(0(1(0(x1)))) -> 0(1(3(0(2(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 0(3(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(3(0(1(4(x1)))))) 2(0(1(0(x1)))) -> 2(0(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(x1))))) 2(4(1(0(x1)))) -> 1(3(0(4(2(x1))))) 2(4(1(0(x1)))) -> 4(0(1(2(3(x1))))) 2(4(1(0(x1)))) -> 0(4(1(2(3(x1))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(x1))))) 2(4(1(0(x1)))) -> 2(3(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(4(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(5(0(1(4(x1))))) 2(4(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(4(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(0(4(1(3(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 3(0(1(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(1(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 2(3(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(4(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(1(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(3(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(3(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(4(x1)))))) 2(0(1(0(0(x1))))) -> 4(0(2(0(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 2(0(0(4(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 0(0(3(0(1(2(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(0(1(3(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 1(3(0(4(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(0(1(3(0(x1)))))) 2(4(1(0(0(x1))))) -> 2(0(4(3(0(1(x1)))))) 2(4(1(0(0(x1))))) -> 4(1(3(0(0(2(x1)))))) 2(4(1(0(0(x1))))) -> 0(0(4(0(1(2(x1)))))) 2(0(2(1(0(x1))))) -> 2(4(0(1(0(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(0(1(3(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(2(0(1(4(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(4(1(3(2(x1)))))) 2(4(3(1(0(x1))))) -> 1(4(0(3(4(2(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(0(1(2(3(x1)))))) 2(4(3(1(0(x1))))) -> 2(1(3(3(0(4(x1)))))) 2(4(3(1(0(x1))))) -> 2(3(0(4(1(4(x1)))))) 2(0(4(1(0(x1))))) -> 4(0(1(3(0(2(x1)))))) 2(0(5(1(0(x1))))) -> 5(0(1(3(2(0(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(1(5(0(3(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(0(3(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(1(4(4(0(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(4(0(1(x1)))))) 2(4(5(1(0(x1))))) -> 1(5(4(4(0(2(x1)))))) 2(4(5(1(0(x1))))) -> 5(1(3(2(0(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(5(2(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(3(1(2(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(0(1(3(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(0(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(1(2(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 1(2(5(4(0(5(x1)))))) 2(4(5(1(0(x1))))) -> 3(4(0(2(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 2(4(0(1(3(5(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(1(4(4(0(x1)))))) 2(4(1(2(0(x1))))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(3(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 1(5(3(4(0(2(x1)))))) 2(4(1(0(5(x1))))) -> 5(1(3(0(4(2(x1)))))) 2(4(1(0(5(x1))))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 4(0(1(1(2(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(3(0(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(4(0(5(x1)))))) 2(4(2(0(5(x1))))) -> 2(2(3(0(4(5(x1)))))) 2(0(5(1(5(x1))))) -> 0(5(4(1(2(5(x1)))))) 2(4(5(1(5(x1))))) -> 5(4(0(1(2(5(x1)))))) 2(4(1(4(5(x1))))) -> 5(4(4(1(2(3(x1)))))) 2(4(1(4(5(x1))))) -> 2(1(5(3(4(4(x1)))))) 2(0(2(4(5(x1))))) -> 4(2(2(5(0(3(x1)))))) 2(0(2(4(5(x1))))) -> 2(2(3(5(0(4(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(2(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(3(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(4(5(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(5(4(4(0(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(3(4(4(5(x1)))))) 2(4(5(4(5(x1))))) -> 2(5(4(0(5(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 