68.59/18.27 YES 69.31/18.47 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 69.31/18.47 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 69.31/18.47 69.31/18.47 69.31/18.47 Termination w.r.t. Q of the given QTRS could be proven: 69.31/18.47 69.31/18.47 (0) QTRS 69.31/18.47 (1) QTRS Reverse [EQUIVALENT, 0 ms] 69.31/18.47 (2) QTRS 69.31/18.47 (3) DependencyPairsProof [EQUIVALENT, 732 ms] 69.31/18.47 (4) QDP 69.31/18.47 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 69.31/18.47 (6) QDP 69.31/18.47 (7) QDPOrderProof [EQUIVALENT, 326 ms] 69.31/18.47 (8) QDP 69.31/18.47 (9) DependencyGraphProof [EQUIVALENT, 0 ms] 69.31/18.47 (10) QDP 69.31/18.47 (11) QDPOrderProof [EQUIVALENT, 126 ms] 69.31/18.47 (12) QDP 69.31/18.47 (13) DependencyGraphProof [EQUIVALENT, 0 ms] 69.31/18.47 (14) TRUE 69.31/18.47 69.31/18.47 69.31/18.47 ---------------------------------------- 69.31/18.47 69.31/18.47 (0) 69.31/18.47 Obligation: 69.31/18.47 Q restricted rewrite system: 69.31/18.47 The TRS R consists of the following rules: 69.31/18.47 69.31/18.47 0(0(1(x1))) -> 0(1(2(0(3(x1))))) 69.31/18.47 0(0(1(x1))) -> 0(1(2(0(3(2(x1)))))) 69.31/18.47 0(0(1(x1))) -> 1(2(0(0(3(2(x1)))))) 69.31/18.47 0(1(0(x1))) -> 2(0(3(1(2(0(x1)))))) 69.31/18.47 0(4(1(x1))) -> 4(0(3(1(x1)))) 69.31/18.47 0(4(1(x1))) -> 2(4(0(3(2(1(x1)))))) 69.31/18.47 0(4(1(x1))) -> 2(4(0(5(3(1(x1)))))) 69.31/18.47 0(4(1(x1))) -> 4(2(1(2(0(3(x1)))))) 69.31/18.47 4(1(0(x1))) -> 1(2(0(4(x1)))) 69.31/18.47 0(0(1(1(x1)))) -> 1(2(0(0(3(1(x1)))))) 69.31/18.47 0(0(2(1(x1)))) -> 2(2(0(3(0(1(x1)))))) 69.31/18.47 0(0(4(1(x1)))) -> 1(4(0(3(0(3(x1)))))) 69.31/18.47 0(0(5(1(x1)))) -> 0(0(3(1(2(5(x1)))))) 69.31/18.47 0(1(1(0(x1)))) -> 1(2(0(0(1(2(x1)))))) 69.31/18.47 0(1(4(1(x1)))) -> 0(1(2(4(1(2(x1)))))) 69.31/18.47 0(1(5(0(x1)))) -> 1(2(5(0(0(3(x1)))))) 69.31/18.47 0(2(4(1(x1)))) -> 4(0(5(3(1(2(x1)))))) 69.31/18.47 0(4(1(0(x1)))) -> 0(4(0(1(2(x1))))) 69.31/18.47 0(4(2(1(x1)))) -> 4(0(3(2(2(1(x1)))))) 69.31/18.47 0(4(2(1(x1)))) -> 4(0(3(5(1(2(x1)))))) 69.31/18.47 0(4(4(1(x1)))) -> 2(4(0(3(4(1(x1)))))) 69.31/18.47 