278.50/71.01 YES 299.06/76.16 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 299.06/76.16 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 299.06/76.16 299.06/76.16 299.06/76.16 Termination w.r.t. Q of the given QTRS could be proven: 299.06/76.16 299.06/76.16 (0) QTRS 299.06/76.16 (1) QTRS Reverse [EQUIVALENT, 0 ms] 299.06/76.16 (2) QTRS 299.06/76.16 (3) DependencyPairsProof [EQUIVALENT, 139 ms] 299.06/76.16 (4) QDP 299.06/76.16 (5) DependencyGraphProof [EQUIVALENT, 5 ms] 299.06/76.16 (6) QDP 299.06/76.16 (7) QDPOrderProof [EQUIVALENT, 5733 ms] 299.06/76.16 (8) QDP 299.06/76.16 (9) QDPOrderProof [EQUIVALENT, 5565 ms] 299.06/76.16 (10) QDP 299.06/76.16 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 299.06/76.16 (12) QDP 299.06/76.16 (13) QDPOrderProof [EQUIVALENT, 3287 ms] 299.06/76.16 (14) QDP 299.06/76.16 (15) QDPOrderProof [EQUIVALENT, 5236 ms] 299.06/76.16 (16) QDP 299.06/76.16 (17) QDPOrderProof [EQUIVALENT, 24 ms] 299.06/76.16 (18) QDP 299.06/76.16 (19) QDPOrderProof [EQUIVALENT, 24 ms] 299.06/76.16 (20) QDP 299.06/76.16 (21) PisEmptyProof [EQUIVALENT, 0 ms] 299.06/76.16 (22) YES 299.06/76.16 299.06/76.16 299.06/76.16 ---------------------------------------- 299.06/76.16 299.06/76.16 (0) 299.06/76.16 Obligation: 299.06/76.16 Q restricted rewrite system: 299.06/76.16 The TRS R consists of the following rules: 299.06/76.16 299.06/76.16 0(1(0(2(x1)))) -> 3(0(1(2(0(x1))))) 299.06/76.16 1(0(0(2(x1)))) -> 2(3(0(0(1(x1))))) 299.06/76.16 1(0(0(2(x1)))) -> 1(0(1(1(2(0(x1)))))) 299.06/76.16 1(0(0(2(x1)))) -> 1(3(3(0(2(0(x1)))))) 299.06/76.16 1(0(0(2(x1)))) -> 3(0(0(1(1(2(x1)))))) 299.06/76.16 1(0(2(2(x1)))) -> 2(1(2(3(0(x1))))) 299.06/76.16 1(0(2(2(x1)))) -> 1(1(1(2(2(0(x1)))))) 299.06/76.16 1(3(4(2(x1)))) -> 1(1(2(3(4(x1))))) 299.06/76.16 1(3(4(2(x1)))) -> 1(2(3(3(4(5(x1)))))) 299.06/76.16 1(3(4(2(x1)))) -> 1(2(3(4(4(4(x1)))))) 299.06/76.16 1(3(4(2(x1)))) -> 1(2(3(4(4(5(x1)))))) 299.06/76.16 1(3(4(2(x1)))) -> 1(2(3(5(4(5(x1)))))) 299.06/76.16 1(3(4(2(x1)))) -> 1(2(3(5(5(4(x1)))))) 299.06/76.16 1(3(5(2(x1)))) -> 2(3(1(0(5(x1))))) 299.06/76.16 1(3(5(2(x1)))) -> 4(5(1(2(3(x1))))) 299.06/76.16 3(1(0(2(x1)))) -> 3(0(1(2(0(x1))))) 299.06/76.16 3(4(0(2(x1)))) -> 3(0(0(2(4(x1))))) 299.06/76.16 3(4(0(2(x1)))) -> 3(0(1(2(4(x1))))) 299.06/76.16 3(4(0(2(x1)))) -> 3(0(5(2(4(x1))))) 299.06/76.16 3(4(0(2(x1)))) -> 3(1(0(2(4(x1))))) 299.06/76.16 3(4(0(2(x1)))) -> 4(1(2(3(0(x1))))) 299.06/76.16 3(4(0(2(x1)))) -> 2(0(4(3(3(0(x1)))))) 299.06/76.16 3(4(0(2(x1)))) -> 3(0(2(1(1(4(x1)))))) 299.06/76.16 3(4(0(2(x1)))) -> 3(3(0(5(2(4(x1)))))) 299.06/76.16 3(4(0(2(x1)))) -> 4(2(3(3(3(0(x1)))))) 299.06/76.16 3(5(0(2(x1)))) -> 3(0(3(2(5(x1))))) 299.06/76.16 3(5(0(2(x1)))) -> 2(5(3(0(0(0(x1)))))) 299.06/76.16 3(5(0(2(x1)))) -> 3(0(0(2(5(4(x1)))))) 299.06/76.16 3(5(0(2(x1)))) -> 5(5(2(3(0(3(x1)))))) 299.06/76.16 0(2(3(4(2(x1))))) -> 3(0(2(2(4(5(x1)))))) 299.06/76.16 0(3(4(0(2(x1))))) -> 0(3(1(0(4(2(x1)))))) 299.06/76.16 0(3(5(0(2(x1))))) -> 2(1(0(5(3(0(x1)))))) 299.06/76.16 1(0(0(4(2(x1))))) -> 0(4(5(1(0(2(x1)))))) 299.06/76.16 1(0(3(5(2(x1))))) -> 4(5(3(1(2(0(x1)))))) 299.06/76.16 1(1(0(2(2(x1))))) -> 1(1(2(0(1(2(x1)))))) 299.06/76.16 1(3(4(0(2(x1))))) -> 1(2(3(0(1(4(x1)))))) 299.06/76.16 1(3(5(0(2(x1))))) -> 1(0(2(5(2(3(x1)))))) 299.06/76.16 3(1(0(0(2(x1))))) -> 3(0(1(2(2(0(x1)))))) 299.06/76.16 3(1(3(5(2(x1))))) -> 3(0(3(5(1(2(x1)))))) 299.06/76.16 3(3(4(0(2(x1))))) -> 3(0(0(4(2(3(x1)))))) 299.06/76.16 3(4(0(0(2(x1))))) -> 3(0(0(4(4(2(x1)))))) 299.06/76.16 3(4(1(0(2(x1))))) -> 3(1(2(3(0(4(x1)))))) 299.06/76.16 3(4(1(5(2(x1))))) -> 2(3(0(1(4(5(x1)))))) 299.06/76.16 3(4(2(0(2(x1))))) -> 3(2(0(4(2(3(x1)))))) 299.06/76.16 3(4(2(0(2(x1))))) -> 4(2(2(3(0(3(x1)))))) 299.06/76.16 3(5(0(0(2(x1))))) -> 0(3(0(2(4(5(x1)))))) 299.06/76.16 3(5(0(4(2(x1))))) -> 2(0(4(4(5(3(x1)))))) 299.06/76.16 3(5(0(4(2(x1))))) -> 2(5(4(3(0(0(x1)))))) 299.06/76.16 3(5(3(4(2(x1))))) -> 0(5(4(2(3(3(x1)))))) 299.06/76.16 299.06/76.16 Q is empty. 299.06/76.16 299.06/76.16 ---------------------------------------- 299.06/76.16 299.06/76.16 (1) QTRS Reverse (EQUIVALENT) 299.06/76.16 We applied the QTRS Reverse Processor [REVERSE]. 299.06/76.16 ---------------------------------------- 299.06/76.16 299.06/76.16 (2) 299.06/76.16 Obligation: 299.06/76.16 Q restricted rewrite system: 299.06/76.16 The TRS R consists of the following rules: 299.06/76.16 299.06/76.16 2(0(1(0(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.16 2(0(0(1(x1)))) -> 1(0(0(3(2(x1))))) 299.06/76.16 2(0(0(1(x1)))) -> 0(2(1(1(0(1(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 0(2(0(3(3(1(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 2(1(1(0(0(3(x1)))))) 299.06/76.16 2(2(0(1(x1)))) -> 0(3(2(1(2(x1))))) 299.06/76.16 2(2(0(1(x1)))) -> 0(2(2(1(1(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(3(2(1(1(x1))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(3(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(4(4(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(4(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(5(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(5(5(3(2(1(x1)))))) 299.06/76.16 2(5(3(1(x1)))) -> 5(0(1(3(2(x1))))) 299.06/76.16 2(5(3(1(x1)))) -> 3(2(1(5(4(x1))))) 299.06/76.16 2(0(1(3(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(0(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(1(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(5(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(0(1(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(2(1(4(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(3(4(0(2(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(1(1(2(0(3(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(5(0(3(3(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(3(3(2(4(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 5(2(3(0(3(x1))))) 299.06/76.16 2(0(5(3(x1)))) -> 0(0(0(3(5(2(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 4(5(2(0(0(3(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 3(0(3(2(5(5(x1)))))) 299.06/76.16 2(4(3(2(0(x1))))) -> 5(4(2(2(0(3(x1)))))) 299.06/76.16 2(0(4(3(0(x1))))) -> 2(4(0(1(3(0(x1)))))) 299.06/76.16 2(0(5(3(0(x1))))) -> 0(3(5(0(1(2(x1)))))) 299.06/76.16 2(4(0(0(1(x1))))) -> 2(0(1(5(4(0(x1)))))) 299.06/76.16 2(5(3(0(1(x1))))) -> 0(2(1(3(5(4(x1)))))) 299.06/76.16 2(2(0(1(1(x1))))) -> 2(1(0(2(1(1(x1)))))) 299.06/76.16 2(0(4(3(1(x1))))) -> 4(1(0(3(2(1(x1)))))) 299.06/76.16 2(0(5(3(1(x1))))) -> 3(2(5(2(0(1(x1)))))) 299.06/76.16 2(0(0(1(3(x1))))) -> 0(2(2(1(0(3(x1)))))) 299.06/76.16 2(5(3(1(3(x1))))) -> 2(1(5(3(0(3(x1)))))) 299.06/76.16 2(0(4(3(3(x1))))) -> 3(2(4(0(0(3(x1)))))) 299.06/76.16 2(0(0(4(3(x1))))) -> 2(4(4(0(0(3(x1)))))) 299.06/76.16 2(0(1(4(3(x1))))) -> 4(0(3(2(1(3(x1)))))) 299.06/76.16 2(5(1(4(3(x1))))) -> 5(4(1(0(3(2(x1)))))) 299.06/76.16 2(0(2(4(3(x1))))) -> 3(2(4(0(2(3(x1)))))) 299.06/76.16 2(0(2(4(3(x1))))) -> 3(0(3(2(2(4(x1)))))) 299.06/76.16 2(0(0(5(3(x1))))) -> 5(4(2(0(3(0(x1)))))) 299.06/76.16 2(4(0(5(3(x1))))) -> 3(5(4(4(0(2(x1)))))) 299.06/76.16 2(4(0(5(3(x1))))) -> 0(0(3(4(5(2(x1)))))) 299.06/76.16 2(4(3(5(3(x1))))) -> 3(3(2(4(5(0(x1)))))) 299.06/76.16 299.06/76.16 Q is empty. 