44.42/12.21 YES 44.79/12.28 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 44.79/12.28 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 44.79/12.28 44.79/12.28 44.79/12.28 Termination w.r.t. Q of the given QTRS could be proven: 44.79/12.28 44.79/12.28 (0) QTRS 44.79/12.28 (1) QTRS Reverse [EQUIVALENT, 0 ms] 44.79/12.28 (2) QTRS 44.79/12.28 (3) DependencyPairsProof [EQUIVALENT, 220 ms] 44.79/12.28 (4) QDP 44.79/12.28 (5) DependencyGraphProof [EQUIVALENT, 1 ms] 44.79/12.28 (6) QDP 44.79/12.28 (7) QDPOrderProof [EQUIVALENT, 278 ms] 44.79/12.28 (8) QDP 44.79/12.28 (9) DependencyGraphProof [EQUIVALENT, 0 ms] 44.79/12.28 (10) QDP 44.79/12.28 (11) QDPOrderProof [EQUIVALENT, 56 ms] 44.79/12.28 (12) QDP 44.79/12.28 (13) PisEmptyProof [EQUIVALENT, 0 ms] 44.79/12.28 (14) YES 44.79/12.28 44.79/12.28 44.79/12.28 ---------------------------------------- 44.79/12.28 44.79/12.28 (0) 44.79/12.28 Obligation: 44.79/12.28 Q restricted rewrite system: 44.79/12.28 The TRS R consists of the following rules: 44.79/12.28 44.79/12.28 0(0(1(2(x1)))) -> 2(3(1(0(0(x1))))) 44.79/12.28 0(0(1(2(x1)))) -> 3(1(0(0(2(x1))))) 44.79/12.28 0(0(1(2(x1)))) -> 0(0(2(3(1(1(x1)))))) 44.79/12.28 0(1(0(4(x1)))) -> 3(1(4(0(0(x1))))) 44.79/12.28 0(1(0(4(x1)))) -> 3(4(0(0(3(1(x1)))))) 44.79/12.28 0(1(0(4(x1)))) -> 4(3(1(0(5(0(x1)))))) 44.79/12.28 0(1(0(4(x1)))) -> 5(0(4(3(1(0(x1)))))) 44.79/12.28 0(1(0(5(x1)))) -> 3(5(3(1(0(0(x1)))))) 44.79/12.28 0(1(0(5(x1)))) -> 5(3(1(0(0(2(x1)))))) 44.79/12.28 0(1(4(2(x1)))) -> 2(3(1(4(0(x1))))) 44.79/12.28 0(1(4(2(x1)))) -> 4(0(3(1(2(4(x1)))))) 44.79/12.28 0(1(4(2(x1)))) -> 4(3(1(1(0(2(x1)))))) 44.79/12.28 0(5(0(4(x1)))) -> 4(0(0(3(5(x1))))) 44.79/12.28 0(5(0(4(x1)))) -> 5(0(0(3(4(x1))))) 44.79/12.28 0(5(0(5(x1)))) -> 3(5(2(0(0(5(x1)))))) 44.79/12.28 0(5(0(5(x1)))) -> 5(3(5(4(0(0(x1)))))) 44.79/12.28 0(5(0(5(x1)))) -> 5(5(3(4(0(0(x1)))))) 44.79/12.28 2(0(1(2(x1)))) -> 0(2(4(1(2(x1))))) 44.79/12.28 2(0(1(2(x1)))) -> 2(5(0(2(2(1(x1)))))) 44.79/12.28 2(0(1(2(x1)))) -> 3(1(3(2(2(0(x1)))))) 44.79/12.28 2(0(4(2(x1)))) -> 4(0(2(2(1(x1))))) 44.79/12.28 2(0(4(2(x1)))) -> 4(0(2(1(2(1(x1)))))) 44.79/12.28 2(0(5(2(x1)))) -> 0(2(1(5(2(2(x1)))))) 44.79/12.28 3(0(4(2(x1)))) -> 0(3(4(1(1(2(x1)))))) 44.79/12.28 3(0(4(2(x1)))) -> 4(3(3(1(0(2(x1)))))) 44.79/12.28 0(0(1(2(2(x1))))) -> 0(0(2(1(2(1(x1)))))) 44.79/12.28 0(0(1(3(2(x1))))) -> 0(3(1(2(0(4(x1)))))) 44.79/12.28 0(0(1(3(2(x1))))) -> 2(5(0(3(1(0(x1)))))) 44.79/12.28 0(0(5(2(2(x1))))) -> 0(0(2(5(2(1(x1)))))) 44.79/12.28 0(1(0(5(2(x1))))) -> 0(2(1(0(3(5(x1)))))) 44.79/12.28 0(1(0(5(5(x1))))) -> 1(3(5(0(0(5(x1)))))) 44.79/12.28 0(1(2(0(5(x1))))) -> 4(0(5(2(1(0(x1)))))) 44.79/12.28 0(1(3(0(4(x1))))) -> 1(0(3(4(4(0(x1)))))) 44.79/12.28 0(1(3(4(2(x1))))) -> 4(2(1(0(3(4(x1)))))) 44.79/12.28 0(1(4(2(2(x1))))) -> 1(1(0(2(4(2(x1)))))) 44.79/12.28 0(1(4(2(4(x1))))) -> 4(0(2(3(1(4(x1)))))) 44.79/12.28 0(1(4(3(2(x1))))) -> 3(1(2(0(4(4(x1)))))) 44.79/12.28 0(3(0(0(5(x1))))) -> 5(3(1(0(0(0(x1)))))) 44.79/12.28 0(4(1(0(4(x1))))) -> 4(0(2(1(4(0(x1)))))) 44.79/12.28 0(4(3(2(2(x1))))) -> 4(2(0(3(5(2(x1)))))) 44.79/12.28 0(5(1(0(4(x1))))) -> 5(2(4(1(0(0(x1)))))) 44.79/12.28 0(5(1(4(2(x1))))) -> 4(0(3(5(1(2(x1)))))) 44.79/12.28 2(1(0(1(2(x1))))) -> 2(1(1(0(2(3(x1)))))) 44.79/12.28 2(4(0(5(2(x1))))) -> 0(4(2(3(5(2(x1)))))) 44.79/12.28 3(0(0(1(2(x1))))) -> 0(3(5(1(0(2(x1)))))) 44.79/12.28 3(0(0(5(2(x1))))) -> 2(0(0(3(5(1(x1)))))) 44.79/12.28 3(0(5(4(2(x1))))) -> 4(0(3(5(2(1(x1)))))) 44.79/12.28 3(2(0(1(2(x1))))) -> 2(1(3(1(0(2(x1)))))) 44.79/12.28 3(2(0(5(2(x1))))) -> 2(0(2(3(4(5(x1)))))) 44.79/12.28 3(2(0(5(2(x1))))) -> 5(3(0(2(2(2(x1)))))) 44.79/12.28 44.79/12.28 Q is empty. 44.79/12.28 44.79/12.28 ---------------------------------------- 44.79/12.28 44.79/12.28 (1) QTRS Reverse (EQUIVALENT) 44.79/12.28 We applied the QTRS Reverse Processor [REVERSE]. 