373.80/95.58 YES 394.29/100.82 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 394.29/100.82 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 394.29/100.82 394.29/100.82 394.29/100.82 Termination w.r.t. Q of the given QTRS could be proven: 394.29/100.82 394.29/100.82 (0) QTRS 394.29/100.82 (1) DependencyPairsProof [EQUIVALENT, 460 ms] 394.29/100.82 (2) QDP 394.29/100.82 (3) DependencyGraphProof [EQUIVALENT, 9 ms] 394.29/100.82 (4) QDP 394.29/100.82 (5) QDPOrderProof [EQUIVALENT, 3438 ms] 394.29/100.82 (6) QDP 394.29/100.82 (7) QDPOrderProof [EQUIVALENT, 2607 ms] 394.29/100.82 (8) QDP 394.29/100.82 (9) QDPOrderProof [EQUIVALENT, 2590 ms] 394.29/100.82 (10) QDP 394.29/100.82 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 394.29/100.82 (12) QDP 394.29/100.82 (13) QDPOrderProof [EQUIVALENT, 3050 ms] 394.29/100.82 (14) QDP 394.29/100.82 (15) QDPOrderProof [EQUIVALENT, 4059 ms] 394.29/100.82 (16) QDP 394.29/100.82 (17) QDPOrderProof [EQUIVALENT, 3131 ms] 394.29/100.82 (18) QDP 394.29/100.82 (19) QDPOrderProof [EQUIVALENT, 3146 ms] 394.29/100.82 (20) QDP 394.29/100.82 (21) QDPOrderProof [EQUIVALENT, 2763 ms] 394.29/100.82 (22) QDP 394.29/100.82 (23) DependencyGraphProof [EQUIVALENT, 0 ms] 394.29/100.82 (24) QDP 394.29/100.82 (25) UsableRulesProof [EQUIVALENT, 0 ms] 394.29/100.82 (26) QDP 394.29/100.82 (27) QDPSizeChangeProof [EQUIVALENT, 0 ms] 394.29/100.82 (28) YES 394.29/100.82 394.29/100.82 394.29/100.82 ---------------------------------------- 394.29/100.82 394.29/100.82 (0) 394.29/100.82 Obligation: 394.29/100.82 Q restricted rewrite system: 394.29/100.82 The TRS R consists of the following rules: 394.29/100.82 394.29/100.82 0(1(2(x1))) -> 0(3(1(2(x1)))) 394.29/100.82 0(1(2(x1))) -> 1(0(2(1(x1)))) 394.29/100.82 0(1(2(x1))) -> 2(1(4(0(x1)))) 394.29/100.82 0(1(2(x1))) -> 0(2(1(3(3(x1))))) 394.29/100.82 0(1(2(x1))) -> 0(2(4(1(3(x1))))) 394.29/100.82 0(1(2(x1))) -> 1(0(2(5(4(x1))))) 394.29/100.82 0(1(2(x1))) -> 2(1(1(0(3(x1))))) 394.29/100.82 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 394.29/100.82 0(1(2(x1))) -> 2(1(4(3(0(x1))))) 394.29/100.82 0(1(2(x1))) -> 4(0(3(1(2(x1))))) 394.29/100.82 0(1(2(x1))) -> 4(1(0(3(2(x1))))) 394.29/100.82 0(1(2(x1))) -> 4(4(2(1(0(x1))))) 394.29/100.82 0(1(2(x1))) -> 0(0(4(1(2(5(x1)))))) 394.29/100.82 0(1(2(x1))) -> 0(4(5(1(2(5(x1)))))) 394.29/100.82 0(1(2(x1))) -> 0(5(2(3(1(4(x1)))))) 394.29/100.82 0(1(2(x1))) -> 1(0(2(2(3(4(x1)))))) 394.29/100.82 0(1(2(x1))) -> 1(1(4(0(3(2(x1)))))) 394.29/100.82 0(1(2(x1))) -> 2(0(0(3(3(1(x1)))))) 394.29/100.82 0(1(2(x1))) -> 2(0(3(4(1(5(x1)))))) 394.29/100.82 0(1(2(x1))) -> 2(3(2(0(4(1(x1)))))) 394.29/100.82 0(1(2(x1))) -> 2(5(3(0(3(1(x1)))))) 394.29/100.82 0(1(2(x1))) -> 3(2(5(3(1(0(x1)))))) 394.29/100.82 0(1(2(x1))) -> 3(4(0(5(1(2(x1)))))) 394.29/100.82 0(1(2(x1))) -> 5(5(4(2(1(0(x1)))))) 394.29/100.82 0(0(1(2(x1)))) -> 0(0(2(4(1(x1))))) 394.29/100.82 0(0(1(2(x1)))) -> 0(2(1(5(0(4(x1)))))) 394.29/100.82 0(0(1(2(x1)))) -> 1(0(2(3(5(0(x1)))))) 394.29/100.82 0(0(1(2(x1)))) -> 5(1(2(0(4(0(x1)))))) 394.29/100.82 0(0(5(2(x1)))) -> 5(3(0(3(0(2(x1)))))) 394.29/100.82 0(1(2(0(x1)))) -> 1(2(4(0(0(x1))))) 394.29/100.82 0(1(2(0(x1)))) -> 0(4(1(0(0(2(x1)))))) 394.29/100.82 0(1(2(2(x1)))) -> 3(3(0(2(1(2(x1)))))) 394.29/100.82 0(1(5(2(x1)))) -> 2(5(1(0(3(x1))))) 394.29/100.82 0(1(5(2(x1)))) -> 0(5(5(3(1(2(x1)))))) 394.29/100.82 0(1(5(2(x1)))) -> 1(3(5(0(2(2(x1)))))) 394.29/100.82 0(5(2(2(x1)))) -> 3(0(2(1(5(2(x1)))))) 394.29/100.82 1(0(1(2(x1)))) -> 0(4(1(2(1(x1))))) 394.29/100.82 1(0(1(2(x1)))) -> 1(2(1(3(0(x1))))) 394.29/100.82 1(0(1(2(x1)))) -> 0(3(1(4(1(2(x1)))))) 394.29/100.82 1(0(1(2(x1)))) -> 4(1(4(1(0(2(x1)))))) 394.29/100.82 5(0(1(2(x1)))) -> 0(2(1(3(5(x1))))) 394.29/100.82 5(0(1(2(x1)))) -> 2(0(3(1(5(x1))))) 394.29/100.82 5(0(1(2(x1)))) -> 5(3(1(0(3(2(x1)))))) 394.29/100.82 5(0(1(2(x1)))) -> 5(5(1(0(2(4(x1)))))) 394.29/100.82 0(1(0(2(2(x1))))) -> 3(0(2(1(0(2(x1)))))) 394.29/100.82 0(1(2(5(2(x1))))) -> 3(5(1(2(2(0(x1)))))) 394.29/100.82 0(1(3(0(0(x1))))) -> 0(0(4(1(3(0(x1)))))) 394.29/100.82 0(1(4(2(2(x1))))) -> 0(2(4(1(5(2(x1)))))) 394.29/100.82 0(1(4(2(2(x1))))) -> 0(2(4(2(1(1(x1)))))) 394.29/100.82 1(3(0(1(2(x1))))) -> 1(3(4(1(0(2(x1)))))) 394.29/100.82 394.29/100.82 Q is empty. 394.29/100.82 394.29/100.82 ---------------------------------------- 394.29/100.82 394.29/100.82 (1) DependencyPairsProof (EQUIVALENT) 394.29/100.82 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 394.29/100.82 ---------------------------------------- 394.29/100.82 394.29/100.82 (2) 394.29/100.82 Obligation: 394.29/100.82 Q DP problem: 394.29/100.82 The TRS P consists of the following rules: 394.29/100.82 394.29/100.82 0^1(1(2(x1))) -> 0^1(3(1(2(x1)))) 394.29/100.82 0^1(1(2(x1))) -> 1^1(0(2(1(x1)))) 394.29/100.82 0^1(1(2(x1))) -> 0^1(2(1(x1))) 394.29/100.82 0^1(1(2(x1))) -> 1^1(x1) 394.29/100.82 0^1(1(2(x1))) -> 1^1(4(0(x1))) 394.29/100.82 0^1(1(2(x1))) -> 0^1(x1) 394.29/100.82 0^1(1(2(x1))) -> 0^1(2(1(3(3(x1))))) 394.29/100.82 0^1(1(2(x1))) -> 1^1(3(3(x1))) 394.29/100.82 0^1(1(2(x1))) -> 0^1(2(4(1(3(x1))))) 394.29/100.82 0^1(1(2(x1))) -> 1^1(3(x1)) 394.29/100.82 0^1(1(2(x1))) -> 1^1(0(2(5(4(x1))))) 394.29/100.82 0^1(1(2(x1))) -> 0^1(2(5(4(x1)))) 394.29/100.82 0^1(1(2(x1))) -> 5^1(4(x1)) 394.29/100.82 0^1(1(2(x1))) -> 1^1(1(0(3(x1)))) 394.29/100.82 0^1(1(2(x1))) -> 1^1(0(3(x1))) 394.29/100.82 0^1(1(2(x1))) -> 0^1(3(x1)) 394.29/100.82 0^1(1(2(x1))) -> 1^1(3(4(0(x1)))) 394.29/100.82 0^1(1(2(x1))) -> 1^1(4(3(0(x1)))) 394.29/100.82 0^1(1(2(x1))) -> 1^1(0(3(2(x1)))) 394.29/100.82 0^1(1(2(x1))) -> 0^1(3(2(x1))) 394.29/100.82 0^1(1(2(x1))) -> 1^1(0(x1)) 394.29/100.82 0^1(1(2(x1))) -> 0^1(0(4(1(2(5(x1)))))) 394.29/100.82 0^1(1(2(x1))) -> 0^1(4(1(2(5(x1))))) 394.29/100.82 0^1(1(2(x1))) -> 1^1(2(5(x1))) 394.29/100.82 0^1(1(2(x1))) -> 5^1(x1) 394.29/100.82 0^1(1(2(x1))) -> 0^1(4(5(1(2(5(x1)))))) 394.29/100.82 0^1(1(2(x1))) -> 5^1(1(2(5(x1)))) 394.29/100.82 0^1(1(2(x1))) -> 0^1(5(2(3(1(4(x1)))))) 394.29/100.82 0^1(1(2(x1))) -> 5^1(2(3(1(4(x1))))) 394.29/100.82 0^1(1(2(x1))) -> 1^1(4(x1)) 394.29/100.82 0^1(1(2(x1))) -> 1^1(0(2(2(3(4(x1)))))) 394.29/100.82 0^1(1(2(x1))) -> 0^1(2(2(3(4(x1))))) 394.29/100.82 0^1(1(2(x1))) -> 1^1(1(4(0(3(2(x1)))))) 394.29/100.82 0^1(1(2(x1))) -> 1^1(4(0(3(2(x1))))) 394.29/100.82 0^1(1(2(x1))) -> 0^1(0(3(3(1(x1))))) 394.29/100.82 0^1(1(2(x1))) -> 0^1(3(3(1(x1)))) 394.29/100.82 0^1(1(2(x1))) -> 0^1(3(4(1(5(x1))))) 394.29/100.82 0^1(1(2(x1))) -> 1^1(5(x1)) 394.29/100.82 0^1(1(2(x1))) -> 0^1(4(1(x1))) 394.29/100.82 0^1(1(2(x1))) -> 5^1(3(0(3(1(x1))))) 394.29/100.82 0^1(1(2(x1))) -> 0^1(3(1(x1))) 394.29/100.82 0^1(1(2(x1))) -> 5^1(3(1(0(x1)))) 394.29/100.82 0^1(1(2(x1))) -> 0^1(5(1(2(x1)))) 394.29/100.82 0^1(1(2(x1))) -> 5^1(1(2(x1))) 394.29/100.82 0^1(1(2(x1))) -> 5^1(5(4(2(1(0(x1)))))) 394.29/100.82 0^1(1(2(x1))) -> 5^1(4(2(1(0(x1))))) 394.29/100.82 0^1(0(1(2(x1)))) -> 0^1(0(2(4(1(x1))))) 394.29/100.82 0^1(0(1(2(x1)))) -> 0^1(2(4(1(x1)))) 394.29/100.82 0^1(0(1(2(x1)))) -> 1^1(x1) 394.29/100.82 0^1(0(1(2(x1)))) -> 0^1(2(1(5(0(4(x1)))))) 394.29/100.82 0^1(0(1(2(x1)))) -> 1^1(5(0(4(x1)))) 394.29/100.82 0^1(0(1(2(x1)))) -> 5^1(0(4(x1))) 394.29/100.82 0^1(0(1(2(x1)))) -> 0^1(4(x1)) 394.29/100.82 0^1(0(1(2(x1)))) -> 1^1(0(2(3(5(0(x1)))))) 394.29/100.82 0^1(0(1(2(x1)))) -> 0^1(2(3(5(0(x1))))) 394.29/100.82 0^1(0(1(2(x1)))) -> 5^1(0(x1)) 394.29/100.82 0^1(0(1(2(x1)))) -> 0^1(x1) 394.29/100.82 0^1(0(1(2(x1)))) -> 5^1(1(2(0(4(0(x1)))))) 394.29/100.82 0^1(0(1(2(x1)))) -> 1^1(2(0(4(0(x1))))) 394.29/100.82 0^1(0(1(2(x1)))) -> 0^1(4(0(x1))) 394.29/100.82 0^1(0(5(2(x1)))) -> 5^1(3(0(3(0(2(x1)))))) 394.29/100.82 0^1(0(5(2(x1)))) -> 0^1(3(0(2(x1)))) 394.29/100.82 0^1(0(5(2(x1)))) -> 0^1(2(x1)) 394.29/100.82 0^1(1(2(0(x1)))) -> 1^1(2(4(0(0(x1))))) 394.29/100.82 0^1(1(2(0(x1)))) -> 0^1(0(x1)) 394.29/100.82 0^1(1(2(0(x1)))) -> 0^1(4(1(0(0(2(x1)))))) 394.29/100.82 0^1(1(2(0(x1)))) -> 1^1(0(0(2(x1)))) 394.29/100.82 0^1(1(2(0(x1)))) -> 0^1(0(2(x1))) 394.29/100.82 0^1(1(2(0(x1)))) -> 0^1(2(x1)) 394.29/100.82 0^1(1(2(2(x1)))) -> 0^1(2(1(2(x1)))) 394.29/100.82 0^1(1(2(2(x1)))) -> 1^1(2(x1)) 394.29/100.82 0^1(1(5(2(x1)))) -> 5^1(1(0(3(x1)))) 394.29/100.82 0^1(1(5(2(x1)))) -> 1^1(0(3(x1))) 394.29/100.82 0^1(1(5(2(x1)))) -> 0^1(3(x1)) 394.29/100.82 0^1(1(5(2(x1)))) -> 0^1(5(5(3(1(2(x1)))))) 394.29/100.82 0^1(1(5(2(x1)))) -> 5^1(5(3(1(2(x1))))) 394.29/100.82 0^1(1(5(2(x1)))) -> 5^1(3(1(2(x1)))) 394.29/100.82 0^1(1(5(2(x1)))) -> 1^1(2(x1)) 394.29/100.82 0^1(1(5(2(x1)))) -> 1^1(3(5(0(2(2(x1)))))) 394.29/100.82 0^1(1(5(2(x1)))) -> 5^1(0(2(2(x1)))) 394.29/100.82 0^1(1(5(2(x1)))) -> 0^1(2(2(x1))) 394.29/100.82 0^1(5(2(2(x1)))) -> 0^1(2(1(5(2(x1))))) 394.29/100.82 0^1(5(2(2(x1)))) -> 1^1(5(2(x1))) 394.29/100.82 0^1(5(2(2(x1)))) -> 5^1(2(x1)) 394.29/100.82 1^1(0(1(2(x1)))) -> 0^1(4(1(2(1(x1))))) 394.29/100.82 1^1(0(1(2(x1)))) -> 1^1(2(1(x1))) 394.29/100.82 1^1(0(1(2(x1)))) -> 1^1(x1) 394.29/100.82 1^1(0(1(2(x1)))) -> 1^1(2(1(3(0(x1))))) 394.29/100.82 1^1(0(1(2(x1)))) -> 1^1(3(0(x1))) 394.29/100.82 1^1(0(1(2(x1)))) -> 0^1(x1) 394.29/100.82 1^1(0(1(2(x1)))) -> 0^1(3(1(4(1(2(x1)))))) 394.29/100.82 1^1(0(1(2(x1)))) -> 1^1(4(1(2(x1)))) 394.29/100.82 1^1(0(1(2(x1)))) -> 1^1(4(1(0(2(x1))))) 394.29/100.82 1^1(0(1(2(x1)))) -> 1^1(0(2(x1))) 394.29/100.82 1^1(0(1(2(x1)))) -> 0^1(2(x1)) 394.29/100.82 5^1(0(1(2(x1)))) -> 0^1(2(1(3(5(x1))))) 394.29/100.82 5^1(0(1(2(x1)))) -> 1^1(3(5(x1))) 394.29/100.82 5^1(0(1(2(x1)))) -> 5^1(x1) 394.29/100.82 5^1(0(1(2(x1)))) -> 0^1(3(1(5(x1)))) 394.29/100.82 5^1(0(1(2(x1)))) -> 1^1(5(x1)) 394.29/100.82 5^1(0(1(2(x1)))) -> 5^1(3(1(0(3(2(x1)))))) 394.29/100.82 5^1(0(1(2(x1)))) -> 1^1(0(3(2(x1)))) 394.29/100.82 5^1(0(1(2(x1)))) -> 0^1(3(2(x1))) 394.29/100.82 5^1(0(1(2(x1)))) -> 5^1(5(1(0(2(4(x1)))))) 394.29/100.82 5^1(0(1(2(x1)))) -> 5^1(1(0(2(4(x1))))) 394.29/100.82 5^1(0(1(2(x1)))) -> 1^1(0(2(4(x1)))) 394.29/100.82 5^1(0(1(2(x1)))) -> 0^1(2(4(x1))) 394.29/100.82 0^1(1(0(2(2(x1))))) -> 0^1(2(1(0(2(x1))))) 394.29/100.82 0^1(1(0(2(2(x1))))) -> 1^1(0(2(x1))) 394.29/100.82 0^1(1(0(2(2(x1))))) -> 0^1(2(x1)) 394.29/100.82 0^1(1(2(5(2(x1))))) -> 5^1(1(2(2(0(x1))))) 394.29/100.82 0^1(1(2(5(2(x1))))) -> 1^1(2(2(0(x1)))) 394.29/100.82 0^1(1(2(5(2(x1))))) -> 0^1(x1) 394.29/100.82 0^1(1(3(0(0(x1))))) -> 0^1(0(4(1(3(0(x1)))))) 394.29/100.82 0^1(1(3(0(0(x1))))) -> 0^1(4(1(3(0(x1))))) 394.29/100.82 0^1(1(3(0(0(x1))))) -> 1^1(3(0(x1))) 394.29/100.82 0^1(1(4(2(2(x1))))) -> 0^1(2(4(1(5(2(x1)))))) 394.29/100.82 0^1(1(4(2(2(x1))))) -> 1^1(5(2(x1))) 394.29/100.82 0^1(1(4(2(2(x1))))) -> 5^1(2(x1)) 394.29/100.82 0^1(1(4(2(2(x1))))) -> 0^1(2(4(2(1(1(x1)))))) 394.29/100.82 0^1(1(4(2(2(x1))))) -> 1^1(1(x1)) 394.29/100.82 0^1(1(4(2(2(x1))))) -> 1^1(x1) 394.29/100.82 1^1(3(0(1(2(x1))))) -> 1^1(3(4(1(0(2(x1)))))) 394.29/100.82 1^1(3(0(1(2(x1))))) -> 1^1(0(2(x1))) 394.29/100.82 1^1(3(0(1(2(x1))))) -> 0^1(2(x1)) 394.29/100.82 394.29/100.82 The TRS R consists of the following rules: 394.70/100.85 394.70/100.85 0(1(2(x1))) -> 0(3(1(2(x1)))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(1(x1)))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(0(x1)))) 394.70/100.85 0(1(2(x1))) -> 