2^1(4(1(0(0(x1))))) -> 2^1(x1) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(2^1(x_1)) = [[-I]] + [[0A, -I, 0A]] * x_1 >>> <<< POL(0(x_1)) = [[-I], [-I], [0A]] + [[0A, 0A, -I], [0A, 0A, 0A], [0A, -I, -I]] * x_1 >>> <<< POL(1(x_1)) = [[1A], [0A], [0A]] + [[0A, 0A, -I], [0A, 0A, 0A], [1A, 0A, 0A]] * x_1 >>> <<< POL(2(x_1)) = [[0A], [0A], [1A]] + [[0A, 0A, -I], [-I, 0A, -I], [0A, 0A, -I]] * x_1 >>> <<< POL(4(x_1)) = [[0A], [1A], [0A]] + [[-I, -I, -I], [0A, -I, -I], [-I, -I, 0A]] * x_1 >>> <<< POL(3(x_1)) = [[0A], [0A], [1A]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 >>> <<< POL(5(x_1)) = [[0A], [0A], [1A]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 2(0(1(0(x1)))) -> 0(1(3(0(2(x1))))) 2(0(1(0(x1)))) -> 2(1(3(0(0(2(x1)))))) 2(0(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(0(1(0(x1)))) -> 0(1(3(0(2(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 0(3(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(3(0(1(4(x1)))))) 2(0(1(0(x1)))) -> 2(0(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(x1))))) 2(4(1(0(x1)))) -> 1(3(0(4(2(x1))))) 2(4(1(0(x1)))) -> 4(0(1(2(3(x1))))) 2(4(1(0(x1)))) -> 0(4(1(2(3(x1))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(x1))))) 2(4(1(0(x1)))) -> 2(3(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(4(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(5(0(1(4(x1))))) 2(4(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(4(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(0(4(1(3(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 3(0(1(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(1(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 2(3(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(4(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(1(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(3(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(3(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(4(x1)))))) 2(0(1(0(0(x1))))) -> 4(0(2(0(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 2(0(0(4(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 0(0(3(0(1(2(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(0(1(3(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 1(3(0(4(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(0(1(3(0(x1)))))) 2(4(1(0(0(x1))))) -> 2(0(4(3(0(1(x1)))))) 2(4(1(0(0(x1))))) -> 4(1(3(0(0(2(x1)))))) 2(4(1(0(0(x1))))) -> 0(0(4(0(1(2(x1)))))) 2(0(2(1(0(x1))))) -> 2(4(0(1(0(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(0(1(3(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(2(0(1(4(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(4(1(3(2(x1)))))) 2(4(3(1(0(x1))))) -> 1(4(0(3(4(2(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(0(1(2(3(x1)))))) 2(4(3(1(0(x1))))) -> 2(1(3(3(0(4(x1)))))) 2(4(3(1(0(x1))))) -> 2(3(0(4(1(4(x1)))))) 2(0(4(1(0(x1))))) -> 4(0(1(3(0(2(x1)))))) 2(0(5(1(0(x1))))) -> 5(0(1(3(2(0(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(1(5(0(3(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(0(3(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(1(4(4(0(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(4(0(1(x1)))))) 2(4(5(1(0(x1))))) -> 1(5(4(4(0(2(x1)))))) 2(4(5(1(0(x1))))) -> 5(1(3(2(0(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(5(2(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(3(1(2(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(0(1(3(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(0(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(1(2(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 1(2(5(4(0(5(x1)))))) 2(4(5(1(0(x1))))) -> 3(4(0(2(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 2(4(0(1(3(5(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(1(4(4(0(x1)))))) 2(4(1(2(0(x1))))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(3(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 1(5(3(4(0(2(x1)))))) 2(4(1(0(5(x1))))) -> 5(1(3(0(4(2(x1)))))) 2(4(1(0(5(x1))))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 