/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) DependencyPairsProof [EQUIVALENT, 129 ms] (4) QDP (5) DependencyGraphProof [EQUIVALENT, 5 ms] (6) QDP (7) QDPOrderProof [EQUIVALENT, 4521 ms] (8) QDP (9) QDPOrderProof [EQUIVALENT, 25.9 s] (10) QDP (11) QDPOrderProof [EQUIVALENT, 971 ms] (12) QDP (13) QDPOrderProof [EQUIVALENT, 5012 ms] (14) QDP (15) QDPOrderProof [EQUIVALENT, 5703 ms] (16) QDP (17) QDPOrderProof [EQUIVALENT, 4420 ms] (18) QDP (19) PisEmptyProof [EQUIVALENT, 0 ms] (20) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0(1(2(3(x1)))) -> 2(0(3(1(2(x1))))) 0(1(2(3(x1)))) -> 2(0(3(1(3(x1))))) 0(1(2(3(x1)))) -> 2(0(3(2(1(x1))))) 0(1(2(3(x1)))) -> 2(1(0(3(0(x1))))) 0(1(2(3(x1)))) -> 2(1(3(0(2(x1))))) 0(1(2(3(x1)))) -> 2(1(4(3(0(x1))))) 0(1(2(3(x1)))) -> 2(3(0(2(1(x1))))) 0(1(2(3(x1)))) -> 2(0(3(2(1(1(x1)))))) 0(1(2(3(x1)))) -> 2(0(3(3(2(1(x1)))))) 0(1(2(3(x1)))) -> 2(1(3(0(3(0(x1)))))) 0(1(5(3(x1)))) -> 2(1(0(3(3(5(x1)))))) 0(1(5(3(x1)))) -> 2(1(0(3(4(5(x1)))))) 0(5(2(3(x1)))) -> 2(0(3(3(5(x1))))) 0(5(2(3(x1)))) -> 2(0(0(3(5(3(x1)))))) 0(0(1(2(3(x1))))) -> 0(2(0(3(1(0(x1)))))) 0(0(1(2(3(x1))))) -> 2(1(1(3(0(0(x1)))))) 0(0(4(2(3(x1))))) -> 0(3(4(0(3(2(x1)))))) 0(0(4(2(3(x1))))) -> 3(0(0(2(4(4(x1)))))) 0(0(5(2(3(x1))))) -> 0(3(5(2(4(0(x1)))))) 0(1(0(2(3(x1))))) -> 3(2(4(1(0(0(x1)))))) 0(1(0(5(3(x1))))) -> 0(1(1(5(0(3(x1)))))) 0(1(0(5(3(x1))))) -> 1(5(0(0(3(4(x1)))))) 0(1(2(1(3(x1))))) -> 2(1(0(0(3(1(x1)))))) 0(1(2(2(3(x1))))) -> 2(0(3(3(1(2(x1)))))) 0(1(2(3(3(x1))))) -> 2(0(3(3(4(1(x1)))))) 0(1(2(5(3(x1))))) -> 2(5(0(3(1(0(x1)))))) 0(4(5(5(3(x1))))) -> 5(0(3(4(1(5(x1)))))) 0(4(5(5(3(x1))))) -> 5(4(0(3(3(5(x1)))))) 0(5(2(1(3(x1))))) -> 2(1(0(3(3(5(x1)))))) 0(5(4(2(3(x1))))) -> 2(0(3(4(5(1(x1)))))) 1(0(0(5(3(x1))))) -> 0(3(1(3(5(0(x1)))))) 1(0(1(2(3(x1))))) -> 1(1(1(2(3(0(x1)))))) 1(0(1(2(3(x1))))) -> 1(2(0(3(5(1(x1)))))) 1(5(3(5(3(x1))))) -> 5(5(0(3(3(1(x1)))))) 4(0(1(2(3(x1))))) -> 2(1(4(4(0(3(x1)))))) 4(0(1(2(3(x1))))) -> 4(3(2(1(0(0(x1)))))) 5(0(0(5(3(x1))))) -> 3(0(3(5(5(0(x1)))))) 5(0(1(2(3(x1))))) -> 3(0(3(1(2(5(x1)))))) 5(0(1(2(3(x1))))) -> 4(5(3(0(2(1(x1)))))) 5(0(1(2(3(x1))))) -> 5(1(0(3(0(2(x1)))))) 5(0(1(2(3(x1))))) -> 5(1(0(3(1(2(x1)))))) 5(0(4(2(3(x1))))) -> 0(2(4(1(5(3(x1)))))) 5(0(4(2(3(x1))))) -> 0(3(3(5(2(4(x1)))))) 5(0(4(2(3(x1))))) -> 2(4(4(0(3(5(x1)))))) 5(0(5(2(3(x1))))) -> 5(0(3(5(4(2(x1)))))) 5(3(0(5(3(x1))))) -> 5(5(3(0(0(3(x1)))))) 5(3(1(2(3(x1))))) -> 2(0(3(1(5(3(x1)))))) 5(3(1(2(3(x1))))) -> 3(3(0(2(1(5(x1)))))) Q is empty. ---------------------------------------- (1) QTRS Reverse (EQUIVALENT) We applied the QTRS Reverse Processor [REVERSE]. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 3(2(1(0(x1)))) -> 2(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 3(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 1(2(3(0(2(x1))))) 3(2(1(0(x1)))) -> 0(3(0(1(2(x1))))) 3(2(1(0(x1)))) -> 2(0(3(1(2(x1))))) 3(2(1(0(x1)))) -> 0(3(4(1(2(x1))))) 3(2(1(0(x1)))) -> 1(2(0(3(2(x1))))) 3(2(1(0(x1)))) -> 1(1(2(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 1(2(3(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 0(3(0(3(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(3(3(0(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(4(3(0(1(2(x1)))))) 3(2(5(0(x1)))) -> 5(3(3(0(2(x1))))) 3(2(5(0(x1)))) -> 3(5(3(0(0(2(x1)))))) 3(2(1(0(0(x1))))) -> 0(1(3(0(2(0(x1)))))) 3(2(1(0(0(x1))))) -> 0(0(3(1(1(2(x1)))))) 3(2(4(0(0(x1))))) -> 2(3(0(4(3(0(x1)))))) 3(2(4(0(0(x1))))) -> 4(4(2(0(0(3(x1)))))) 3(2(5(0(0(x1))))) -> 0(4(2(5(3(0(x1)))))) 3(2(0(1(0(x1))))) -> 0(0(1(4(2(3(x1)))))) 3(5(0(1(0(x1))))) -> 3(0(5(1(1(0(x1)))))) 3(5(0(1(0(x1))))) -> 4(3(0(0(5(1(x1)))))) 3(1(2(1(0(x1))))) -> 1(3(0(0(1(2(x1)))))) 3(2(2(1(0(x1))))) -> 2(1(3(3(0(2(x1)))))) 3(3(2(1(0(x1))))) -> 1(4(3(3(0(2(x1)))))) 3(5(2(1(0(x1))))) -> 0(1(3(0(5(2(x1)))))) 3(5(5(4(0(x1))))) -> 5(1(4(3(0(5(x1)))))) 3(5(5(4(0(x1))))) -> 5(3(3(0(4(5(x1)))))) 3(1(2(5(0(x1))))) -> 5(3(3(0(1(2(x1)))))) 3(2(4(5(0(x1))))) -> 1(5(4(3(0(2(x1)))))) 3(5(0(0(1(x1))))) -> 0(5(3(1(3(0(x1)))))) 3(2(1(0(1(x1))))) -> 0(3(2(1(1(1(x1)))))) 3(2(1(0(1(x1))))) -> 1(5(3(0(2(1(x1)))))) 3(5(3(5(1(x1))))) -> 1(3(3(0(5(5(x1)))))) 3(2(1(0(4(x1))))) -> 3(0(4(4(1(2(x1)))))) 3(2(1(0(4(x1))))) -> 0(0(1(2(3(4(x1)))))) 3(5(0(0(5(x1))))) -> 0(5(5(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 5(2(1(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 1(2(0(3(5(4(x1)))))) 3(2(1(0(5(x1))))) -> 2(0(3(0(1(5(x1)))))) 3(2(1(0(5(x1))))) -> 2(1(3(0(1(5(x1)))))) 3(2(4(0(5(x1))))) -> 3(5(1(4(2(0(x1)))))) 3(2(4(0(5(x1))))) -> 4(2(5(3(3(0(x1)))))) 3(2(4(0(5(x1))))) -> 5(3(0(4(4(2(x1)))))) 3(2(5(0(5(x1))))) -> 2(4(5(3(0(5(x1)))))) 3(5(0(3(5(x1))))) -> 3(0(0(3(5(5(x1)))))) 3(2(1(3(5(x1))))) -> 3(5(1(3(0(2(x1)))))) 3(2(1(3(5(x1))))) -> 5(1(2(0(3(3(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: 3^1(2(1(0(x1)))) -> 3^1(0(2(x1))) 3^1(2(1(0(x1)))) -> 3^1(1(3(0(2(x1))))) 3^1(2(1(0(x1)))) -> 3^1(0(1(2(x1)))) 3^1(2(1(0(x1)))) -> 3^1(1(2(x1))) 3^1(2(1(0(x1)))) -> 3^1(4(1(2(x1)))) 3^1(2(1(0(x1)))) -> 3^1(2(x1)) 3^1(2(1(0(x1)))) -> 3^1(3(0(2(x1)))) 3^1(2(1(0(x1)))) -> 3^1(0(3(1(2(x1))))) 3^1(5(1(0(x1)))) -> 3^1(3(0(1(2(x1))))) 3^1(5(1(0(x1)))) -> 3^1(0(1(2(x1)))) 3^1(2(5(0(x1)))) -> 3^1(3(0(2(x1)))) 3^1(2(5(0(x1)))) -> 3^1(0(2(x1))) 3^1(2(5(0(x1)))) -> 3^1(5(3(0(0(2(x1)))))) 3^1(2(5(0(x1)))) -> 3^1(0(0(2(x1)))) 3^1(2(1(0(0(x1))))) -> 3^1(0(2(0(x1)))) 3^1(2(1(0(0(x1))))) -> 3^1(1(1(2(x1)))) 