0(2(1(3(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(2(4(1(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(5(4(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(1(0(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(3(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(0(3(1(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(1(0(3(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(4(2(1(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(0(4(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(4(5(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(5(2(3(1(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(2(3(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(1(4(0(3(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(0(3(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(3(4(1(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(3(2(0(4(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(5(3(0(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(2(5(3(1(0(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(4(0(5(1(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 5(5(4(2(1(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(0(2(4(1(x1))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(2(1(5(0(4(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 1(0(2(3(5(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 5(1(2(0(4(0(x1)))))) 394.70/100.85 0(0(5(2(x1)))) -> 5(3(0(3(0(2(x1)))))) 394.70/100.85 0(1(2(0(x1)))) -> 1(2(4(0(0(x1))))) 394.70/100.85 0(1(2(0(x1)))) -> 0(4(1(0(0(2(x1)))))) 394.70/100.85 0(1(2(2(x1)))) -> 3(3(0(2(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 2(5(1(0(3(x1))))) 394.70/100.85 0(1(5(2(x1)))) -> 0(5(5(3(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 1(3(5(0(2(2(x1)))))) 394.70/100.85 0(5(2(2(x1)))) -> 3(0(2(1(5(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(4(1(2(1(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 1(2(1(3(0(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(3(1(4(1(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 4(1(4(1(0(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 0(2(1(3(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 2(0(3(1(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(3(1(0(3(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(5(1(0(2(4(x1)))))) 394.70/100.85 0(1(0(2(2(x1))))) -> 3(0(2(1(0(2(x1)))))) 394.70/100.85 0(1(2(5(2(x1))))) -> 3(5(1(2(2(0(x1)))))) 394.70/100.85 0(1(3(0(0(x1))))) -> 0(0(4(1(3(0(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(1(5(2(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(2(1(1(x1)))))) 394.70/100.85 1(3(0(1(2(x1))))) -> 1(3(4(1(0(2(x1)))))) 394.70/100.85 394.70/100.85 Q is empty. 394.70/100.85 We have to consider all minimal (P,Q,R)-chains. 394.70/100.85 ---------------------------------------- 394.70/100.85 394.70/100.85 (3) DependencyGraphProof (EQUIVALENT) 394.70/100.85 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 109 less nodes. 394.70/100.85 ---------------------------------------- 394.70/100.85 394.70/100.85 (4) 394.70/100.85 Obligation: 394.70/100.85 Q DP problem: 394.70/100.85 The TRS P consists of the following rules: 394.70/100.85 394.70/100.85 0^1(1(2(x1))) -> 1^1(x1) 394.70/100.85 1^1(0(1(2(x1)))) -> 1^1(x1) 394.70/100.85 1^1(0(1(2(x1)))) -> 0^1(x1) 394.70/100.85 0^1(1(2(x1))) -> 0^1(x1) 394.70/100.85 0^1(1(2(x1))) -> 1^1(0(x1)) 394.70/100.85 0^1(1(2(x1))) -> 5^1(x1) 394.70/100.85 5^1(0(1(2(x1)))) -> 5^1(x1) 394.70/100.85 5^1(0(1(2(x1)))) -> 1^1(5(x1)) 394.70/100.85 0^1(1(2(x1))) -> 1^1(5(x1)) 394.70/100.85 0^1(0(1(2(x1)))) -> 1^1(x1) 394.70/100.85 0^1(0(1(2(x1)))) -> 5^1(0(x1)) 394.70/100.85 0^1(0(1(2(x1)))) -> 0^1(x1) 394.70/100.85 0^1(1(2(0(x1)))) -> 0^1(0(x1)) 394.70/100.85 0^1(1(2(5(2(x1))))) -> 0^1(x1) 394.70/100.85 0^1(1(4(2(2(x1))))) -> 1^1(1(x1)) 394.70/100.85 0^1(1(4(2(2(x1))))) -> 1^1(x1) 394.70/100.85 394.70/100.85 The TRS R consists of the following rules: 394.70/100.85 394.70/100.85 0(1(2(x1))) -> 0(3(1(2(x1)))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(1(x1)))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(0(x1)))) 394.70/100.85 0(1(2(x1))) -> 0(2(1(3(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(2(4(1(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(5(4(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(1(0(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(3(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(0(3(1(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(1(0(3(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(4(2(1(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(0(4(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(4(5(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(5(2(3(1(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(2(3(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(1(4(0(3(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(0(3(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(3(4(1(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(3(2(0(4(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(5(3(0(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(2(5(3(1(0(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(4(0(5(1(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 5(5(4(2(1(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(0(2(4(1(x1))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(2(1(5(0(4(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 1(0(2(3(5(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 5(1(2(0(4(0(x1)))))) 394.70/100.85 0(0(5(2(x1)))) -> 5(3(0(3(0(2(x1)))))) 394.70/100.85 0(1(2(0(x1)))) -> 1(2(4(0(0(x1))))) 394.70/100.85 0(1(2(0(x1)))) -> 0(4(1(0(0(2(x1)))))) 394.70/100.85 0(1(2(2(x1)))) -> 3(3(0(2(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 2(5(1(0(3(x1))))) 394.70/100.85 0(1(5(2(x1)))) -> 0(5(5(3(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 1(3(5(0(2(2(x1)))))) 394.70/100.85 0(5(2(2(x1)))) -> 3(0(2(1(5(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(4(1(2(1(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 1(2(1(3(0(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(3(1(4(1(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 4(1(4(1(0(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 0(2(1(3(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 2(0(3(1(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(3(1(0(3(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(5(1(0(2(4(x1)))))) 394.70/100.85 0(1(0(2(2(x1))))) -> 3(0(2(1(0(2(x1)))))) 394.70/100.85 0(1(2(5(2(x1))))) -> 3(5(1(2(2(0(x1)))))) 394.70/100.85 0(1(3(0(0(x1))))) -> 0(0(4(1(3(0(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(1(5(2(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(2(1(1(x1)))))) 394.70/100.85 1(3(0(1(2(x1))))) -> 1(3(4(1(0(2(x1)))))) 394.70/100.85 394.70/100.85 Q is empty. 394.70/100.85 We have to consider all minimal (P,Q,R)-chains. 394.70/100.85 ---------------------------------------- 394.70/100.85 394.70/100.85 (5) QDPOrderProof (EQUIVALENT) 394.70/100.85 We use the reduction pair processor [LPAR04,JAR06]. 394.70/100.85 394.70/100.85 394.70/100.85 The following pairs can be oriented strictly and are deleted. 394.70/100.85 394.70/100.85 0^1(1(2(5(2(x1))))) -> 0^1(x1) 394.70/100.85 The remaining pairs can at least be oriented weakly. 394.70/100.85 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(0^1(x_1)) = [[0A]] + [[0A, 0A, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(1(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, 0A], [-I, 0A, 1A], [0A, -I, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(2(x_1)) = [[-I], [0A], [-I]] + [[0A, -I, -I], [0A, 0A, -I], [1A, 0A, 1A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(1^1(x_1)) = [[0A]] + [[0A, -I, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(0(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [-I, 0A, -I], [0A, -I, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(5^1(x_1)) = [[0A]] + [[0A, -I, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(5(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, 0A], [-I, 0A, 0A], [0A, -I, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(4(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, 0A], [0A, -I, 0A], [0A, -I, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(3(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [0A, -I, 0A], [-I, -I, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 394.70/100.85 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 394.70/100.85 394.70/100.85 0(1(2(x1))) -> 0(3(1(2(x1)))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(1(x1)))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(0(x1)))) 394.70/100.85 0(1(2(x1))) -> 0(2(1(3(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(2(4(1(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(5(4(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(1(0(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(3(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(0(3(1(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(1(0(3(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(4(2(1(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(0(4(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(4(5(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(5(2(3(1(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(2(3(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(1(4(0(3(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(0(3(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(3(4(1(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(3(2(0(4(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(5(3(0(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(2(5(3(1(0(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(4(0(5(1(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 5(5(4(2(1(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(0(2(4(1(x1))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(2(1(5(0(4(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 1(0(2(3(5(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 5(1(2(0(4(0(x1)))))) 394.70/100.85 0(0(5(2(x1)))) -> 5(3(0(3(0(2(x1)))))) 394.70/100.85 0(1(2(0(x1)))) -> 1(2(4(0(0(x1))))) 394.70/100.85 0(1(2(0(x1)))) -> 0(4(1(0(0(2(x1)))))) 394.70/100.85 0(1(2(2(x1)))) -> 3(3(0(2(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 2(5(1(0(3(x1))))) 394.70/100.85 0(1(5(2(x1)))) -> 0(5(5(3(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 1(3(5(0(2(2(x1)))))) 394.70/100.85 0(5(2(2(x1)))) -> 3(0(2(1(5(2(x1)))))) 394.70/100.85 0(1(0(2(2(x1))))) -> 3(0(2(1(0(2(x1)))))) 394.70/100.85 0(1(2(5(2(x1))))) -> 3(5(1(2(2(0(x1)))))) 394.70/100.85 0(1(3(0(0(x1))))) -> 0(0(4(1(3(0(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(1(5(2(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(2(1(1(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 0(2(1(3(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 2(0(3(1(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(3(1(0(3(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(5(1(0(2(4(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(4(1(2(1(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 1(2(1(3(0(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(3(1(4(1(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 4(1(4(1(0(2(x1)))))) 394.70/100.85 1(3(0(1(2(x1))))) -> 1(3(4(1(0(2(x1)))))) 394.70/100.85 394.70/100.85 394.70/100.85 ---------------------------------------- 394.70/100.85 394.70/100.85 (6) 394.70/100.85 Obligation: 394.70/100.85 