4(0(1(1(2(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(3(0(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(4(0(5(x1)))))) 2(4(2(0(5(x1))))) -> 2(2(3(0(4(5(x1)))))) 2(0(5(1(5(x1))))) -> 0(5(4(1(2(5(x1)))))) 2(4(5(1(5(x1))))) -> 5(4(0(1(2(5(x1)))))) 2(4(1(4(5(x1))))) -> 5(4(4(1(2(3(x1)))))) 2(4(1(4(5(x1))))) -> 2(1(5(3(4(4(x1)))))) 2(0(2(4(5(x1))))) -> 4(2(2(5(0(3(x1)))))) 2(0(2(4(5(x1))))) -> 2(2(3(5(0(4(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(2(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(3(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(4(5(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(5(4(4(0(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(3(4(4(5(x1)))))) 2(4(5(4(5(x1))))) -> 2(5(4(0(5(4(x1)))))) ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(0(1(0(x1)))) -> 2^1(x1) 2^1(0(1(0(x1)))) -> 2^1(2(x1)) 2^1(4(1(0(x1)))) -> 2^1(x1) 2^1(4(1(0(0(x1))))) -> 2^1(0(x1)) 2^1(0(2(1(0(x1))))) -> 2^1(x1) 2^1(4(2(1(0(x1))))) -> 2^1(x1) The TRS R consists of the following rules: 2(0(1(0(x1)))) -> 0(1(3(0(2(x1))))) 2(0(1(0(x1)))) -> 2(1(3(0(0(2(x1)))))) 2(0(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(0(1(0(x1)))) -> 0(1(3(0(2(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 0(3(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(3(0(1(4(x1)))))) 2(0(1(0(x1)))) -> 2(0(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(x1))))) 2(4(1(0(x1)))) -> 1(3(0(4(2(x1))))) 2(4(1(0(x1)))) -> 4(0(1(2(3(x1))))) 2(4(1(0(x1)))) -> 0(4(1(2(3(x1))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(x1))))) 2(4(1(0(x1)))) -> 2(3(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(4(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(5(0(1(4(x1))))) 2(4(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(4(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(0(4(1(3(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 3(0(1(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(1(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 2(3(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(4(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(1(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(3(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(3(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(4(x1)))))) 2(0(1(0(0(x1))))) -> 4(0(2(0(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 2(0(0(4(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 0(0(3(0(1(2(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(0(1(3(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 1(3(0(4(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(0(1(3(0(x1)))))) 2(4(1(0(0(x1))))) -> 2(0(4(3(0(1(x1)))))) 2(4(1(0(0(x1))))) -> 4(1(3(0(0(2(x1)))))) 2(4(1(0(0(x1))))) -> 0(0(4(0(1(2(x1)))))) 2(0(2(1(0(x1))))) -> 2(4(0(1(0(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(0(1(3(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(2(0(1(4(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(4(1(3(2(x1)))))) 2(4(3(1(0(x1))))) -> 1(4(0(3(4(2(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(0(1(2(3(x1)))))) 2(4(3(1(0(x1))))) -> 2(1(3(3(0(4(x1)))))) 2(4(3(1(0(x1))))) -> 2(3(0(4(1(4(x1)))))) 2(0(4(1(0(x1))))) -> 4(0(1(3(0(2(x1)))))) 2(0(5(1(0(x1))))) -> 5(0(1(3(2(0(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(1(5(0(3(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(0(3(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(1(4(4(0(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(4(0(1(x1)))))) 2(4(5(1(0(x1))))) -> 1(5(4(4(0(2(x1)))))) 2(4(5(1(0(x1))))) -> 5(1(3(2(0(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(5(2(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(3(1(2(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(0(1(3(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(0(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(1(2(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 1(2(5(4(0(5(x1)))))) 2(4(5(1(0(x1))))) -> 