299.06/76.16 299.06/76.16 ---------------------------------------- 299.06/76.16 299.06/76.16 (3) DependencyPairsProof (EQUIVALENT) 299.06/76.16 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 299.06/76.16 ---------------------------------------- 299.06/76.16 299.06/76.16 (4) 299.06/76.16 Obligation: 299.06/76.16 Q DP problem: 299.06/76.16 The TRS P consists of the following rules: 299.06/76.16 299.06/76.16 2^1(0(1(0(x1)))) -> 2^1(1(0(3(x1)))) 299.06/76.16 2^1(0(0(1(x1)))) -> 2^1(x1) 299.06/76.16 2^1(0(0(1(x1)))) -> 2^1(1(1(0(1(x1))))) 299.06/76.16 2^1(0(0(1(x1)))) -> 2^1(0(3(3(1(x1))))) 299.06/76.16 2^1(0(0(1(x1)))) -> 2^1(1(1(0(0(3(x1)))))) 299.06/76.16 2^1(2(0(1(x1)))) -> 2^1(1(2(x1))) 299.06/76.16 2^1(2(0(1(x1)))) -> 2^1(x1) 299.06/76.16 2^1(2(0(1(x1)))) -> 2^1(2(1(1(1(x1))))) 299.06/76.16 2^1(2(0(1(x1)))) -> 2^1(1(1(1(x1)))) 299.06/76.16 2^1(4(3(1(x1)))) -> 2^1(1(1(x1))) 299.06/76.16 2^1(4(3(1(x1)))) -> 2^1(1(x1)) 299.06/76.16 2^1(5(3(1(x1)))) -> 2^1(x1) 299.06/76.16 2^1(5(3(1(x1)))) -> 2^1(1(5(4(x1)))) 299.06/76.16 2^1(0(1(3(x1)))) -> 2^1(1(0(3(x1)))) 299.06/76.16 2^1(0(4(3(x1)))) -> 2^1(0(0(3(x1)))) 299.06/76.16 2^1(0(4(3(x1)))) -> 2^1(1(0(3(x1)))) 299.06/76.16 2^1(0(4(3(x1)))) -> 2^1(5(0(3(x1)))) 299.06/76.16 2^1(0(4(3(x1)))) -> 2^1(0(1(3(x1)))) 299.06/76.16 2^1(0(4(3(x1)))) -> 2^1(1(4(x1))) 299.06/76.16 2^1(0(4(3(x1)))) -> 2^1(x1) 299.06/76.16 2^1(0(4(3(x1)))) -> 2^1(0(3(x1))) 299.06/76.16 2^1(0(4(3(x1)))) -> 2^1(5(0(3(3(x1))))) 299.06/76.16 2^1(0(4(3(x1)))) -> 2^1(4(x1)) 299.06/76.16 2^1(0(5(3(x1)))) -> 2^1(3(0(3(x1)))) 299.06/76.16 2^1(0(5(3(x1)))) -> 2^1(x1) 299.06/76.16 2^1(0(5(3(x1)))) -> 2^1(0(0(3(x1)))) 299.06/76.16 2^1(0(5(3(x1)))) -> 2^1(5(5(x1))) 299.06/76.16 2^1(4(3(2(0(x1))))) -> 2^1(2(0(3(x1)))) 299.06/76.16 2^1(4(3(2(0(x1))))) -> 2^1(0(3(x1))) 299.06/76.16 2^1(0(4(3(0(x1))))) -> 2^1(4(0(1(3(0(x1)))))) 299.06/76.16 2^1(0(5(3(0(x1))))) -> 2^1(x1) 299.06/76.16 2^1(4(0(0(1(x1))))) -> 2^1(0(1(5(4(0(x1)))))) 299.06/76.16 2^1(5(3(0(1(x1))))) -> 2^1(1(3(5(4(x1))))) 299.06/76.16 2^1(2(0(1(1(x1))))) -> 2^1(1(0(2(1(1(x1)))))) 299.06/76.16 2^1(2(0(1(1(x1))))) -> 2^1(1(1(x1))) 299.06/76.16 2^1(0(4(3(1(x1))))) -> 2^1(1(x1)) 299.06/76.16 2^1(0(5(3(1(x1))))) -> 2^1(5(2(0(1(x1))))) 299.06/76.16 2^1(0(5(3(1(x1))))) -> 2^1(0(1(x1))) 299.06/76.16 2^1(0(0(1(3(x1))))) -> 2^1(2(1(0(3(x1))))) 299.06/76.16 2^1(0(0(1(3(x1))))) -> 2^1(1(0(3(x1)))) 299.06/76.16 2^1(5(3(1(3(x1))))) -> 2^1(1(5(3(0(3(x1)))))) 299.06/76.16 2^1(0(4(3(3(x1))))) -> 2^1(4(0(0(3(x1))))) 299.06/76.16 2^1(0(0(4(3(x1))))) -> 2^1(4(4(0(0(3(x1)))))) 299.06/76.16 2^1(0(1(4(3(x1))))) -> 2^1(1(3(x1))) 299.06/76.16 2^1(5(1(4(3(x1))))) -> 2^1(x1) 299.06/76.16 2^1(0(2(4(3(x1))))) -> 2^1(4(0(2(3(x1))))) 299.06/76.16 2^1(0(2(4(3(x1))))) -> 2^1(3(x1)) 299.06/76.16 2^1(0(2(4(3(x1))))) -> 2^1(2(4(x1))) 299.06/76.16 2^1(0(2(4(3(x1))))) -> 2^1(4(x1)) 299.06/76.16 2^1(0(0(5(3(x1))))) -> 2^1(0(3(0(x1)))) 299.06/76.16 2^1(4(0(5(3(x1))))) -> 2^1(x1) 299.06/76.16 2^1(4(3(5(3(x1))))) -> 2^1(4(5(0(x1)))) 299.06/76.16 299.06/76.16 The TRS R consists of the following rules: 299.06/76.16 299.06/76.16 2(0(1(0(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.16 2(0(0(1(x1)))) -> 1(0(0(3(2(x1))))) 299.06/76.16 2(0(0(1(x1)))) -> 0(2(1(1(0(1(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 0(2(0(3(3(1(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 2(1(1(0(0(3(x1)))))) 299.06/76.16 2(2(0(1(x1)))) -> 0(3(2(1(2(x1))))) 299.06/76.16 2(2(0(1(x1)))) -> 0(2(2(1(1(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(3(2(1(1(x1))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(3(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(4(4(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(4(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(5(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(5(5(3(2(1(x1)))))) 299.06/76.16 2(5(3(1(x1)))) -> 5(0(1(3(2(x1))))) 299.06/76.16 2(5(3(1(x1)))) -> 3(2(1(5(4(x1))))) 299.06/76.16 2(0(1(3(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(0(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(1(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(5(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(0(1(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(2(1(4(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(3(4(0(2(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(1(1(2(0(3(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(5(0(3(3(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(3(3(2(4(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 5(2(3(0(3(x1))))) 299.06/76.16 2(0(5(3(x1)))) -> 0(0(0(3(5(2(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 4(5(2(0(0(3(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 3(0(3(2(5(5(x1)))))) 299.06/76.16 2(4(3(2(0(x1))))) -> 5(4(2(2(0(3(x1)))))) 299.06/76.16 2(0(4(3(0(x1))))) -> 2(4(0(1(3(0(x1)))))) 299.06/76.16 2(0(5(3(0(x1))))) -> 0(3(5(0(1(2(x1)))))) 299.06/76.16 2(4(0(0(1(x1))))) -> 2(0(1(5(4(0(x1)))))) 299.06/76.16 2(5(3(0(1(x1))))) -> 0(2(1(3(5(4(x1)))))) 299.06/76.16 2(2(0(1(1(x1))))) -> 2(1(0(2(1(1(x1)))))) 299.06/76.16 2(0(4(3(1(x1))))) -> 4(1(0(3(2(1(x1)))))) 299.06/76.16 2(0(5(3(1(x1))))) -> 3(2(5(2(0(1(x1)))))) 299.06/76.16 2(0(0(1(3(x1))))) -> 0(2(2(1(0(3(x1)))))) 299.06/76.16 2(5(3(1(3(x1))))) -> 2(1(5(3(0(3(x1)))))) 299.06/76.16 2(0(4(3(3(x1))))) -> 3(2(4(0(0(3(x1)))))) 299.06/76.16 2(0(0(4(3(x1))))) -> 2(4(4(0(0(3(x1)))))) 299.06/76.16 2(0(1(4(3(x1))))) -> 4(0(3(2(1(3(x1)))))) 299.06/76.16 2(5(1(4(3(x1))))) -> 5(4(1(0(3(2(x1)))))) 299.06/76.16 2(0(2(4(3(x1))))) -> 3(2(4(0(2(3(x1)))))) 299.06/76.16 2(0(2(4(3(x1))))) -> 3(0(3(2(2(4(x1)))))) 299.06/76.16 2(0(0(5(3(x1))))) -> 5(4(2(0(3(0(x1)))))) 299.06/76.16 2(4(0(5(3(x1))))) -> 3(5(4(4(0(2(x1)))))) 299.06/76.16 2(4(0(5(3(x1))))) -> 0(0(3(4(5(2(x1)))))) 299.06/76.16 2(4(3(5(3(x1))))) -> 3(3(2(4(5(0(x1)))))) 299.06/76.16 299.06/76.16 Q is empty. 299.06/76.16 We have to consider all minimal (P,Q,R)-chains. 299.06/76.16 ---------------------------------------- 299.06/76.16 299.06/76.16 (5) DependencyGraphProof (EQUIVALENT) 299.06/76.16 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 40 less nodes. 299.06/76.16 ---------------------------------------- 299.06/76.16 299.06/76.16 (6) 299.06/76.16 Obligation: 299.06/76.16 Q DP problem: 299.06/76.16 The TRS P consists of the following rules: 299.06/76.16 299.06/76.16 2^1(2(0(1(x1)))) -> 2^1(x1) 299.06/76.16 2^1(0(0(1(x1)))) -> 2^1(x1) 299.06/76.16 2^1(5(3(1(x1)))) -> 2^1(x1) 299.06/76.16 2^1(0(4(3(x1)))) -> 2^1(x1) 299.06/76.16 2^1(0(4(3(x1)))) -> 2^1(4(x1)) 299.06/76.16 2^1(4(0(5(3(x1))))) -> 2^1(x1) 299.06/76.16 2^1(0(5(3(x1)))) -> 2^1(x1) 299.06/76.16 2^1(0(5(3(0(x1))))) -> 2^1(x1) 299.06/76.16 2^1(0(5(3(1(x1))))) -> 2^1(5(2(0(1(x1))))) 299.06/76.16 2^1(5(1(4(3(x1))))) -> 2^1(x1) 299.06/76.16 2^1(0(2(4(3(x1))))) -> 2^1(2(4(x1))) 299.06/76.16 2^1(0(2(4(3(x1))))) -> 2^1(4(x1)) 299.06/76.16 299.06/76.16 The TRS R consists of the following rules: 299.06/76.16 299.06/76.16 2(0(1(0(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.16 2(0(0(1(x1)))) -> 1(0(0(3(2(x1))))) 299.06/76.16 2(0(0(1(x1)))) -> 0(2(1(1(0(1(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 0(2(0(3(3(1(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 2(1(1(0(0(3(x1)))))) 299.06/76.16 2(2(0(1(x1)))) -> 0(3(2(1(2(x1))))) 299.06/76.16 2(2(0(1(x1)))) -> 0(2(2(1(1(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(3(2(1(1(x1))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(3(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(4(4(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(4(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(5(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(5(5(3(2(1(x1)))))) 299.06/76.16 2(5(3(1(x1)))) -> 5(0(1(3(2(x1))))) 299.06/76.16 2(5(3(1(x1)))) -> 3(2(1(5(4(x1))))) 299.06/76.16 