Q DP problem: 394.70/100.85 The TRS P consists of the following rules: 394.70/100.85 394.70/100.85 0^1(1(2(x1))) -> 1^1(x1) 394.70/100.85 1^1(0(1(2(x1)))) -> 1^1(x1) 394.70/100.85 1^1(0(1(2(x1)))) -> 0^1(x1) 394.70/100.85 0^1(1(2(x1))) -> 0^1(x1) 394.70/100.85 0^1(1(2(x1))) -> 1^1(0(x1)) 394.70/100.85 0^1(1(2(x1))) -> 5^1(x1) 394.70/100.85 5^1(0(1(2(x1)))) -> 5^1(x1) 394.70/100.85 5^1(0(1(2(x1)))) -> 1^1(5(x1)) 394.70/100.85 0^1(1(2(x1))) -> 1^1(5(x1)) 394.70/100.85 0^1(0(1(2(x1)))) -> 1^1(x1) 394.70/100.85 0^1(0(1(2(x1)))) -> 5^1(0(x1)) 394.70/100.85 0^1(0(1(2(x1)))) -> 0^1(x1) 394.70/100.85 0^1(1(2(0(x1)))) -> 0^1(0(x1)) 394.70/100.85 0^1(1(4(2(2(x1))))) -> 1^1(1(x1)) 394.70/100.85 0^1(1(4(2(2(x1))))) -> 1^1(x1) 394.70/100.85 394.70/100.85 The TRS R consists of the following rules: 394.70/100.85 394.70/100.85 0(1(2(x1))) -> 0(3(1(2(x1)))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(1(x1)))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(0(x1)))) 394.70/100.85 0(1(2(x1))) -> 0(2(1(3(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(2(4(1(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(5(4(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(1(0(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(3(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(0(3(1(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(1(0(3(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(4(2(1(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(0(4(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(4(5(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(5(2(3(1(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(2(3(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(1(4(0(3(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(0(3(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(3(4(1(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(3(2(0(4(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(5(3(0(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(2(5(3(1(0(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(4(0(5(1(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 5(5(4(2(1(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(0(2(4(1(x1))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(2(1(5(0(4(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 1(0(2(3(5(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 5(1(2(0(4(0(x1)))))) 394.70/100.85 0(0(5(2(x1)))) -> 5(3(0(3(0(2(x1)))))) 394.70/100.85 0(1(2(0(x1)))) -> 1(2(4(0(0(x1))))) 394.70/100.85 0(1(2(0(x1)))) -> 0(4(1(0(0(2(x1)))))) 394.70/100.85 0(1(2(2(x1)))) -> 3(3(0(2(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 2(5(1(0(3(x1))))) 394.70/100.85 0(1(5(2(x1)))) -> 0(5(5(3(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 1(3(5(0(2(2(x1)))))) 394.70/100.85 0(5(2(2(x1)))) -> 3(0(2(1(5(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(4(1(2(1(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 1(2(1(3(0(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(3(1(4(1(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 4(1(4(1(0(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 0(2(1(3(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 2(0(3(1(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(3(1(0(3(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(5(1(0(2(4(x1)))))) 394.70/100.85 0(1(0(2(2(x1))))) -> 3(0(2(1(0(2(x1)))))) 394.70/100.85 0(1(2(5(2(x1))))) -> 3(5(1(2(2(0(x1)))))) 394.70/100.85 0(1(3(0(0(x1))))) -> 0(0(4(1(3(0(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(1(5(2(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(2(1(1(x1)))))) 394.70/100.85 1(3(0(1(2(x1))))) -> 1(3(4(1(0(2(x1)))))) 394.70/100.85 394.70/100.85 Q is empty. 394.70/100.85 We have to consider all minimal (P,Q,R)-chains. 394.70/100.85 ---------------------------------------- 394.70/100.85 394.70/100.85 (7) QDPOrderProof (EQUIVALENT) 394.70/100.85 We use the reduction pair processor [LPAR04,JAR06]. 394.70/100.85 394.70/100.85 394.70/100.85 The following pairs can be oriented strictly and are deleted. 394.70/100.85 394.70/100.85 0^1(0(1(2(x1)))) -> 5^1(0(x1)) 394.70/100.85 The remaining pairs can at least be oriented weakly. 394.70/100.85 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(0^1(x_1)) = [[0A]] + [[-I, 0A, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(1(x_1)) = [[1A], [0A], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [-I, 0A, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(2(x_1)) = [[0A], [-I], [-I]] + [[-I, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(1^1(x_1)) = [[-I]] + [[0A, -I, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(0(x_1)) = [[-I], [0A], [-I]] + [[0A, -I, -I], [0A, 0A, 0A], [0A, -I, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(5^1(x_1)) = [[0A]] + [[-I, -I, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(5(x_1)) = [[0A], [-I], [0A]] + [[-I, -I, -I], [-I, -I, 0A], [-I, -I, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(4(x_1)) = [[1A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [-I, 0A, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(3(x_1)) = [[0A], [-I], [-I]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 394.70/100.85 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 394.70/100.85 394.70/100.85 0(1(2(x1))) -> 0(3(1(2(x1)))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(1(x1)))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(0(x1)))) 394.70/100.85 0(1(2(x1))) -> 0(2(1(3(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(2(4(1(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(5(4(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(1(0(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(3(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(0(3(1(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(1(0(3(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(4(2(1(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(0(4(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(4(5(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(5(2(3(1(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(2(3(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(1(4(0(3(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(0(3(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(3(4(1(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(3(2(0(4(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(5(3(0(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(2(5(3(1(0(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(4(0(5(1(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 5(5(4(2(1(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(0(2(4(1(x1))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(2(1(5(0(4(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 1(0(2(3(5(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 5(1(2(0(4(0(x1)))))) 394.70/100.85 0(0(5(2(x1)))) -> 5(3(0(3(0(2(x1)))))) 394.70/100.85 0(1(2(0(x1)))) -> 1(2(4(0(0(x1))))) 394.70/100.85 0(1(2(0(x1)))) -> 0(4(1(0(0(2(x1)))))) 394.70/100.85 0(1(2(2(x1)))) -> 3(3(0(2(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 2(5(1(0(3(x1))))) 394.70/100.85 0(1(5(2(x1)))) -> 0(5(5(3(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 1(3(5(0(2(2(x1)))))) 394.70/100.85 0(5(2(2(x1)))) -> 3(0(2(1(5(2(x1)))))) 394.70/100.85 0(1(0(2(2(x1))))) -> 3(0(2(1(0(2(x1)))))) 394.70/100.85 0(1(2(5(2(x1))))) -> 3(5(1(2(2(0(x1)))))) 394.70/100.85 0(1(3(0(0(x1))))) -> 0(0(4(1(3(0(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(1(5(2(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(2(1(1(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 0(2(1(3(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 2(0(3(1(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(3(1(0(3(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(5(1(0(2(4(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(4(1(2(1(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 1(2(1(3(0(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(3(1(4(1(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 4(1(4(1(0(2(x1)))))) 394.70/100.85 1(3(0(1(2(x1))))) -> 1(3(4(1(0(2(x1)))))) 394.70/100.85 394.70/100.85 394.70/100.85 ---------------------------------------- 394.70/100.85 394.70/100.85 (8) 394.70/100.85 Obligation: 394.70/100.85 Q DP problem: 394.70/100.85 The TRS P consists of the following rules: 394.70/100.85 394.70/100.85 0^1(1(2(x1))) -> 1^1(x1) 394.70/100.85 1^1(0(1(2(x1)))) -> 1^1(x1) 394.70/100.85 1^1(0(1(2(x1)))) -> 0^1(x1) 394.70/100.85 0^1(1(2(x1))) -> 0^1(x1) 394.70/100.85 0^1(1(2(x1))) -> 1^1(0(x1)) 394.70/100.85 0^1(1(2(x1))) -> 5^1(x1) 394.70/100.85 5^1(0(1(2(x1)))) -> 5^1(x1) 394.70/100.85 5^1(0(1(2(x1)))) -> 1^1(5(x1)) 394.70/100.85 0^1(1(2(x1))) -> 1^1(5(x1)) 394.70/100.85 0^1(0(1(2(x1)))) -> 1^1(x1) 394.70/100.85 0^1(0(1(2(x1)))) -> 0^1(x1) 394.70/100.85 0^1(1(2(0(x1)))) -> 0^1(0(x1)) 394.70/100.85 0^1(1(4(2(2(x1))))) -> 1^1(1(x1)) 394.70/100.85 0^1(1(4(2(2(x1))))) -> 1^1(x1) 394.70/100.85 394.70/100.85 The TRS R consists of the following rules: 394.70/100.85 394.70/100.85 0(1(2(x1))) -> 0(3(1(2(x1)))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(1(x1)))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(0(x1)))) 394.70/100.85 0(1(2(x1))) -> 0(2(1(3(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(2(4(1(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(5(4(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(1(0(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(3(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(0(3(1(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(1(0(3(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(4(2(1(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(0(4(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(4(5(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(5(2(3(1(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(2(3(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(1(4(0(3(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(0(3(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(3(4(1(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(3(2(0(4(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(5(3(0(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(2(5(3(1(0(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(4(0(5(1(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 5(5(4(2(1(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(0(2(4(1(x1))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(2(1(5(0(4(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 1(0(2(3(5(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 5(1(2(0(4(0(x1)))))) 394.70/100.85 0(0(5(2(x1)))) -> 5(3(0(3(0(2(x1)))))) 394.70/100.85 0(1(2(0(x1)))) -> 1(2(4(0(0(x1))))) 394.70/100.85 0(1(2(0(x1)))) -> 0(4(1(0(0(2(x1)))))) 394.70/100.85 0(1(2(2(x1)))) -> 3(3(0(2(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 2(5(1(0(3(x1))))) 394.70/100.85 0(1(5(2(x1)))) -> 0(5(5(3(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 1(3(5(0(2(2(x1)))))) 394.70/100.85 0(5(2(2(x1)))) -> 3(0(2(1(5(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(4(1(2(1(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 1(2(1(3(0(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(3(1(4(1(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 4(1(4(1(0(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 0(2(1(3(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 2(0(3(1(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(3(1(0(3(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(5(1(0(2(4(x1)))))) 394.70/100.85 0(1(0(2(2(x1))))) -> 3(0(2(1(0(2(x1)))))) 394.70/100.85 0(1(2(5(2(x1))))) -> 3(5(1(2(2(0(x1)))))) 394.70/100.85 0(1(3(0(0(x1))))) -> 0(0(4(1(3(0(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(1(5(2(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(2(1(1(x1)))))) 394.70/100.85 1(3(0(1(2(x1))))) -> 1(3(4(1(0(2(x1)))))) 394.70/100.85 394.70/100.85 Q is empty. 