3(4(0(2(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 2(4(0(1(3(5(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(1(4(4(0(x1)))))) 2(4(1(2(0(x1))))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(3(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 1(5(3(4(0(2(x1)))))) 2(4(1(0(5(x1))))) -> 5(1(3(0(4(2(x1)))))) 2(4(1(0(5(x1))))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 4(0(1(1(2(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(3(0(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(4(0(5(x1)))))) 2(4(2(0(5(x1))))) -> 2(2(3(0(4(5(x1)))))) 2(0(5(1(5(x1))))) -> 0(5(4(1(2(5(x1)))))) 2(4(5(1(5(x1))))) -> 5(4(0(1(2(5(x1)))))) 2(4(1(4(5(x1))))) -> 5(4(4(1(2(3(x1)))))) 2(4(1(4(5(x1))))) -> 2(1(5(3(4(4(x1)))))) 2(0(2(4(5(x1))))) -> 4(2(2(5(0(3(x1)))))) 2(0(2(4(5(x1))))) -> 2(2(3(5(0(4(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(2(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(3(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(4(5(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(5(4(4(0(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(3(4(4(5(x1)))))) 2(4(5(4(5(x1))))) -> 2(5(4(0(5(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 2^1(4(1(0(0(x1))))) -> 2^1(0(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(2^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(0(x_1)) = [[0A], [0A], [1A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [-I, 0A, -I]] * x_1 >>> <<< POL(1(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 1A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(2(x_1)) = [[0A], [1A], [-I]] + [[0A, 0A, 0A], [-I, 0A, -I], [0A, 0A, 0A]] * x_1 >>> <<< POL(4(x_1)) = [[1A], [0A], [0A]] + [[-I, 0A, -I], [0A, 0A, 0A], [-I, 0A, -I]] * x_1 >>> <<< POL(3(x_1)) = [[0A], [-I], [0A]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 >>> <<< POL(5(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 2(0(1(0(x1)))) -> 0(1(3(0(2(x1))))) 2(0(1(0(x1)))) -> 2(1(3(0(0(2(x1)))))) 2(0(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(0(1(0(x1)))) -> 0(1(3(0(2(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 0(3(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(3(0(1(4(x1)))))) 2(0(1(0(x1)))) -> 2(0(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(x1))))) 2(4(1(0(x1)))) -> 1(3(0(4(2(x1))))) 2(4(1(0(x1)))) -> 4(0(1(2(3(x1))))) 2(4(1(0(x1)))) -> 0(4(1(2(3(x1))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(x1))))) 2(4(1(0(x1)))) -> 2(3(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(4(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(5(0(1(4(x1))))) 2(4(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(4(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(0(4(1(3(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 3(0(1(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(1(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 2(3(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(4(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(1(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(3(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(3(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(4(x1)))))) 2(0(1(0(0(x1))))) -> 4(0(2(0(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 2(0(0(4(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 0(0(3(0(1(2(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(0(1(3(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 1(3(0(4(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(0(1(3(0(x1)))))) 2(4(1(0(0(x1))))) -> 2(0(4(3(0(1(x1)))))) 2(4(1(0(0(x1))))) -> 4(1(3(0(0(2(x1)))))) 2(4(1(0(0(x1))))) -> 0(0(4(0(1(2(x1)))))) 2(0(2(1(0(x1))))) -> 2(4(0(1(0(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(0(1(3(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(2(0(1(4(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(4(1(3(2(x1)))))) 2(4(3(1(0(x1))))) -> 1(4(0(3(4(2(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(0(1(2(3(x1)))))) 2(4(3(1(0(x1))))) -> 2(1(3(3(0(4(x1)))))) 2(4(3(1(0(x1))))) -> 