3^1(2(4(0(0(x1))))) -> 3^1(0(4(3(0(x1))))) 3^1(2(4(0(0(x1))))) -> 3^1(0(x1)) 3^1(2(4(0(0(x1))))) -> 3^1(x1) 3^1(2(5(0(0(x1))))) -> 3^1(0(x1)) 3^1(2(0(1(0(x1))))) -> 3^1(x1) 3^1(5(0(1(0(x1))))) -> 3^1(0(5(1(1(0(x1)))))) 3^1(5(0(1(0(x1))))) -> 3^1(0(0(5(1(x1))))) 3^1(1(2(1(0(x1))))) -> 3^1(0(0(1(2(x1))))) 3^1(2(2(1(0(x1))))) -> 3^1(3(0(2(x1)))) 3^1(2(2(1(0(x1))))) -> 3^1(0(2(x1))) 3^1(3(2(1(0(x1))))) -> 3^1(3(0(2(x1)))) 3^1(3(2(1(0(x1))))) -> 3^1(0(2(x1))) 3^1(5(2(1(0(x1))))) -> 3^1(0(5(2(x1)))) 3^1(5(5(4(0(x1))))) -> 3^1(0(5(x1))) 3^1(5(5(4(0(x1))))) -> 3^1(3(0(4(5(x1))))) 3^1(5(5(4(0(x1))))) -> 3^1(0(4(5(x1)))) 3^1(1(2(5(0(x1))))) -> 3^1(3(0(1(2(x1))))) 3^1(1(2(5(0(x1))))) -> 3^1(0(1(2(x1)))) 3^1(2(4(5(0(x1))))) -> 3^1(0(2(x1))) 3^1(5(0(0(1(x1))))) -> 3^1(1(3(0(x1)))) 3^1(5(0(0(1(x1))))) -> 3^1(0(x1)) 3^1(2(1(0(1(x1))))) -> 3^1(2(1(1(1(x1))))) 3^1(2(1(0(1(x1))))) -> 3^1(0(2(1(x1)))) 3^1(5(3(5(1(x1))))) -> 3^1(3(0(5(5(x1))))) 3^1(5(3(5(1(x1))))) -> 3^1(0(5(5(x1)))) 3^1(2(1(0(4(x1))))) -> 3^1(0(4(4(1(2(x1)))))) 3^1(2(1(0(4(x1))))) -> 3^1(4(x1)) 3^1(5(0(0(5(x1))))) -> 3^1(0(3(x1))) 3^1(5(0(0(5(x1))))) -> 3^1(x1) 3^1(2(1(0(5(x1))))) -> 3^1(0(3(x1))) 3^1(2(1(0(5(x1))))) -> 3^1(x1) 3^1(2(1(0(5(x1))))) -> 3^1(5(4(x1))) 3^1(2(1(0(5(x1))))) -> 3^1(0(1(5(x1)))) 3^1(2(4(0(5(x1))))) -> 3^1(5(1(4(2(0(x1)))))) 3^1(2(4(0(5(x1))))) -> 3^1(3(0(x1))) 3^1(2(4(0(5(x1))))) -> 3^1(0(x1)) 3^1(2(4(0(5(x1))))) -> 3^1(0(4(4(2(x1))))) 3^1(2(5(0(5(x1))))) -> 3^1(0(5(x1))) 3^1(5(0(3(5(x1))))) -> 3^1(0(0(3(5(5(x1)))))) 3^1(5(0(3(5(x1))))) -> 3^1(5(5(x1))) 3^1(2(1(3(5(x1))))) -> 3^1(5(1(3(0(2(x1)))))) 3^1(2(1(3(5(x1))))) -> 3^1(0(2(x1))) 3^1(2(1(3(5(x1))))) -> 3^1(3(x1)) 3^1(2(1(3(5(x1))))) -> 3^1(x1) The TRS R consists of the following rules: 3(2(1(0(x1)))) -> 2(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 3(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 1(2(3(0(2(x1))))) 3(2(1(0(x1)))) -> 0(3(0(1(2(x1))))) 3(2(1(0(x1)))) -> 2(0(3(1(2(x1))))) 3(2(1(0(x1)))) -> 0(3(4(1(2(x1))))) 3(2(1(0(x1)))) -> 1(2(0(3(2(x1))))) 3(2(1(0(x1)))) -> 1(1(2(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 1(2(3(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 0(3(0(3(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(3(3(0(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(4(3(0(1(2(x1)))))) 3(2(5(0(x1)))) -> 5(3(3(0(2(x1))))) 3(2(5(0(x1)))) -> 3(5(3(0(0(2(x1)))))) 3(2(1(0(0(x1))))) -> 0(1(3(0(2(0(x1)))))) 3(2(1(0(0(x1))))) -> 0(0(3(1(1(2(x1)))))) 3(2(4(0(0(x1))))) -> 2(3(0(4(3(0(x1)))))) 3(2(4(0(0(x1))))) -> 4(4(2(0(0(3(x1)))))) 3(2(5(0(0(x1))))) -> 0(4(2(5(3(0(x1)))))) 3(2(0(1(0(x1))))) -> 0(0(1(4(2(3(x1)))))) 3(5(0(1(0(x1))))) -> 3(0(5(1(1(0(x1)))))) 3(5(0(1(0(x1))))) -> 4(3(0(0(5(1(x1)))))) 3(1(2(1(0(x1))))) -> 1(3(0(0(1(2(x1)))))) 3(2(2(1(0(x1))))) -> 2(1(3(3(0(2(x1)))))) 3(3(2(1(0(x1))))) -> 1(4(3(3(0(2(x1)))))) 3(5(2(1(0(x1))))) -> 0(1(3(0(5(2(x1)))))) 3(5(5(4(0(x1))))) -> 5(1(4(3(0(5(x1)))))) 3(5(5(4(0(x1))))) -> 5(3(3(0(4(5(x1)))))) 3(1(2(5(0(x1))))) -> 5(3(3(0(1(2(x1)))))) 3(2(4(5(0(x1))))) -> 1(5(4(3(0(2(x1)))))) 3(5(0(0(1(x1))))) -> 0(5(3(1(3(0(x1)))))) 3(2(1(0(1(x1))))) -> 0(3(2(1(1(1(x1)))))) 3(2(1(0(1(x1))))) -> 1(5(3(0(2(1(x1)))))) 3(5(3(5(1(x1))))) -> 1(3(3(0(5(5(x1)))))) 3(2(1(0(4(x1))))) -> 3(0(4(4(1(2(x1)))))) 3(2(1(0(4(x1))))) -> 0(0(1(2(3(4(x1)))))) 3(5(0(0(5(x1))))) -> 0(5(5(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 5(2(1(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 1(2(0(3(5(4(x1)))))) 3(2(1(0(5(x1))))) -> 2(0(3(0(1(5(x1)))))) 3(2(1(0(5(x1))))) -> 2(1(3(0(1(5(x1)))))) 3(2(4(0(5(x1))))) -> 3(5(1(4(2(0(x1)))))) 3(2(4(0(5(x1))))) -> 4(2(5(3(3(0(x1)))))) 3(2(4(0(5(x1))))) -> 5(3(0(4(4(2(x1)))))) 3(2(5(0(5(x1))))) -> 2(4(5(3(0(5(x1)))))) 3(5(0(3(5(x1))))) -> 3(0(0(3(5(5(x1)))))) 3(2(1(3(5(x1))))) -> 3(5(1(3(0(2(x1)))))) 3(2(1(3(5(x1))))) -> 5(1(2(0(3(3(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 53 less nodes. ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(2(4(0(0(x1))))) -> 3^1(x1) 3^1(2(1(0(x1)))) -> 3^1(2(x1)) 3^1(2(0(1(0(x1))))) -> 3^1(x1) 3^1(5(0(0(5(x1))))) -> 3^1(x1) 3^1(2(1(0(5(x1))))) -> 3^1(x1) 3^1(2(1(3(5(x1))))) -> 3^1(3(x1)) 3^1(2(1(3(5(x1))))) -> 3^1(x1) The TRS R consists of the following rules: 3(2(1(0(x1)))) -> 2(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 3(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 1(2(3(0(2(x1))))) 3(2(1(0(x1)))) -> 0(3(0(1(2(x1))))) 3(2(1(0(x1)))) -> 2(0(3(1(2(x1))))) 3(2(1(0(x1)))) -> 0(3(4(1(2(x1))))) 3(2(1(0(x1)))) -> 1(2(0(3(2(x1))))) 3(2(1(0(x1)))) -> 1(1(2(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 1(2(3(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 0(3(0(3(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(3(3(0(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(4(3(0(1(2(x1)))))) 3(2(5(0(x1)))) -> 5(3(3(0(2(x1))))) 3(2(5(0(x1)))) -> 3(5(3(0(0(2(x1)))))) 3(2(1(0(0(x1))))) -> 0(1(3(0(2(0(x1)))))) 3(2(1(0(0(x1))))) -> 0(0(3(1(1(2(x1)))))) 3(2(4(0(0(x1))))) -> 2(3(0(4(3(0(x1)))))) 3(2(4(0(0(x1))))) -> 4(4(2(0(0(3(x1)))))) 3(2(5(0(0(x1))))) -> 0(4(2(5(3(0(x1)))))) 3(2(0(1(0(x1))))) -> 0(0(1(4(2(3(x1)))))) 3(5(0(1(0(x1))))) -> 3(0(5(1(1(0(x1)))))) 3(5(0(1(0(x1))))) -> 4(3(0(0(5(1(x1)))))) 3(1(2(1(0(x1))))) -> 1(3(0(0(1(2(x1)))))) 3(2(2(1(0(x1))))) -> 2(1(3(3(0(2(x1)))))) 3(3(2(1(0(x1))))) -> 1(4(3(3(0(2(x1)))))) 3(5(2(1(0(x1))))) -> 0(1(3(0(5(2(x1)))))) 3(5(5(4(0(x1))))) -> 5(1(4(3(0(5(x1)))))) 3(5(5(4(0(x1))))) -> 