44.79/12.28 ---------------------------------------- 44.79/12.28 44.79/12.28 (2) 44.79/12.28 Obligation: 44.79/12.28 Q restricted rewrite system: 44.79/12.28 The TRS R consists of the following rules: 44.79/12.28 44.79/12.28 2(1(0(0(x1)))) -> 0(0(1(3(2(x1))))) 44.79/12.28 2(1(0(0(x1)))) -> 2(0(0(1(3(x1))))) 44.79/12.28 2(1(0(0(x1)))) -> 1(1(3(2(0(0(x1)))))) 44.79/12.28 4(0(1(0(x1)))) -> 0(0(4(1(3(x1))))) 44.79/12.28 4(0(1(0(x1)))) -> 1(3(0(0(4(3(x1)))))) 44.79/12.28 4(0(1(0(x1)))) -> 0(5(0(1(3(4(x1)))))) 44.79/12.28 4(0(1(0(x1)))) -> 0(1(3(4(0(5(x1)))))) 44.79/12.28 5(0(1(0(x1)))) -> 0(0(1(3(5(3(x1)))))) 44.79/12.28 5(0(1(0(x1)))) -> 2(0(0(1(3(5(x1)))))) 44.79/12.28 2(4(1(0(x1)))) -> 0(4(1(3(2(x1))))) 44.79/12.28 2(4(1(0(x1)))) -> 4(2(1(3(0(4(x1)))))) 44.79/12.28 2(4(1(0(x1)))) -> 2(0(1(1(3(4(x1)))))) 44.79/12.28 4(0(5(0(x1)))) -> 5(3(0(0(4(x1))))) 44.79/12.28 4(0(5(0(x1)))) -> 4(3(0(0(5(x1))))) 44.79/12.28 5(0(5(0(x1)))) -> 5(0(0(2(5(3(x1)))))) 44.79/12.28 5(0(5(0(x1)))) -> 0(0(4(5(3(5(x1)))))) 44.79/12.28 5(0(5(0(x1)))) -> 0(0(4(3(5(5(x1)))))) 44.79/12.28 2(1(0(2(x1)))) -> 2(1(4(2(0(x1))))) 44.79/12.28 2(1(0(2(x1)))) -> 1(2(2(0(5(2(x1)))))) 44.79/12.28 2(1(0(2(x1)))) -> 0(2(2(3(1(3(x1)))))) 44.79/12.28 2(4(0(2(x1)))) -> 1(2(2(0(4(x1))))) 44.79/12.28 2(4(0(2(x1)))) -> 1(2(1(2(0(4(x1)))))) 44.79/12.28 2(5(0(2(x1)))) -> 2(2(5(1(2(0(x1)))))) 44.79/12.28 2(4(0(3(x1)))) -> 2(1(1(4(3(0(x1)))))) 44.79/12.28 2(4(0(3(x1)))) -> 2(0(1(3(3(4(x1)))))) 44.79/12.28 2(2(1(0(0(x1))))) -> 1(2(1(2(0(0(x1)))))) 44.79/12.28 2(3(1(0(0(x1))))) -> 4(0(2(1(3(0(x1)))))) 44.79/12.28 2(3(1(0(0(x1))))) -> 0(1(3(0(5(2(x1)))))) 44.79/12.28 2(2(5(0(0(x1))))) -> 1(2(5(2(0(0(x1)))))) 44.79/12.28 2(5(0(1(0(x1))))) -> 5(3(0(1(2(0(x1)))))) 44.79/12.28 5(5(0(1(0(x1))))) -> 5(0(0(5(3(1(x1)))))) 44.79/12.28 5(0(2(1(0(x1))))) -> 0(1(2(5(0(4(x1)))))) 44.79/12.28 4(0(3(1(0(x1))))) -> 0(4(4(3(0(1(x1)))))) 44.79/12.28 2(4(3(1(0(x1))))) -> 4(3(0(1(2(4(x1)))))) 44.79/12.28 2(2(4(1(0(x1))))) -> 2(4(2(0(1(1(x1)))))) 44.79/12.28 4(2(4(1(0(x1))))) -> 4(1(3(2(0(4(x1)))))) 44.79/12.28 2(3(4(1(0(x1))))) -> 4(4(0(2(1(3(x1)))))) 44.79/12.28 5(0(0(3(0(x1))))) -> 0(0(0(1(3(5(x1)))))) 44.79/12.28 4(0(1(4(0(x1))))) -> 0(4(1(2(0(4(x1)))))) 44.79/12.28 2(2(3(4(0(x1))))) -> 2(5(3(0(2(4(x1)))))) 44.79/12.28 4(0(1(5(0(x1))))) -> 0(0(1(4(2(5(x1)))))) 44.79/12.28 2(4(1(5(0(x1))))) -> 2(1(5(3(0(4(x1)))))) 44.79/12.28 2(1(0(1(2(x1))))) -> 3(2(0(1(1(2(x1)))))) 44.79/12.28 2(5(0(4(2(x1))))) -> 2(5(3(2(4(0(x1)))))) 44.79/12.28 2(1(0(0(3(x1))))) -> 2(0(1(5(3(0(x1)))))) 44.79/12.28 2(5(0(0(3(x1))))) -> 1(5(3(0(0(2(x1)))))) 44.79/12.28 2(4(5(0(3(x1))))) -> 1(2(5(3(0(4(x1)))))) 44.79/12.28 2(1(0(2(3(x1))))) -> 2(0(1(3(1(2(x1)))))) 44.79/12.28 2(5(0(2(3(x1))))) -> 5(4(3(2(0(2(x1)))))) 44.79/12.28 2(5(0(2(3(x1))))) -> 2(2(2(0(3(5(x1)))))) 44.79/12.28 44.79/12.28 Q is empty. 44.79/12.28 44.79/12.28 ---------------------------------------- 44.79/12.28 44.79/12.28 (3) DependencyPairsProof (EQUIVALENT) 44.79/12.28 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 44.79/12.28 ---------------------------------------- 44.79/12.28 44.79/12.28 (4) 44.79/12.28 Obligation: 44.79/12.28 Q DP problem: 44.79/12.28 The TRS P consists of the following rules: 44.79/12.28 44.79/12.28 2^1(1(0(0(x1)))) -> 2^1(x1) 44.79/12.28 2^1(1(0(0(x1)))) -> 2^1(0(0(1(3(x1))))) 44.79/12.28 2^1(1(0(0(x1)))) -> 2^1(0(0(x1))) 44.79/12.28 4^1(0(1(0(x1)))) -> 4^1(1(3(x1))) 44.79/12.28 4^1(0(1(0(x1)))) -> 4^1(3(x1)) 44.79/12.28 4^1(0(1(0(x1)))) -> 5^1(0(1(3(4(x1))))) 44.79/12.28 4^1(0(1(0(x1)))) -> 4^1(x1) 44.79/12.28 4^1(0(1(0(x1)))) -> 4^1(0(5(x1))) 44.79/12.28 4^1(0(1(0(x1)))) -> 5^1(x1) 44.79/12.28 5^1(0(1(0(x1)))) -> 5^1(3(x1)) 44.79/12.28 5^1(0(1(0(x1)))) -> 2^1(0(0(1(3(5(x1)))))) 44.79/12.28 5^1(0(1(0(x1)))) -> 5^1(x1) 44.79/12.28 2^1(4(1(0(x1)))) -> 4^1(1(3(2(x1)))) 44.79/12.28 2^1(4(1(0(x1)))) -> 2^1(x1) 44.79/12.28 2^1(4(1(0(x1)))) -> 4^1(2(1(3(0(4(x1)))))) 44.79/12.28 2^1(4(1(0(x1)))) -> 2^1(1(3(0(4(x1))))) 44.79/12.28 2^1(4(1(0(x1)))) -> 4^1(x1) 44.79/12.28 2^1(4(1(0(x1)))) -> 2^1(0(1(1(3(4(x1)))))) 44.79/12.28 4^1(0(5(0(x1)))) -> 5^1(3(0(0(4(x1))))) 44.79/12.28 4^1(0(5(0(x1)))) -> 4^1(x1) 44.79/12.28 4^1(0(5(0(x1)))) -> 4^1(3(0(0(5(x1))))) 44.79/12.28 4^1(0(5(0(x1)))) -> 5^1(x1) 44.79/12.28 5^1(0(5(0(x1)))) -> 5^1(0(0(2(5(3(x1)))))) 44.79/12.28 5^1(0(5(0(x1)))) -> 2^1(5(3(x1))) 44.79/12.28 5^1(0(5(0(x1)))) -> 5^1(3(x1)) 44.79/12.28 5^1(0(5(0(x1)))) -> 4^1(5(3(5(x1)))) 44.79/12.28 5^1(0(5(0(x1)))) -> 5^1(3(5(x1))) 44.79/12.28 5^1(0(5(0(x1)))) -> 5^1(x1) 44.79/12.28 5^1(0(5(0(x1)))) -> 4^1(3(5(5(x1)))) 44.79/12.28 5^1(0(5(0(x1)))) -> 5^1(5(x1)) 44.79/12.28 2^1(1(0(2(x1)))) -> 2^1(1(4(2(0(x1))))) 44.79/12.28 2^1(1(0(2(x1)))) -> 4^1(2(0(x1))) 44.79/12.28 2^1(1(0(2(x1)))) -> 2^1(0(x1)) 44.79/12.28 2^1(1(0(2(x1)))) -> 2^1(2(0(5(2(x1))))) 44.79/12.28 2^1(1(0(2(x1)))) -> 2^1(0(5(2(x1)))) 44.79/12.28 2^1(1(0(2(x1)))) -> 5^1(2(x1)) 44.79/12.28 2^1(1(0(2(x1)))) -> 2^1(2(3(1(3(x1))))) 44.79/12.28 2^1(1(0(2(x1)))) -> 2^1(3(1(3(x1)))) 44.79/12.28 2^1(4(0(2(x1)))) -> 2^1(2(0(4(x1)))) 44.79/12.28 2^1(4(0(2(x1)))) -> 2^1(0(4(x1))) 44.79/12.28 2^1(4(0(2(x1)))) -> 4^1(x1) 44.79/12.28 