0(5(0(1(x1)))) -> 0(2(0(3(5(1(x1)))))) 69.31/18.47 0(5(0(1(x1)))) -> 5(2(0(3(0(1(x1)))))) 69.31/18.47 0(5(1(0(x1)))) -> 0(1(2(0(3(5(x1)))))) 69.31/18.47 0(5(1(0(x1)))) -> 2(0(1(3(5(0(x1)))))) 69.31/18.47 4(0(5(1(x1)))) -> 1(4(0(5(3(2(x1)))))) 69.31/18.47 4(1(0(0(x1)))) -> 0(4(1(2(0(x1))))) 69.31/18.47 4(1(0(1(x1)))) -> 1(1(2(0(4(x1))))) 69.31/18.47 4(1(5(0(x1)))) -> 1(2(0(5(4(x1))))) 69.31/18.47 4(1(5(0(x1)))) -> 1(4(0(3(5(x1))))) 69.31/18.47 4(3(1(0(x1)))) -> 1(4(2(0(3(x1))))) 69.31/18.47 4(3(1(0(x1)))) -> 2(0(2(1(3(4(x1)))))) 69.31/18.47 4(3(1(0(x1)))) -> 2(1(4(2(0(3(x1)))))) 69.31/18.47 4(3(1(0(x1)))) -> 2(2(4(0(1(3(x1)))))) 69.31/18.47 5(0(1(0(x1)))) -> 0(3(5(1(2(0(x1)))))) 69.31/18.47 5(0(1(0(x1)))) -> 1(5(2(0(3(0(x1)))))) 69.31/18.47 5(4(1(0(x1)))) -> 0(1(2(4(5(2(x1)))))) 69.31/18.47 0(0(5(5(1(x1))))) -> 1(0(0(3(5(5(x1)))))) 69.31/18.47 0(1(0(5(0(x1))))) -> 0(1(5(2(0(0(x1)))))) 69.31/18.47 0(2(5(0(1(x1))))) -> 2(0(3(5(0(1(x1)))))) 69.31/18.47 0(3(1(0(0(x1))))) -> 0(0(3(0(1(2(x1)))))) 69.31/18.47 0(4(1(4(1(x1))))) -> 4(4(0(1(2(1(x1)))))) 69.31/18.47 0(5(5(4(1(x1))))) -> 4(1(0(5(5(3(x1)))))) 69.31/18.47 4(1(5(0(0(x1))))) -> 1(2(0(4(5(0(x1)))))) 69.31/18.47 4(1(5(5(0(x1))))) -> 5(1(3(0(5(4(x1)))))) 69.31/18.47 4(3(1(0(1(x1))))) -> 1(1(2(3(0(4(x1)))))) 69.31/18.47 4(3(1(1(0(x1))))) -> 1(2(0(1(4(3(x1)))))) 69.31/18.47 4(3(1(5(0(x1))))) -> 5(1(0(3(2(4(x1)))))) 69.31/18.47 4(4(1(0(5(x1))))) -> 4(0(5(3(4(1(x1)))))) 69.31/18.47 69.31/18.47 Q is empty. 69.31/18.47 69.31/18.47 ---------------------------------------- 69.31/18.47 69.31/18.47 (1) QTRS Reverse (EQUIVALENT) 69.31/18.47 We applied the QTRS Reverse Processor [REVERSE]. 69.31/18.47 ---------------------------------------- 69.31/18.47 69.31/18.47 (2) 69.31/18.47 Obligation: 69.31/18.47 Q restricted rewrite system: 69.31/18.47 The TRS R consists of the following rules: 69.31/18.47 69.31/18.47 1(0(0(x1))) -> 3(0(2(1(0(x1))))) 69.31/18.47 1(0(0(x1))) -> 2(3(0(2(1(0(x1)))))) 69.31/18.47 1(0(0(x1))) -> 2(3(0(0(2(1(x1)))))) 69.31/18.47 0(1(0(x1))) -> 0(2(1(3(0(2(x1)))))) 69.31/18.47 1(4(0(x1))) -> 