5(3(3(0(4(5(x1)))))) 3(1(2(5(0(x1))))) -> 5(3(3(0(1(2(x1)))))) 3(2(4(5(0(x1))))) -> 1(5(4(3(0(2(x1)))))) 3(5(0(0(1(x1))))) -> 0(5(3(1(3(0(x1)))))) 3(2(1(0(1(x1))))) -> 0(3(2(1(1(1(x1)))))) 3(2(1(0(1(x1))))) -> 1(5(3(0(2(1(x1)))))) 3(5(3(5(1(x1))))) -> 1(3(3(0(5(5(x1)))))) 3(2(1(0(4(x1))))) -> 3(0(4(4(1(2(x1)))))) 3(2(1(0(4(x1))))) -> 0(0(1(2(3(4(x1)))))) 3(5(0(0(5(x1))))) -> 0(5(5(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 5(2(1(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 1(2(0(3(5(4(x1)))))) 3(2(1(0(5(x1))))) -> 2(0(3(0(1(5(x1)))))) 3(2(1(0(5(x1))))) -> 2(1(3(0(1(5(x1)))))) 3(2(4(0(5(x1))))) -> 3(5(1(4(2(0(x1)))))) 3(2(4(0(5(x1))))) -> 4(2(5(3(3(0(x1)))))) 3(2(4(0(5(x1))))) -> 5(3(0(4(4(2(x1)))))) 3(2(5(0(5(x1))))) -> 2(4(5(3(0(5(x1)))))) 3(5(0(3(5(x1))))) -> 3(0(0(3(5(5(x1)))))) 3(2(1(3(5(x1))))) -> 3(5(1(3(0(2(x1)))))) 3(2(1(3(5(x1))))) -> 5(1(2(0(3(3(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. 3^1(2(1(0(5(x1))))) -> 3^1(x1) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(3^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(2(x_1)) = [[0A], [0A], [-I]] + [[0A, 0A, 1A], [-I, 0A, -I], [-I, 0A, 0A]] * x_1 >>> <<< POL(4(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, 0A], [-I, 0A, 0A], [-I, -I, -I]] * x_1 >>> <<< POL(0(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, 0A], [0A, 0A, 0A], [-I, -I, -I]] * x_1 >>> <<< POL(1(x_1)) = [[0A], [-I], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [-I, 0A, 0A]] * x_1 >>> <<< POL(5(x_1)) = [[-I], [0A], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(3(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, 0A], [0A, -I, 0A], [0A, -I, 0A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 3(2(1(0(x1)))) -> 2(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 3(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 1(2(3(0(2(x1))))) 3(2(1(0(x1)))) -> 0(3(0(1(2(x1))))) 3(2(1(0(x1)))) -> 2(0(3(1(2(x1))))) 3(2(1(0(x1)))) -> 0(3(4(1(2(x1))))) 3(2(1(0(x1)))) -> 1(2(0(3(2(x1))))) 3(2(1(0(x1)))) -> 1(1(2(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 1(2(3(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 0(3(0(3(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(3(3(0(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(4(3(0(1(2(x1)))))) 3(2(5(0(x1)))) -> 5(3(3(0(2(x1))))) 3(2(5(0(x1)))) -> 3(5(3(0(0(2(x1)))))) 3(2(1(0(0(x1))))) -> 0(1(3(0(2(0(x1)))))) 3(2(1(0(0(x1))))) -> 0(0(3(1(1(2(x1)))))) 3(2(4(0(0(x1))))) -> 2(3(0(4(3(0(x1)))))) 3(2(4(0(0(x1))))) -> 4(4(2(0(0(3(x1)))))) 3(2(5(0(0(x1))))) -> 0(4(2(5(3(0(x1)))))) 3(2(0(1(0(x1))))) -> 0(0(1(4(2(3(x1)))))) 3(5(0(1(0(x1))))) -> 3(0(5(1(1(0(x1)))))) 3(5(0(1(0(x1))))) -> 4(3(0(0(5(1(x1)))))) 3(1(2(1(0(x1))))) -> 1(3(0(0(1(2(x1)))))) 3(2(2(1(0(x1))))) -> 2(1(3(3(0(2(x1)))))) 3(3(2(1(0(x1))))) -> 1(4(3(3(0(2(x1)))))) 3(5(2(1(0(x1))))) -> 0(1(3(0(5(2(x1)))))) 3(5(5(4(0(x1))))) -> 5(1(4(3(0(5(x1)))))) 3(5(5(4(0(x1))))) -> 5(3(3(0(4(5(x1)))))) 3(1(2(5(0(x1))))) -> 5(3(3(0(1(2(x1)))))) 3(2(4(5(0(x1))))) -> 1(5(4(3(0(2(x1)))))) 3(5(0(0(1(x1))))) -> 0(5(3(1(3(0(x1)))))) 3(2(1(0(1(x1))))) -> 0(3(2(1(1(1(x1)))))) 3(2(1(0(1(x1))))) -> 1(5(3(0(2(1(x1)))))) 3(5(3(5(1(x1))))) -> 1(3(3(0(5(5(x1)))))) 3(2(1(0(4(x1))))) -> 3(0(4(4(1(2(x1)))))) 3(2(1(0(4(x1))))) -> 0(0(1(2(3(4(x1)))))) 3(5(0(0(5(x1))))) -> 0(5(5(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 5(2(1(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 1(2(0(3(5(4(x1)))))) 3(2(1(0(5(x1))))) -> 2(0(3(0(1(5(x1)))))) 3(2(1(0(5(x1))))) -> 2(1(3(0(1(5(x1)))))) 3(2(4(0(5(x1))))) -> 3(5(1(4(2(0(x1)))))) 3(2(4(0(5(x1))))) -> 4(2(5(3(3(0(x1)))))) 3(2(4(0(5(x1))))) -> 5(3(0(4(4(2(x1)))))) 3(2(5(0(5(x1))))) -> 2(4(5(3(0(5(x1)))))) 3(5(0(3(5(x1))))) -> 3(0(0(3(5(5(x1)))))) 3(2(1(3(5(x1))))) -> 3(5(1(3(0(2(x1)))))) 3(2(1(3(5(x1))))) -> 5(1(2(0(3(3(x1)))))) ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(2(4(0(0(x1))))) -> 3^1(x1) 3^1(2(1(0(x1)))) -> 3^1(2(x1)) 3^1(2(0(1(0(x1))))) -> 3^1(x1) 3^1(5(0(0(5(x1))))) -> 3^1(x1) 3^1(2(1(3(5(x1))))) -> 3^1(3(x1)) 3^1(2(1(3(5(x1))))) -> 3^1(x1) The TRS R consists of the following rules: 3(2(1(0(x1)))) -> 2(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 3(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 1(2(3(0(2(x1))))) 3(2(1(0(x1)))) -> 0(3(0(1(2(x1))))) 3(2(1(0(x1)))) -> 2(0(3(1(2(x1))))) 3(2(1(0(x1)))) -> 0(3(4(1(2(x1))))) 3(2(1(0(x1)))) -> 1(2(0(3(2(x1))))) 3(2(1(0(x1)))) -> 1(1(2(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 1(2(3(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 0(3(0(3(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(3(3(0(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(4(3(0(1(2(x1)))))) 3(2(5(0(x1)))) -> 5(3(3(0(2(x1))))) 3(2(5(0(x1)))) -> 3(5(3(0(0(2(x1)))))) 3(2(1(0(0(x1))))) -> 0(1(3(0(2(0(x1)))))) 3(2(1(0(0(x1))))) -> 0(0(3(1(1(2(x1)))))) 3(2(4(0(0(x1))))) -> 2(3(0(4(3(0(x1)))))) 3(2(4(0(0(x1))))) -> 4(4(2(0(0(3(x1)))))) 3(2(5(0(0(x1))))) -> 0(4(2(5(3(0(x1)))))) 3(2(0(1(0(x1))))) -> 0(0(1(4(2(3(x1)))))) 3(5(0(1(0(x1))))) -> 3(0(5(1(1(0(x1)))))) 3(5(0(1(0(x1))))) -> 4(3(0(0(5(1(x1)))))) 3(1(2(1(0(x1))))) -> 1(3(0(0(1(2(x1)))))) 3(2(2(1(0(x1))))) -> 2(1(3(3(0(2(x1)))))) 3(3(2(1(0(x1))))) -> 1(4(3(3(0(2(x1)))))) 3(5(2(1(0(x1))))) -> 0(1(3(0(5(2(x1)))))) 3(5(5(4(0(x1))))) -> 5(1(4(3(0(5(x1)))))) 3(5(5(4(0(x1))))) -> 5(3(3(0(4(5(x1)))))) 3(1(2(5(0(x1))))) -> 5(3(3(0(1(2(x1)))))) 3(2(4(5(0(x1))))) -> 1(5(4(3(0(2(x1)))))) 3(5(0(0(1(x1))))) -> 