2^1(4(0(2(x1)))) -> 2^1(1(2(0(4(x1))))) 44.79/12.28 2^1(5(0(2(x1)))) -> 2^1(2(5(1(2(0(x1)))))) 44.79/12.28 2^1(5(0(2(x1)))) -> 2^1(5(1(2(0(x1))))) 44.79/12.28 2^1(5(0(2(x1)))) -> 5^1(1(2(0(x1)))) 44.79/12.28 2^1(5(0(2(x1)))) -> 2^1(0(x1)) 44.79/12.28 2^1(4(0(3(x1)))) -> 2^1(1(1(4(3(0(x1)))))) 44.79/12.28 2^1(4(0(3(x1)))) -> 4^1(3(0(x1))) 44.79/12.28 2^1(4(0(3(x1)))) -> 2^1(0(1(3(3(4(x1)))))) 44.79/12.28 2^1(4(0(3(x1)))) -> 4^1(x1) 44.79/12.28 2^1(2(1(0(0(x1))))) -> 2^1(1(2(0(0(x1))))) 44.79/12.28 2^1(2(1(0(0(x1))))) -> 2^1(0(0(x1))) 44.79/12.28 2^1(3(1(0(0(x1))))) -> 4^1(0(2(1(3(0(x1)))))) 44.79/12.28 2^1(3(1(0(0(x1))))) -> 2^1(1(3(0(x1)))) 44.79/12.28 2^1(3(1(0(0(x1))))) -> 5^1(2(x1)) 44.79/12.28 2^1(3(1(0(0(x1))))) -> 2^1(x1) 44.79/12.28 2^1(2(5(0(0(x1))))) -> 2^1(5(2(0(0(x1))))) 44.79/12.28 2^1(2(5(0(0(x1))))) -> 5^1(2(0(0(x1)))) 44.79/12.28 2^1(2(5(0(0(x1))))) -> 2^1(0(0(x1))) 44.79/12.28 2^1(5(0(1(0(x1))))) -> 5^1(3(0(1(2(0(x1)))))) 44.79/12.28 2^1(5(0(1(0(x1))))) -> 2^1(0(x1)) 44.79/12.28 5^1(5(0(1(0(x1))))) -> 5^1(0(0(5(3(1(x1)))))) 44.79/12.28 5^1(5(0(1(0(x1))))) -> 5^1(3(1(x1))) 44.79/12.28 5^1(0(2(1(0(x1))))) -> 2^1(5(0(4(x1)))) 44.79/12.28 5^1(0(2(1(0(x1))))) -> 5^1(0(4(x1))) 44.79/12.28 5^1(0(2(1(0(x1))))) -> 4^1(x1) 44.79/12.28 4^1(0(3(1(0(x1))))) -> 4^1(4(3(0(1(x1))))) 44.79/12.28 4^1(0(3(1(0(x1))))) -> 4^1(3(0(1(x1)))) 45.03/12.29 2^1(4(3(1(0(x1))))) -> 4^1(3(0(1(2(4(x1)))))) 45.03/12.29 2^1(4(3(1(0(x1))))) -> 2^1(4(x1)) 45.03/12.29 2^1(4(3(1(0(x1))))) -> 4^1(x1) 45.03/12.29 2^1(2(4(1(0(x1))))) -> 2^1(4(2(0(1(1(x1)))))) 45.03/12.29 2^1(2(4(1(0(x1))))) -> 4^1(2(0(1(1(x1))))) 45.03/12.29 2^1(2(4(1(0(x1))))) -> 2^1(0(1(1(x1)))) 45.03/12.29 4^1(2(4(1(0(x1))))) -> 4^1(1(3(2(0(4(x1)))))) 45.03/12.29 4^1(2(4(1(0(x1))))) -> 2^1(0(4(x1))) 45.03/12.29 4^1(2(4(1(0(x1))))) -> 4^1(x1) 45.03/12.29 2^1(3(4(1(0(x1))))) -> 4^1(4(0(2(1(3(x1)))))) 45.03/12.29 2^1(3(4(1(0(x1))))) -> 4^1(0(2(1(3(x1))))) 45.03/12.29 2^1(3(4(1(0(x1))))) -> 2^1(1(3(x1))) 45.03/12.29 5^1(0(0(3(0(x1))))) -> 5^1(x1) 45.03/12.29 4^1(0(1(4(0(x1))))) -> 4^1(1(2(0(4(x1))))) 45.03/12.29 4^1(0(1(4(0(x1))))) -> 2^1(0(4(x1))) 45.03/12.29 4^1(0(1(4(0(x1))))) -> 4^1(x1) 45.03/12.29 2^1(2(3(4(0(x1))))) -> 2^1(5(3(0(2(4(x1)))))) 45.03/12.29 2^1(2(3(4(0(x1))))) -> 5^1(3(0(2(4(x1))))) 45.03/12.29 2^1(2(3(4(0(x1))))) -> 2^1(4(x1)) 45.03/12.29 2^1(2(3(4(0(x1))))) -> 4^1(x1) 45.03/12.29 4^1(0(1(5(0(x1))))) -> 4^1(2(5(x1))) 45.03/12.29 4^1(0(1(5(0(x1))))) -> 2^1(5(x1)) 45.03/12.29 4^1(0(1(5(0(x1))))) -> 5^1(x1) 45.03/12.29 2^1(4(1(5(0(x1))))) -> 2^1(1(5(3(0(4(x1)))))) 45.03/12.29 2^1(4(1(5(0(x1))))) -> 5^1(3(0(4(x1)))) 45.03/12.29 2^1(4(1(5(0(x1))))) -> 4^1(x1) 45.03/12.29 2^1(1(0(1(2(x1))))) -> 2^1(0(1(1(2(x1))))) 45.03/12.29 2^1(5(0(4(2(x1))))) -> 2^1(5(3(2(4(0(x1)))))) 45.03/12.29 2^1(5(0(4(2(x1))))) -> 5^1(3(2(4(0(x1))))) 45.03/12.29 2^1(5(0(4(2(x1))))) -> 2^1(4(0(x1))) 45.03/12.29 2^1(5(0(4(2(x1))))) -> 4^1(0(x1)) 45.03/12.29 2^1(1(0(0(3(x1))))) -> 2^1(0(1(5(3(0(x1)))))) 45.03/12.29 2^1(1(0(0(3(x1))))) -> 5^1(3(0(x1))) 45.03/12.29 2^1(5(0(0(3(x1))))) -> 5^1(3(0(0(2(x1))))) 45.03/12.29 2^1(5(0(0(3(x1))))) -> 2^1(x1) 45.03/12.29 2^1(4(5(0(3(x1))))) -> 2^1(5(3(0(4(x1))))) 45.03/12.29 2^1(4(5(0(3(x1))))) -> 5^1(3(0(4(x1)))) 45.03/12.29 2^1(4(5(0(3(x1))))) -> 4^1(x1) 45.03/12.29 2^1(1(0(2(3(x1))))) -> 2^1(0(1(3(1(2(x1)))))) 45.03/12.29 2^1(1(0(2(3(x1))))) -> 2^1(x1) 45.03/12.29 2^1(5(0(2(3(x1))))) -> 5^1(4(3(2(0(2(x1)))))) 45.03/12.29 2^1(5(0(2(3(x1))))) -> 4^1(3(2(0(2(x1))))) 45.03/12.29 2^1(5(0(2(3(x1))))) -> 2^1(0(2(x1))) 45.03/12.29 2^1(5(0(2(3(x1))))) -> 2^1(x1) 45.03/12.29 2^1(5(0(2(3(x1))))) -> 2^1(2(2(0(3(5(x1)))))) 45.03/12.29 2^1(5(0(2(3(x1))))) -> 2^1(2(0(3(5(x1))))) 45.03/12.29 2^1(5(0(2(3(x1))))) -> 2^1(0(3(5(x1)))) 45.03/12.29 2^1(5(0(2(3(x1))))) -> 5^1(x1) 45.03/12.29 45.03/12.29 The TRS R consists of the following rules: 45.03/12.29 45.03/12.29 2(1(0(0(x1)))) -> 0(0(1(3(2(x1))))) 45.03/12.29 2(1(0(0(x1)))) -> 2(0(0(1(3(x1))))) 45.03/12.29 2(1(0(0(x1)))) -> 1(1(3(2(0(0(x1)))))) 45.03/12.29 4(0(1(0(x1)))) -> 0(0(4(1(3(x1))))) 45.03/12.29 4(0(1(0(x1)))) -> 1(3(0(0(4(3(x1)))))) 45.03/12.29 4(0(1(0(x1)))) -> 0(5(0(1(3(4(x1)))))) 45.03/12.29 4(0(1(0(x1)))) -> 0(1(3(4(0(5(x1)))))) 45.03/12.29 5(0(1(0(x1)))) -> 0(0(1(3(5(3(x1)))))) 45.03/12.29 5(0(1(0(x1)))) -> 2(0(0(1(3(5(x1)))))) 45.03/12.29 2(4(1(0(x1)))) -> 0(4(1(3(2(x1))))) 45.03/12.29 2(4(1(0(x1)))) -> 4(2(1(3(0(4(x1)))))) 45.03/12.29 2(4(1(0(x1)))) -> 2(0(1(1(3(4(x1)))))) 45.03/12.29 4(0(5(0(x1)))) -> 