2(3(0(4(1(4(x1)))))) 2(0(4(1(0(x1))))) -> 4(0(1(3(0(2(x1)))))) 2(0(5(1(0(x1))))) -> 5(0(1(3(2(0(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(1(5(0(3(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(0(3(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(1(4(4(0(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(4(0(1(x1)))))) 2(4(5(1(0(x1))))) -> 1(5(4(4(0(2(x1)))))) 2(4(5(1(0(x1))))) -> 5(1(3(2(0(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(5(2(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(3(1(2(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(0(1(3(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(0(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(1(2(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 1(2(5(4(0(5(x1)))))) 2(4(5(1(0(x1))))) -> 3(4(0(2(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 2(4(0(1(3(5(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(1(4(4(0(x1)))))) 2(4(1(2(0(x1))))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(3(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 1(5(3(4(0(2(x1)))))) 2(4(1(0(5(x1))))) -> 5(1(3(0(4(2(x1)))))) 2(4(1(0(5(x1))))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 4(0(1(1(2(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(3(0(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(4(0(5(x1)))))) 2(4(2(0(5(x1))))) -> 2(2(3(0(4(5(x1)))))) 2(0(5(1(5(x1))))) -> 0(5(4(1(2(5(x1)))))) 2(4(5(1(5(x1))))) -> 5(4(0(1(2(5(x1)))))) 2(4(1(4(5(x1))))) -> 5(4(4(1(2(3(x1)))))) 2(4(1(4(5(x1))))) -> 2(1(5(3(4(4(x1)))))) 2(0(2(4(5(x1))))) -> 4(2(2(5(0(3(x1)))))) 2(0(2(4(5(x1))))) -> 2(2(3(5(0(4(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(2(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(3(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(4(5(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(5(4(4(0(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(3(4(4(5(x1)))))) 2(4(5(4(5(x1))))) -> 2(5(4(0(5(4(x1)))))) ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(0(1(0(x1)))) -> 2^1(x1) 2^1(0(1(0(x1)))) -> 2^1(2(x1)) 2^1(4(1(0(x1)))) -> 2^1(x1) 2^1(0(2(1(0(x1))))) -> 2^1(x1) 2^1(4(2(1(0(x1))))) -> 2^1(x1) The TRS R consists of the following rules: 2(0(1(0(x1)))) -> 0(1(3(0(2(x1))))) 2(0(1(0(x1)))) -> 2(1(3(0(0(2(x1)))))) 2(0(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(0(1(0(x1)))) -> 0(1(3(0(2(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 0(3(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(3(0(1(4(x1)))))) 2(0(1(0(x1)))) -> 2(0(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(x1))))) 2(4(1(0(x1)))) -> 1(3(0(4(2(x1))))) 2(4(1(0(x1)))) -> 4(0(1(2(3(x1))))) 2(4(1(0(x1)))) -> 0(4(1(2(3(x1))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(x1))))) 2(4(1(0(x1)))) -> 2(3(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(4(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(5(0(1(4(x1))))) 2(4(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(4(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(0(4(1(3(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 3(0(1(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(1(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 2(3(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(4(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(1(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(3(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(3(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(4(x1)))))) 2(0(1(0(0(x1))))) -> 4(0(2(0(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 2(0(0(4(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 0(0(3(0(1(2(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(0(1(3(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 1(3(0(4(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(0(1(3(0(x1)))))) 2(4(1(0(0(x1))))) -> 2(0(4(3(0(1(x1)))))) 2(4(1(0(0(x1))))) -> 4(1(3(0(0(2(x1)))))) 2(4(1(0(0(x1))))) -> 0(0(4(0(1(2(x1)))))) 2(0(2(1(0(x1))))) -> 