2(0(1(3(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(0(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(1(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(5(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(0(1(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(2(1(4(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(3(4(0(2(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(1(1(2(0(3(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(5(0(3(3(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(3(3(2(4(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 5(2(3(0(3(x1))))) 299.06/76.16 2(0(5(3(x1)))) -> 0(0(0(3(5(2(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 4(5(2(0(0(3(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 3(0(3(2(5(5(x1)))))) 299.06/76.16 2(4(3(2(0(x1))))) -> 5(4(2(2(0(3(x1)))))) 299.06/76.16 2(0(4(3(0(x1))))) -> 2(4(0(1(3(0(x1)))))) 299.06/76.16 2(0(5(3(0(x1))))) -> 0(3(5(0(1(2(x1)))))) 299.06/76.16 2(4(0(0(1(x1))))) -> 2(0(1(5(4(0(x1)))))) 299.06/76.16 2(5(3(0(1(x1))))) -> 0(2(1(3(5(4(x1)))))) 299.06/76.16 2(2(0(1(1(x1))))) -> 2(1(0(2(1(1(x1)))))) 299.06/76.16 2(0(4(3(1(x1))))) -> 4(1(0(3(2(1(x1)))))) 299.06/76.16 2(0(5(3(1(x1))))) -> 3(2(5(2(0(1(x1)))))) 299.06/76.16 2(0(0(1(3(x1))))) -> 0(2(2(1(0(3(x1)))))) 299.06/76.16 2(5(3(1(3(x1))))) -> 2(1(5(3(0(3(x1)))))) 299.06/76.16 2(0(4(3(3(x1))))) -> 3(2(4(0(0(3(x1)))))) 299.06/76.16 2(0(0(4(3(x1))))) -> 2(4(4(0(0(3(x1)))))) 299.06/76.16 2(0(1(4(3(x1))))) -> 4(0(3(2(1(3(x1)))))) 299.06/76.16 2(5(1(4(3(x1))))) -> 5(4(1(0(3(2(x1)))))) 299.06/76.16 2(0(2(4(3(x1))))) -> 3(2(4(0(2(3(x1)))))) 299.06/76.16 2(0(2(4(3(x1))))) -> 3(0(3(2(2(4(x1)))))) 299.06/76.16 2(0(0(5(3(x1))))) -> 5(4(2(0(3(0(x1)))))) 299.06/76.16 2(4(0(5(3(x1))))) -> 3(5(4(4(0(2(x1)))))) 299.06/76.16 2(4(0(5(3(x1))))) -> 0(0(3(4(5(2(x1)))))) 299.06/76.16 2(4(3(5(3(x1))))) -> 3(3(2(4(5(0(x1)))))) 299.06/76.16 299.06/76.16 Q is empty. 299.06/76.16 We have to consider all minimal (P,Q,R)-chains. 299.06/76.16 ---------------------------------------- 299.06/76.16 299.06/76.16 (7) QDPOrderProof (EQUIVALENT) 299.06/76.16 We use the reduction pair processor [LPAR04,JAR06]. 299.06/76.16 299.06/76.16 299.06/76.16 The following pairs can be oriented strictly and are deleted. 299.06/76.16 299.06/76.16 2^1(0(5(3(1(x1))))) -> 2^1(5(2(0(1(x1))))) 299.06/76.16 The remaining pairs can at least be oriented weakly. 299.06/76.16 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(2^1(x_1)) = [[1A]] + [[-I, -I, 0A]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(2(x_1)) = [[-I], [1A], [0A]] + [[0A, 0A, 0A], [-I, -I, 0A], [0A, 0A, 0A]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(0(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [0A, 0A, 0A], [0A, 0A, -I]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(1(x_1)) = [[-I], [-I], [1A]] + [[0A, -I, 0A], [0A, 0A, 0A], [0A, -I, 0A]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(5(x_1)) = [[0A], [-I], [0A]] + [[-I, -I, -I], [1A, -I, -I], [0A, -I, 0A]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(3(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(4(x_1)) = [[0A], [0A], [1A]] + [[-I, -I, -I], [0A, 0A, 0A], [0A, -I, 0A]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 299.06/76.16 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 299.06/76.16 299.06/76.16 2(0(1(0(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.16 2(0(1(3(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.16 2(0(1(4(3(x1))))) -> 4(0(3(2(1(3(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(3(2(1(1(x1))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(3(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(4(4(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(4(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(5(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(5(5(3(2(1(x1)))))) 299.06/76.16 2(4(3(2(0(x1))))) -> 5(4(2(2(0(3(x1)))))) 299.06/76.16 2(4(0(0(1(x1))))) -> 2(0(1(5(4(0(x1)))))) 299.06/76.16 2(4(0(5(3(x1))))) -> 3(5(4(4(0(2(x1)))))) 299.06/76.16 2(4(0(5(3(x1))))) -> 0(0(3(4(5(2(x1)))))) 299.06/76.16 2(4(3(5(3(x1))))) -> 3(3(2(4(5(0(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 1(0(0(3(2(x1))))) 299.06/76.16 2(2(0(1(x1)))) -> 0(3(2(1(2(x1))))) 299.06/76.16 2(5(3(1(x1)))) -> 5(0(1(3(2(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(3(4(0(2(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(3(3(2(4(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 0(0(0(3(5(2(x1)))))) 299.06/76.16 2(0(5(3(0(x1))))) -> 0(3(5(0(1(2(x1)))))) 299.06/76.16 2(0(5(3(1(x1))))) -> 3(2(5(2(0(1(x1)))))) 299.06/76.16 2(5(1(4(3(x1))))) -> 5(4(1(0(3(2(x1)))))) 299.06/76.16 2(0(2(4(3(x1))))) -> 3(0(3(2(2(4(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 0(2(1(1(0(1(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 0(2(0(3(3(1(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 2(1(1(0(0(3(x1)))))) 299.06/76.16 2(2(0(1(x1)))) -> 0(2(2(1(1(1(x1)))))) 299.06/76.16 2(5(3(1(x1)))) -> 3(2(1(5(4(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(0(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(1(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(5(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(0(1(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(2(1(4(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(1(1(2(0(3(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(5(0(3(3(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 5(2(3(0(3(x1))))) 299.06/76.16 2(0(5(3(x1)))) -> 4(5(2(0(0(3(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 3(0(3(2(5(5(x1)))))) 299.06/76.16 2(0(4(3(0(x1))))) -> 2(4(0(1(3(0(x1)))))) 299.06/76.16 2(5(3(0(1(x1))))) -> 0(2(1(3(5(4(x1)))))) 299.06/76.16 2(2(0(1(1(x1))))) -> 2(1(0(2(1(1(x1)))))) 299.06/76.16 2(0(4(3(1(x1))))) -> 4(1(0(3(2(1(x1)))))) 299.06/76.16 2(0(0(1(3(x1))))) -> 0(2(2(1(0(3(x1)))))) 299.06/76.16 2(5(3(1(3(x1))))) -> 2(1(5(3(0(3(x1)))))) 299.06/76.16 2(0(4(3(3(x1))))) -> 3(2(4(0(0(3(x1)))))) 299.06/76.16 2(0(0(4(3(x1))))) -> 2(4(4(0(0(3(x1)))))) 299.06/76.16 2(0(2(4(3(x1))))) -> 3(2(4(0(2(3(x1)))))) 299.06/76.16 2(0(0(5(3(x1))))) -> 5(4(2(0(3(0(x1)))))) 299.06/76.16 299.06/76.16 299.06/76.16 ---------------------------------------- 299.06/76.16 299.06/76.16 (8) 299.06/76.16 Obligation: 299.06/76.16 Q DP problem: 299.06/76.16 The TRS P consists of the following rules: 299.06/76.16 299.06/76.16 2^1(2(0(1(x1)))) -> 2^1(x1) 299.06/76.16 2^1(0(0(1(x1)))) -> 2^1(x1) 299.06/76.16 2^1(5(3(1(x1)))) -> 2^1(x1) 299.06/76.16 2^1(0(4(3(x1)))) -> 2^1(x1) 299.06/76.16 2^1(0(4(3(x1)))) -> 2^1(4(x1)) 299.06/76.16 2^1(4(0(5(3(x1))))) -> 2^1(x1) 299.06/76.16 2^1(0(5(3(x1)))) -> 2^1(x1) 299.06/76.16 2^1(0(5(3(0(x1))))) -> 2^1(x1) 299.06/76.16 2^1(5(1(4(3(x1))))) -> 2^1(x1) 299.06/76.16 2^1(0(2(4(3(x1))))) -> 2^1(2(4(x1))) 299.06/76.16 2^1(0(2(4(3(x1))))) -> 2^1(4(x1)) 299.06/76.16 299.06/76.16 The TRS R consists of the following rules: 299.06/76.16 299.06/76.16 2(0(1(0(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.16 2(0(0(1(x1)))) -> 1(0(0(3(2(x1))))) 299.06/76.16 2(0(0(1(x1)))) -> 0(2(1(1(0(1(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 0(2(0(3(3(1(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 2(1(1(0(0(3(x1)))))) 299.06/76.16 2(2(0(1(x1)))) -> 0(3(2(1(2(x1))))) 299.06/76.16 2(2(0(1(x1)))) -> 0(2(2(1(1(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(3(2(1(1(x1))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(3(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(4(4(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(4(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(5(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(5(5(3(2(1(x1)))))) 