5(3(0(0(4(x1))))) 45.03/12.29 4(0(5(0(x1)))) -> 4(3(0(0(5(x1))))) 45.03/12.29 5(0(5(0(x1)))) -> 5(0(0(2(5(3(x1)))))) 45.03/12.29 5(0(5(0(x1)))) -> 0(0(4(5(3(5(x1)))))) 45.03/12.29 5(0(5(0(x1)))) -> 0(0(4(3(5(5(x1)))))) 45.03/12.29 2(1(0(2(x1)))) -> 2(1(4(2(0(x1))))) 45.03/12.29 2(1(0(2(x1)))) -> 1(2(2(0(5(2(x1)))))) 45.03/12.29 2(1(0(2(x1)))) -> 0(2(2(3(1(3(x1)))))) 45.03/12.29 2(4(0(2(x1)))) -> 1(2(2(0(4(x1))))) 45.03/12.29 2(4(0(2(x1)))) -> 1(2(1(2(0(4(x1)))))) 45.03/12.29 2(5(0(2(x1)))) -> 2(2(5(1(2(0(x1)))))) 45.03/12.29 2(4(0(3(x1)))) -> 2(1(1(4(3(0(x1)))))) 45.03/12.29 2(4(0(3(x1)))) -> 2(0(1(3(3(4(x1)))))) 45.03/12.29 2(2(1(0(0(x1))))) -> 1(2(1(2(0(0(x1)))))) 45.03/12.29 2(3(1(0(0(x1))))) -> 4(0(2(1(3(0(x1)))))) 45.03/12.29 2(3(1(0(0(x1))))) -> 0(1(3(0(5(2(x1)))))) 45.03/12.29 2(2(5(0(0(x1))))) -> 1(2(5(2(0(0(x1)))))) 45.03/12.29 2(5(0(1(0(x1))))) -> 5(3(0(1(2(0(x1)))))) 45.03/12.29 5(5(0(1(0(x1))))) -> 5(0(0(5(3(1(x1)))))) 45.03/12.29 5(0(2(1(0(x1))))) -> 0(1(2(5(0(4(x1)))))) 45.03/12.29 4(0(3(1(0(x1))))) -> 0(4(4(3(0(1(x1)))))) 45.03/12.29 2(4(3(1(0(x1))))) -> 4(3(0(1(2(4(x1)))))) 45.03/12.29 2(2(4(1(0(x1))))) -> 2(4(2(0(1(1(x1)))))) 45.03/12.29 4(2(4(1(0(x1))))) -> 4(1(3(2(0(4(x1)))))) 45.03/12.29 2(3(4(1(0(x1))))) -> 4(4(0(2(1(3(x1)))))) 45.03/12.29 5(0(0(3(0(x1))))) -> 0(0(0(1(3(5(x1)))))) 45.03/12.29 4(0(1(4(0(x1))))) -> 0(4(1(2(0(4(x1)))))) 45.03/12.29 2(2(3(4(0(x1))))) -> 2(5(3(0(2(4(x1)))))) 45.03/12.29 4(0(1(5(0(x1))))) -> 0(0(1(4(2(5(x1)))))) 45.03/12.29 2(4(1(5(0(x1))))) -> 2(1(5(3(0(4(x1)))))) 45.03/12.29 2(1(0(1(2(x1))))) -> 3(2(0(1(1(2(x1)))))) 45.03/12.29 2(5(0(4(2(x1))))) -> 2(5(3(2(4(0(x1)))))) 45.03/12.29 2(1(0(0(3(x1))))) -> 2(0(1(5(3(0(x1)))))) 45.03/12.29 2(5(0(0(3(x1))))) -> 1(5(3(0(0(2(x1)))))) 45.03/12.29 2(4(5(0(3(x1))))) -> 1(2(5(3(0(4(x1)))))) 45.03/12.29 2(1(0(2(3(x1))))) -> 2(0(1(3(1(2(x1)))))) 45.03/12.29 2(5(0(2(3(x1))))) -> 5(4(3(2(0(2(x1)))))) 45.03/12.29 2(5(0(2(3(x1))))) -> 2(2(2(0(3(5(x1)))))) 45.03/12.29 45.03/12.29 Q is empty. 45.03/12.29 We have to consider all minimal (P,Q,R)-chains. 45.03/12.29 ---------------------------------------- 45.03/12.29 45.03/12.29 (5) DependencyGraphProof (EQUIVALENT) 45.03/12.29 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 79 less nodes. 45.03/12.29 ---------------------------------------- 45.03/12.29 45.03/12.29 (6) 45.03/12.29 Obligation: 45.03/12.29 Q DP problem: 45.03/12.29 The TRS P consists of the following rules: 45.03/12.29 45.03/12.29 2^1(4(1(0(x1)))) -> 2^1(x1) 45.03/12.29 2^1(1(0(0(x1)))) -> 2^1(x1) 45.03/12.29 2^1(4(1(0(x1)))) -> 4^1(x1) 45.03/12.29 4^1(0(1(0(x1)))) -> 4^1(x1) 45.03/12.29 4^1(0(1(0(x1)))) -> 4^1(0(5(x1))) 45.03/12.29 4^1(0(1(0(x1)))) -> 5^1(x1) 45.03/12.29 5^1(0(1(0(x1)))) -> 5^1(x1) 45.03/12.29 5^1(0(5(0(x1)))) -> 5^1(x1) 45.03/12.29 5^1(0(5(0(x1)))) -> 5^1(5(x1)) 45.03/12.29 5^1(0(2(1(0(x1))))) -> 2^1(5(0(4(x1)))) 45.03/12.29 2^1(1(0(2(x1)))) -> 5^1(2(x1)) 45.03/12.29 5^1(0(2(1(0(x1))))) -> 5^1(0(4(x1))) 45.03/12.29 5^1(0(2(1(0(x1))))) -> 4^1(x1) 45.03/12.29 4^1(0(5(0(x1)))) -> 4^1(x1) 45.03/12.29 4^1(0(5(0(x1)))) -> 5^1(x1) 45.03/12.29 5^1(0(0(3(0(x1))))) -> 5^1(x1) 45.03/12.29 4^1(2(4(1(0(x1))))) -> 4^1(x1) 45.03/12.29 4^1(0(1(4(0(x1))))) -> 4^1(x1) 45.03/12.29 4^1(0(1(5(0(x1))))) -> 4^1(2(5(x1))) 45.03/12.29 4^1(0(1(5(0(x1))))) -> 2^1(5(x1)) 45.03/12.29 2^1(4(0(2(x1)))) -> 4^1(x1) 45.03/12.29 4^1(0(1(5(0(x1))))) -> 5^1(x1) 45.03/12.29 2^1(4(0(3(x1)))) -> 4^1(x1) 45.03/12.29 2^1(3(1(0(0(x1))))) -> 5^1(2(x1)) 45.03/12.29 2^1(3(1(0(0(x1))))) -> 2^1(x1) 45.03/12.29 2^1(4(3(1(0(x1))))) -> 2^1(4(x1)) 45.03/12.29 2^1(4(3(1(0(x1))))) -> 4^1(x1) 45.03/12.29 2^1(2(3(4(0(x1))))) -> 2^1(4(x1)) 45.03/12.29 2^1(2(3(4(0(x1))))) -> 4^1(x1) 45.03/12.29 2^1(4(1(5(0(x1))))) -> 4^1(x1) 45.03/12.29 2^1(5(0(4(2(x1))))) -> 2^1(4(0(x1))) 45.03/12.29 2^1(5(0(4(2(x1))))) -> 4^1(0(x1)) 45.03/12.29 2^1(5(0(0(3(x1))))) -> 2^1(x1) 45.03/12.29 2^1(4(5(0(3(x1))))) -> 4^1(x1) 45.03/12.29 2^1(1(0(2(3(x1))))) -> 2^1(x1) 45.03/12.29 2^1(5(0(2(3(x1))))) -> 2^1(x1) 45.03/12.29 2^1(5(0(2(3(x1))))) -> 5^1(x1) 45.03/12.29 45.03/12.29 The TRS R consists of the following rules: 45.03/12.29 45.03/12.29 2(1(0(0(x1)))) -> 0(0(1(3(2(x1))))) 45.03/12.29 2(1(0(0(x1)))) -> 2(0(0(1(3(x1))))) 45.03/12.29 2(1(0(0(x1)))) -> 1(1(3(2(0(0(x1)))))) 45.03/12.29 4(0(1(0(x1)))) -> 0(0(4(1(3(x1))))) 45.03/12.29 4(0(1(0(x1)))) -> 1(3(0(0(4(3(x1)))))) 45.03/12.29 4(0(1(0(x1)))) -> 0(5(0(1(3(4(x1)))))) 45.03/12.29 4(0(1(0(x1)))) -> 0(1(3(4(0(5(x1)))))) 45.03/12.29 