2(4(0(1(0(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(0(1(3(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(2(0(1(4(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(4(1(3(2(x1)))))) 2(4(3(1(0(x1))))) -> 1(4(0(3(4(2(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(0(1(2(3(x1)))))) 2(4(3(1(0(x1))))) -> 2(1(3(3(0(4(x1)))))) 2(4(3(1(0(x1))))) -> 2(3(0(4(1(4(x1)))))) 2(0(4(1(0(x1))))) -> 4(0(1(3(0(2(x1)))))) 2(0(5(1(0(x1))))) -> 5(0(1(3(2(0(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(1(5(0(3(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(0(3(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(1(4(4(0(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(4(0(1(x1)))))) 2(4(5(1(0(x1))))) -> 1(5(4(4(0(2(x1)))))) 2(4(5(1(0(x1))))) -> 5(1(3(2(0(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(5(2(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(3(1(2(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(0(1(3(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(0(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(1(2(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 1(2(5(4(0(5(x1)))))) 2(4(5(1(0(x1))))) -> 3(4(0(2(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 2(4(0(1(3(5(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(1(4(4(0(x1)))))) 2(4(1(2(0(x1))))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(3(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 1(5(3(4(0(2(x1)))))) 2(4(1(0(5(x1))))) -> 5(1(3(0(4(2(x1)))))) 2(4(1(0(5(x1))))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 4(0(1(1(2(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(3(0(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(4(0(5(x1)))))) 2(4(2(0(5(x1))))) -> 2(2(3(0(4(5(x1)))))) 2(0(5(1(5(x1))))) -> 0(5(4(1(2(5(x1)))))) 2(4(5(1(5(x1))))) -> 5(4(0(1(2(5(x1)))))) 2(4(1(4(5(x1))))) -> 5(4(4(1(2(3(x1)))))) 2(4(1(4(5(x1))))) -> 2(1(5(3(4(4(x1)))))) 2(0(2(4(5(x1))))) -> 4(2(2(5(0(3(x1)))))) 2(0(2(4(5(x1))))) -> 2(2(3(5(0(4(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(2(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(3(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(4(5(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(5(4(4(0(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(3(4(4(5(x1)))))) 2(4(5(4(5(x1))))) -> 2(5(4(0(5(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 2^1(0(1(0(x1)))) -> 2^1(2(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(2^1(x_1)) = [[0A]] + [[-I, 0A, 0A]] * x_1 >>> <<< POL(0(x_1)) = [[-I], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [-I, 0A, -I]] * x_1 >>> <<< POL(1(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [-I, 0A, 1A], [0A, -I, -I]] * x_1 >>> <<< POL(2(x_1)) = [[0A], [0A], [-I]] + [[0A, 0A, 0A], [-I, 0A, -I], [-I, 0A, -I]] * x_1 >>> <<< POL(4(x_1)) = [[0A], [0A], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [-I, -I, -I]] * x_1 >>> <<< POL(3(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, 0A], [-I, -I, -I], [-I, -I, -I]] * x_1 >>> <<< POL(5(x_1)) = [[0A], [-I], [-I]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 2(0(1(0(x1)))) -> 0(1(3(0(2(x1))))) 2(0(1(0(x1)))) -> 2(1(3(0(0(2(x1)))))) 2(0(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(0(1(0(x1)))) -> 0(1(3(0(2(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 0(3(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(3(0(1(4(x1)))))) 2(0(1(0(x1)))) -> 2(0(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(x1))))) 2(4(1(0(x1)))) -> 1(3(0(4(2(x1))))) 2(4(1(0(x1)))) -> 4(0(1(2(3(x1))))) 2(4(1(0(x1)))) -> 0(4(1(2(3(x1))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(x1))))) 2(4(1(0(x1)))) -> 2(3(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(4(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(5(0(1(4(x1))))) 2(4(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(4(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(0(4(1(3(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 3(0(1(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(1(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 