394.70/100.85 We have to consider all minimal (P,Q,R)-chains. 394.70/100.85 ---------------------------------------- 394.70/100.85 394.70/100.85 (9) QDPOrderProof (EQUIVALENT) 394.70/100.85 We use the reduction pair processor [LPAR04,JAR06]. 394.70/100.85 394.70/100.85 394.70/100.85 The following pairs can be oriented strictly and are deleted. 394.70/100.85 394.70/100.85 5^1(0(1(2(x1)))) -> 5^1(x1) 394.70/100.85 5^1(0(1(2(x1)))) -> 1^1(5(x1)) 394.70/100.85 The remaining pairs can at least be oriented weakly. 394.70/100.85 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(0^1(x_1)) = [[0A]] + [[-I, 0A, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(1(x_1)) = [[0A], [0A], [1A]] + [[-I, 0A, -I], [-I, -I, -I], [0A, 0A, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(2(x_1)) = [[0A], [-I], [-I]] + [[0A, 0A, 0A], [-I, 0A, -I], [-I, -I, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(1^1(x_1)) = [[1A]] + [[-I, 0A, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(0(x_1)) = [[0A], [-I], [-I]] + [[0A, 0A, 1A], [-I, 0A, 0A], [-I, -I, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(5^1(x_1)) = [[1A]] + [[0A, 0A, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(5(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(4(x_1)) = [[0A], [-I], [-I]] + [[-I, -I, -I], [-I, 0A, -I], [-I, -I, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(3(x_1)) = [[0A], [-I], [-I]] + [[0A, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 394.70/100.85 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 394.70/100.85 394.70/100.85 0(1(2(x1))) -> 0(3(1(2(x1)))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(1(x1)))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(0(x1)))) 394.70/100.85 0(1(2(x1))) -> 0(2(1(3(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(2(4(1(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(5(4(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(1(0(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(3(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(0(3(1(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(1(0(3(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(4(2(1(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(0(4(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(4(5(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(5(2(3(1(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(2(3(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(1(4(0(3(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(0(3(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(3(4(1(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(3(2(0(4(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(5(3(0(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(2(5(3(1(0(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(4(0(5(1(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 5(5(4(2(1(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(0(2(4(1(x1))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(2(1(5(0(4(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 1(0(2(3(5(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 5(1(2(0(4(0(x1)))))) 394.70/100.85 0(0(5(2(x1)))) -> 5(3(0(3(0(2(x1)))))) 394.70/100.85 0(1(2(0(x1)))) -> 1(2(4(0(0(x1))))) 394.70/100.85 0(1(2(0(x1)))) -> 0(4(1(0(0(2(x1)))))) 394.70/100.85 0(1(2(2(x1)))) -> 3(3(0(2(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 2(5(1(0(3(x1))))) 394.70/100.85 0(1(5(2(x1)))) -> 0(5(5(3(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 1(3(5(0(2(2(x1)))))) 394.70/100.85 0(5(2(2(x1)))) -> 3(0(2(1(5(2(x1)))))) 394.70/100.85 0(1(0(2(2(x1))))) -> 3(0(2(1(0(2(x1)))))) 394.70/100.85 0(1(2(5(2(x1))))) -> 3(5(1(2(2(0(x1)))))) 394.70/100.85 0(1(3(0(0(x1))))) -> 0(0(4(1(3(0(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(1(5(2(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(2(1(1(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 0(2(1(3(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 2(0(3(1(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(3(1(0(3(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(5(1(0(2(4(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(4(1(2(1(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 1(2(1(3(0(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(3(1(4(1(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 4(1(4(1(0(2(x1)))))) 394.70/100.85 1(3(0(1(2(x1))))) -> 1(3(4(1(0(2(x1)))))) 394.70/100.85 394.70/100.85 394.70/100.85 ---------------------------------------- 394.70/100.85 394.70/100.85 (10) 394.70/100.85 Obligation: 394.70/100.85 Q DP problem: 394.70/100.85 The TRS P consists of the following rules: 394.70/100.85 394.70/100.85 0^1(1(2(x1))) -> 1^1(x1) 394.70/100.85 1^1(0(1(2(x1)))) -> 1^1(x1) 394.70/100.85 1^1(0(1(2(x1)))) -> 0^1(x1) 394.70/100.85 0^1(1(2(x1))) -> 0^1(x1) 394.70/100.85 0^1(1(2(x1))) -> 1^1(0(x1)) 394.70/100.85 0^1(1(2(x1))) -> 5^1(x1) 394.70/100.85 0^1(1(2(x1))) -> 1^1(5(x1)) 394.70/100.85 0^1(0(1(2(x1)))) -> 1^1(x1) 394.70/100.85 0^1(0(1(2(x1)))) -> 0^1(x1) 394.70/100.85 0^1(1(2(0(x1)))) -> 0^1(0(x1)) 394.70/100.85 0^1(1(4(2(2(x1))))) -> 1^1(1(x1)) 394.70/100.85 0^1(1(4(2(2(x1))))) -> 1^1(x1) 394.70/100.85 394.70/100.85 The TRS R consists of the following rules: 394.70/100.85 394.70/100.85 0(1(2(x1))) -> 0(3(1(2(x1)))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(1(x1)))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(0(x1)))) 394.70/100.85 0(1(2(x1))) -> 0(2(1(3(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(2(4(1(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(5(4(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(1(0(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(3(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(0(3(1(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(1(0(3(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(4(2(1(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(0(4(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(4(5(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(5(2(3(1(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(2(3(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(1(4(0(3(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(0(3(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(3(4(1(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(3(2(0(4(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(5(3(0(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(2(5(3(1(0(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(4(0(5(1(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 5(5(4(2(1(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(0(2(4(1(x1))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(2(1(5(0(4(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 1(0(2(3(5(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 5(1(2(0(4(0(x1)))))) 394.70/100.85 0(0(5(2(x1)))) -> 5(3(0(3(0(2(x1)))))) 394.70/100.85 0(1(2(0(x1)))) -> 1(2(4(0(0(x1))))) 394.70/100.85 0(1(2(0(x1)))) -> 0(4(1(0(0(2(x1)))))) 394.70/100.85 0(1(2(2(x1)))) -> 3(3(0(2(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 2(5(1(0(3(x1))))) 394.70/100.85 0(1(5(2(x1)))) -> 0(5(5(3(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 1(3(5(0(2(2(x1)))))) 394.70/100.85 0(5(2(2(x1)))) -> 3(0(2(1(5(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(4(1(2(1(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 1(2(1(3(0(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(3(1(4(1(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 4(1(4(1(0(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 0(2(1(3(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 2(0(3(1(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(3(1(0(3(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(5(1(0(2(4(x1)))))) 394.70/100.85 0(1(0(2(2(x1))))) -> 3(0(2(1(0(2(x1)))))) 394.70/100.85 0(1(2(5(2(x1))))) -> 3(5(1(2(2(0(x1)))))) 394.70/100.85 0(1(3(0(0(x1))))) -> 0(0(4(1(3(0(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(1(5(2(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(2(1(1(x1)))))) 394.70/100.85 1(3(0(1(2(x1))))) -> 1(3(4(1(0(2(x1)))))) 394.70/100.85 394.70/100.85 Q is empty. 394.70/100.85 We have to consider all minimal (P,Q,R)-chains. 394.70/100.85 ---------------------------------------- 394.70/100.85 394.70/100.85 (11) DependencyGraphProof (EQUIVALENT) 394.70/100.85 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 394.70/100.85 ---------------------------------------- 394.70/100.85 394.70/100.85 (12) 394.70/100.85 Obligation: 394.70/100.85 Q DP problem: 394.70/100.85 The TRS P consists of the following rules: 394.70/100.85 394.70/100.85 1^1(0(1(2(x1)))) -> 1^1(x1) 394.70/100.85 1^1(0(1(2(x1)))) -> 0^1(x1) 394.70/100.85 0^1(1(2(x1))) -> 1^1(x1) 394.70/100.85 0^1(1(2(x1))) -> 0^1(x1) 394.70/100.85 0^1(1(2(x1))) -> 1^1(0(x1)) 394.70/100.85 0^1(1(2(x1))) -> 1^1(5(x1)) 394.70/100.85 0^1(0(1(2(x1)))) -> 1^1(x1) 394.70/100.85 0^1(0(1(2(x1)))) -> 0^1(x1) 394.70/100.85 0^1(1(2(0(x1)))) -> 0^1(0(x1)) 394.70/100.85 0^1(1(4(2(2(x1))))) -> 1^1(1(x1)) 394.70/100.85 0^1(1(4(2(2(x1))))) -> 1^1(x1) 394.70/100.85 394.70/100.85 The TRS R consists of the following rules: 394.70/100.85 394.70/100.85 0(1(2(x1))) -> 0(3(1(2(x1)))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(1(x1)))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(0(x1)))) 394.70/100.85 0(1(2(x1))) -> 0(2(1(3(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(2(4(1(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(5(4(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(1(0(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(3(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(0(3(1(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(1(0(3(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(4(2(1(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(0(4(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(4(5(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(5(2(3(1(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(2(3(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(1(4(0(3(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(0(3(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(3(4(1(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(3(2(0(4(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(5(3(0(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(2(5(3(1(0(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(4(0(5(1(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 5(5(4(2(1(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(0(2(4(1(x1))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(2(1(5(0(4(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 1(0(2(3(5(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 5(1(2(0(4(0(x1)))))) 394.70/100.85 0(0(5(2(x1)))) -> 5(3(0(3(0(2(x1)))))) 394.70/100.85 0(1(2(0(x1)))) -> 1(2(4(0(0(x1))))) 394.70/100.85 0(1(2(0(x1)))) -> 0(4(1(0(0(2(x1)))))) 394.70/100.85 0(1(2(2(x1)))) -> 3(3(0(2(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 2(5(1(0(3(x1))))) 394.70/100.85 0(1(5(2(x1)))) -> 0(5(5(3(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 1(3(5(0(2(2(x1)))))) 394.70/100.85 0(5(2(2(x1)))) -> 3(0(2(1(5(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(4(1(2(1(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 1(2(1(3(0(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(3(1(4(1(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 4(1(4(1(0(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 0(2(1(3(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 2(0(3(1(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(3(1(0(3(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(5(1(0(2(4(x1)))))) 394.70/100.85 0(1(0(2(2(x1))))) -> 3(0(2(1(0(2(x1)))))) 394.70/100.85 0(1(2(5(2(x1))))) -> 3(5(1(2(2(0(x1)))))) 394.70/100.85 0(1(3(0(0(x1))))) -> 0(0(4(1(3(0(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(1(5(2(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(2(1(1(x1)))))) 394.70/100.85 1(3(0(1(2(x1))))) -> 1(3(4(1(0(2(x1)))))) 394.70/100.85 394.70/100.85 Q is empty. 