5(0(1(0(x1)))) -> 0(0(1(3(5(3(x1)))))) 45.03/12.29 5(0(1(0(x1)))) -> 2(0(0(1(3(5(x1)))))) 45.03/12.29 2(4(1(0(x1)))) -> 0(4(1(3(2(x1))))) 45.03/12.29 2(4(1(0(x1)))) -> 4(2(1(3(0(4(x1)))))) 45.03/12.29 2(4(1(0(x1)))) -> 2(0(1(1(3(4(x1)))))) 45.03/12.29 4(0(5(0(x1)))) -> 5(3(0(0(4(x1))))) 45.03/12.29 4(0(5(0(x1)))) -> 4(3(0(0(5(x1))))) 45.03/12.29 5(0(5(0(x1)))) -> 5(0(0(2(5(3(x1)))))) 45.03/12.29 5(0(5(0(x1)))) -> 0(0(4(5(3(5(x1)))))) 45.03/12.29 5(0(5(0(x1)))) -> 0(0(4(3(5(5(x1)))))) 45.03/12.29 2(1(0(2(x1)))) -> 2(1(4(2(0(x1))))) 45.03/12.29 2(1(0(2(x1)))) -> 1(2(2(0(5(2(x1)))))) 45.03/12.29 2(1(0(2(x1)))) -> 0(2(2(3(1(3(x1)))))) 45.03/12.29 2(4(0(2(x1)))) -> 1(2(2(0(4(x1))))) 45.03/12.29 2(4(0(2(x1)))) -> 1(2(1(2(0(4(x1)))))) 45.03/12.29 2(5(0(2(x1)))) -> 2(2(5(1(2(0(x1)))))) 45.03/12.29 2(4(0(3(x1)))) -> 2(1(1(4(3(0(x1)))))) 45.03/12.29 2(4(0(3(x1)))) -> 2(0(1(3(3(4(x1)))))) 45.03/12.29 2(2(1(0(0(x1))))) -> 1(2(1(2(0(0(x1)))))) 45.03/12.29 2(3(1(0(0(x1))))) -> 4(0(2(1(3(0(x1)))))) 45.03/12.29 2(3(1(0(0(x1))))) -> 0(1(3(0(5(2(x1)))))) 45.03/12.29 2(2(5(0(0(x1))))) -> 1(2(5(2(0(0(x1)))))) 45.03/12.29 2(5(0(1(0(x1))))) -> 5(3(0(1(2(0(x1)))))) 45.03/12.29 5(5(0(1(0(x1))))) -> 5(0(0(5(3(1(x1)))))) 45.03/12.29 5(0(2(1(0(x1))))) -> 0(1(2(5(0(4(x1)))))) 45.03/12.29 4(0(3(1(0(x1))))) -> 0(4(4(3(0(1(x1)))))) 45.03/12.29 2(4(3(1(0(x1))))) -> 4(3(0(1(2(4(x1)))))) 45.03/12.29 2(2(4(1(0(x1))))) -> 2(4(2(0(1(1(x1)))))) 45.03/12.29 4(2(4(1(0(x1))))) -> 4(1(3(2(0(4(x1)))))) 45.03/12.29 2(3(4(1(0(x1))))) -> 4(4(0(2(1(3(x1)))))) 45.03/12.29 5(0(0(3(0(x1))))) -> 0(0(0(1(3(5(x1)))))) 45.03/12.29 4(0(1(4(0(x1))))) -> 0(4(1(2(0(4(x1)))))) 45.03/12.29 2(2(3(4(0(x1))))) -> 2(5(3(0(2(4(x1)))))) 45.03/12.29 4(0(1(5(0(x1))))) -> 0(0(1(4(2(5(x1)))))) 45.03/12.29 2(4(1(5(0(x1))))) -> 2(1(5(3(0(4(x1)))))) 45.03/12.29 2(1(0(1(2(x1))))) -> 3(2(0(1(1(2(x1)))))) 45.03/12.29 2(5(0(4(2(x1))))) -> 2(5(3(2(4(0(x1)))))) 45.03/12.29 2(1(0(0(3(x1))))) -> 2(0(1(5(3(0(x1)))))) 45.03/12.29 2(5(0(0(3(x1))))) -> 1(5(3(0(0(2(x1)))))) 45.03/12.29 2(4(5(0(3(x1))))) -> 1(2(5(3(0(4(x1)))))) 45.03/12.29 2(1(0(2(3(x1))))) -> 2(0(1(3(1(2(x1)))))) 45.03/12.29 2(5(0(2(3(x1))))) -> 5(4(3(2(0(2(x1)))))) 45.03/12.29 2(5(0(2(3(x1))))) -> 2(2(2(0(3(5(x1)))))) 45.03/12.29 45.03/12.29 Q is empty. 45.03/12.29 We have to consider all minimal (P,Q,R)-chains. 45.03/12.29 ---------------------------------------- 45.03/12.29 45.03/12.29 (7) QDPOrderProof (EQUIVALENT) 45.03/12.29 We use the reduction pair processor [LPAR04,JAR06]. 45.03/12.29 45.03/12.29 45.03/12.29 The following pairs can be oriented strictly and are deleted. 45.03/12.29 45.03/12.29 2^1(4(1(0(x1)))) -> 2^1(x1) 45.03/12.29 2^1(1(0(0(x1)))) -> 2^1(x1) 45.03/12.29 2^1(4(1(0(x1)))) -> 4^1(x1) 45.03/12.29 4^1(0(1(0(x1)))) -> 4^1(x1) 45.03/12.29 4^1(0(1(0(x1)))) -> 4^1(0(5(x1))) 45.03/12.29 4^1(0(1(0(x1)))) -> 5^1(x1) 45.03/12.29 5^1(0(1(0(x1)))) -> 5^1(x1) 45.03/12.29 5^1(0(5(0(x1)))) -> 5^1(x1) 45.03/12.29 5^1(0(5(0(x1)))) -> 5^1(5(x1)) 45.03/12.29 5^1(0(2(1(0(x1))))) -> 2^1(5(0(4(x1)))) 45.03/12.29 5^1(0(2(1(0(x1))))) -> 5^1(0(4(x1))) 45.03/12.29 5^1(0(2(1(0(x1))))) -> 4^1(x1) 45.03/12.29 4^1(0(5(0(x1)))) -> 4^1(x1) 45.03/12.29 4^1(0(5(0(x1)))) -> 5^1(x1) 45.03/12.29 5^1(0(0(3(0(x1))))) -> 5^1(x1) 45.03/12.29 4^1(2(4(1(0(x1))))) -> 4^1(x1) 45.03/12.29 4^1(0(1(4(0(x1))))) -> 4^1(x1) 45.03/12.29 4^1(0(1(5(0(x1))))) -> 4^1(2(5(x1))) 45.03/12.29 4^1(0(1(5(0(x1))))) -> 2^1(5(x1)) 45.03/12.29 2^1(4(0(2(x1)))) -> 4^1(x1) 45.03/12.29 4^1(0(1(5(0(x1))))) -> 5^1(x1) 45.03/12.29 2^1(4(0(3(x1)))) -> 4^1(x1) 45.03/12.29 2^1(3(1(0(0(x1))))) -> 5^1(2(x1)) 45.03/12.29 2^1(3(1(0(0(x1))))) -> 2^1(x1) 45.03/12.29 2^1(4(3(1(0(x1))))) -> 2^1(4(x1)) 45.03/12.29 2^1(4(3(1(0(x1))))) -> 4^1(x1) 45.03/12.29 2^1(2(3(4(0(x1))))) -> 2^1(4(x1)) 45.03/12.29 2^1(2(3(4(0(x1))))) -> 4^1(x1) 45.03/12.29 2^1(4(1(5(0(x1))))) -> 4^1(x1) 45.03/12.29 2^1(5(0(0(3(x1))))) -> 2^1(x1) 45.03/12.29 2^1(4(5(0(3(x1))))) -> 4^1(x1) 45.03/12.29 2^1(1(0(2(3(x1))))) -> 2^1(x1) 45.03/12.29 2^1(5(0(2(3(x1))))) -> 2^1(x1) 45.03/12.29 The remaining pairs can at least be oriented weakly. 