299.06/76.16 2(5(3(1(x1)))) -> 5(0(1(3(2(x1))))) 299.06/76.16 2(5(3(1(x1)))) -> 3(2(1(5(4(x1))))) 299.06/76.16 2(0(1(3(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(0(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(1(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(5(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(0(1(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(2(1(4(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(3(4(0(2(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(1(1(2(0(3(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(5(0(3(3(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(3(3(2(4(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 5(2(3(0(3(x1))))) 299.06/76.16 2(0(5(3(x1)))) -> 0(0(0(3(5(2(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 4(5(2(0(0(3(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 3(0(3(2(5(5(x1)))))) 299.06/76.16 2(4(3(2(0(x1))))) -> 5(4(2(2(0(3(x1)))))) 299.06/76.16 2(0(4(3(0(x1))))) -> 2(4(0(1(3(0(x1)))))) 299.06/76.16 2(0(5(3(0(x1))))) -> 0(3(5(0(1(2(x1)))))) 299.06/76.16 2(4(0(0(1(x1))))) -> 2(0(1(5(4(0(x1)))))) 299.06/76.16 2(5(3(0(1(x1))))) -> 0(2(1(3(5(4(x1)))))) 299.06/76.16 2(2(0(1(1(x1))))) -> 2(1(0(2(1(1(x1)))))) 299.06/76.16 2(0(4(3(1(x1))))) -> 4(1(0(3(2(1(x1)))))) 299.06/76.16 2(0(5(3(1(x1))))) -> 3(2(5(2(0(1(x1)))))) 299.06/76.16 2(0(0(1(3(x1))))) -> 0(2(2(1(0(3(x1)))))) 299.06/76.16 2(5(3(1(3(x1))))) -> 2(1(5(3(0(3(x1)))))) 299.06/76.16 2(0(4(3(3(x1))))) -> 3(2(4(0(0(3(x1)))))) 299.06/76.16 2(0(0(4(3(x1))))) -> 2(4(4(0(0(3(x1)))))) 299.06/76.16 2(0(1(4(3(x1))))) -> 4(0(3(2(1(3(x1)))))) 299.06/76.16 2(5(1(4(3(x1))))) -> 5(4(1(0(3(2(x1)))))) 299.06/76.16 2(0(2(4(3(x1))))) -> 3(2(4(0(2(3(x1)))))) 299.06/76.16 2(0(2(4(3(x1))))) -> 3(0(3(2(2(4(x1)))))) 299.06/76.16 2(0(0(5(3(x1))))) -> 5(4(2(0(3(0(x1)))))) 299.06/76.16 2(4(0(5(3(x1))))) -> 3(5(4(4(0(2(x1)))))) 299.06/76.16 2(4(0(5(3(x1))))) -> 0(0(3(4(5(2(x1)))))) 299.06/76.16 2(4(3(5(3(x1))))) -> 3(3(2(4(5(0(x1)))))) 299.06/76.16 299.06/76.16 Q is empty. 299.06/76.16 We have to consider all minimal (P,Q,R)-chains. 299.06/76.16 ---------------------------------------- 299.06/76.16 299.06/76.16 (9) QDPOrderProof (EQUIVALENT) 299.06/76.16 We use the reduction pair processor [LPAR04,JAR06]. 299.06/76.16 299.06/76.16 299.06/76.16 The following pairs can be oriented strictly and are deleted. 299.06/76.16 299.06/76.16 2^1(0(4(3(x1)))) -> 2^1(x1) 299.06/76.16 2^1(4(0(5(3(x1))))) -> 2^1(x1) 299.06/76.16 The remaining pairs can at least be oriented weakly. 299.06/76.16 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(2^1(x_1)) = [[-I]] + [[0A, 0A, -I]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(2(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(0(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [0A, -I, 0A], [0A, -I, 0A]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(1(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, 0A], [0A, 0A, 0A], [0A, -I, 0A]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(5(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, 0A], [0A, -I, 0A], [-I, -I, -I]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(3(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(4(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [0A, 0A, 1A], [0A, -I, 0A]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 299.06/76.16 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 299.06/76.16 299.06/76.16 2(4(3(1(x1)))) -> 4(3(2(1(1(x1))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(3(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(4(4(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(4(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(5(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(5(5(3(2(1(x1)))))) 299.06/76.16 2(4(3(2(0(x1))))) -> 5(4(2(2(0(3(x1)))))) 299.06/76.16 2(4(0(0(1(x1))))) -> 2(0(1(5(4(0(x1)))))) 299.06/76.16 2(4(0(5(3(x1))))) -> 3(5(4(4(0(2(x1)))))) 299.06/76.16 2(4(0(5(3(x1))))) -> 0(0(3(4(5(2(x1)))))) 299.06/76.16 2(4(3(5(3(x1))))) -> 3(3(2(4(5(0(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 1(0(0(3(2(x1))))) 299.06/76.16 2(2(0(1(x1)))) -> 0(3(2(1(2(x1))))) 299.06/76.16 2(5(3(1(x1)))) -> 5(0(1(3(2(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(3(4(0(2(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(3(3(2(4(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 0(0(0(3(5(2(x1)))))) 299.06/76.16 2(0(5(3(0(x1))))) -> 0(3(5(0(1(2(x1)))))) 299.06/76.16 2(0(5(3(1(x1))))) -> 3(2(5(2(0(1(x1)))))) 299.06/76.16 2(5(1(4(3(x1))))) -> 5(4(1(0(3(2(x1)))))) 299.06/76.16 2(0(2(4(3(x1))))) -> 3(0(3(2(2(4(x1)))))) 299.06/76.16 2(0(1(0(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.16 2(0(0(1(x1)))) -> 0(2(1(1(0(1(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 0(2(0(3(3(1(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 2(1(1(0(0(3(x1)))))) 299.06/76.16 2(2(0(1(x1)))) -> 0(2(2(1(1(1(x1)))))) 299.06/76.16 2(5(3(1(x1)))) -> 3(2(1(5(4(x1))))) 299.06/76.16 2(0(1(3(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(0(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(1(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(5(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(0(1(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(2(1(4(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(1(1(2(0(3(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(5(0(3(3(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 5(2(3(0(3(x1))))) 299.06/76.16 2(0(5(3(x1)))) -> 4(5(2(0(0(3(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 3(0(3(2(5(5(x1)))))) 299.06/76.16 2(0(4(3(0(x1))))) -> 2(4(0(1(3(0(x1)))))) 299.06/76.16 2(5(3(0(1(x1))))) -> 0(2(1(3(5(4(x1)))))) 299.06/76.16 2(2(0(1(1(x1))))) -> 2(1(0(2(1(1(x1)))))) 299.06/76.16 2(0(4(3(1(x1))))) -> 4(1(0(3(2(1(x1)))))) 299.06/76.16 2(0(0(1(3(x1))))) -> 0(2(2(1(0(3(x1)))))) 299.06/76.16 2(5(3(1(3(x1))))) -> 2(1(5(3(0(3(x1)))))) 299.06/76.16 2(0(4(3(3(x1))))) -> 3(2(4(0(0(3(x1)))))) 299.06/76.16 2(0(0(4(3(x1))))) -> 2(4(4(0(0(3(x1)))))) 299.06/76.16 2(0(1(4(3(x1))))) -> 4(0(3(2(1(3(x1)))))) 299.06/76.16 2(0(2(4(3(x1))))) -> 3(2(4(0(2(3(x1)))))) 299.06/76.16 2(0(0(5(3(x1))))) -> 5(4(2(0(3(0(x1)))))) 299.06/76.16 299.06/76.16 299.06/76.16 ---------------------------------------- 299.06/76.16 299.06/76.16 (10) 299.06/76.16 Obligation: 299.06/76.16 Q DP problem: 299.06/76.16 The TRS P consists of the following rules: 299.06/76.16 299.06/76.16 2^1(2(0(1(x1)))) -> 2^1(x1) 299.06/76.16 2^1(0(0(1(x1)))) -> 2^1(x1) 299.06/76.16 2^1(5(3(1(x1)))) -> 2^1(x1) 299.06/76.16 2^1(0(4(3(x1)))) -> 2^1(4(x1)) 299.06/76.16 2^1(0(5(3(x1)))) -> 2^1(x1) 299.06/76.16 2^1(0(5(3(0(x1))))) -> 2^1(x1) 299.06/76.16 2^1(5(1(4(3(x1))))) -> 2^1(x1) 299.06/76.16 2^1(0(2(4(3(x1))))) -> 2^1(2(4(x1))) 299.06/76.16 2^1(0(2(4(3(x1))))) -> 2^1(4(x1)) 299.06/76.16 299.06/76.16 The TRS R consists of the following rules: 299.06/76.16 299.06/76.16 2(0(1(0(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.16 2(0(0(1(x1)))) -> 1(0(0(3(2(x1))))) 299.06/76.16 2(0(0(1(x1)))) -> 0(2(1(1(0(1(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 0(2(0(3(3(1(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 2(1(1(0(0(3(x1)))))) 299.06/76.16 2(2(0(1(x1)))) -> 0(3(2(1(2(x1))))) 299.06/76.16 2(2(0(1(x1)))) -> 