0(5(3(1(3(0(x1)))))) 3(2(1(0(1(x1))))) -> 0(3(2(1(1(1(x1)))))) 3(2(1(0(1(x1))))) -> 1(5(3(0(2(1(x1)))))) 3(5(3(5(1(x1))))) -> 1(3(3(0(5(5(x1)))))) 3(2(1(0(4(x1))))) -> 3(0(4(4(1(2(x1)))))) 3(2(1(0(4(x1))))) -> 0(0(1(2(3(4(x1)))))) 3(5(0(0(5(x1))))) -> 0(5(5(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 5(2(1(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 1(2(0(3(5(4(x1)))))) 3(2(1(0(5(x1))))) -> 2(0(3(0(1(5(x1)))))) 3(2(1(0(5(x1))))) -> 2(1(3(0(1(5(x1)))))) 3(2(4(0(5(x1))))) -> 3(5(1(4(2(0(x1)))))) 3(2(4(0(5(x1))))) -> 4(2(5(3(3(0(x1)))))) 3(2(4(0(5(x1))))) -> 5(3(0(4(4(2(x1)))))) 3(2(5(0(5(x1))))) -> 2(4(5(3(0(5(x1)))))) 3(5(0(3(5(x1))))) -> 3(0(0(3(5(5(x1)))))) 3(2(1(3(5(x1))))) -> 3(5(1(3(0(2(x1)))))) 3(2(1(3(5(x1))))) -> 5(1(2(0(3(3(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 3^1(2(1(3(5(x1))))) -> 3^1(x1) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(3^1(x_1)) = [[0A]] + [[-I, 0A, -I, 0A, 0A]] * x_1 >>> <<< POL(2(x_1)) = [[0A], [0A], [0A], [0A], [0A]] + [[0A, 0A, -I, -I, -I], [-I, -I, -I, -I, -I], [0A, -I, -I, 0A, -I], [0A, 0A, 0A, 0A, -I], [0A, 0A, -I, -I, 0A]] * x_1 >>> <<< POL(4(x_1)) = [[0A], [-I], [-I], [0A], [0A]] + [[-I, 0A, -I, -I, 0A], [-I, -I, -I, -I, -I], [0A, 0A, 0A, 0A, 0A], [-I, -I, -I, -I, 0A], [-I, 0A, -I, -I, 0A]] * x_1 >>> <<< POL(0(x_1)) = [[1A], [0A], [0A], [0A], [0A]] + [[0A, 0A, 0A, 0A, 0A], [0A, 0A, -I, -I, 0A], [0A, 0A, 0A, 0A, 0A], [0A, 0A, 0A, 0A, 0A], [-I, -I, -I, -I, -I]] * x_1 >>> <<< POL(1(x_1)) = [[1A], [0A], [1A], [0A], [0A]] + [[0A, 0A, -I, -I, 0A], [-I, 0A, -I, -I, 0A], [0A, 0A, -I, -I, 0A], [0A, 0A, -I, -I, 0A], [0A, 1A, -I, -I, 0A]] * x_1 >>> <<< POL(5(x_1)) = [[1A], [0A], [0A], [0A], [0A]] + [[0A, 0A, 0A, 0A, 0A], [0A, 0A, -I, 0A, 0A], [0A, 0A, 0A, 0A, 0A], [0A, 0A, 0A, 0A, 0A], [0A, 0A, -I, 0A, 0A]] * x_1 >>> <<< POL(3(x_1)) = [[0A], [0A], [0A], [1A], [0A]] + [[-I, 0A, -I, -I, 0A], [-I, -I, -I, -I, 0A], [0A, 0A, -I, 0A, 0A], [0A, 0A, -I, 0A, 0A], [-I, -I, -I, -I, 0A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 3(2(1(0(x1)))) -> 2(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 3(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 1(2(3(0(2(x1))))) 3(2(1(0(x1)))) -> 0(3(0(1(2(x1))))) 3(2(1(0(x1)))) -> 2(0(3(1(2(x1))))) 3(2(1(0(x1)))) -> 0(3(4(1(2(x1))))) 3(2(1(0(x1)))) -> 1(2(0(3(2(x1))))) 3(2(1(0(x1)))) -> 1(1(2(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 1(2(3(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 0(3(0(3(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(3(3(0(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(4(3(0(1(2(x1)))))) 3(2(5(0(x1)))) -> 5(3(3(0(2(x1))))) 3(2(5(0(x1)))) -> 3(5(3(0(0(2(x1)))))) 3(2(1(0(0(x1))))) -> 0(1(3(0(2(0(x1)))))) 3(2(1(0(0(x1))))) -> 0(0(3(1(1(2(x1)))))) 3(2(4(0(0(x1))))) -> 2(3(0(4(3(0(x1)))))) 3(2(4(0(0(x1))))) -> 4(4(2(0(0(3(x1)))))) 3(2(5(0(0(x1))))) -> 0(4(2(5(3(0(x1)))))) 3(2(0(1(0(x1))))) -> 0(0(1(4(2(3(x1)))))) 3(5(0(1(0(x1))))) -> 3(0(5(1(1(0(x1)))))) 3(5(0(1(0(x1))))) -> 4(3(0(0(5(1(x1)))))) 3(1(2(1(0(x1))))) -> 1(3(0(0(1(2(x1)))))) 3(2(2(1(0(x1))))) -> 2(1(3(3(0(2(x1)))))) 3(3(2(1(0(x1))))) -> 1(4(3(3(0(2(x1)))))) 3(5(2(1(0(x1))))) -> 0(1(3(0(5(2(x1)))))) 3(5(5(4(0(x1))))) -> 5(1(4(3(0(5(x1)))))) 3(5(5(4(0(x1))))) -> 5(3(3(0(4(5(x1)))))) 3(1(2(5(0(x1))))) -> 5(3(3(0(1(2(x1)))))) 3(2(4(5(0(x1))))) -> 1(5(4(3(0(2(x1)))))) 3(5(0(0(1(x1))))) -> 0(5(3(1(3(0(x1)))))) 3(2(1(0(1(x1))))) -> 0(3(2(1(1(1(x1)))))) 3(2(1(0(1(x1))))) -> 1(5(3(0(2(1(x1)))))) 3(5(3(5(1(x1))))) -> 1(3(3(0(5(5(x1)))))) 3(2(1(0(4(x1))))) -> 3(0(4(4(1(2(x1)))))) 3(2(1(0(4(x1))))) -> 0(0(1(2(3(4(x1)))))) 3(5(0(0(5(x1))))) -> 0(5(5(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 5(2(1(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 1(2(0(3(5(4(x1)))))) 3(2(1(0(5(x1))))) -> 2(0(3(0(1(5(x1)))))) 3(2(1(0(5(x1))))) -> 2(1(3(0(1(5(x1)))))) 3(2(4(0(5(x1))))) -> 3(5(1(4(2(0(x1)))))) 3(2(4(0(5(x1))))) -> 4(2(5(3(3(0(x1)))))) 3(2(4(0(5(x1))))) -> 5(3(0(4(4(2(x1)))))) 3(2(5(0(5(x1))))) -> 2(4(5(3(0(5(x1)))))) 3(5(0(3(5(x1))))) -> 3(0(0(3(5(5(x1)))))) 3(2(1(3(5(x1))))) -> 3(5(1(3(0(2(x1)))))) 3(2(1(3(5(x1))))) -> 5(1(2(0(3(3(x1)))))) ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(2(4(0(0(x1))))) -> 3^1(x1) 3^1(2(1(0(x1)))) -> 3^1(2(x1)) 3^1(2(0(1(0(x1))))) -> 3^1(x1) 3^1(5(0(0(5(x1))))) -> 3^1(x1) 3^1(2(1(3(5(x1))))) -> 3^1(3(x1)) The TRS R consists of the following rules: 3(2(1(0(x1)))) -> 2(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 3(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 1(2(3(0(2(x1))))) 3(2(1(0(x1)))) -> 0(3(0(1(2(x1))))) 3(2(1(0(x1)))) -> 2(0(3(1(2(x1))))) 3(2(1(0(x1)))) -> 0(3(4(1(2(x1))))) 3(2(1(0(x1)))) -> 1(2(0(3(2(x1))))) 3(2(1(0(x1)))) -> 1(1(2(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 1(2(3(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 0(3(0(3(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(3(3(0(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(4(3(0(1(2(x1)))))) 3(2(5(0(x1)))) -> 5(3(3(0(2(x1))))) 3(2(5(0(x1)))) -> 3(5(3(0(0(2(x1)))))) 3(2(1(0(0(x1))))) -> 0(1(3(0(2(0(x1)))))) 3(2(1(0(0(x1))))) -> 0(0(3(1(1(2(x1)))))) 3(2(4(0(0(x1))))) -> 2(3(0(4(3(0(x1)))))) 3(2(4(0(0(x1))))) -> 4(4(2(0(0(3(x1)))))) 3(2(5(0(0(x1))))) -> 0(4(2(5(3(0(x1)))))) 3(2(0(1(0(x1))))) -> 0(0(1(4(2(3(x1)))))) 3(5(0(1(0(x1))))) -> 3(0(5(1(1(0(x1)))))) 3(5(0(1(0(x1))))) -> 4(3(0(0(5(1(x1)))))) 3(1(2(1(0(x1))))) -> 1(3(0(0(1(2(x1)))))) 3(2(2(1(0(x1))))) -> 