45.03/12.29 Used ordering: Polynomial interpretation [POLO]: 45.03/12.29 45.03/12.29 POL(0(x_1)) = 1 + x_1 45.03/12.29 POL(1(x_1)) = x_1 45.03/12.29 POL(2(x_1)) = x_1 45.03/12.29 POL(2^1(x_1)) = x_1 45.03/12.29 POL(3(x_1)) = x_1 45.03/12.29 POL(4(x_1)) = x_1 45.03/12.29 POL(4^1(x_1)) = x_1 45.03/12.29 POL(5(x_1)) = x_1 45.03/12.29 POL(5^1(x_1)) = 1 + x_1 45.03/12.29 45.03/12.29 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 45.03/12.29 45.03/12.29 5(0(1(0(x1)))) -> 0(0(1(3(5(3(x1)))))) 45.03/12.29 5(0(1(0(x1)))) -> 2(0(0(1(3(5(x1)))))) 45.03/12.29 5(0(5(0(x1)))) -> 5(0(0(2(5(3(x1)))))) 45.03/12.29 5(0(5(0(x1)))) -> 0(0(4(5(3(5(x1)))))) 45.03/12.29 5(0(5(0(x1)))) -> 0(0(4(3(5(5(x1)))))) 45.03/12.29 5(5(0(1(0(x1))))) -> 5(0(0(5(3(1(x1)))))) 45.03/12.29 5(0(2(1(0(x1))))) -> 0(1(2(5(0(4(x1)))))) 45.03/12.29 5(0(0(3(0(x1))))) -> 0(0(0(1(3(5(x1)))))) 45.03/12.29 4(0(1(0(x1)))) -> 0(0(4(1(3(x1))))) 45.03/12.29 4(0(1(0(x1)))) -> 1(3(0(0(4(3(x1)))))) 45.03/12.29 4(0(1(0(x1)))) -> 0(5(0(1(3(4(x1)))))) 45.03/12.29 4(0(1(0(x1)))) -> 0(1(3(4(0(5(x1)))))) 45.03/12.29 4(0(5(0(x1)))) -> 5(3(0(0(4(x1))))) 45.03/12.29 4(0(5(0(x1)))) -> 4(3(0(0(5(x1))))) 45.03/12.29 4(0(3(1(0(x1))))) -> 0(4(4(3(0(1(x1)))))) 45.03/12.29 4(2(4(1(0(x1))))) -> 4(1(3(2(0(4(x1)))))) 45.03/12.29 4(0(1(4(0(x1))))) -> 0(4(1(2(0(4(x1)))))) 45.03/12.29 4(0(1(5(0(x1))))) -> 0(0(1(4(2(5(x1)))))) 45.03/12.29 2(1(0(0(x1)))) -> 0(0(1(3(2(x1))))) 45.03/12.29 2(1(0(0(x1)))) -> 2(0(0(1(3(x1))))) 45.03/12.29 2(1(0(0(x1)))) -> 1(1(3(2(0(0(x1)))))) 45.03/12.29 2(4(1(0(x1)))) -> 0(4(1(3(2(x1))))) 45.03/12.29 2(4(1(0(x1)))) -> 4(2(1(3(0(4(x1)))))) 45.03/12.29 2(4(1(0(x1)))) -> 2(0(1(1(3(4(x1)))))) 45.03/12.29 2(1(0(2(x1)))) -> 2(1(4(2(0(x1))))) 45.03/12.29 2(1(0(2(x1)))) -> 1(2(2(0(5(2(x1)))))) 45.03/12.29 2(1(0(2(x1)))) -> 0(2(2(3(1(3(x1)))))) 45.03/12.29 2(4(0(2(x1)))) -> 1(2(2(0(4(x1))))) 45.03/12.29 2(4(0(2(x1)))) -> 1(2(1(2(0(4(x1)))))) 45.03/12.29 2(5(0(2(x1)))) -> 2(2(5(1(2(0(x1)))))) 45.03/12.29 2(4(0(3(x1)))) -> 2(1(1(4(3(0(x1)))))) 45.03/12.29 2(4(0(3(x1)))) -> 2(0(1(3(3(4(x1)))))) 45.03/12.29 2(2(1(0(0(x1))))) -> 1(2(1(2(0(0(x1)))))) 45.03/12.29 2(3(1(0(0(x1))))) -> 4(0(2(1(3(0(x1)))))) 45.03/12.29 2(3(1(0(0(x1))))) -> 0(1(3(0(5(2(x1)))))) 45.03/12.29 2(2(5(0(0(x1))))) -> 1(2(5(2(0(0(x1)))))) 45.03/12.29 2(5(0(1(0(x1))))) -> 5(3(0(1(2(0(x1)))))) 45.03/12.29 2(4(3(1(0(x1))))) -> 4(3(0(1(2(4(x1)))))) 45.03/12.29 2(2(4(1(0(x1))))) -> 2(4(2(0(1(1(x1)))))) 45.03/12.29 2(3(4(1(0(x1))))) -> 4(4(0(2(1(3(x1)))))) 45.03/12.29 2(2(3(4(0(x1))))) -> 2(5(3(0(2(4(x1)))))) 45.03/12.29 2(4(1(5(0(x1))))) -> 2(1(5(3(0(4(x1)))))) 45.03/12.29 2(1(0(1(2(x1))))) -> 3(2(0(1(1(2(x1)))))) 45.03/12.29 2(5(0(4(2(x1))))) -> 2(5(3(2(4(0(x1)))))) 45.03/12.29 2(1(0(0(3(x1))))) -> 2(0(1(5(3(0(x1)))))) 45.03/12.29 2(5(0(0(3(x1))))) -> 1(5(3(0(0(2(x1)))))) 45.03/12.29 2(4(5(0(3(x1))))) -> 1(2(5(3(0(4(x1)))))) 45.03/12.29 2(1(0(2(3(x1))))) -> 2(0(1(3(1(2(x1)))))) 45.03/12.29 2(5(0(2(3(x1))))) -> 5(4(3(2(0(2(x1)))))) 45.03/12.29 2(5(0(2(3(x1))))) -> 2(2(2(0(3(5(x1)))))) 45.03/12.29 45.03/12.29 45.03/12.29 ---------------------------------------- 45.03/12.29 45.03/12.29 (8) 45.03/12.29 Obligation: 45.03/12.29 Q DP problem: 45.03/12.29 The TRS P consists of the following rules: 45.03/12.29 45.03/12.29 2^1(1(0(2(x1)))) -> 5^1(2(x1)) 45.03/12.29 2^1(5(0(4(2(x1))))) -> 2^1(4(0(x1))) 45.03/12.29 2^1(5(0(4(2(x1))))) -> 4^1(0(x1)) 45.03/12.29 2^1(5(0(2(3(x1))))) -> 5^1(x1) 45.03/12.29 45.03/12.29 The TRS R consists of the following rules: 45.03/12.29 45.03/12.29 2(1(0(0(x1)))) -> 0(0(1(3(2(x1))))) 45.03/12.29 2(1(0(0(x1)))) -> 2(0(0(1(3(x1))))) 45.03/12.29 2(1(0(0(x1)))) -> 1(1(3(2(0(0(x1)))))) 45.03/12.29 4(0(1(0(x1)))) -> 0(0(4(1(3(x1))))) 45.03/12.29 4(0(1(0(x1)))) -> 1(3(0(0(4(3(x1)))))) 45.03/12.29 4(0(1(0(x1)))) -> 0(5(0(1(3(4(x1)))))) 45.03/12.29 4(0(1(0(x1)))) -> 0(1(3(4(0(5(x1)))))) 45.03/12.29 5(0(1(0(x1)))) -> 0(0(1(3(5(3(x1)))))) 45.03/12.29 5(0(1(0(x1)))) -> 2(0(0(1(3(5(x1)))))) 45.03/12.29 2(4(1(0(x1)))) -> 0(4(1(3(2(x1))))) 45.03/12.29 2(4(1(0(x1)))) -> 4(2(1(3(0(4(x1)))))) 45.03/12.29 2(4(1(0(x1)))) -> 2(0(1(1(3(4(x1)))))) 45.03/12.29 4(0(5(0(x1)))) -> 5(3(0(0(4(x1))))) 45.03/12.29 4(0(5(0(x1)))) -> 4(3(0(0(5(x1))))) 45.03/12.29 5(0(5(0(x1)))) -> 5(0(0(2(5(3(x1)))))) 45.03/12.29 5(0(5(0(x1)))) -> 0(0(4(5(3(5(x1)))))) 45.03/12.29 5(0(5(0(x1)))) -> 0(0(4(3(5(5(x1)))))) 45.03/12.29 2(1(0(2(x1)))) -> 2(1(4(2(0(x1))))) 45.03/12.29 2(1(0(2(x1)))) -> 1(2(2(0(5(2(x1)))))) 45.03/12.29 2(1(0(2(x1)))) -> 0(2(2(3(1(3(x1)))))) 45.03/12.29 2(4(0(2(x1)))) -> 1(2(2(0(4(x1))))) 45.03/12.29 2(4(0(2(x1)))) -> 1(2(1(2(0(4(x1)))))) 45.03/12.29 2(5(0(2(x1)))) -> 2(2(5(1(2(0(x1)))))) 45.03/12.29 2(4(0(3(x1)))) -> 2(1(1(4(3(0(x1)))))) 45.03/12.29 2(4(0(3(x1)))) -> 2(0(1(3(3(4(x1)))))) 45.03/12.29 2(2(1(0(0(x1))))) -> 1(2(1(2(0(0(x1)))))) 45.03/12.29 2(3(1(0(0(x1))))) -> 4(0(2(1(3(0(x1)))))) 