0(2(2(1(1(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(3(2(1(1(x1))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(3(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(4(4(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(4(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(5(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(5(5(3(2(1(x1)))))) 299.06/76.16 2(5(3(1(x1)))) -> 5(0(1(3(2(x1))))) 299.06/76.16 2(5(3(1(x1)))) -> 3(2(1(5(4(x1))))) 299.06/76.16 2(0(1(3(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(0(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(1(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(5(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(0(1(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(2(1(4(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(3(4(0(2(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(1(1(2(0(3(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(5(0(3(3(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(3(3(2(4(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 5(2(3(0(3(x1))))) 299.06/76.16 2(0(5(3(x1)))) -> 0(0(0(3(5(2(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 4(5(2(0(0(3(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 3(0(3(2(5(5(x1)))))) 299.06/76.16 2(4(3(2(0(x1))))) -> 5(4(2(2(0(3(x1)))))) 299.06/76.16 2(0(4(3(0(x1))))) -> 2(4(0(1(3(0(x1)))))) 299.06/76.16 2(0(5(3(0(x1))))) -> 0(3(5(0(1(2(x1)))))) 299.06/76.16 2(4(0(0(1(x1))))) -> 2(0(1(5(4(0(x1)))))) 299.06/76.16 2(5(3(0(1(x1))))) -> 0(2(1(3(5(4(x1)))))) 299.06/76.16 2(2(0(1(1(x1))))) -> 2(1(0(2(1(1(x1)))))) 299.06/76.16 2(0(4(3(1(x1))))) -> 4(1(0(3(2(1(x1)))))) 299.06/76.16 2(0(5(3(1(x1))))) -> 3(2(5(2(0(1(x1)))))) 299.06/76.16 2(0(0(1(3(x1))))) -> 0(2(2(1(0(3(x1)))))) 299.06/76.16 2(5(3(1(3(x1))))) -> 2(1(5(3(0(3(x1)))))) 299.06/76.16 2(0(4(3(3(x1))))) -> 3(2(4(0(0(3(x1)))))) 299.06/76.16 2(0(0(4(3(x1))))) -> 2(4(4(0(0(3(x1)))))) 299.06/76.16 2(0(1(4(3(x1))))) -> 4(0(3(2(1(3(x1)))))) 299.06/76.16 2(5(1(4(3(x1))))) -> 5(4(1(0(3(2(x1)))))) 299.06/76.16 2(0(2(4(3(x1))))) -> 3(2(4(0(2(3(x1)))))) 299.06/76.16 2(0(2(4(3(x1))))) -> 3(0(3(2(2(4(x1)))))) 299.06/76.16 2(0(0(5(3(x1))))) -> 5(4(2(0(3(0(x1)))))) 299.06/76.16 2(4(0(5(3(x1))))) -> 3(5(4(4(0(2(x1)))))) 299.06/76.16 2(4(0(5(3(x1))))) -> 0(0(3(4(5(2(x1)))))) 299.06/76.16 2(4(3(5(3(x1))))) -> 3(3(2(4(5(0(x1)))))) 299.06/76.16 299.06/76.16 Q is empty. 299.06/76.16 We have to consider all minimal (P,Q,R)-chains. 299.06/76.16 ---------------------------------------- 299.06/76.16 299.06/76.16 (11) DependencyGraphProof (EQUIVALENT) 299.06/76.16 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 299.06/76.16 ---------------------------------------- 299.06/76.16 299.06/76.16 (12) 299.06/76.16 Obligation: 299.06/76.16 Q DP problem: 299.06/76.16 The TRS P consists of the following rules: 299.06/76.16 299.06/76.16 2^1(0(0(1(x1)))) -> 2^1(x1) 299.06/76.16 2^1(2(0(1(x1)))) -> 2^1(x1) 299.06/76.16 2^1(5(3(1(x1)))) -> 2^1(x1) 299.06/76.16 2^1(0(5(3(x1)))) -> 2^1(x1) 299.06/76.16 2^1(0(5(3(0(x1))))) -> 2^1(x1) 299.06/76.16 2^1(5(1(4(3(x1))))) -> 2^1(x1) 299.06/76.16 2^1(0(2(4(3(x1))))) -> 2^1(2(4(x1))) 299.06/76.16 299.06/76.16 The TRS R consists of the following rules: 299.06/76.16 299.06/76.16 2(0(1(0(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.16 2(0(0(1(x1)))) -> 1(0(0(3(2(x1))))) 299.06/76.16 2(0(0(1(x1)))) -> 0(2(1(1(0(1(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 0(2(0(3(3(1(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 2(1(1(0(0(3(x1)))))) 299.06/76.16 2(2(0(1(x1)))) -> 0(3(2(1(2(x1))))) 299.06/76.16 2(2(0(1(x1)))) -> 0(2(2(1(1(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(3(2(1(1(x1))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(3(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(4(4(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(4(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(5(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(5(5(3(2(1(x1)))))) 299.06/76.16 2(5(3(1(x1)))) -> 5(0(1(3(2(x1))))) 299.06/76.16 2(5(3(1(x1)))) -> 3(2(1(5(4(x1))))) 299.06/76.16 2(0(1(3(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(0(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(1(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(5(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(0(1(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(2(1(4(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(3(4(0(2(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(1(1(2(0(3(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(5(0(3(3(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(3(3(2(4(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 5(2(3(0(3(x1))))) 299.06/76.16 2(0(5(3(x1)))) -> 0(0(0(3(5(2(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 4(5(2(0(0(3(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 3(0(3(2(5(5(x1)))))) 299.06/76.16 2(4(3(2(0(x1))))) -> 5(4(2(2(0(3(x1)))))) 299.06/76.16 2(0(4(3(0(x1))))) -> 2(4(0(1(3(0(x1)))))) 299.06/76.16 2(0(5(3(0(x1))))) -> 0(3(5(0(1(2(x1)))))) 299.06/76.16 2(4(0(0(1(x1))))) -> 2(0(1(5(4(0(x1)))))) 299.06/76.16 2(5(3(0(1(x1))))) -> 0(2(1(3(5(4(x1)))))) 299.06/76.16 2(2(0(1(1(x1))))) -> 2(1(0(2(1(1(x1)))))) 299.06/76.16 2(0(4(3(1(x1))))) -> 4(1(0(3(2(1(x1)))))) 299.06/76.16 2(0(5(3(1(x1))))) -> 3(2(5(2(0(1(x1)))))) 299.06/76.16 2(0(0(1(3(x1))))) -> 0(2(2(1(0(3(x1)))))) 299.06/76.16 2(5(3(1(3(x1))))) -> 2(1(5(3(0(3(x1)))))) 299.06/76.16 2(0(4(3(3(x1))))) -> 3(2(4(0(0(3(x1)))))) 299.06/76.16 2(0(0(4(3(x1))))) -> 2(4(4(0(0(3(x1)))))) 299.06/76.16 2(0(1(4(3(x1))))) -> 4(0(3(2(1(3(x1)))))) 299.06/76.16 2(5(1(4(3(x1))))) -> 5(4(1(0(3(2(x1)))))) 299.06/76.16 2(0(2(4(3(x1))))) -> 3(2(4(0(2(3(x1)))))) 299.06/76.16 2(0(2(4(3(x1))))) -> 3(0(3(2(2(4(x1)))))) 299.06/76.16 2(0(0(5(3(x1))))) -> 5(4(2(0(3(0(x1)))))) 299.06/76.16 2(4(0(5(3(x1))))) -> 3(5(4(4(0(2(x1)))))) 299.06/76.16 2(4(0(5(3(x1))))) -> 0(0(3(4(5(2(x1)))))) 299.06/76.16 2(4(3(5(3(x1))))) -> 3(3(2(4(5(0(x1)))))) 299.06/76.16 299.06/76.16 Q is empty. 299.06/76.16 We have to consider all minimal (P,Q,R)-chains. 299.06/76.16 ---------------------------------------- 299.06/76.16 299.06/76.16 (13) QDPOrderProof (EQUIVALENT) 299.06/76.16 We use the reduction pair processor [LPAR04,JAR06]. 299.06/76.16 299.06/76.16 299.06/76.16 The following pairs can be oriented strictly and are deleted. 299.06/76.16 299.06/76.16 2^1(5(3(1(x1)))) -> 2^1(x1) 299.06/76.16 2^1(0(5(3(x1)))) -> 2^1(x1) 299.06/76.16 2^1(0(5(3(0(x1))))) -> 2^1(x1) 299.06/76.16 The remaining pairs can at least be oriented weakly. 