2(3(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(4(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(1(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(3(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(3(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(4(x1)))))) 2(0(1(0(0(x1))))) -> 4(0(2(0(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 2(0(0(4(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 0(0(3(0(1(2(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(0(1(3(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 1(3(0(4(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(0(1(3(0(x1)))))) 2(4(1(0(0(x1))))) -> 2(0(4(3(0(1(x1)))))) 2(4(1(0(0(x1))))) -> 4(1(3(0(0(2(x1)))))) 2(4(1(0(0(x1))))) -> 0(0(4(0(1(2(x1)))))) 2(0(2(1(0(x1))))) -> 2(4(0(1(0(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(0(1(3(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(2(0(1(4(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(4(1(3(2(x1)))))) 2(4(3(1(0(x1))))) -> 1(4(0(3(4(2(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(0(1(2(3(x1)))))) 2(4(3(1(0(x1))))) -> 2(1(3(3(0(4(x1)))))) 2(4(3(1(0(x1))))) -> 2(3(0(4(1(4(x1)))))) 2(0(4(1(0(x1))))) -> 4(0(1(3(0(2(x1)))))) 2(0(5(1(0(x1))))) -> 5(0(1(3(2(0(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(1(5(0(3(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(0(3(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(1(4(4(0(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(4(0(1(x1)))))) 2(4(5(1(0(x1))))) -> 1(5(4(4(0(2(x1)))))) 2(4(5(1(0(x1))))) -> 5(1(3(2(0(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(5(2(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(3(1(2(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(0(1(3(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(0(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(1(2(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 1(2(5(4(0(5(x1)))))) 2(4(5(1(0(x1))))) -> 3(4(0(2(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 2(4(0(1(3(5(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(1(4(4(0(x1)))))) 2(4(1(2(0(x1))))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(3(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 1(5(3(4(0(2(x1)))))) 2(4(1(0(5(x1))))) -> 5(1(3(0(4(2(x1)))))) 2(4(1(0(5(x1))))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 4(0(1(1(2(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(3(0(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(4(0(5(x1)))))) 2(4(2(0(5(x1))))) -> 2(2(3(0(4(5(x1)))))) 2(0(5(1(5(x1))))) -> 0(5(4(1(2(5(x1)))))) 2(4(5(1(5(x1))))) -> 5(4(0(1(2(5(x1)))))) 2(4(1(4(5(x1))))) -> 5(4(4(1(2(3(x1)))))) 2(4(1(4(5(x1))))) -> 2(1(5(3(4(4(x1)))))) 2(0(2(4(5(x1))))) -> 4(2(2(5(0(3(x1)))))) 2(0(2(4(5(x1))))) -> 2(2(3(5(0(4(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(2(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(3(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(4(5(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(5(4(4(0(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(3(4(4(5(x1)))))) 2(4(5(4(5(x1))))) -> 2(5(4(0(5(4(x1)))))) ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(0(1(0(x1)))) -> 2^1(x1) 2^1(4(1(0(x1)))) -> 2^1(x1) 2^1(0(2(1(0(x1))))) -> 2^1(x1) 2^1(4(2(1(0(x1))))) -> 2^1(x1) The TRS R consists of the following rules: 2(0(1(0(x1)))) -> 0(1(3(0(2(x1))))) 2(0(1(0(x1)))) -> 2(1(3(0(0(2(x1)))))) 2(0(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(0(1(0(x1)))) -> 0(1(3(0(2(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 0(3(0(1(3(2(x1)))))) 2(0(1(0(x1)))) -> 2(0(3(0(1(4(x1)))))) 2(0(1(0(x1)))) -> 2(0(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(x1))))) 2(4(1(0(x1)))) -> 1(3(0(4(2(x1))))) 2(4(1(0(x1)))) -> 4(0(1(2(3(x1))))) 2(4(1(0(x1)))) -> 0(4(1(2(3(x1))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(x1))))) 2(4(1(0(x1)))) -> 2(3(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(4(0(1(4(x1))))) 2(4(1(0(x1)))) -> 2(5(0(1(4(x1))))) 