45.03/12.29 2(3(1(0(0(x1))))) -> 0(1(3(0(5(2(x1)))))) 45.03/12.29 2(2(5(0(0(x1))))) -> 1(2(5(2(0(0(x1)))))) 45.03/12.29 2(5(0(1(0(x1))))) -> 5(3(0(1(2(0(x1)))))) 45.03/12.29 5(5(0(1(0(x1))))) -> 5(0(0(5(3(1(x1)))))) 45.03/12.29 5(0(2(1(0(x1))))) -> 0(1(2(5(0(4(x1)))))) 45.03/12.29 4(0(3(1(0(x1))))) -> 0(4(4(3(0(1(x1)))))) 45.03/12.29 2(4(3(1(0(x1))))) -> 4(3(0(1(2(4(x1)))))) 45.03/12.29 2(2(4(1(0(x1))))) -> 2(4(2(0(1(1(x1)))))) 45.03/12.29 4(2(4(1(0(x1))))) -> 4(1(3(2(0(4(x1)))))) 45.03/12.29 2(3(4(1(0(x1))))) -> 4(4(0(2(1(3(x1)))))) 45.03/12.29 5(0(0(3(0(x1))))) -> 0(0(0(1(3(5(x1)))))) 45.03/12.29 4(0(1(4(0(x1))))) -> 0(4(1(2(0(4(x1)))))) 45.03/12.29 2(2(3(4(0(x1))))) -> 2(5(3(0(2(4(x1)))))) 45.03/12.29 4(0(1(5(0(x1))))) -> 0(0(1(4(2(5(x1)))))) 45.03/12.29 2(4(1(5(0(x1))))) -> 2(1(5(3(0(4(x1)))))) 45.03/12.29 2(1(0(1(2(x1))))) -> 3(2(0(1(1(2(x1)))))) 45.03/12.29 2(5(0(4(2(x1))))) -> 2(5(3(2(4(0(x1)))))) 45.03/12.29 2(1(0(0(3(x1))))) -> 2(0(1(5(3(0(x1)))))) 45.03/12.29 2(5(0(0(3(x1))))) -> 1(5(3(0(0(2(x1)))))) 45.03/12.29 2(4(5(0(3(x1))))) -> 1(2(5(3(0(4(x1)))))) 45.03/12.29 2(1(0(2(3(x1))))) -> 2(0(1(3(1(2(x1)))))) 45.03/12.29 2(5(0(2(3(x1))))) -> 5(4(3(2(0(2(x1)))))) 45.03/12.29 2(5(0(2(3(x1))))) -> 2(2(2(0(3(5(x1)))))) 45.03/12.29 45.03/12.29 Q is empty. 45.03/12.29 We have to consider all minimal (P,Q,R)-chains. 45.03/12.29 ---------------------------------------- 45.03/12.29 45.03/12.29 (9) DependencyGraphProof (EQUIVALENT) 45.03/12.29 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 45.03/12.29 ---------------------------------------- 45.03/12.29 45.03/12.29 (10) 45.03/12.29 Obligation: 45.03/12.29 Q DP problem: 45.03/12.29 The TRS P consists of the following rules: 45.03/12.29 45.03/12.29 2^1(5(0(4(2(x1))))) -> 2^1(4(0(x1))) 45.03/12.29 45.03/12.29 The TRS R consists of the following rules: 45.03/12.29 45.03/12.29 2(1(0(0(x1)))) -> 0(0(1(3(2(x1))))) 45.03/12.29 2(1(0(0(x1)))) -> 2(0(0(1(3(x1))))) 45.03/12.29 2(1(0(0(x1)))) -> 1(1(3(2(0(0(x1)))))) 45.03/12.29 4(0(1(0(x1)))) -> 0(0(4(1(3(x1))))) 45.03/12.29 4(0(1(0(x1)))) -> 1(3(0(0(4(3(x1)))))) 45.03/12.29 4(0(1(0(x1)))) -> 0(5(0(1(3(4(x1)))))) 45.03/12.29 4(0(1(0(x1)))) -> 0(1(3(4(0(5(x1)))))) 45.03/12.29 5(0(1(0(x1)))) -> 0(0(1(3(5(3(x1)))))) 45.03/12.29 5(0(1(0(x1)))) -> 2(0(0(1(3(5(x1)))))) 45.03/12.29 2(4(1(0(x1)))) -> 0(4(1(3(2(x1))))) 45.03/12.29 2(4(1(0(x1)))) -> 4(2(1(3(0(4(x1)))))) 45.03/12.29 2(4(1(0(x1)))) -> 2(0(1(1(3(4(x1)))))) 45.03/12.29 4(0(5(0(x1)))) -> 5(3(0(0(4(x1))))) 45.03/12.29 4(0(5(0(x1)))) -> 4(3(0(0(5(x1))))) 45.03/12.29 5(0(5(0(x1)))) -> 5(0(0(2(5(3(x1)))))) 45.03/12.29 5(0(5(0(x1)))) -> 0(0(4(5(3(5(x1)))))) 45.03/12.29 5(0(5(0(x1)))) -> 0(0(4(3(5(5(x1)))))) 45.03/12.29 2(1(0(2(x1)))) -> 2(1(4(2(0(x1))))) 45.03/12.29 2(1(0(2(x1)))) -> 1(2(2(0(5(2(x1)))))) 45.03/12.29 2(1(0(2(x1)))) -> 0(2(2(3(1(3(x1)))))) 45.03/12.29 2(4(0(2(x1)))) -> 1(2(2(0(4(x1))))) 45.03/12.29 2(4(0(2(x1)))) -> 1(2(1(2(0(4(x1)))))) 45.03/12.29 2(5(0(2(x1)))) -> 2(2(5(1(2(0(x1)))))) 45.03/12.29 2(4(0(3(x1)))) -> 2(1(1(4(3(0(x1)))))) 45.03/12.29 2(4(0(3(x1)))) -> 2(0(1(3(3(4(x1)))))) 45.03/12.29 2(2(1(0(0(x1))))) -> 1(2(1(2(0(0(x1)))))) 45.03/12.29 2(3(1(0(0(x1))))) -> 4(0(2(1(3(0(x1)))))) 45.03/12.29 2(3(1(0(0(x1))))) -> 0(1(3(0(5(2(x1)))))) 45.03/12.29 2(2(5(0(0(x1))))) -> 1(2(5(2(0(0(x1)))))) 45.03/12.29 2(5(0(1(0(x1))))) -> 5(3(0(1(2(0(x1)))))) 45.03/12.29 5(5(0(1(0(x1))))) -> 5(0(0(5(3(1(x1)))))) 45.03/12.29 5(0(2(1(0(x1))))) -> 0(1(2(5(0(4(x1)))))) 45.03/12.29 4(0(3(1(0(x1))))) -> 0(4(4(3(0(1(x1)))))) 45.03/12.29 2(4(3(1(0(x1))))) -> 4(3(0(1(2(4(x1)))))) 45.03/12.29 2(2(4(1(0(x1))))) -> 2(4(2(0(1(1(x1)))))) 45.03/12.29 4(2(4(1(0(x1))))) -> 4(1(3(2(0(4(x1)))))) 45.03/12.29 2(3(4(1(0(x1))))) -> 4(4(0(2(1(3(x1)))))) 45.03/12.29 5(0(0(3(0(x1))))) -> 0(0(0(1(3(5(x1)))))) 45.03/12.29 4(0(1(4(0(x1))))) -> 0(4(1(2(0(4(x1)))))) 45.03/12.29 2(2(3(4(0(x1))))) -> 2(5(3(0(2(4(x1)))))) 45.03/12.29 4(0(1(5(0(x1))))) -> 0(0(1(4(2(5(x1)))))) 45.03/12.29 2(4(1(5(0(x1))))) -> 2(1(5(3(0(4(x1)))))) 45.03/12.29 2(1(0(1(2(x1))))) -> 3(2(0(1(1(2(x1)))))) 45.03/12.29 2(5(0(4(2(x1))))) -> 2(5(3(2(4(0(x1)))))) 45.03/12.29 2(1(0(0(3(x1))))) -> 2(0(1(5(3(0(x1)))))) 45.03/12.29 2(5(0(0(3(x1))))) -> 1(5(3(0(0(2(x1)))))) 45.03/12.29 2(4(5(0(3(x1))))) -> 1(2(5(3(0(4(x1)))))) 45.03/12.29 2(1(0(2(3(x1))))) -> 2(0(1(3(1(2(x1)))))) 45.03/12.29 2(5(0(2(3(x1))))) -> 5(4(3(2(0(2(x1)))))) 45.03/12.29 2(5(0(2(3(x1))))) -> 2(2(2(0(3(5(x1)))))) 45.03/12.29 45.03/12.29 Q is empty. 