394.70/100.85 We have to consider all minimal (P,Q,R)-chains. 394.70/100.85 ---------------------------------------- 394.70/100.85 394.70/100.85 (13) QDPOrderProof (EQUIVALENT) 394.70/100.85 We use the reduction pair processor [LPAR04,JAR06]. 394.70/100.85 394.70/100.85 394.70/100.85 The following pairs can be oriented strictly and are deleted. 394.70/100.85 394.70/100.85 0^1(0(1(2(x1)))) -> 1^1(x1) 394.70/100.85 0^1(0(1(2(x1)))) -> 0^1(x1) 394.70/100.85 The remaining pairs can at least be oriented weakly. 394.70/100.85 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(1^1(x_1)) = [[1A]] + [[0A, -I, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(0(x_1)) = [[0A], [0A], [-I]] + [[0A, 0A, -I], [1A, 0A, 0A], [-I, 0A, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(1(x_1)) = [[1A], [1A], [0A]] + [[0A, -I, 0A], [0A, -I, 0A], [0A, -I, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(2(x_1)) = [[0A], [1A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(0^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(5(x_1)) = [[1A], [0A], [1A]] + [[-I, -I, -I], [0A, 0A, -I], [-I, -I, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(4(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(3(x_1)) = [[0A], [-I], [0A]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 394.70/100.85 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 394.70/100.85 394.70/100.85 0(1(2(x1))) -> 0(3(1(2(x1)))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(1(x1)))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(0(x1)))) 394.70/100.85 0(1(2(x1))) -> 0(2(1(3(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(2(4(1(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(5(4(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(1(0(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(3(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(0(3(1(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(1(0(3(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(4(2(1(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(0(4(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(4(5(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(5(2(3(1(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(2(3(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(1(4(0(3(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(0(3(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(3(4(1(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(3(2(0(4(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(5(3(0(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(2(5(3(1(0(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(4(0(5(1(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 5(5(4(2(1(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(0(2(4(1(x1))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(2(1(5(0(4(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 1(0(2(3(5(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 5(1(2(0(4(0(x1)))))) 394.70/100.85 0(0(5(2(x1)))) -> 5(3(0(3(0(2(x1)))))) 394.70/100.85 0(1(2(0(x1)))) -> 1(2(4(0(0(x1))))) 394.70/100.85 0(1(2(0(x1)))) -> 0(4(1(0(0(2(x1)))))) 394.70/100.85 0(1(2(2(x1)))) -> 3(3(0(2(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 2(5(1(0(3(x1))))) 394.70/100.85 0(1(5(2(x1)))) -> 0(5(5(3(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 1(3(5(0(2(2(x1)))))) 394.70/100.85 0(5(2(2(x1)))) -> 3(0(2(1(5(2(x1)))))) 394.70/100.85 0(1(0(2(2(x1))))) -> 3(0(2(1(0(2(x1)))))) 394.70/100.85 0(1(2(5(2(x1))))) -> 3(5(1(2(2(0(x1)))))) 394.70/100.85 0(1(3(0(0(x1))))) -> 0(0(4(1(3(0(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(1(5(2(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(2(1(1(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 0(2(1(3(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 2(0(3(1(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(3(1(0(3(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(5(1(0(2(4(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(4(1(2(1(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 1(2(1(3(0(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(3(1(4(1(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 4(1(4(1(0(2(x1)))))) 394.70/100.85 1(3(0(1(2(x1))))) -> 1(3(4(1(0(2(x1)))))) 394.70/100.85 394.70/100.85 394.70/100.85 ---------------------------------------- 394.70/100.85 394.70/100.85 (14) 394.70/100.85 Obligation: 394.70/100.85 Q DP problem: 394.70/100.85 The TRS P consists of the following rules: 394.70/100.85 394.70/100.85 1^1(0(1(2(x1)))) -> 1^1(x1) 394.70/100.85 1^1(0(1(2(x1)))) -> 0^1(x1) 394.70/100.85 0^1(1(2(x1))) -> 1^1(x1) 394.70/100.85 0^1(1(2(x1))) -> 0^1(x1) 394.70/100.85 0^1(1(2(x1))) -> 1^1(0(x1)) 394.70/100.85 0^1(1(2(x1))) -> 1^1(5(x1)) 394.70/100.85 0^1(1(2(0(x1)))) -> 0^1(0(x1)) 394.70/100.85 0^1(1(4(2(2(x1))))) -> 1^1(1(x1)) 394.70/100.85 0^1(1(4(2(2(x1))))) -> 1^1(x1) 394.70/100.85 394.70/100.85 The TRS R consists of the following rules: 394.70/100.85 394.70/100.85 0(1(2(x1))) -> 0(3(1(2(x1)))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(1(x1)))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(0(x1)))) 394.70/100.85 0(1(2(x1))) -> 0(2(1(3(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(2(4(1(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(5(4(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(1(0(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(3(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(0(3(1(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(1(0(3(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(4(2(1(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(0(4(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(4(5(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(5(2(3(1(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(2(3(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(1(4(0(3(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(0(3(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(3(4(1(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(3(2(0(4(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(5(3(0(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(2(5(3(1(0(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(4(0(5(1(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 5(5(4(2(1(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(0(2(4(1(x1))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(2(1(5(0(4(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 1(0(2(3(5(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 5(1(2(0(4(0(x1)))))) 394.70/100.85 0(0(5(2(x1)))) -> 5(3(0(3(0(2(x1)))))) 394.70/100.85 0(1(2(0(x1)))) -> 1(2(4(0(0(x1))))) 394.70/100.85 0(1(2(0(x1)))) -> 0(4(1(0(0(2(x1)))))) 394.70/100.85 0(1(2(2(x1)))) -> 3(3(0(2(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 2(5(1(0(3(x1))))) 394.70/100.85 0(1(5(2(x1)))) -> 0(5(5(3(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 1(3(5(0(2(2(x1)))))) 394.70/100.85 0(5(2(2(x1)))) -> 3(0(2(1(5(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(4(1(2(1(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 1(2(1(3(0(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(3(1(4(1(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 4(1(4(1(0(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 0(2(1(3(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 2(0(3(1(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(3(1(0(3(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(5(1(0(2(4(x1)))))) 394.70/100.85 0(1(0(2(2(x1))))) -> 3(0(2(1(0(2(x1)))))) 394.70/100.85 0(1(2(5(2(x1))))) -> 3(5(1(2(2(0(x1)))))) 394.70/100.85 0(1(3(0(0(x1))))) -> 0(0(4(1(3(0(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(1(5(2(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(2(1(1(x1)))))) 394.70/100.85 1(3(0(1(2(x1))))) -> 1(3(4(1(0(2(x1)))))) 394.70/100.85 394.70/100.85 Q is empty. 394.70/100.85 We have to consider all minimal (P,Q,R)-chains. 394.70/100.85 ---------------------------------------- 394.70/100.85 394.70/100.85 (15) QDPOrderProof (EQUIVALENT) 394.70/100.85 We use the reduction pair processor [LPAR04,JAR06]. 394.70/100.85 394.70/100.85 394.70/100.85 The following pairs can be oriented strictly and are deleted. 394.70/100.85 394.70/100.85 1^1(0(1(2(x1)))) -> 1^1(x1) 394.70/100.85 The remaining pairs can at least be oriented weakly. 