2(4(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(4(1(3(0(2(x1)))))) 2(4(1(0(x1)))) -> 4(0(4(1(3(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 3(0(1(3(4(2(x1)))))) 2(4(1(0(x1)))) -> 1(4(0(1(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(0(x1)))) -> 4(0(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 0(4(1(3(2(3(x1)))))) 2(4(1(0(x1)))) -> 2(3(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(4(1(3(0(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(4(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(1(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(5(0(3(1(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(3(4(x1)))))) 2(4(1(0(x1)))) -> 2(1(3(0(4(4(x1)))))) 2(0(1(0(0(x1))))) -> 4(0(2(0(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 2(0(0(4(0(1(x1)))))) 2(0(1(0(0(x1))))) -> 0(0(3(0(1(2(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(0(1(3(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 1(3(0(4(2(0(x1)))))) 2(4(1(0(0(x1))))) -> 4(2(0(1(3(0(x1)))))) 2(4(1(0(0(x1))))) -> 2(0(4(3(0(1(x1)))))) 2(4(1(0(0(x1))))) -> 4(1(3(0(0(2(x1)))))) 2(4(1(0(0(x1))))) -> 0(0(4(0(1(2(x1)))))) 2(0(2(1(0(x1))))) -> 2(4(0(1(0(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(0(1(3(2(x1)))))) 2(4(2(1(0(x1))))) -> 4(2(2(0(1(4(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(4(1(3(2(x1)))))) 2(4(3(1(0(x1))))) -> 1(4(0(3(4(2(x1)))))) 2(4(3(1(0(x1))))) -> 0(4(0(1(2(3(x1)))))) 2(4(3(1(0(x1))))) -> 2(1(3(3(0(4(x1)))))) 2(4(3(1(0(x1))))) -> 2(3(0(4(1(4(x1)))))) 2(0(4(1(0(x1))))) -> 4(0(1(3(0(2(x1)))))) 2(0(5(1(0(x1))))) -> 5(0(1(3(2(0(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(1(5(0(3(x1)))))) 2(0(5(1(0(x1))))) -> 2(0(0(3(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(1(4(4(0(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(4(0(1(x1)))))) 2(4(5(1(0(x1))))) -> 1(5(4(4(0(2(x1)))))) 2(4(5(1(0(x1))))) -> 5(1(3(2(0(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(5(2(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(3(1(2(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(2(0(1(3(4(x1)))))) 2(4(5(1(0(x1))))) -> 2(1(5(0(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 5(0(1(2(4(4(x1)))))) 2(4(5(1(0(x1))))) -> 1(2(5(4(0(5(x1)))))) 2(4(5(1(0(x1))))) -> 3(4(0(2(1(5(x1)))))) 2(4(5(1(0(x1))))) -> 2(4(0(1(3(5(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(1(4(4(0(x1)))))) 2(4(1(2(0(x1))))) -> 0(4(1(2(2(3(x1)))))) 2(4(1(2(0(x1))))) -> 2(2(3(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 1(5(3(4(0(2(x1)))))) 2(4(1(0(5(x1))))) -> 5(1(3(0(4(2(x1)))))) 2(4(1(0(5(x1))))) -> 2(3(5(0(1(4(x1)))))) 2(4(1(0(5(x1))))) -> 4(0(1(1(2(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(3(0(5(x1)))))) 2(0(2(0(5(x1))))) -> 2(2(0(4(0(5(x1)))))) 2(4(2(0(5(x1))))) -> 2(2(3(0(4(5(x1)))))) 2(0(5(1(5(x1))))) -> 0(5(4(1(2(5(x1)))))) 2(4(5(1(5(x1))))) -> 5(4(0(1(2(5(x1)))))) 2(4(1(4(5(x1))))) -> 5(4(4(1(2(3(x1)))))) 2(4(1(4(5(x1))))) -> 2(1(5(3(4(4(x1)))))) 2(0(2(4(5(x1))))) -> 4(2(2(5(0(3(x1)))))) 2(0(2(4(5(x1))))) -> 2(2(3(5(0(4(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(2(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(3(5(x1)))))) 2(0(2(4(5(x1))))) -> 0(4(2(2(4(5(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(5(4(4(0(x1)))))) 2(4(2(4(5(x1))))) -> 2(2(3(4(4(5(x1)))))) 2(4(5(4(5(x1))))) -> 2(5(4(0(5(4(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) 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. ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(0(1(0(x1)))) -> 2^1(x1) 2^1(4(1(0(x1)))) -> 2^1(x1) 2^1(0(2(1(0(x1))))) -> 2^1(x1) 2^1(4(2(1(0(x1))))) -> 2^1(x1) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *2^1(0(1(0(x1)))) -> 2^1(x1) The graph contains the following edges 1 > 1 *2^1(4(1(0(x1)))) -> 2^1(x1) The graph contains the following edges 1 > 1 *2^1(0(2(1(0(x1))))) -> 2^1(x1) The graph contains the following edges 1 > 1 *2^1(4(2(1(0(x1))))) -> 2^1(x1) The graph contains the following edges 1 > 1 ---------------------------------------- (28) YES