2(1(3(3(0(2(x1)))))) 3(3(2(1(0(x1))))) -> 1(4(3(3(0(2(x1)))))) 3(5(2(1(0(x1))))) -> 0(1(3(0(5(2(x1)))))) 3(5(5(4(0(x1))))) -> 5(1(4(3(0(5(x1)))))) 3(5(5(4(0(x1))))) -> 5(3(3(0(4(5(x1)))))) 3(1(2(5(0(x1))))) -> 5(3(3(0(1(2(x1)))))) 3(2(4(5(0(x1))))) -> 1(5(4(3(0(2(x1)))))) 3(5(0(0(1(x1))))) -> 0(5(3(1(3(0(x1)))))) 3(2(1(0(1(x1))))) -> 0(3(2(1(1(1(x1)))))) 3(2(1(0(1(x1))))) -> 1(5(3(0(2(1(x1)))))) 3(5(3(5(1(x1))))) -> 1(3(3(0(5(5(x1)))))) 3(2(1(0(4(x1))))) -> 3(0(4(4(1(2(x1)))))) 3(2(1(0(4(x1))))) -> 0(0(1(2(3(4(x1)))))) 3(5(0(0(5(x1))))) -> 0(5(5(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 5(2(1(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 1(2(0(3(5(4(x1)))))) 3(2(1(0(5(x1))))) -> 2(0(3(0(1(5(x1)))))) 3(2(1(0(5(x1))))) -> 2(1(3(0(1(5(x1)))))) 3(2(4(0(5(x1))))) -> 3(5(1(4(2(0(x1)))))) 3(2(4(0(5(x1))))) -> 4(2(5(3(3(0(x1)))))) 3(2(4(0(5(x1))))) -> 5(3(0(4(4(2(x1)))))) 3(2(5(0(5(x1))))) -> 2(4(5(3(0(5(x1)))))) 3(5(0(3(5(x1))))) -> 3(0(0(3(5(5(x1)))))) 3(2(1(3(5(x1))))) -> 3(5(1(3(0(2(x1)))))) 3(2(1(3(5(x1))))) -> 5(1(2(0(3(3(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 3^1(5(0(0(5(x1))))) -> 3^1(x1) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(3^1(x_1)) = [[1A]] + [[0A, 0A, -I]] * x_1 >>> <<< POL(2(x_1)) = [[-I], [0A], [0A]] + [[0A, -I, -I], [0A, 0A, -I], [-I, -I, -I]] * x_1 >>> <<< POL(4(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, 0A], [0A, -I, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(0(x_1)) = [[-I], [-I], [1A]] + [[0A, -I, 0A], [-I, -I, 0A], [-I, 0A, -I]] * x_1 >>> <<< POL(1(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(5(x_1)) = [[0A], [1A], [1A]] + [[1A, 0A, 1A], [-I, 1A, 0A], [-I, 0A, 0A]] * x_1 >>> <<< POL(3(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, -I], [-I, 0A, -I], [-I, 0A, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 3(2(1(0(x1)))) -> 2(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 3(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 1(2(3(0(2(x1))))) 3(2(1(0(x1)))) -> 0(3(0(1(2(x1))))) 3(2(1(0(x1)))) -> 2(0(3(1(2(x1))))) 3(2(1(0(x1)))) -> 0(3(4(1(2(x1))))) 3(2(1(0(x1)))) -> 1(2(0(3(2(x1))))) 3(2(1(0(x1)))) -> 1(1(2(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 1(2(3(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 0(3(0(3(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(3(3(0(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(4(3(0(1(2(x1)))))) 3(2(5(0(x1)))) -> 5(3(3(0(2(x1))))) 3(2(5(0(x1)))) -> 3(5(3(0(0(2(x1)))))) 3(2(1(0(0(x1))))) -> 0(1(3(0(2(0(x1)))))) 3(2(1(0(0(x1))))) -> 0(0(3(1(1(2(x1)))))) 3(2(4(0(0(x1))))) -> 2(3(0(4(3(0(x1)))))) 3(2(4(0(0(x1))))) -> 4(4(2(0(0(3(x1)))))) 3(2(5(0(0(x1))))) -> 0(4(2(5(3(0(x1)))))) 3(2(0(1(0(x1))))) -> 0(0(1(4(2(3(x1)))))) 3(5(0(1(0(x1))))) -> 3(0(5(1(1(0(x1)))))) 3(5(0(1(0(x1))))) -> 4(3(0(0(5(1(x1)))))) 3(1(2(1(0(x1))))) -> 1(3(0(0(1(2(x1)))))) 3(2(2(1(0(x1))))) -> 2(1(3(3(0(2(x1)))))) 3(3(2(1(0(x1))))) -> 1(4(3(3(0(2(x1)))))) 3(5(2(1(0(x1))))) -> 0(1(3(0(5(2(x1)))))) 3(5(5(4(0(x1))))) -> 5(1(4(3(0(5(x1)))))) 3(5(5(4(0(x1))))) -> 5(3(3(0(4(5(x1)))))) 3(1(2(5(0(x1))))) -> 5(3(3(0(1(2(x1)))))) 3(2(4(5(0(x1))))) -> 1(5(4(3(0(2(x1)))))) 3(5(0(0(1(x1))))) -> 0(5(3(1(3(0(x1)))))) 3(2(1(0(1(x1))))) -> 0(3(2(1(1(1(x1)))))) 3(2(1(0(1(x1))))) -> 1(5(3(0(2(1(x1)))))) 3(5(3(5(1(x1))))) -> 1(3(3(0(5(5(x1)))))) 3(2(1(0(4(x1))))) -> 3(0(4(4(1(2(x1)))))) 3(2(1(0(4(x1))))) -> 0(0(1(2(3(4(x1)))))) 3(5(0(0(5(x1))))) -> 0(5(5(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 5(2(1(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 1(2(0(3(5(4(x1)))))) 3(2(1(0(5(x1))))) -> 2(0(3(0(1(5(x1)))))) 3(2(1(0(5(x1))))) -> 2(1(3(0(1(5(x1)))))) 3(2(4(0(5(x1))))) -> 3(5(1(4(2(0(x1)))))) 3(2(4(0(5(x1))))) -> 4(2(5(3(3(0(x1)))))) 3(2(4(0(5(x1))))) -> 5(3(0(4(4(2(x1)))))) 3(2(5(0(5(x1))))) -> 2(4(5(3(0(5(x1)))))) 3(5(0(3(5(x1))))) -> 3(0(0(3(5(5(x1)))))) 3(2(1(3(5(x1))))) -> 3(5(1(3(0(2(x1)))))) 3(2(1(3(5(x1))))) -> 5(1(2(0(3(3(x1)))))) ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(2(4(0(0(x1))))) -> 3^1(x1) 3^1(2(1(0(x1)))) -> 3^1(2(x1)) 3^1(2(0(1(0(x1))))) -> 3^1(x1) 3^1(2(1(3(5(x1))))) -> 3^1(3(x1)) The TRS R consists of the following rules: 3(2(1(0(x1)))) -> 2(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 3(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 1(2(3(0(2(x1))))) 3(2(1(0(x1)))) -> 0(3(0(1(2(x1))))) 3(2(1(0(x1)))) -> 2(0(3(1(2(x1))))) 3(2(1(0(x1)))) -> 0(3(4(1(2(x1))))) 3(2(1(0(x1)))) -> 1(2(0(3(2(x1))))) 3(2(1(0(x1)))) -> 1(1(2(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 1(2(3(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 0(3(0(3(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(3(3(0(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(4(3(0(1(2(x1)))))) 3(2(5(0(x1)))) -> 5(3(3(0(2(x1))))) 3(2(5(0(x1)))) -> 3(5(3(0(0(2(x1)))))) 3(2(1(0(0(x1))))) -> 0(1(3(0(2(0(x1)))))) 3(2(1(0(0(x1))))) -> 0(0(3(1(1(2(x1)))))) 3(2(4(0(0(x1))))) -> 2(3(0(4(3(0(x1)))))) 3(2(4(0(0(x1))))) -> 4(4(2(0(0(3(x1)))))) 3(2(5(0(0(x1))))) -> 0(4(2(5(3(0(x1)))))) 3(2(0(1(0(x1))))) -> 0(0(1(4(2(3(x1)))))) 3(5(0(1(0(x1))))) -> 3(0(5(1(1(0(x1)))))) 3(5(0(1(0(x1))))) -> 4(3(0(0(5(1(x1)))))) 3(1(2(1(0(x1))))) -> 1(3(0(0(1(2(x1)))))) 3(2(2(1(0(x1))))) -> 2(1(3(3(0(2(x1)))))) 3(3(2(1(0(x1))))) -> 1(4(3(3(0(2(x1)))))) 3(5(2(1(0(x1))))) -> 0(1(3(0(5(2(x1)))))) 3(5(5(4(0(x1))))) -> 5(1(4(3(0(5(x1)))))) 3(5(5(4(0(x1))))) -> 5(3(3(0(4(5(x1)))))) 3(1(2(5(0(x1))))) -> 5(3(3(0(1(2(x1)))))) 