299.06/76.16 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(2^1(x_1)) = [[0A]] + [[-I, 0A, 0A]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(0(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [-I, 0A, 0A], [-I, 0A, -I]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(1(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [0A, 0A, -I], [0A, -I, 0A]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(2(x_1)) = [[0A], [-I], [0A]] + [[-I, -I, 0A], [-I, 0A, -I], [0A, -I, 0A]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(5(x_1)) = [[0A], [0A], [1A]] + [[0A, -I, -I], [1A, -I, -I], [0A, -I, 0A]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(3(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [-I, -I, -I], [-I, 0A, 0A]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(4(x_1)) = [[0A], [1A], [1A]] + [[-I, -I, -I], [0A, -I, 0A], [0A, -I, 0A]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 299.06/76.16 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 299.06/76.16 299.06/76.16 2(4(3(1(x1)))) -> 4(3(2(1(1(x1))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(3(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(4(4(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(4(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(5(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(5(5(3(2(1(x1)))))) 299.06/76.16 2(4(3(2(0(x1))))) -> 5(4(2(2(0(3(x1)))))) 299.06/76.16 2(4(0(0(1(x1))))) -> 2(0(1(5(4(0(x1)))))) 299.06/76.16 2(4(0(5(3(x1))))) -> 3(5(4(4(0(2(x1)))))) 299.06/76.16 2(4(0(5(3(x1))))) -> 0(0(3(4(5(2(x1)))))) 299.06/76.16 2(4(3(5(3(x1))))) -> 3(3(2(4(5(0(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 1(0(0(3(2(x1))))) 299.06/76.16 2(2(0(1(x1)))) -> 0(3(2(1(2(x1))))) 299.06/76.16 2(5(3(1(x1)))) -> 5(0(1(3(2(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(3(4(0(2(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(3(3(2(4(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 0(0(0(3(5(2(x1)))))) 299.06/76.16 2(0(5(3(0(x1))))) -> 0(3(5(0(1(2(x1)))))) 299.06/76.16 2(0(5(3(1(x1))))) -> 3(2(5(2(0(1(x1)))))) 299.06/76.16 2(5(1(4(3(x1))))) -> 5(4(1(0(3(2(x1)))))) 299.06/76.16 2(0(2(4(3(x1))))) -> 3(0(3(2(2(4(x1)))))) 299.06/76.16 2(0(1(0(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.16 2(0(0(1(x1)))) -> 0(2(1(1(0(1(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 0(2(0(3(3(1(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 2(1(1(0(0(3(x1)))))) 299.06/76.16 2(2(0(1(x1)))) -> 0(2(2(1(1(1(x1)))))) 299.06/76.16 2(5(3(1(x1)))) -> 3(2(1(5(4(x1))))) 299.06/76.16 2(0(1(3(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(0(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(1(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(5(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(0(1(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(2(1(4(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(1(1(2(0(3(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(5(0(3(3(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 5(2(3(0(3(x1))))) 299.06/76.16 2(0(5(3(x1)))) -> 4(5(2(0(0(3(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 3(0(3(2(5(5(x1)))))) 299.06/76.16 2(0(4(3(0(x1))))) -> 2(4(0(1(3(0(x1)))))) 299.06/76.16 2(5(3(0(1(x1))))) -> 0(2(1(3(5(4(x1)))))) 299.06/76.16 2(2(0(1(1(x1))))) -> 2(1(0(2(1(1(x1)))))) 299.06/76.16 2(0(4(3(1(x1))))) -> 4(1(0(3(2(1(x1)))))) 299.06/76.16 2(0(0(1(3(x1))))) -> 0(2(2(1(0(3(x1)))))) 299.06/76.16 2(5(3(1(3(x1))))) -> 2(1(5(3(0(3(x1)))))) 299.06/76.16 2(0(4(3(3(x1))))) -> 3(2(4(0(0(3(x1)))))) 299.06/76.16 2(0(0(4(3(x1))))) -> 2(4(4(0(0(3(x1)))))) 299.06/76.16 2(0(1(4(3(x1))))) -> 4(0(3(2(1(3(x1)))))) 299.06/76.16 2(0(2(4(3(x1))))) -> 3(2(4(0(2(3(x1)))))) 299.06/76.16 2(0(0(5(3(x1))))) -> 5(4(2(0(3(0(x1)))))) 299.06/76.16 299.06/76.16 299.06/76.16 ---------------------------------------- 299.06/76.16 299.06/76.16 (14) 299.06/76.16 Obligation: 299.06/76.16 Q DP problem: 299.06/76.16 The TRS P consists of the following rules: 299.06/76.16 299.06/76.16 2^1(0(0(1(x1)))) -> 2^1(x1) 299.06/76.16 2^1(2(0(1(x1)))) -> 2^1(x1) 299.06/76.16 2^1(5(1(4(3(x1))))) -> 2^1(x1) 299.06/76.16 2^1(0(2(4(3(x1))))) -> 2^1(2(4(x1))) 299.06/76.16 299.06/76.16 The TRS R consists of the following rules: 299.06/76.16 299.06/76.16 2(0(1(0(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.16 2(0(0(1(x1)))) -> 1(0(0(3(2(x1))))) 299.06/76.16 2(0(0(1(x1)))) -> 0(2(1(1(0(1(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 0(2(0(3(3(1(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 2(1(1(0(0(3(x1)))))) 299.06/76.16 2(2(0(1(x1)))) -> 0(3(2(1(2(x1))))) 299.06/76.16 2(2(0(1(x1)))) -> 0(2(2(1(1(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(3(2(1(1(x1))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(3(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(4(4(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(4(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(5(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(5(5(3(2(1(x1)))))) 299.06/76.16 2(5(3(1(x1)))) -> 5(0(1(3(2(x1))))) 299.06/76.16 2(5(3(1(x1)))) -> 3(2(1(5(4(x1))))) 299.06/76.16 2(0(1(3(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(0(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(1(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(5(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(0(1(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(2(1(4(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(3(4(0(2(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(1(1(2(0(3(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(5(0(3(3(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(3(3(2(4(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 5(2(3(0(3(x1))))) 299.06/76.16 2(0(5(3(x1)))) -> 0(0(0(3(5(2(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 4(5(2(0(0(3(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 3(0(3(2(5(5(x1)))))) 299.06/76.16 2(4(3(2(0(x1))))) -> 5(4(2(2(0(3(x1)))))) 299.06/76.16 2(0(4(3(0(x1))))) -> 2(4(0(1(3(0(x1)))))) 299.06/76.16 2(0(5(3(0(x1))))) -> 0(3(5(0(1(2(x1)))))) 299.06/76.16 2(4(0(0(1(x1))))) -> 2(0(1(5(4(0(x1)))))) 299.06/76.16 2(5(3(0(1(x1))))) -> 0(2(1(3(5(4(x1)))))) 299.06/76.16 2(2(0(1(1(x1))))) -> 2(1(0(2(1(1(x1)))))) 299.06/76.16 2(0(4(3(1(x1))))) -> 4(1(0(3(2(1(x1)))))) 299.06/76.16 2(0(5(3(1(x1))))) -> 3(2(5(2(0(1(x1)))))) 299.06/76.16 2(0(0(1(3(x1))))) -> 0(2(2(1(0(3(x1)))))) 299.06/76.16 2(5(3(1(3(x1))))) -> 2(1(5(3(0(3(x1)))))) 299.06/76.16 2(0(4(3(3(x1))))) -> 3(2(4(0(0(3(x1)))))) 299.06/76.16 2(0(0(4(3(x1))))) -> 2(4(4(0(0(3(x1)))))) 299.06/76.16 2(0(1(4(3(x1))))) -> 4(0(3(2(1(3(x1)))))) 299.06/76.16 2(5(1(4(3(x1))))) -> 5(4(1(0(3(2(x1)))))) 299.06/76.16 2(0(2(4(3(x1))))) -> 3(2(4(0(2(3(x1)))))) 299.06/76.16 2(0(2(4(3(x1))))) -> 3(0(3(2(2(4(x1)))))) 299.06/76.16 2(0(0(5(3(x1))))) -> 5(4(2(0(3(0(x1)))))) 299.06/76.16 2(4(0(5(3(x1))))) -> 3(5(4(4(0(2(x1)))))) 299.06/76.16 2(4(0(5(3(x1))))) -> 0(0(3(4(5(2(x1)))))) 299.06/76.16 2(4(3(5(3(x1))))) -> 3(3(2(4(5(0(x1)))))) 299.06/76.16 299.06/76.16 Q is empty. 299.06/76.16 We have to consider all minimal (P,Q,R)-chains. 299.06/76.16 ---------------------------------------- 299.06/76.16 299.06/76.16 (15) QDPOrderProof (EQUIVALENT) 299.06/76.16 We use the reduction pair processor [LPAR04,JAR06]. 299.06/76.16 299.06/76.16 299.06/76.16 The following pairs can be oriented strictly and are deleted. 299.06/76.16 299.06/76.16 2^1(5(1(4(3(x1))))) -> 2^1(x1) 299.06/76.16 The remaining pairs can at least be oriented weakly. 