394.70/100.85 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(1^1(x_1)) = [[1A]] + [[0A, 0A, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(0(x_1)) = [[-I], [-I], [0A]] + [[0A, 1A, 0A], [-I, -I, 0A], [-I, -I, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(1(x_1)) = [[0A], [1A], [1A]] + [[0A, 0A, 0A], [-I, 0A, 0A], [0A, 0A, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(2(x_1)) = [[0A], [-I], [0A]] + [[0A, 0A, 0A], [-I, -I, -I], [0A, 0A, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(0^1(x_1)) = [[1A]] + [[1A, 0A, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(5(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, -I], [-I, 0A, -I], [-I, 0A, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(4(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, 0A], [-I, -I, -I], [-I, 0A, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(3(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, 0A], [-I, -I, -I], [-I, -I, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 394.70/100.85 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 394.70/100.85 394.70/100.85 0(1(2(x1))) -> 0(3(1(2(x1)))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(1(x1)))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(0(x1)))) 394.70/100.85 0(1(2(x1))) -> 0(2(1(3(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(2(4(1(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(5(4(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(1(0(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(3(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(0(3(1(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(1(0(3(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(4(2(1(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(0(4(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(4(5(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(5(2(3(1(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(2(3(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(1(4(0(3(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(0(3(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(3(4(1(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(3(2(0(4(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(5(3(0(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(2(5(3(1(0(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(4(0(5(1(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 5(5(4(2(1(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(0(2(4(1(x1))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(2(1(5(0(4(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 1(0(2(3(5(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 5(1(2(0(4(0(x1)))))) 394.70/100.85 0(0(5(2(x1)))) -> 5(3(0(3(0(2(x1)))))) 394.70/100.85 0(1(2(0(x1)))) -> 1(2(4(0(0(x1))))) 394.70/100.85 0(1(2(0(x1)))) -> 0(4(1(0(0(2(x1)))))) 394.70/100.85 0(1(2(2(x1)))) -> 3(3(0(2(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 2(5(1(0(3(x1))))) 394.70/100.85 0(1(5(2(x1)))) -> 0(5(5(3(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 1(3(5(0(2(2(x1)))))) 394.70/100.85 0(5(2(2(x1)))) -> 3(0(2(1(5(2(x1)))))) 394.70/100.85 0(1(0(2(2(x1))))) -> 3(0(2(1(0(2(x1)))))) 394.70/100.85 0(1(2(5(2(x1))))) -> 3(5(1(2(2(0(x1)))))) 394.70/100.85 0(1(3(0(0(x1))))) -> 0(0(4(1(3(0(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(1(5(2(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(2(1(1(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 0(2(1(3(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 2(0(3(1(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(3(1(0(3(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(5(1(0(2(4(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(4(1(2(1(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 1(2(1(3(0(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(3(1(4(1(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 4(1(4(1(0(2(x1)))))) 394.70/100.85 1(3(0(1(2(x1))))) -> 1(3(4(1(0(2(x1)))))) 394.70/100.85 394.70/100.85 394.70/100.85 ---------------------------------------- 394.70/100.85 394.70/100.85 (16) 394.70/100.85 Obligation: 394.70/100.85 Q DP problem: 394.70/100.85 The TRS P consists of the following rules: 394.70/100.85 394.70/100.85 1^1(0(1(2(x1)))) -> 0^1(x1) 394.70/100.85 0^1(1(2(x1))) -> 1^1(x1) 394.70/100.85 0^1(1(2(x1))) -> 0^1(x1) 394.70/100.85 0^1(1(2(x1))) -> 1^1(0(x1)) 394.70/100.85 0^1(1(2(x1))) -> 1^1(5(x1)) 394.70/100.85 0^1(1(2(0(x1)))) -> 0^1(0(x1)) 394.70/100.85 0^1(1(4(2(2(x1))))) -> 1^1(1(x1)) 394.70/100.85 0^1(1(4(2(2(x1))))) -> 1^1(x1) 394.70/100.85 394.70/100.85 The TRS R consists of the following rules: 394.70/100.85 394.70/100.85 0(1(2(x1))) -> 0(3(1(2(x1)))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(1(x1)))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(0(x1)))) 394.70/100.85 0(1(2(x1))) -> 0(2(1(3(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(2(4(1(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(5(4(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(1(0(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(3(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(0(3(1(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(1(0(3(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(4(2(1(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(0(4(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(4(5(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(5(2(3(1(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(2(3(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(1(4(0(3(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(0(3(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(3(4(1(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(3(2(0(4(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(5(3(0(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(2(5(3(1(0(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(4(0(5(1(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 5(5(4(2(1(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(0(2(4(1(x1))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(2(1(5(0(4(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 1(0(2(3(5(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 5(1(2(0(4(0(x1)))))) 394.70/100.85 0(0(5(2(x1)))) -> 5(3(0(3(0(2(x1)))))) 394.70/100.85 0(1(2(0(x1)))) -> 1(2(4(0(0(x1))))) 394.70/100.85 0(1(2(0(x1)))) -> 0(4(1(0(0(2(x1)))))) 394.70/100.85 0(1(2(2(x1)))) -> 3(3(0(2(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 2(5(1(0(3(x1))))) 394.70/100.85 0(1(5(2(x1)))) -> 0(5(5(3(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 1(3(5(0(2(2(x1)))))) 394.70/100.85 0(5(2(2(x1)))) -> 3(0(2(1(5(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(4(1(2(1(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 1(2(1(3(0(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(3(1(4(1(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 4(1(4(1(0(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 0(2(1(3(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 2(0(3(1(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(3(1(0(3(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(5(1(0(2(4(x1)))))) 394.70/100.85 0(1(0(2(2(x1))))) -> 3(0(2(1(0(2(x1)))))) 394.70/100.85 0(1(2(5(2(x1))))) -> 3(5(1(2(2(0(x1)))))) 394.70/100.85 0(1(3(0(0(x1))))) -> 0(0(4(1(3(0(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(1(5(2(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(2(1(1(x1)))))) 394.70/100.85 1(3(0(1(2(x1))))) -> 1(3(4(1(0(2(x1)))))) 394.70/100.85 394.70/100.85 Q is empty. 394.70/100.85 We have to consider all minimal (P,Q,R)-chains. 394.70/100.85 ---------------------------------------- 394.70/100.85 394.70/100.85 (17) QDPOrderProof (EQUIVALENT) 394.70/100.85 We use the reduction pair processor [LPAR04,JAR06]. 394.70/100.85 394.70/100.85 394.70/100.85 The following pairs can be oriented strictly and are deleted. 394.70/100.85 394.70/100.85 0^1(1(2(0(x1)))) -> 0^1(0(x1)) 394.70/100.85 The remaining pairs can at least be oriented weakly. 394.70/100.85 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(1^1(x_1)) = [[0A]] + [[0A, -I, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(0(x_1)) = [[-I], [0A], [0A]] + [[0A, -I, -I], [0A, -I, -I], [0A, -I, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(1(x_1)) = [[-I], [0A], [0A]] + [[0A, -I, 0A], [0A, -I, 0A], [0A, 0A, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(2(x_1)) = [[0A], [-I], [1A]] + [[0A, -I, 0A], [-I, -I, -I], [0A, 1A, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(0^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(5(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, 0A], [0A, -I, 0A], [0A, -I, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(4(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [-I, -I, 0A], [-I, -I, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(3(x_1)) = [[0A], [-I], [-I]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 394.70/100.85 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 394.70/100.85 394.70/100.85 0(1(2(x1))) -> 0(3(1(2(x1)))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(1(x1)))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(0(x1)))) 394.70/100.85 0(1(2(x1))) -> 0(2(1(3(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(2(4(1(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(5(4(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(1(0(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(3(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(0(3(1(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(1(0(3(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(4(2(1(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(0(4(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(4(5(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(5(2(3(1(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(2(3(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(1(4(0(3(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(0(3(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(3(4(1(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(3(2(0(4(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(5(3(0(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(2(5(3(1(0(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(4(0(5(1(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 5(5(4(2(1(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(0(2(4(1(x1))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(2(1(5(0(4(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 1(0(2(3(5(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 5(1(2(0(4(0(x1)))))) 394.70/100.85 0(0(5(2(x1)))) -> 5(3(0(3(0(2(x1)))))) 394.70/100.85 0(1(2(0(x1)))) -> 1(2(4(0(0(x1))))) 394.70/100.85 0(1(2(0(x1)))) -> 0(4(1(0(0(2(x1)))))) 394.70/100.85 0(1(2(2(x1)))) -> 3(3(0(2(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 2(5(1(0(3(x1))))) 394.70/100.85 0(1(5(2(x1)))) -> 0(5(5(3(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 1(3(5(0(2(2(x1)))))) 394.70/100.85 0(5(2(2(x1)))) -> 3(0(2(1(5(2(x1)))))) 394.70/100.85 0(1(0(2(2(x1))))) -> 3(0(2(1(0(2(x1)))))) 394.70/100.85 0(1(2(5(2(x1))))) -> 3(5(1(2(2(0(x1)))))) 394.70/100.85 0(1(3(0(0(x1))))) -> 0(0(4(1(3(0(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(1(5(2(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(2(1(1(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 0(2(1(3(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 2(0(3(1(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(3(1(0(3(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(5(1(0(2(4(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(4(1(2(1(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 1(2(1(3(0(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(3(1(4(1(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 4(1(4(1(0(2(x1)))))) 394.70/100.85 1(3(0(1(2(x1))))) -> 1(3(4(1(0(2(x1)))))) 394.70/100.85 394.70/100.85 394.70/100.85 ---------------------------------------- 394.70/100.85 394.70/100.85 (18) 394.70/100.85 Obligation: 394.70/100.85 Q DP problem: 394.70/100.85 The TRS P consists of the following rules: 394.70/100.85 394.70/100.85 1^1(0(1(2(x1)))) -> 0^1(x1) 394.70/100.85 0^1(1(2(x1))) -> 1^1(x1) 394.70/100.85 0^1(1(2(x1))) -> 0^1(x1) 394.70/100.85 0^1(1(2(x1))) -> 1^1(0(x1)) 394.70/100.85 0^1(1(2(x1))) -> 1^1(5(x1)) 394.70/100.85 0^1(1(4(2(2(x1))))) -> 1^1(1(x1)) 394.70/100.85 0^1(1(4(2(2(x1))))) -> 1^1(x1) 394.70/100.85 394.70/100.85 The TRS R consists of the following rules: 394.70/100.85 394.70/100.85 0(1(2(x1))) -> 0(3(1(2(x1)))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(1(x1)))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(0(x1)))) 394.70/100.85 0(1(2(x1))) -> 