3(2(4(5(0(x1))))) -> 1(5(4(3(0(2(x1)))))) 3(5(0(0(1(x1))))) -> 0(5(3(1(3(0(x1)))))) 3(2(1(0(1(x1))))) -> 0(3(2(1(1(1(x1)))))) 3(2(1(0(1(x1))))) -> 1(5(3(0(2(1(x1)))))) 3(5(3(5(1(x1))))) -> 1(3(3(0(5(5(x1)))))) 3(2(1(0(4(x1))))) -> 3(0(4(4(1(2(x1)))))) 3(2(1(0(4(x1))))) -> 0(0(1(2(3(4(x1)))))) 3(5(0(0(5(x1))))) -> 0(5(5(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 5(2(1(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 1(2(0(3(5(4(x1)))))) 3(2(1(0(5(x1))))) -> 2(0(3(0(1(5(x1)))))) 3(2(1(0(5(x1))))) -> 2(1(3(0(1(5(x1)))))) 3(2(4(0(5(x1))))) -> 3(5(1(4(2(0(x1)))))) 3(2(4(0(5(x1))))) -> 4(2(5(3(3(0(x1)))))) 3(2(4(0(5(x1))))) -> 5(3(0(4(4(2(x1)))))) 3(2(5(0(5(x1))))) -> 2(4(5(3(0(5(x1)))))) 3(5(0(3(5(x1))))) -> 3(0(0(3(5(5(x1)))))) 3(2(1(3(5(x1))))) -> 3(5(1(3(0(2(x1)))))) 3(2(1(3(5(x1))))) -> 5(1(2(0(3(3(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. 3^1(2(4(0(0(x1))))) -> 3^1(x1) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(3^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(2(x_1)) = [[0A], [1A], [0A]] + [[0A, -I, -I], [-I, -I, 0A], [-I, 0A, 0A]] * x_1 >>> <<< POL(4(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, -I], [1A, -I, -I], [-I, -I, -I]] * x_1 >>> <<< POL(0(x_1)) = [[1A], [0A], [0A]] + [[0A, 0A, 0A], [0A, -I, -I], [0A, -I, -I]] * x_1 >>> <<< POL(1(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [0A, -I, 0A], [0A, -I, 0A]] * x_1 >>> <<< POL(3(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, 0A], [-I, 0A, 0A], [-I, 0A, 0A]] * x_1 >>> <<< POL(5(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, 0A], [-I, 0A, 0A], [1A, -I, 0A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 3(2(1(0(x1)))) -> 2(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 3(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 1(2(3(0(2(x1))))) 3(2(1(0(x1)))) -> 0(3(0(1(2(x1))))) 3(2(1(0(x1)))) -> 2(0(3(1(2(x1))))) 3(2(1(0(x1)))) -> 0(3(4(1(2(x1))))) 3(2(1(0(x1)))) -> 1(2(0(3(2(x1))))) 3(2(1(0(x1)))) -> 1(1(2(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 1(2(3(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 0(3(0(3(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(3(3(0(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(4(3(0(1(2(x1)))))) 3(2(5(0(x1)))) -> 5(3(3(0(2(x1))))) 3(2(5(0(x1)))) -> 3(5(3(0(0(2(x1)))))) 3(2(1(0(0(x1))))) -> 0(1(3(0(2(0(x1)))))) 3(2(1(0(0(x1))))) -> 0(0(3(1(1(2(x1)))))) 3(2(4(0(0(x1))))) -> 2(3(0(4(3(0(x1)))))) 3(2(4(0(0(x1))))) -> 4(4(2(0(0(3(x1)))))) 3(2(5(0(0(x1))))) -> 0(4(2(5(3(0(x1)))))) 3(2(0(1(0(x1))))) -> 0(0(1(4(2(3(x1)))))) 3(5(0(1(0(x1))))) -> 3(0(5(1(1(0(x1)))))) 3(5(0(1(0(x1))))) -> 4(3(0(0(5(1(x1)))))) 3(1(2(1(0(x1))))) -> 1(3(0(0(1(2(x1)))))) 3(2(2(1(0(x1))))) -> 2(1(3(3(0(2(x1)))))) 3(3(2(1(0(x1))))) -> 1(4(3(3(0(2(x1)))))) 3(5(2(1(0(x1))))) -> 0(1(3(0(5(2(x1)))))) 3(5(5(4(0(x1))))) -> 5(1(4(3(0(5(x1)))))) 3(5(5(4(0(x1))))) -> 5(3(3(0(4(5(x1)))))) 3(1(2(5(0(x1))))) -> 5(3(3(0(1(2(x1)))))) 3(2(4(5(0(x1))))) -> 1(5(4(3(0(2(x1)))))) 3(5(0(0(1(x1))))) -> 0(5(3(1(3(0(x1)))))) 3(2(1(0(1(x1))))) -> 0(3(2(1(1(1(x1)))))) 3(2(1(0(1(x1))))) -> 1(5(3(0(2(1(x1)))))) 3(5(3(5(1(x1))))) -> 1(3(3(0(5(5(x1)))))) 3(2(1(0(4(x1))))) -> 3(0(4(4(1(2(x1)))))) 3(2(1(0(4(x1))))) -> 0(0(1(2(3(4(x1)))))) 3(5(0(0(5(x1))))) -> 0(5(5(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 5(2(1(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 1(2(0(3(5(4(x1)))))) 3(2(1(0(5(x1))))) -> 2(0(3(0(1(5(x1)))))) 3(2(1(0(5(x1))))) -> 2(1(3(0(1(5(x1)))))) 3(2(4(0(5(x1))))) -> 3(5(1(4(2(0(x1)))))) 3(2(4(0(5(x1))))) -> 4(2(5(3(3(0(x1)))))) 3(2(4(0(5(x1))))) -> 5(3(0(4(4(2(x1)))))) 3(2(5(0(5(x1))))) -> 2(4(5(3(0(5(x1)))))) 3(5(0(3(5(x1))))) -> 3(0(0(3(5(5(x1)))))) 3(2(1(3(5(x1))))) -> 3(5(1(3(0(2(x1)))))) 3(2(1(3(5(x1))))) -> 5(1(2(0(3(3(x1)))))) ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(2(1(0(x1)))) -> 3^1(2(x1)) 3^1(2(0(1(0(x1))))) -> 3^1(x1) 3^1(2(1(3(5(x1))))) -> 3^1(3(x1)) The TRS R consists of the following rules: 3(2(1(0(x1)))) -> 2(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 3(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 1(2(3(0(2(x1))))) 3(2(1(0(x1)))) -> 0(3(0(1(2(x1))))) 3(2(1(0(x1)))) -> 2(0(3(1(2(x1))))) 3(2(1(0(x1)))) -> 0(3(4(1(2(x1))))) 3(2(1(0(x1)))) -> 1(2(0(3(2(x1))))) 3(2(1(0(x1)))) -> 1(1(2(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 1(2(3(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 0(3(0(3(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(3(3(0(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(4(3(0(1(2(x1)))))) 3(2(5(0(x1)))) -> 5(3(3(0(2(x1))))) 3(2(5(0(x1)))) -> 3(5(3(0(0(2(x1)))))) 3(2(1(0(0(x1))))) -> 0(1(3(0(2(0(x1)))))) 3(2(1(0(0(x1))))) -> 0(0(3(1(1(2(x1)))))) 3(2(4(0(0(x1))))) -> 2(3(0(4(3(0(x1)))))) 3(2(4(0(0(x1))))) -> 4(4(2(0(0(3(x1)))))) 3(2(5(0(0(x1))))) -> 0(4(2(5(3(0(x1)))))) 3(2(0(1(0(x1))))) -> 0(0(1(4(2(3(x1)))))) 3(5(0(1(0(x1))))) -> 3(0(5(1(1(0(x1)))))) 3(5(0(1(0(x1))))) -> 4(3(0(0(5(1(x1)))))) 3(1(2(1(0(x1))))) -> 1(3(0(0(1(2(x1)))))) 3(2(2(1(0(x1))))) -> 2(1(3(3(0(2(x1)))))) 3(3(2(1(0(x1))))) -> 1(4(3(3(0(2(x1)))))) 3(5(2(1(0(x1))))) -> 0(1(3(0(5(2(x1)))))) 3(5(5(4(0(x1))))) -> 5(1(4(3(0(5(x1)))))) 3(5(5(4(0(x1))))) -> 5(3(3(0(4(5(x1)))))) 3(1(2(5(0(x1))))) -> 5(3(3(0(1(2(x1)))))) 3(2(4(5(0(x1))))) -> 1(5(4(3(0(2(x1)))))) 3(5(0(0(1(x1))))) -> 0(5(3(1(3(0(x1)))))) 3(2(1(0(1(x1))))) -> 0(3(2(1(1(1(x1)))))) 3(2(1(0(1(x1))))) -> 1(5(3(0(2(1(x1)))))) 3(5(3(5(1(x1))))) -> 1(3(3(0(5(5(x1)))))) 3(2(1(0(4(x1))))) -> 3(0(4(4(1(2(x1)))))) 3(2(1(0(4(x1))))) -> 0(0(1(2(3(4(x1)))))) 3(5(0(0(5(x1))))) -> 0(5(5(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 5(2(1(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 1(2(0(3(5(4(x1)))))) 3(2(1(0(5(x1))))) -> 2(0(3(0(1(5(x1)))))) 3(2(1(0(5(x1))))) -> 2(1(3(0(1(5(x1)))))) 3(2(4(0(5(x1))))) -> 3(5(1(4(2(0(x1)))))) 3(2(4(0(5(x1))))) -> 4(2(5(3(3(0(x1)))))) 3(2(4(0(5(x1))))) -> 5(3(0(4(4(2(x1)))))) 3(2(5(0(5(x1))))) -> 2(4(5(3(0(5(x1)))))) 3(5(0(3(5(x1))))) -> 3(0(0(3(5(5(x1)))))) 3(2(1(3(5(x1))))) -> 3(5(1(3(0(2(x1)))))) 3(2(1(3(5(x1))))) -> 5(1(2(0(3(3(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. 