299.06/76.16 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(2^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(0(x_1)) = [[-I], [-I], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [-I, 0A, 0A]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(1(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [0A, 0A, -I], [0A, 0A, -I]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(2(x_1)) = [[0A], [0A], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, -I]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(5(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, -I], [-I, -I, 0A], [-I, -I, -I]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(4(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 1A], [0A, 0A, 0A], [-I, 0A, -I]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 <<< 299.06/76.16 POL(3(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, 0A], [-I, 0A, 0A], [0A, 0A, 0A]] * x_1 299.06/76.16 >>> 299.06/76.16 299.06/76.16 299.06/76.16 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 299.06/76.16 299.06/76.16 2(4(3(1(x1)))) -> 4(3(2(1(1(x1))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(3(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(4(4(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(4(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(5(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(5(5(3(2(1(x1)))))) 299.06/76.16 2(4(3(2(0(x1))))) -> 5(4(2(2(0(3(x1)))))) 299.06/76.16 2(4(0(0(1(x1))))) -> 2(0(1(5(4(0(x1)))))) 299.06/76.16 2(4(0(5(3(x1))))) -> 3(5(4(4(0(2(x1)))))) 299.06/76.16 2(4(0(5(3(x1))))) -> 0(0(3(4(5(2(x1)))))) 299.06/76.16 2(4(3(5(3(x1))))) -> 3(3(2(4(5(0(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 1(0(0(3(2(x1))))) 299.06/76.16 2(2(0(1(x1)))) -> 0(3(2(1(2(x1))))) 299.06/76.16 2(5(3(1(x1)))) -> 5(0(1(3(2(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(3(4(0(2(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(3(3(2(4(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 0(0(0(3(5(2(x1)))))) 299.06/76.16 2(0(5(3(0(x1))))) -> 0(3(5(0(1(2(x1)))))) 299.06/76.16 2(0(5(3(1(x1))))) -> 3(2(5(2(0(1(x1)))))) 299.06/76.16 2(5(1(4(3(x1))))) -> 5(4(1(0(3(2(x1)))))) 299.06/76.16 2(0(2(4(3(x1))))) -> 3(0(3(2(2(4(x1)))))) 299.06/76.16 2(0(1(0(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.16 2(0(0(1(x1)))) -> 0(2(1(1(0(1(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 0(2(0(3(3(1(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 2(1(1(0(0(3(x1)))))) 299.06/76.16 2(2(0(1(x1)))) -> 0(2(2(1(1(1(x1)))))) 299.06/76.16 2(5(3(1(x1)))) -> 3(2(1(5(4(x1))))) 299.06/76.16 2(0(1(3(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(0(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(1(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(5(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(0(1(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(2(1(4(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(1(1(2(0(3(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(5(0(3(3(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 5(2(3(0(3(x1))))) 299.06/76.16 2(0(5(3(x1)))) -> 4(5(2(0(0(3(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 3(0(3(2(5(5(x1)))))) 299.06/76.16 2(0(4(3(0(x1))))) -> 2(4(0(1(3(0(x1)))))) 299.06/76.16 2(5(3(0(1(x1))))) -> 0(2(1(3(5(4(x1)))))) 299.06/76.16 2(2(0(1(1(x1))))) -> 2(1(0(2(1(1(x1)))))) 299.06/76.16 2(0(4(3(1(x1))))) -> 4(1(0(3(2(1(x1)))))) 299.06/76.16 2(0(0(1(3(x1))))) -> 0(2(2(1(0(3(x1)))))) 299.06/76.16 2(5(3(1(3(x1))))) -> 2(1(5(3(0(3(x1)))))) 299.06/76.16 2(0(4(3(3(x1))))) -> 3(2(4(0(0(3(x1)))))) 299.06/76.16 2(0(0(4(3(x1))))) -> 2(4(4(0(0(3(x1)))))) 299.06/76.16 2(0(1(4(3(x1))))) -> 4(0(3(2(1(3(x1)))))) 299.06/76.16 2(0(2(4(3(x1))))) -> 3(2(4(0(2(3(x1)))))) 299.06/76.16 2(0(0(5(3(x1))))) -> 5(4(2(0(3(0(x1)))))) 299.06/76.16 299.06/76.16 299.06/76.16 ---------------------------------------- 299.06/76.16 299.06/76.16 (16) 299.06/76.16 Obligation: 299.06/76.16 Q DP problem: 299.06/76.16 The TRS P consists of the following rules: 299.06/76.16 299.06/76.16 2^1(0(0(1(x1)))) -> 2^1(x1) 299.06/76.16 2^1(2(0(1(x1)))) -> 2^1(x1) 299.06/76.16 2^1(0(2(4(3(x1))))) -> 2^1(2(4(x1))) 299.06/76.16 299.06/76.16 The TRS R consists of the following rules: 299.06/76.16 299.06/76.16 2(0(1(0(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.16 2(0(0(1(x1)))) -> 1(0(0(3(2(x1))))) 299.06/76.16 2(0(0(1(x1)))) -> 0(2(1(1(0(1(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 0(2(0(3(3(1(x1)))))) 299.06/76.16 2(0(0(1(x1)))) -> 2(1(1(0(0(3(x1)))))) 299.06/76.16 2(2(0(1(x1)))) -> 0(3(2(1(2(x1))))) 299.06/76.16 2(2(0(1(x1)))) -> 0(2(2(1(1(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(3(2(1(1(x1))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(3(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(4(4(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(4(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 5(4(5(3(2(1(x1)))))) 299.06/76.16 2(4(3(1(x1)))) -> 4(5(5(3(2(1(x1)))))) 299.06/76.16 2(5(3(1(x1)))) -> 5(0(1(3(2(x1))))) 299.06/76.16 2(5(3(1(x1)))) -> 3(2(1(5(4(x1))))) 299.06/76.16 2(0(1(3(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(0(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(1(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(5(0(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(0(1(3(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(2(1(4(x1))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(3(4(0(2(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(1(1(2(0(3(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 4(2(5(0(3(3(x1)))))) 299.06/76.16 2(0(4(3(x1)))) -> 0(3(3(3(2(4(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 5(2(3(0(3(x1))))) 299.06/76.16 2(0(5(3(x1)))) -> 0(0(0(3(5(2(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 4(5(2(0(0(3(x1)))))) 299.06/76.16 2(0(5(3(x1)))) -> 3(0(3(2(5(5(x1)))))) 299.06/76.16 2(4(3(2(0(x1))))) -> 5(4(2(2(0(3(x1)))))) 299.06/76.16 2(0(4(3(0(x1))))) -> 2(4(0(1(3(0(x1)))))) 299.06/76.16 2(0(5(3(0(x1))))) -> 0(3(5(0(1(2(x1)))))) 299.06/76.16 2(4(0(0(1(x1))))) -> 2(0(1(5(4(0(x1)))))) 299.06/76.16 2(5(3(0(1(x1))))) -> 0(2(1(3(5(4(x1)))))) 299.06/76.16 2(2(0(1(1(x1))))) -> 2(1(0(2(1(1(x1)))))) 299.06/76.16 2(0(4(3(1(x1))))) -> 4(1(0(3(2(1(x1)))))) 299.06/76.16 2(0(5(3(1(x1))))) -> 3(2(5(2(0(1(x1)))))) 299.06/76.16 2(0(0(1(3(x1))))) -> 0(2(2(1(0(3(x1)))))) 299.06/76.16 2(5(3(1(3(x1))))) -> 2(1(5(3(0(3(x1)))))) 299.06/76.16 2(0(4(3(3(x1))))) -> 3(2(4(0(0(3(x1)))))) 299.06/76.16 2(0(0(4(3(x1))))) -> 2(4(4(0(0(3(x1)))))) 299.06/76.16 2(0(1(4(3(x1))))) -> 4(0(3(2(1(3(x1)))))) 299.06/76.16 2(5(1(4(3(x1))))) -> 5(4(1(0(3(2(x1)))))) 299.06/76.16 2(0(2(4(3(x1))))) -> 3(2(4(0(2(3(x1)))))) 299.06/76.16 2(0(2(4(3(x1))))) -> 3(0(3(2(2(4(x1)))))) 299.06/76.16 2(0(0(5(3(x1))))) -> 5(4(2(0(3(0(x1)))))) 299.06/76.16 2(4(0(5(3(x1))))) -> 3(5(4(4(0(2(x1)))))) 299.06/76.16 2(4(0(5(3(x1))))) -> 0(0(3(4(5(2(x1)))))) 299.06/76.16 2(4(3(5(3(x1))))) -> 3(3(2(4(5(0(x1)))))) 299.06/76.16 299.06/76.16 Q is empty. 299.06/76.16 We have to consider all minimal (P,Q,R)-chains. 299.06/76.16 ---------------------------------------- 299.06/76.16 299.06/76.16 (17) QDPOrderProof (EQUIVALENT) 299.06/76.16 We use the reduction pair processor [LPAR04,JAR06]. 299.06/76.16 299.06/76.16 299.06/76.16 The following pairs can be oriented strictly and are deleted. 299.06/76.17 299.06/76.17 2^1(0(0(1(x1)))) -> 2^1(x1) 299.06/76.17 2^1(2(0(1(x1)))) -> 2^1(x1) 299.06/76.17 The remaining pairs can at least be oriented weakly. 