0(2(1(3(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(2(4(1(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(5(4(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(1(0(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(3(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(0(3(1(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(1(0(3(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(4(2(1(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(0(4(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(4(5(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(5(2(3(1(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(2(3(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(1(4(0(3(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(0(3(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(3(4(1(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(3(2(0(4(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(5(3(0(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(2(5(3(1(0(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(4(0(5(1(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 5(5(4(2(1(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(0(2(4(1(x1))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(2(1(5(0(4(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 1(0(2(3(5(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 5(1(2(0(4(0(x1)))))) 394.70/100.85 0(0(5(2(x1)))) -> 5(3(0(3(0(2(x1)))))) 394.70/100.85 0(1(2(0(x1)))) -> 1(2(4(0(0(x1))))) 394.70/100.85 0(1(2(0(x1)))) -> 0(4(1(0(0(2(x1)))))) 394.70/100.85 0(1(2(2(x1)))) -> 3(3(0(2(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 2(5(1(0(3(x1))))) 394.70/100.85 0(1(5(2(x1)))) -> 0(5(5(3(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 1(3(5(0(2(2(x1)))))) 394.70/100.85 0(5(2(2(x1)))) -> 3(0(2(1(5(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(4(1(2(1(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 1(2(1(3(0(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(3(1(4(1(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 4(1(4(1(0(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 0(2(1(3(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 2(0(3(1(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(3(1(0(3(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(5(1(0(2(4(x1)))))) 394.70/100.85 0(1(0(2(2(x1))))) -> 3(0(2(1(0(2(x1)))))) 394.70/100.85 0(1(2(5(2(x1))))) -> 3(5(1(2(2(0(x1)))))) 394.70/100.85 0(1(3(0(0(x1))))) -> 0(0(4(1(3(0(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(1(5(2(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(2(1(1(x1)))))) 394.70/100.85 1(3(0(1(2(x1))))) -> 1(3(4(1(0(2(x1)))))) 394.70/100.85 394.70/100.85 Q is empty. 394.70/100.85 We have to consider all minimal (P,Q,R)-chains. 394.70/100.85 ---------------------------------------- 394.70/100.85 394.70/100.85 (19) QDPOrderProof (EQUIVALENT) 394.70/100.85 We use the reduction pair processor [LPAR04,JAR06]. 394.70/100.85 394.70/100.85 394.70/100.85 The following pairs can be oriented strictly and are deleted. 394.70/100.85 394.70/100.85 0^1(1(4(2(2(x1))))) -> 1^1(1(x1)) 394.70/100.85 The remaining pairs can at least be oriented weakly. 394.70/100.85 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(1^1(x_1)) = [[0A]] + [[0A, -I, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(0(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, -I], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(1(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(2(x_1)) = [[-I], [1A], [0A]] + [[0A, 0A, 0A], [-I, 0A, 0A], [0A, 0A, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(0^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(5(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [-I, -I, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(4(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, -I], [0A, -I, -I], [-I, -I, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(3(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 394.70/100.85 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 394.70/100.85 394.70/100.85 0(1(2(x1))) -> 0(3(1(2(x1)))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(1(x1)))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(0(x1)))) 394.70/100.85 0(1(2(x1))) -> 0(2(1(3(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(2(4(1(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(5(4(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(1(0(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(3(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(0(3(1(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(1(0(3(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(4(2(1(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(0(4(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(4(5(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(5(2(3(1(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(2(3(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(1(4(0(3(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(0(3(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(3(4(1(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(3(2(0(4(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(5(3(0(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(2(5(3(1(0(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(4(0(5(1(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 5(5(4(2(1(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(0(2(4(1(x1))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(2(1(5(0(4(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 1(0(2(3(5(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 5(1(2(0(4(0(x1)))))) 394.70/100.85 0(0(5(2(x1)))) -> 5(3(0(3(0(2(x1)))))) 394.70/100.85 0(1(2(0(x1)))) -> 1(2(4(0(0(x1))))) 394.70/100.85 0(1(2(0(x1)))) -> 0(4(1(0(0(2(x1)))))) 394.70/100.85 0(1(2(2(x1)))) -> 3(3(0(2(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 2(5(1(0(3(x1))))) 394.70/100.85 0(1(5(2(x1)))) -> 0(5(5(3(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 1(3(5(0(2(2(x1)))))) 394.70/100.85 0(5(2(2(x1)))) -> 3(0(2(1(5(2(x1)))))) 394.70/100.85 0(1(0(2(2(x1))))) -> 3(0(2(1(0(2(x1)))))) 394.70/100.85 0(1(2(5(2(x1))))) -> 3(5(1(2(2(0(x1)))))) 394.70/100.85 0(1(3(0(0(x1))))) -> 0(0(4(1(3(0(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(1(5(2(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(2(1(1(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 0(2(1(3(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 2(0(3(1(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(3(1(0(3(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(5(1(0(2(4(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(4(1(2(1(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 1(2(1(3(0(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(3(1(4(1(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 4(1(4(1(0(2(x1)))))) 394.70/100.85 1(3(0(1(2(x1))))) -> 1(3(4(1(0(2(x1)))))) 394.70/100.85 394.70/100.85 394.70/100.85 ---------------------------------------- 394.70/100.85 394.70/100.85 (20) 394.70/100.85 Obligation: 394.70/100.85 Q DP problem: 394.70/100.85 The TRS P consists of the following rules: 394.70/100.85 394.70/100.85 1^1(0(1(2(x1)))) -> 0^1(x1) 394.70/100.85 0^1(1(2(x1))) -> 1^1(x1) 394.70/100.85 0^1(1(2(x1))) -> 0^1(x1) 394.70/100.85 0^1(1(2(x1))) -> 1^1(0(x1)) 394.70/100.85 0^1(1(2(x1))) -> 1^1(5(x1)) 394.70/100.85 0^1(1(4(2(2(x1))))) -> 1^1(x1) 394.70/100.85 394.70/100.85 The TRS R consists of the following rules: 394.70/100.85 394.70/100.85 0(1(2(x1))) -> 0(3(1(2(x1)))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(1(x1)))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(0(x1)))) 394.70/100.85 0(1(2(x1))) -> 0(2(1(3(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(2(4(1(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(5(4(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(1(0(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(3(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(0(3(1(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(1(0(3(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(4(2(1(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(0(4(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(4(5(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(5(2(3(1(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(2(3(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(1(4(0(3(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(0(3(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(3(4(1(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(3(2(0(4(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(5(3(0(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(2(5(3(1(0(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(4(0(5(1(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 5(5(4(2(1(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(0(2(4(1(x1))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(2(1(5(0(4(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 1(0(2(3(5(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 5(1(2(0(4(0(x1)))))) 394.70/100.85 0(0(5(2(x1)))) -> 5(3(0(3(0(2(x1)))))) 394.70/100.85 0(1(2(0(x1)))) -> 1(2(4(0(0(x1))))) 394.70/100.85 0(1(2(0(x1)))) -> 0(4(1(0(0(2(x1)))))) 394.70/100.85 0(1(2(2(x1)))) -> 3(3(0(2(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 2(5(1(0(3(x1))))) 394.70/100.85 0(1(5(2(x1)))) -> 0(5(5(3(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 1(3(5(0(2(2(x1)))))) 394.70/100.85 0(5(2(2(x1)))) -> 3(0(2(1(5(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(4(1(2(1(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 1(2(1(3(0(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(3(1(4(1(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 4(1(4(1(0(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 0(2(1(3(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 2(0(3(1(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(3(1(0(3(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(5(1(0(2(4(x1)))))) 394.70/100.85 0(1(0(2(2(x1))))) -> 3(0(2(1(0(2(x1)))))) 394.70/100.85 0(1(2(5(2(x1))))) -> 3(5(1(2(2(0(x1)))))) 394.70/100.85 0(1(3(0(0(x1))))) -> 0(0(4(1(3(0(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(1(5(2(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(2(1(1(x1)))))) 394.70/100.85 1(3(0(1(2(x1))))) -> 1(3(4(1(0(2(x1)))))) 394.70/100.85 394.70/100.85 Q is empty. 394.70/100.85 We have to consider all minimal (P,Q,R)-chains. 394.70/100.85 ---------------------------------------- 394.70/100.85 394.70/100.85 (21) QDPOrderProof (EQUIVALENT) 394.70/100.85 We use the reduction pair processor [LPAR04,JAR06]. 394.70/100.85 394.70/100.85 394.70/100.85 The following pairs can be oriented strictly and are deleted. 394.70/100.85 394.70/100.85 1^1(0(1(2(x1)))) -> 0^1(x1) 394.70/100.85 The remaining pairs can at least be oriented weakly. 