3^1(2(1(0(x1)))) -> 3^1(2(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(3^1(x_1)) = [[0A]] + [[-I, 0A, 0A]] * x_1 >>> <<< POL(2(x_1)) = [[1A], [0A], [0A]] + [[0A, -I, -I], [-I, -I, -I], [-I, 0A, -I]] * x_1 >>> <<< POL(1(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 1A], [-I, -I, 0A], [-I, -I, 0A]] * x_1 >>> <<< POL(0(x_1)) = [[0A], [0A], [1A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 1A, -I]] * x_1 >>> <<< POL(3(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, -I], [0A, 0A, -I], [0A, 0A, -I]] * x_1 >>> <<< POL(5(x_1)) = [[1A], [0A], [0A]] + [[0A, 0A, 1A], [0A, 0A, 0A], [-I, -I, 0A]] * x_1 >>> <<< POL(4(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: 3(2(1(0(x1)))) -> 2(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 3(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 1(2(3(0(2(x1))))) 3(2(1(0(x1)))) -> 0(3(0(1(2(x1))))) 3(2(1(0(x1)))) -> 2(0(3(1(2(x1))))) 3(2(1(0(x1)))) -> 0(3(4(1(2(x1))))) 3(2(1(0(x1)))) -> 1(2(0(3(2(x1))))) 3(2(1(0(x1)))) -> 1(1(2(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 1(2(3(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 0(3(0(3(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(3(3(0(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(4(3(0(1(2(x1)))))) 3(2(5(0(x1)))) -> 5(3(3(0(2(x1))))) 3(2(5(0(x1)))) -> 3(5(3(0(0(2(x1)))))) 3(2(1(0(0(x1))))) -> 0(1(3(0(2(0(x1)))))) 3(2(1(0(0(x1))))) -> 0(0(3(1(1(2(x1)))))) 3(2(4(0(0(x1))))) -> 2(3(0(4(3(0(x1)))))) 3(2(4(0(0(x1))))) -> 4(4(2(0(0(3(x1)))))) 3(2(5(0(0(x1))))) -> 0(4(2(5(3(0(x1)))))) 3(2(0(1(0(x1))))) -> 0(0(1(4(2(3(x1)))))) 3(5(0(1(0(x1))))) -> 3(0(5(1(1(0(x1)))))) 3(5(0(1(0(x1))))) -> 4(3(0(0(5(1(x1)))))) 3(1(2(1(0(x1))))) -> 1(3(0(0(1(2(x1)))))) 3(2(2(1(0(x1))))) -> 2(1(3(3(0(2(x1)))))) 3(3(2(1(0(x1))))) -> 1(4(3(3(0(2(x1)))))) 3(5(2(1(0(x1))))) -> 0(1(3(0(5(2(x1)))))) 3(5(5(4(0(x1))))) -> 5(1(4(3(0(5(x1)))))) 3(5(5(4(0(x1))))) -> 5(3(3(0(4(5(x1)))))) 3(1(2(5(0(x1))))) -> 5(3(3(0(1(2(x1)))))) 3(2(4(5(0(x1))))) -> 1(5(4(3(0(2(x1)))))) 3(5(0(0(1(x1))))) -> 0(5(3(1(3(0(x1)))))) 3(2(1(0(1(x1))))) -> 0(3(2(1(1(1(x1)))))) 3(2(1(0(1(x1))))) -> 1(5(3(0(2(1(x1)))))) 3(5(3(5(1(x1))))) -> 1(3(3(0(5(5(x1)))))) 3(2(1(0(4(x1))))) -> 3(0(4(4(1(2(x1)))))) 3(2(1(0(4(x1))))) -> 0(0(1(2(3(4(x1)))))) 3(5(0(0(5(x1))))) -> 0(5(5(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 5(2(1(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 1(2(0(3(5(4(x1)))))) 3(2(1(0(5(x1))))) -> 2(0(3(0(1(5(x1)))))) 3(2(1(0(5(x1))))) -> 2(1(3(0(1(5(x1)))))) 3(2(4(0(5(x1))))) -> 3(5(1(4(2(0(x1)))))) 3(2(4(0(5(x1))))) -> 4(2(5(3(3(0(x1)))))) 3(2(4(0(5(x1))))) -> 5(3(0(4(4(2(x1)))))) 3(2(5(0(5(x1))))) -> 2(4(5(3(0(5(x1)))))) 3(5(0(3(5(x1))))) -> 3(0(0(3(5(5(x1)))))) 3(2(1(3(5(x1))))) -> 3(5(1(3(0(2(x1)))))) 3(2(1(3(5(x1))))) -> 5(1(2(0(3(3(x1)))))) ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(2(0(1(0(x1))))) -> 3^1(x1) 3^1(2(1(3(5(x1))))) -> 3^1(3(x1)) The TRS R consists of the following rules: 3(2(1(0(x1)))) -> 2(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 3(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 1(2(3(0(2(x1))))) 3(2(1(0(x1)))) -> 0(3(0(1(2(x1))))) 3(2(1(0(x1)))) -> 2(0(3(1(2(x1))))) 3(2(1(0(x1)))) -> 0(3(4(1(2(x1))))) 3(2(1(0(x1)))) -> 1(2(0(3(2(x1))))) 3(2(1(0(x1)))) -> 1(1(2(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 1(2(3(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 0(3(0(3(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(3(3(0(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(4(3(0(1(2(x1)))))) 3(2(5(0(x1)))) -> 5(3(3(0(2(x1))))) 3(2(5(0(x1)))) -> 3(5(3(0(0(2(x1)))))) 3(2(1(0(0(x1))))) -> 0(1(3(0(2(0(x1)))))) 3(2(1(0(0(x1))))) -> 0(0(3(1(1(2(x1)))))) 3(2(4(0(0(x1))))) -> 2(3(0(4(3(0(x1)))))) 3(2(4(0(0(x1))))) -> 4(4(2(0(0(3(x1)))))) 3(2(5(0(0(x1))))) -> 0(4(2(5(3(0(x1)))))) 3(2(0(1(0(x1))))) -> 0(0(1(4(2(3(x1)))))) 3(5(0(1(0(x1))))) -> 3(0(5(1(1(0(x1)))))) 3(5(0(1(0(x1))))) -> 4(3(0(0(5(1(x1)))))) 3(1(2(1(0(x1))))) -> 1(3(0(0(1(2(x1)))))) 3(2(2(1(0(x1))))) -> 2(1(3(3(0(2(x1)))))) 3(3(2(1(0(x1))))) -> 1(4(3(3(0(2(x1)))))) 3(5(2(1(0(x1))))) -> 0(1(3(0(5(2(x1)))))) 3(5(5(4(0(x1))))) -> 5(1(4(3(0(5(x1)))))) 3(5(5(4(0(x1))))) -> 5(3(3(0(4(5(x1)))))) 3(1(2(5(0(x1))))) -> 5(3(3(0(1(2(x1)))))) 3(2(4(5(0(x1))))) -> 1(5(4(3(0(2(x1)))))) 3(5(0(0(1(x1))))) -> 0(5(3(1(3(0(x1)))))) 3(2(1(0(1(x1))))) -> 0(3(2(1(1(1(x1)))))) 3(2(1(0(1(x1))))) -> 1(5(3(0(2(1(x1)))))) 3(5(3(5(1(x1))))) -> 1(3(3(0(5(5(x1)))))) 3(2(1(0(4(x1))))) -> 3(0(4(4(1(2(x1)))))) 3(2(1(0(4(x1))))) -> 0(0(1(2(3(4(x1)))))) 3(5(0(0(5(x1))))) -> 0(5(5(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 5(2(1(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 1(2(0(3(5(4(x1)))))) 3(2(1(0(5(x1))))) -> 2(0(3(0(1(5(x1)))))) 3(2(1(0(5(x1))))) -> 2(1(3(0(1(5(x1)))))) 3(2(4(0(5(x1))))) -> 3(5(1(4(2(0(x1)))))) 3(2(4(0(5(x1))))) -> 4(2(5(3(3(0(x1)))))) 3(2(4(0(5(x1))))) -> 5(3(0(4(4(2(x1)))))) 3(2(5(0(5(x1))))) -> 2(4(5(3(0(5(x1)))))) 3(5(0(3(5(x1))))) -> 3(0(0(3(5(5(x1)))))) 3(2(1(3(5(x1))))) -> 3(5(1(3(0(2(x1)))))) 3(2(1(3(5(x1))))) -> 5(1(2(0(3(3(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. 