299.06/76.17 Used ordering: Polynomial interpretation [POLO]: 299.06/76.17 299.06/76.17 POL(0(x_1)) = x_1 299.06/76.17 POL(1(x_1)) = 1 + x_1 299.06/76.17 POL(2(x_1)) = x_1 299.06/76.17 POL(2^1(x_1)) = x_1 299.06/76.17 POL(3(x_1)) = 0 299.06/76.17 POL(4(x_1)) = 1 299.06/76.17 POL(5(x_1)) = 0 299.06/76.17 299.06/76.17 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 299.06/76.17 299.06/76.17 2(4(3(1(x1)))) -> 4(3(2(1(1(x1))))) 299.06/76.17 2(4(3(1(x1)))) -> 5(4(3(3(2(1(x1)))))) 299.06/76.17 2(4(3(1(x1)))) -> 4(4(4(3(2(1(x1)))))) 299.06/76.17 2(4(3(1(x1)))) -> 5(4(4(3(2(1(x1)))))) 299.06/76.17 2(4(3(1(x1)))) -> 5(4(5(3(2(1(x1)))))) 299.06/76.17 2(4(3(1(x1)))) -> 4(5(5(3(2(1(x1)))))) 299.06/76.17 2(4(3(2(0(x1))))) -> 5(4(2(2(0(3(x1)))))) 299.06/76.17 2(4(0(0(1(x1))))) -> 2(0(1(5(4(0(x1)))))) 299.06/76.17 2(4(0(5(3(x1))))) -> 3(5(4(4(0(2(x1)))))) 299.06/76.17 2(4(0(5(3(x1))))) -> 0(0(3(4(5(2(x1)))))) 299.06/76.17 2(4(3(5(3(x1))))) -> 3(3(2(4(5(0(x1)))))) 299.06/76.17 299.06/76.17 299.06/76.17 ---------------------------------------- 299.06/76.17 299.06/76.17 (18) 299.06/76.17 Obligation: 299.06/76.17 Q DP problem: 299.06/76.17 The TRS P consists of the following rules: 299.06/76.17 299.06/76.17 2^1(0(2(4(3(x1))))) -> 2^1(2(4(x1))) 299.06/76.17 299.06/76.17 The TRS R consists of the following rules: 299.06/76.17 299.06/76.17 2(0(1(0(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.17 2(0(0(1(x1)))) -> 1(0(0(3(2(x1))))) 299.06/76.17 2(0(0(1(x1)))) -> 0(2(1(1(0(1(x1)))))) 299.06/76.17 2(0(0(1(x1)))) -> 0(2(0(3(3(1(x1)))))) 299.06/76.17 2(0(0(1(x1)))) -> 2(1(1(0(0(3(x1)))))) 299.06/76.17 2(2(0(1(x1)))) -> 0(3(2(1(2(x1))))) 299.06/76.17 2(2(0(1(x1)))) -> 0(2(2(1(1(1(x1)))))) 299.06/76.17 2(4(3(1(x1)))) -> 4(3(2(1(1(x1))))) 299.06/76.17 2(4(3(1(x1)))) -> 5(4(3(3(2(1(x1)))))) 299.06/76.17 2(4(3(1(x1)))) -> 4(4(4(3(2(1(x1)))))) 299.06/76.17 2(4(3(1(x1)))) -> 5(4(4(3(2(1(x1)))))) 299.06/76.17 2(4(3(1(x1)))) -> 5(4(5(3(2(1(x1)))))) 299.06/76.17 2(4(3(1(x1)))) -> 4(5(5(3(2(1(x1)))))) 299.06/76.17 2(5(3(1(x1)))) -> 5(0(1(3(2(x1))))) 299.06/76.17 2(5(3(1(x1)))) -> 3(2(1(5(4(x1))))) 299.06/76.17 2(0(1(3(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.17 2(0(4(3(x1)))) -> 4(2(0(0(3(x1))))) 299.06/76.17 2(0(4(3(x1)))) -> 4(2(1(0(3(x1))))) 299.06/76.17 2(0(4(3(x1)))) -> 4(2(5(0(3(x1))))) 299.06/76.17 2(0(4(3(x1)))) -> 4(2(0(1(3(x1))))) 299.06/76.17 2(0(4(3(x1)))) -> 0(3(2(1(4(x1))))) 299.06/76.17 2(0(4(3(x1)))) -> 0(3(3(4(0(2(x1)))))) 299.06/76.17 2(0(4(3(x1)))) -> 4(1(1(2(0(3(x1)))))) 299.06/76.17 2(0(4(3(x1)))) -> 4(2(5(0(3(3(x1)))))) 299.06/76.17 2(0(4(3(x1)))) -> 0(3(3(3(2(4(x1)))))) 299.06/76.17 2(0(5(3(x1)))) -> 5(2(3(0(3(x1))))) 299.06/76.17 2(0(5(3(x1)))) -> 0(0(0(3(5(2(x1)))))) 299.06/76.17 2(0(5(3(x1)))) -> 4(5(2(0(0(3(x1)))))) 299.06/76.17 2(0(5(3(x1)))) -> 3(0(3(2(5(5(x1)))))) 299.06/76.17 2(4(3(2(0(x1))))) -> 5(4(2(2(0(3(x1)))))) 299.06/76.17 2(0(4(3(0(x1))))) -> 2(4(0(1(3(0(x1)))))) 299.06/76.17 2(0(5(3(0(x1))))) -> 0(3(5(0(1(2(x1)))))) 299.06/76.17 2(4(0(0(1(x1))))) -> 2(0(1(5(4(0(x1)))))) 299.06/76.17 2(5(3(0(1(x1))))) -> 0(2(1(3(5(4(x1)))))) 299.06/76.17 2(2(0(1(1(x1))))) -> 2(1(0(2(1(1(x1)))))) 299.06/76.17 2(0(4(3(1(x1))))) -> 4(1(0(3(2(1(x1)))))) 299.06/76.17 2(0(5(3(1(x1))))) -> 3(2(5(2(0(1(x1)))))) 299.06/76.17 2(0(0(1(3(x1))))) -> 0(2(2(1(0(3(x1)))))) 299.06/76.17 2(5(3(1(3(x1))))) -> 2(1(5(3(0(3(x1)))))) 299.06/76.17 2(0(4(3(3(x1))))) -> 3(2(4(0(0(3(x1)))))) 299.06/76.17 2(0(0(4(3(x1))))) -> 2(4(4(0(0(3(x1)))))) 299.06/76.17 2(0(1(4(3(x1))))) -> 4(0(3(2(1(3(x1)))))) 299.06/76.17 2(5(1(4(3(x1))))) -> 5(4(1(0(3(2(x1)))))) 299.06/76.17 2(0(2(4(3(x1))))) -> 3(2(4(0(2(3(x1)))))) 299.06/76.17 2(0(2(4(3(x1))))) -> 3(0(3(2(2(4(x1)))))) 299.06/76.17 2(0(0(5(3(x1))))) -> 5(4(2(0(3(0(x1)))))) 299.06/76.17 2(4(0(5(3(x1))))) -> 3(5(4(4(0(2(x1)))))) 299.06/76.17 2(4(0(5(3(x1))))) -> 0(0(3(4(5(2(x1)))))) 299.06/76.17 2(4(3(5(3(x1))))) -> 3(3(2(4(5(0(x1)))))) 299.06/76.17 299.06/76.17 Q is empty. 299.06/76.17 We have to consider all minimal (P,Q,R)-chains. 299.06/76.17 ---------------------------------------- 299.06/76.17 299.06/76.17 (19) QDPOrderProof (EQUIVALENT) 299.06/76.17 We use the reduction pair processor [LPAR04,JAR06]. 299.06/76.17 299.06/76.17 299.06/76.17 The following pairs can be oriented strictly and are deleted. 299.06/76.17 299.06/76.17 2^1(0(2(4(3(x1))))) -> 2^1(2(4(x1))) 299.06/76.17 The remaining pairs can at least be oriented weakly. 299.06/76.17 Used ordering: Polynomial interpretation [POLO]: 299.06/76.17 299.06/76.17 POL(0(x_1)) = 1 + x_1 299.06/76.17 POL(1(x_1)) = 0 299.06/76.17 POL(2(x_1)) = 1 + x_1 299.06/76.17 POL(2^1(x_1)) = x_1 299.06/76.17 POL(3(x_1)) = 0 299.06/76.17 POL(4(x_1)) = 1 299.06/76.17 POL(5(x_1)) = 0 299.06/76.17 299.06/76.17 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 299.06/76.17 299.06/76.17 2(4(3(1(x1)))) -> 4(3(2(1(1(x1))))) 299.06/76.17 2(4(3(1(x1)))) -> 5(4(3(3(2(1(x1)))))) 299.06/76.17 2(4(3(1(x1)))) -> 4(4(4(3(2(1(x1)))))) 299.06/76.17 2(4(3(1(x1)))) -> 5(4(4(3(2(1(x1)))))) 299.06/76.17 2(4(3(1(x1)))) -> 5(4(5(3(2(1(x1)))))) 299.06/76.17 2(4(3(1(x1)))) -> 4(5(5(3(2(1(x1)))))) 299.06/76.17 2(4(3(2(0(x1))))) -> 5(4(2(2(0(3(x1)))))) 299.06/76.17 2(4(0(0(1(x1))))) -> 2(0(1(5(4(0(x1)))))) 299.06/76.17 2(4(0(5(3(x1))))) -> 3(5(4(4(0(2(x1)))))) 299.06/76.17 2(4(0(5(3(x1))))) -> 0(0(3(4(5(2(x1)))))) 299.06/76.17 2(4(3(5(3(x1))))) -> 3(3(2(4(5(0(x1)))))) 299.06/76.17 299.06/76.17 299.06/76.17 ---------------------------------------- 299.06/76.17 299.06/76.17 (20) 299.06/76.17 Obligation: 299.06/76.17 Q DP problem: 299.06/76.17 P is empty. 299.06/76.17 The TRS R consists of the following rules: 299.06/76.17 299.06/76.17 2(0(1(0(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.17 2(0(0(1(x1)))) -> 1(0(0(3(2(x1))))) 299.06/76.17 2(0(0(1(x1)))) -> 0(2(1(1(0(1(x1)))))) 299.06/76.17 2(0(0(1(x1)))) -> 0(2(0(3(3(1(x1)))))) 299.06/76.17 2(0(0(1(x1)))) -> 2(1(1(0(0(3(x1)))))) 299.06/76.17 2(2(0(1(x1)))) -> 0(3(2(1(2(x1))))) 299.06/76.17 2(2(0(1(x1)))) -> 0(2(2(1(1(1(x1)))))) 299.06/76.17 2(4(3(1(x1)))) -> 4(3(2(1(1(x1))))) 299.06/76.17 2(4(3(1(x1)))) -> 5(4(3(3(2(1(x1)))))) 299.06/76.17 2(4(3(1(x1)))) -> 4(4(4(3(2(1(x1)))))) 299.06/76.17 2(4(3(1(x1)))) -> 5(4(4(3(2(1(x1)))))) 299.06/76.17 2(4(3(1(x1)))) -> 5(4(5(3(2(1(x1)))))) 299.06/76.17 2(4(3(1(x1)))) -> 4(5(5(3(2(1(x1)))))) 299.06/76.17 2(5(3(1(x1)))) -> 5(0(1(3(2(x1))))) 299.06/76.17 2(5(3(1(x1)))) -> 3(2(1(5(4(x1))))) 299.06/76.17 2(0(1(3(x1)))) -> 0(2(1(0(3(x1))))) 299.06/76.17 2(0(4(3(x1)))) -> 4(2(0(0(3(x1))))) 299.06/76.17 2(0(4(3(x1)))) -> 4(2(1(0(3(x1))))) 299.06/76.17 2(0(4(3(x1)))) -> 4(2(5(0(3(x1))))) 299.06/76.17 2(0(4(3(x1)))) -> 4(2(0(1(3(x1))))) 299.06/76.17 2(0(4(3(x1)))) -> 0(3(2(1(4(x1))))) 299.06/76.17 2(0(4(3(x1)))) -> 0(3(3(4(0(2(x1)))))) 299.06/76.17 2(0(4(3(x1)))) -> 4(1(1(2(0(3(x1)))))) 299.06/76.17 2(0(4(3(x1)))) -> 4(2(5(0(3(3(x1)))))) 299.06/76.17 2(0(4(3(x1)))) -> 0(3(3(3(2(4(x1)))))) 299.06/76.17 2(0(5(3(x1)))) -> 5(2(3(0(3(x1))))) 299.06/76.17 2(0(5(3(x1)))) -> 0(0(0(3(5(2(x1)))))) 299.06/76.17 2(0(5(3(x1)))) -> 4(5(2(0(0(3(x1)))))) 299.06/76.17 2(0(5(3(x1)))) -> 3(0(3(2(5(5(x1)))))) 299.06/76.17 2(4(3(2(0(x1))))) -> 5(4(2(2(0(3(x1)))))) 299.06/76.17 2(0(4(3(0(x1))))) -> 2(4(0(1(3(0(x1)))))) 299.06/76.17 2(0(5(3(0(x1))))) -> 0(3(5(0(1(2(x1)))))) 299.06/76.17 2(4(0(0(1(x1))))) -> 2(0(1(5(4(0(x1)))))) 299.06/76.17 2(5(3(0(1(x1))))) -> 0(2(1(3(5(4(x1)))))) 299.06/76.17 2(2(0(1(1(x1))))) -> 2(1(0(2(1(1(x1)))))) 299.06/76.17 2(0(4(3(1(x1))))) -> 4(1(0(3(2(1(x1)))))) 299.06/76.17 2(0(5(3(1(x1))))) -> 3(2(5(2(0(1(x1)))))) 299.06/76.17 2(0(0(1(3(x1))))) -> 0(2(2(1(0(3(x1)))))) 299.06/76.17 2(5(3(1(3(x1))))) -> 2(1(5(3(0(3(x1)))))) 299.06/76.17 2(0(4(3(3(x1))))) -> 3(2(4(0(0(3(x1)))))) 299.06/76.17 2(0(0(4(3(x1))))) -> 2(4(4(0(0(3(x1)))))) 299.06/76.17 2(0(1(4(3(x1))))) -> 4(0(3(2(1(3(x1)))))) 299.06/76.17 2(5(1(4(3(x1))))) -> 5(4(1(0(3(2(x1)))))) 299.06/76.17 2(0(2(4(3(x1))))) -> 3(2(4(0(2(3(x1)))))) 299.06/76.17 2(0(2(4(3(x1))))) -> 3(0(3(2(2(4(x1)))))) 299.06/76.17 2(0(0(5(3(x1))))) -> 5(4(2(0(3(0(x1)))))) 299.06/76.17 2(4(0(5(3(x1))))) -> 3(5(4(4(0(2(x1)))))) 299.06/76.17 2(4(0(5(3(x1))))) -> 0(0(3(4(5(2(x1)))))) 299.06/76.17 2(4(3(5(3(x1))))) -> 3(3(2(4(5(0(x1)))))) 299.06/76.17 299.06/76.17 Q is empty. 299.06/76.17 We have to consider all minimal (P,Q,R)-chains. 299.06/76.17 ---------------------------------------- 299.06/76.17 299.06/76.17 (21) PisEmptyProof (EQUIVALENT) 299.06/76.17 The TRS P is empty. Hence, there is no (P,Q,R) chain. 299.06/76.17 ---------------------------------------- 299.06/76.17 299.06/76.17 (22) 299.06/76.17 YES 299.11/76.22 EOF