394.70/100.85 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(1^1(x_1)) = [[-I]] + [[0A, -I, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(0(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, -I], [-I, 1A, 0A], [0A, 1A, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(1(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [-I, -I, 0A], [0A, -I, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(2(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, 0A], [-I, -I, 0A], [0A, 1A, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(0^1(x_1)) = [[-I]] + [[0A, 0A, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(5(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [-I, 0A, 0A], [-I, 0A, 0A]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(4(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 <<< 394.70/100.85 POL(3(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 394.70/100.85 >>> 394.70/100.85 394.70/100.85 394.70/100.85 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 394.70/100.85 394.70/100.85 0(1(2(x1))) -> 0(3(1(2(x1)))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(1(x1)))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(0(x1)))) 394.70/100.85 0(1(2(x1))) -> 0(2(1(3(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(2(4(1(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(5(4(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(1(0(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(3(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(0(3(1(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(1(0(3(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(4(2(1(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(0(4(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(4(5(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(5(2(3(1(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(2(3(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(1(4(0(3(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(0(3(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(3(4(1(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(3(2(0(4(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(5(3(0(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(2(5(3(1(0(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(4(0(5(1(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 5(5(4(2(1(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(0(2(4(1(x1))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(2(1(5(0(4(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 1(0(2(3(5(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 5(1(2(0(4(0(x1)))))) 394.70/100.85 0(0(5(2(x1)))) -> 5(3(0(3(0(2(x1)))))) 394.70/100.85 0(1(2(0(x1)))) -> 1(2(4(0(0(x1))))) 394.70/100.85 0(1(2(0(x1)))) -> 0(4(1(0(0(2(x1)))))) 394.70/100.85 0(1(2(2(x1)))) -> 3(3(0(2(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 2(5(1(0(3(x1))))) 394.70/100.85 0(1(5(2(x1)))) -> 0(5(5(3(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 1(3(5(0(2(2(x1)))))) 394.70/100.85 0(5(2(2(x1)))) -> 3(0(2(1(5(2(x1)))))) 394.70/100.85 0(1(0(2(2(x1))))) -> 3(0(2(1(0(2(x1)))))) 394.70/100.85 0(1(2(5(2(x1))))) -> 3(5(1(2(2(0(x1)))))) 394.70/100.85 0(1(3(0(0(x1))))) -> 0(0(4(1(3(0(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(1(5(2(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(2(1(1(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 0(2(1(3(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 2(0(3(1(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(3(1(0(3(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(5(1(0(2(4(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(4(1(2(1(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 1(2(1(3(0(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(3(1(4(1(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 4(1(4(1(0(2(x1)))))) 394.70/100.85 1(3(0(1(2(x1))))) -> 1(3(4(1(0(2(x1)))))) 394.70/100.85 394.70/100.85 394.70/100.85 ---------------------------------------- 394.70/100.85 394.70/100.85 (22) 394.70/100.85 Obligation: 394.70/100.85 Q DP problem: 394.70/100.85 The TRS P consists of the following rules: 394.70/100.85 394.70/100.85 0^1(1(2(x1))) -> 1^1(x1) 394.70/100.85 0^1(1(2(x1))) -> 0^1(x1) 394.70/100.85 0^1(1(2(x1))) -> 1^1(0(x1)) 394.70/100.85 0^1(1(2(x1))) -> 1^1(5(x1)) 394.70/100.85 0^1(1(4(2(2(x1))))) -> 1^1(x1) 394.70/100.85 394.70/100.85 The TRS R consists of the following rules: 394.70/100.85 394.70/100.85 0(1(2(x1))) -> 0(3(1(2(x1)))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(1(x1)))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(0(x1)))) 394.70/100.85 0(1(2(x1))) -> 0(2(1(3(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(2(4(1(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(5(4(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(1(0(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(3(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(0(3(1(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(1(0(3(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(4(2(1(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(0(4(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(4(5(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(5(2(3(1(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(2(3(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(1(4(0(3(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(0(3(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(3(4(1(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(3(2(0(4(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(5(3(0(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(2(5(3(1(0(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(4(0(5(1(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 5(5(4(2(1(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(0(2(4(1(x1))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(2(1(5(0(4(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 1(0(2(3(5(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 5(1(2(0(4(0(x1)))))) 394.70/100.85 0(0(5(2(x1)))) -> 5(3(0(3(0(2(x1)))))) 394.70/100.85 0(1(2(0(x1)))) -> 1(2(4(0(0(x1))))) 394.70/100.85 0(1(2(0(x1)))) -> 0(4(1(0(0(2(x1)))))) 394.70/100.85 0(1(2(2(x1)))) -> 3(3(0(2(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 2(5(1(0(3(x1))))) 394.70/100.85 0(1(5(2(x1)))) -> 0(5(5(3(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 1(3(5(0(2(2(x1)))))) 394.70/100.85 0(5(2(2(x1)))) -> 3(0(2(1(5(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(4(1(2(1(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 1(2(1(3(0(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(3(1(4(1(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 4(1(4(1(0(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 0(2(1(3(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 2(0(3(1(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(3(1(0(3(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(5(1(0(2(4(x1)))))) 394.70/100.85 0(1(0(2(2(x1))))) -> 3(0(2(1(0(2(x1)))))) 394.70/100.85 0(1(2(5(2(x1))))) -> 3(5(1(2(2(0(x1)))))) 394.70/100.85 0(1(3(0(0(x1))))) -> 0(0(4(1(3(0(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(1(5(2(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(2(1(1(x1)))))) 394.70/100.85 1(3(0(1(2(x1))))) -> 1(3(4(1(0(2(x1)))))) 394.70/100.85 394.70/100.85 Q is empty. 394.70/100.85 We have to consider all minimal (P,Q,R)-chains. 394.70/100.85 ---------------------------------------- 394.70/100.85 394.70/100.85 (23) DependencyGraphProof (EQUIVALENT) 394.70/100.85 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. 394.70/100.85 ---------------------------------------- 394.70/100.85 394.70/100.85 (24) 394.70/100.85 Obligation: 394.70/100.85 Q DP problem: 394.70/100.85 The TRS P consists of the following rules: 394.70/100.85 394.70/100.85 0^1(1(2(x1))) -> 0^1(x1) 394.70/100.85 394.70/100.85 The TRS R consists of the following rules: 394.70/100.85 394.70/100.85 0(1(2(x1))) -> 0(3(1(2(x1)))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(1(x1)))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(0(x1)))) 394.70/100.85 0(1(2(x1))) -> 0(2(1(3(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(2(4(1(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(5(4(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(1(0(3(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 2(1(4(3(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(0(3(1(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(1(0(3(2(x1))))) 394.70/100.85 0(1(2(x1))) -> 4(4(2(1(0(x1))))) 394.70/100.85 0(1(2(x1))) -> 0(0(4(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(4(5(1(2(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 0(5(2(3(1(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(0(2(2(3(4(x1)))))) 394.70/100.85 0(1(2(x1))) -> 1(1(4(0(3(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(0(3(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(0(3(4(1(5(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(3(2(0(4(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 2(5(3(0(3(1(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(2(5(3(1(0(x1)))))) 394.70/100.85 0(1(2(x1))) -> 3(4(0(5(1(2(x1)))))) 394.70/100.85 0(1(2(x1))) -> 5(5(4(2(1(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(0(2(4(1(x1))))) 394.70/100.85 0(0(1(2(x1)))) -> 0(2(1(5(0(4(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 1(0(2(3(5(0(x1)))))) 394.70/100.85 0(0(1(2(x1)))) -> 5(1(2(0(4(0(x1)))))) 394.70/100.85 0(0(5(2(x1)))) -> 5(3(0(3(0(2(x1)))))) 394.70/100.85 0(1(2(0(x1)))) -> 1(2(4(0(0(x1))))) 394.70/100.85 0(1(2(0(x1)))) -> 0(4(1(0(0(2(x1)))))) 394.70/100.85 0(1(2(2(x1)))) -> 3(3(0(2(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 2(5(1(0(3(x1))))) 394.70/100.85 0(1(5(2(x1)))) -> 0(5(5(3(1(2(x1)))))) 394.70/100.85 0(1(5(2(x1)))) -> 1(3(5(0(2(2(x1)))))) 394.70/100.85 0(5(2(2(x1)))) -> 3(0(2(1(5(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(4(1(2(1(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 1(2(1(3(0(x1))))) 394.70/100.85 1(0(1(2(x1)))) -> 0(3(1(4(1(2(x1)))))) 394.70/100.85 1(0(1(2(x1)))) -> 4(1(4(1(0(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 0(2(1(3(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 2(0(3(1(5(x1))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(3(1(0(3(2(x1)))))) 394.70/100.85 5(0(1(2(x1)))) -> 5(5(1(0(2(4(x1)))))) 394.70/100.85 0(1(0(2(2(x1))))) -> 3(0(2(1(0(2(x1)))))) 394.70/100.85 0(1(2(5(2(x1))))) -> 3(5(1(2(2(0(x1)))))) 394.70/100.85 0(1(3(0(0(x1))))) -> 0(0(4(1(3(0(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(1(5(2(x1)))))) 394.70/100.85 0(1(4(2(2(x1))))) -> 0(2(4(2(1(1(x1)))))) 394.70/100.85 1(3(0(1(2(x1))))) -> 1(3(4(1(0(2(x1)))))) 394.70/100.85 394.70/100.85 Q is empty. 394.70/100.85 We have to consider all minimal (P,Q,R)-chains. 394.70/100.85 ---------------------------------------- 394.70/100.85 394.70/100.85 (25) UsableRulesProof (EQUIVALENT) 394.70/100.85 We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. 394.70/100.85 ---------------------------------------- 394.70/100.85 394.70/100.85 (26) 394.70/100.85 Obligation: 394.70/100.85 Q DP problem: 394.70/100.85 The TRS P consists of the following rules: 394.70/100.85 394.70/100.85 0^1(1(2(x1))) -> 0^1(x1) 394.70/100.85 394.70/100.85 R is empty. 394.70/100.85 Q is empty. 394.70/100.85 We have to consider all minimal (P,Q,R)-chains. 394.70/100.85 ---------------------------------------- 394.70/100.85 394.70/100.85 (27) QDPSizeChangeProof (EQUIVALENT) 394.70/100.85 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 394.70/100.85 394.70/100.85 From the DPs we obtained the following set of size-change graphs: 394.70/100.85 *0^1(1(2(x1))) -> 0^1(x1) 394.70/100.85 The graph contains the following edges 1 > 1 394.70/100.85 394.70/100.85 394.70/100.85 ---------------------------------------- 394.70/100.85 394.70/100.85 (28) 394.70/100.85 YES 394.84/100.93 EOF