3^1(2(0(1(0(x1))))) -> 3^1(x1) 3^1(2(1(3(5(x1))))) -> 3^1(3(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(3^1(x_1)) = [[0A]] + [[0A, 0A, 1A]] * x_1 >>> <<< POL(2(x_1)) = [[0A], [-I], [0A]] + [[-I, -I, 0A], [-I, -I, -I], [-I, 0A, 1A]] * x_1 >>> <<< POL(0(x_1)) = [[1A], [1A], [-I]] + [[1A, 0A, 1A], [0A, 0A, -I], [-I, -I, -I]] * x_1 >>> <<< POL(1(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, -I], [0A, -I, -I], [-I, 0A, -I]] * x_1 >>> <<< POL(3(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, 1A], [-I, -I, 0A], [-I, 0A, 0A]] * x_1 >>> <<< POL(5(x_1)) = [[0A], [-I], [0A]] + [[-I, -I, -I], [-I, -I, -I], [0A, 1A, 0A]] * x_1 >>> <<< POL(4(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, -I], [-I, -I, -I], [1A, 0A, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 3(2(1(0(x1)))) -> 2(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 3(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 1(2(3(0(2(x1))))) 3(2(1(0(x1)))) -> 0(3(0(1(2(x1))))) 3(2(1(0(x1)))) -> 2(0(3(1(2(x1))))) 3(2(1(0(x1)))) -> 0(3(4(1(2(x1))))) 3(2(1(0(x1)))) -> 1(2(0(3(2(x1))))) 3(2(1(0(x1)))) -> 1(1(2(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 1(2(3(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 0(3(0(3(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(3(3(0(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(4(3(0(1(2(x1)))))) 3(2(5(0(x1)))) -> 5(3(3(0(2(x1))))) 3(2(5(0(x1)))) -> 3(5(3(0(0(2(x1)))))) 3(2(1(0(0(x1))))) -> 0(1(3(0(2(0(x1)))))) 3(2(1(0(0(x1))))) -> 0(0(3(1(1(2(x1)))))) 3(2(4(0(0(x1))))) -> 2(3(0(4(3(0(x1)))))) 3(2(4(0(0(x1))))) -> 4(4(2(0(0(3(x1)))))) 3(2(5(0(0(x1))))) -> 0(4(2(5(3(0(x1)))))) 3(2(0(1(0(x1))))) -> 0(0(1(4(2(3(x1)))))) 3(5(0(1(0(x1))))) -> 3(0(5(1(1(0(x1)))))) 3(5(0(1(0(x1))))) -> 4(3(0(0(5(1(x1)))))) 3(1(2(1(0(x1))))) -> 1(3(0(0(1(2(x1)))))) 3(2(2(1(0(x1))))) -> 2(1(3(3(0(2(x1)))))) 3(3(2(1(0(x1))))) -> 1(4(3(3(0(2(x1)))))) 3(5(2(1(0(x1))))) -> 0(1(3(0(5(2(x1)))))) 3(5(5(4(0(x1))))) -> 5(1(4(3(0(5(x1)))))) 3(5(5(4(0(x1))))) -> 5(3(3(0(4(5(x1)))))) 3(1(2(5(0(x1))))) -> 5(3(3(0(1(2(x1)))))) 3(2(4(5(0(x1))))) -> 1(5(4(3(0(2(x1)))))) 3(5(0(0(1(x1))))) -> 0(5(3(1(3(0(x1)))))) 3(2(1(0(1(x1))))) -> 0(3(2(1(1(1(x1)))))) 3(2(1(0(1(x1))))) -> 1(5(3(0(2(1(x1)))))) 3(5(3(5(1(x1))))) -> 1(3(3(0(5(5(x1)))))) 3(2(1(0(4(x1))))) -> 3(0(4(4(1(2(x1)))))) 3(2(1(0(4(x1))))) -> 0(0(1(2(3(4(x1)))))) 3(5(0(0(5(x1))))) -> 0(5(5(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 5(2(1(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 1(2(0(3(5(4(x1)))))) 3(2(1(0(5(x1))))) -> 2(0(3(0(1(5(x1)))))) 3(2(1(0(5(x1))))) -> 2(1(3(0(1(5(x1)))))) 3(2(4(0(5(x1))))) -> 3(5(1(4(2(0(x1)))))) 3(2(4(0(5(x1))))) -> 4(2(5(3(3(0(x1)))))) 3(2(4(0(5(x1))))) -> 5(3(0(4(4(2(x1)))))) 3(2(5(0(5(x1))))) -> 2(4(5(3(0(5(x1)))))) 3(5(0(3(5(x1))))) -> 3(0(0(3(5(5(x1)))))) 3(2(1(3(5(x1))))) -> 3(5(1(3(0(2(x1)))))) 3(2(1(3(5(x1))))) -> 5(1(2(0(3(3(x1)))))) ---------------------------------------- (18) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: 3(2(1(0(x1)))) -> 2(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 3(1(3(0(2(x1))))) 3(2(1(0(x1)))) -> 1(2(3(0(2(x1))))) 3(2(1(0(x1)))) -> 0(3(0(1(2(x1))))) 3(2(1(0(x1)))) -> 2(0(3(1(2(x1))))) 3(2(1(0(x1)))) -> 0(3(4(1(2(x1))))) 3(2(1(0(x1)))) -> 1(2(0(3(2(x1))))) 3(2(1(0(x1)))) -> 1(1(2(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 1(2(3(3(0(2(x1)))))) 3(2(1(0(x1)))) -> 0(3(0(3(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(3(3(0(1(2(x1)))))) 3(5(1(0(x1)))) -> 5(4(3(0(1(2(x1)))))) 3(2(5(0(x1)))) -> 5(3(3(0(2(x1))))) 3(2(5(0(x1)))) -> 3(5(3(0(0(2(x1)))))) 3(2(1(0(0(x1))))) -> 0(1(3(0(2(0(x1)))))) 3(2(1(0(0(x1))))) -> 0(0(3(1(1(2(x1)))))) 3(2(4(0(0(x1))))) -> 2(3(0(4(3(0(x1)))))) 3(2(4(0(0(x1))))) -> 4(4(2(0(0(3(x1)))))) 3(2(5(0(0(x1))))) -> 0(4(2(5(3(0(x1)))))) 3(2(0(1(0(x1))))) -> 0(0(1(4(2(3(x1)))))) 3(5(0(1(0(x1))))) -> 3(0(5(1(1(0(x1)))))) 3(5(0(1(0(x1))))) -> 4(3(0(0(5(1(x1)))))) 3(1(2(1(0(x1))))) -> 1(3(0(0(1(2(x1)))))) 3(2(2(1(0(x1))))) -> 2(1(3(3(0(2(x1)))))) 3(3(2(1(0(x1))))) -> 1(4(3(3(0(2(x1)))))) 3(5(2(1(0(x1))))) -> 0(1(3(0(5(2(x1)))))) 3(5(5(4(0(x1))))) -> 5(1(4(3(0(5(x1)))))) 3(5(5(4(0(x1))))) -> 5(3(3(0(4(5(x1)))))) 3(1(2(5(0(x1))))) -> 5(3(3(0(1(2(x1)))))) 3(2(4(5(0(x1))))) -> 1(5(4(3(0(2(x1)))))) 3(5(0(0(1(x1))))) -> 0(5(3(1(3(0(x1)))))) 3(2(1(0(1(x1))))) -> 0(3(2(1(1(1(x1)))))) 3(2(1(0(1(x1))))) -> 1(5(3(0(2(1(x1)))))) 3(5(3(5(1(x1))))) -> 1(3(3(0(5(5(x1)))))) 3(2(1(0(4(x1))))) -> 3(0(4(4(1(2(x1)))))) 3(2(1(0(4(x1))))) -> 0(0(1(2(3(4(x1)))))) 3(5(0(0(5(x1))))) -> 0(5(5(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 5(2(1(3(0(3(x1)))))) 3(2(1(0(5(x1))))) -> 1(2(0(3(5(4(x1)))))) 3(2(1(0(5(x1))))) -> 2(0(3(0(1(5(x1)))))) 3(2(1(0(5(x1))))) -> 2(1(3(0(1(5(x1)))))) 3(2(4(0(5(x1))))) -> 3(5(1(4(2(0(x1)))))) 3(2(4(0(5(x1))))) -> 4(2(5(3(3(0(x1)))))) 3(2(4(0(5(x1))))) -> 5(3(0(4(4(2(x1)))))) 3(2(5(0(5(x1))))) -> 2(4(5(3(0(5(x1)))))) 3(5(0(3(5(x1))))) -> 3(0(0(3(5(5(x1)))))) 3(2(1(3(5(x1))))) -> 3(5(1(3(0(2(x1)))))) 3(2(1(3(5(x1))))) -> 5(1(2(0(3(3(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (20) YES