45.03/12.29 We have to consider all minimal (P,Q,R)-chains. 45.03/12.29 ---------------------------------------- 45.03/12.29 45.03/12.29 (11) QDPOrderProof (EQUIVALENT) 45.03/12.29 We use the reduction pair processor [LPAR04,JAR06]. 45.03/12.29 45.03/12.29 45.03/12.29 The following pairs can be oriented strictly and are deleted. 45.03/12.29 45.03/12.29 2^1(5(0(4(2(x1))))) -> 2^1(4(0(x1))) 45.03/12.29 The remaining pairs can at least be oriented weakly. 45.03/12.29 Used ordering: Polynomial interpretation [POLO]: 45.03/12.29 45.03/12.29 POL(0(x_1)) = x_1 45.03/12.29 POL(1(x_1)) = 0 45.03/12.29 POL(2(x_1)) = 1 + x_1 45.03/12.29 POL(2^1(x_1)) = x_1 45.03/12.29 POL(3(x_1)) = 0 45.03/12.29 POL(4(x_1)) = x_1 45.03/12.29 POL(5(x_1)) = x_1 45.03/12.29 45.03/12.29 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 45.03/12.29 45.03/12.29 4(0(1(0(x1)))) -> 0(0(4(1(3(x1))))) 45.03/12.29 4(0(1(0(x1)))) -> 1(3(0(0(4(3(x1)))))) 45.03/12.29 4(0(1(0(x1)))) -> 0(5(0(1(3(4(x1)))))) 45.03/12.29 4(0(1(0(x1)))) -> 0(1(3(4(0(5(x1)))))) 45.03/12.29 4(0(5(0(x1)))) -> 5(3(0(0(4(x1))))) 45.03/12.29 4(0(5(0(x1)))) -> 4(3(0(0(5(x1))))) 45.03/12.29 4(0(3(1(0(x1))))) -> 0(4(4(3(0(1(x1)))))) 45.03/12.29 4(0(1(4(0(x1))))) -> 0(4(1(2(0(4(x1)))))) 45.03/12.29 4(0(1(5(0(x1))))) -> 0(0(1(4(2(5(x1)))))) 45.03/12.29 45.03/12.29 45.03/12.29 ---------------------------------------- 45.03/12.29 45.03/12.29 (12) 45.03/12.29 Obligation: 45.03/12.29 Q DP problem: 45.03/12.29 P is empty. 45.03/12.29 The TRS R consists of the following rules: 45.03/12.29 45.03/12.29 2(1(0(0(x1)))) -> 0(0(1(3(2(x1))))) 45.03/12.29 2(1(0(0(x1)))) -> 2(0(0(1(3(x1))))) 45.03/12.29 2(1(0(0(x1)))) -> 1(1(3(2(0(0(x1)))))) 45.03/12.29 4(0(1(0(x1)))) -> 0(0(4(1(3(x1))))) 45.03/12.29 4(0(1(0(x1)))) -> 1(3(0(0(4(3(x1)))))) 45.03/12.29 4(0(1(0(x1)))) -> 0(5(0(1(3(4(x1)))))) 45.03/12.29 4(0(1(0(x1)))) -> 0(1(3(4(0(5(x1)))))) 45.03/12.29 5(0(1(0(x1)))) -> 0(0(1(3(5(3(x1)))))) 45.03/12.29 5(0(1(0(x1)))) -> 2(0(0(1(3(5(x1)))))) 45.03/12.29 2(4(1(0(x1)))) -> 0(4(1(3(2(x1))))) 45.03/12.29 2(4(1(0(x1)))) -> 4(2(1(3(0(4(x1)))))) 45.03/12.29 2(4(1(0(x1)))) -> 2(0(1(1(3(4(x1)))))) 45.03/12.29 4(0(5(0(x1)))) -> 5(3(0(0(4(x1))))) 45.03/12.29 4(0(5(0(x1)))) -> 4(3(0(0(5(x1))))) 45.03/12.29 5(0(5(0(x1)))) -> 5(0(0(2(5(3(x1)))))) 45.03/12.29 5(0(5(0(x1)))) -> 0(0(4(5(3(5(x1)))))) 45.03/12.29 5(0(5(0(x1)))) -> 0(0(4(3(5(5(x1)))))) 45.03/12.29 2(1(0(2(x1)))) -> 2(1(4(2(0(x1))))) 45.03/12.29 2(1(0(2(x1)))) -> 1(2(2(0(5(2(x1)))))) 45.03/12.29 2(1(0(2(x1)))) -> 0(2(2(3(1(3(x1)))))) 45.03/12.29 2(4(0(2(x1)))) -> 1(2(2(0(4(x1))))) 45.03/12.29 2(4(0(2(x1)))) -> 1(2(1(2(0(4(x1)))))) 45.03/12.29 2(5(0(2(x1)))) -> 2(2(5(1(2(0(x1)))))) 45.03/12.29 2(4(0(3(x1)))) -> 2(1(1(4(3(0(x1)))))) 45.03/12.29 2(4(0(3(x1)))) -> 2(0(1(3(3(4(x1)))))) 45.03/12.29 2(2(1(0(0(x1))))) -> 1(2(1(2(0(0(x1)))))) 45.03/12.29 2(3(1(0(0(x1))))) -> 4(0(2(1(3(0(x1)))))) 45.03/12.29 2(3(1(0(0(x1))))) -> 0(1(3(0(5(2(x1)))))) 45.03/12.29 2(2(5(0(0(x1))))) -> 1(2(5(2(0(0(x1)))))) 45.03/12.29 2(5(0(1(0(x1))))) -> 5(3(0(1(2(0(x1)))))) 45.03/12.29 5(5(0(1(0(x1))))) -> 5(0(0(5(3(1(x1)))))) 45.03/12.29 5(0(2(1(0(x1))))) -> 0(1(2(5(0(4(x1)))))) 45.03/12.29 4(0(3(1(0(x1))))) -> 0(4(4(3(0(1(x1)))))) 45.03/12.29 2(4(3(1(0(x1))))) -> 4(3(0(1(2(4(x1)))))) 45.03/12.29 2(2(4(1(0(x1))))) -> 2(4(2(0(1(1(x1)))))) 45.03/12.29 4(2(4(1(0(x1))))) -> 4(1(3(2(0(4(x1)))))) 45.03/12.29 2(3(4(1(0(x1))))) -> 4(4(0(2(1(3(x1)))))) 45.03/12.29 5(0(0(3(0(x1))))) -> 0(0(0(1(3(5(x1)))))) 45.03/12.29 4(0(1(4(0(x1))))) -> 0(4(1(2(0(4(x1)))))) 45.03/12.29 2(2(3(4(0(x1))))) -> 2(5(3(0(2(4(x1)))))) 45.03/12.29 4(0(1(5(0(x1))))) -> 0(0(1(4(2(5(x1)))))) 45.03/12.29 2(4(1(5(0(x1))))) -> 2(1(5(3(0(4(x1)))))) 45.03/12.29 2(1(0(1(2(x1))))) -> 3(2(0(1(1(2(x1)))))) 45.03/12.29 2(5(0(4(2(x1))))) -> 2(5(3(2(4(0(x1)))))) 45.03/12.29 2(1(0(0(3(x1))))) -> 2(0(1(5(3(0(x1)))))) 45.03/12.29 2(5(0(0(3(x1))))) -> 1(5(3(0(0(2(x1)))))) 45.03/12.29 2(4(5(0(3(x1))))) -> 1(2(5(3(0(4(x1)))))) 45.03/12.29 2(1(0(2(3(x1))))) -> 2(0(1(3(1(2(x1)))))) 45.03/12.29 2(5(0(2(3(x1))))) -> 5(4(3(2(0(2(x1)))))) 45.03/12.29 2(5(0(2(3(x1))))) -> 2(2(2(0(3(5(x1)))))) 45.03/12.29 45.03/12.29 Q is empty. 45.03/12.29 We have to consider all minimal (P,Q,R)-chains. 45.03/12.29 ---------------------------------------- 45.03/12.29 45.03/12.29 (13) PisEmptyProof (EQUIVALENT) 45.03/12.29 The TRS P is empty. Hence, there is no (P,Q,R) chain. 45.03/12.29 ---------------------------------------- 45.03/12.29 45.03/12.29 (14) 45.03/12.29 YES 45.19/12.43 EOF