1(3(0(4(x1)))) 69.31/18.47 1(4(0(x1))) -> 1(2(3(0(4(2(x1)))))) 69.31/18.47 1(4(0(x1))) -> 1(3(5(0(4(2(x1)))))) 69.31/18.47 1(4(0(x1))) -> 3(0(2(1(2(4(x1)))))) 69.31/18.47 0(1(4(x1))) -> 4(0(2(1(x1)))) 69.31/18.47 1(1(0(0(x1)))) -> 1(3(0(0(2(1(x1)))))) 69.31/18.47 1(2(0(0(x1)))) -> 1(0(3(0(2(2(x1)))))) 69.31/18.47 1(4(0(0(x1)))) -> 3(0(3(0(4(1(x1)))))) 69.31/18.47 1(5(0(0(x1)))) -> 5(2(1(3(0(0(x1)))))) 69.31/18.47 0(1(1(0(x1)))) -> 2(1(0(0(2(1(x1)))))) 69.31/18.47 1(4(1(0(x1)))) -> 2(1(4(2(1(0(x1)))))) 69.31/18.47 0(5(1(0(x1)))) -> 3(0(0(5(2(1(x1)))))) 69.31/18.47 1(4(2(0(x1)))) -> 2(1(3(5(0(4(x1)))))) 69.31/18.47 0(1(4(0(x1)))) -> 2(1(0(4(0(x1))))) 69.31/18.47 1(2(4(0(x1)))) -> 1(2(2(3(0(4(x1)))))) 69.31/18.47 1(2(4(0(x1)))) -> 2(1(5(3(0(4(x1)))))) 69.31/18.47 1(4(4(0(x1)))) -> 1(4(3(0(4(2(x1)))))) 69.31/18.47 1(0(5(0(x1)))) -> 1(5(3(0(2(0(x1)))))) 69.31/18.47 1(0(5(0(x1)))) -> 1(0(3(0(2(5(x1)))))) 69.31/18.47 0(1(5(0(x1)))) -> 5(3(0(2(1(0(x1)))))) 69.31/18.47 0(1(5(0(x1)))) -> 0(5(3(1(0(2(x1)))))) 69.31/18.47 1(5(0(4(x1)))) -> 2(3(5(0(4(1(x1)))))) 69.31/18.47 0(0(1(4(x1)))) -> 0(2(1(4(0(x1))))) 69.31/18.47 1(0(1(4(x1)))) -> 4(0(2(1(1(x1))))) 69.31/18.47 0(5(1(4(x1)))) -> 4(5(0(2(1(x1))))) 69.31/18.47 0(5(1(4(x1)))) -> 5(3(0(4(1(x1))))) 69.31/18.47 0(1(3(4(x1)))) -> 3(0(2(4(1(x1))))) 69.31/18.47 0(1(3(4(x1)))) -> 4(3(1(2(0(2(x1)))))) 69.31/18.47 0(1(3(4(x1)))) -> 3(0(2(4(1(2(x1)))))) 69.31/18.47 0(1(3(4(x1)))) -> 3(1(0(4(2(2(x1)))))) 69.31/18.47 0(1(0(5(x1)))) -> 0(2(1(5(3(0(x1)))))) 69.31/18.47 0(1(0(5(x1)))) -> 0(3(0(2(5(1(x1)))))) 69.31/18.47 0(1(4(5(x1)))) -> 2(5(4(2(1(0(x1)))))) 69.31/18.47 1(5(5(0(0(x1))))) -> 5(5(3(0(0(1(x1)))))) 69.31/18.47 0(5(0(1(0(x1))))) -> 0(0(2(5(1(0(x1)))))) 69.31/18.47 1(0(5(2(0(x1))))) -> 1(0(5(3(0(2(x1)))))) 69.31/18.47 0(0(1(3(0(x1))))) -> 2(1(0(3(0(0(x1)))))) 69.31/18.47 1(4(1(4(0(x1))))) -> 1(2(1(0(4(4(x1)))))) 69.31/18.47 1(4(5(5(0(x1))))) -> 3(5(5(0(1(4(x1)))))) 69.31/18.47 0(0(5(1(4(x1))))) -> 0(5(4(0(2(1(x1)))))) 69.31/18.47 0(5(5(1(4(x1))))) -> 4(5(0(3(1(5(x1)))))) 69.31/18.47 1(0(1(3(4(x1))))) -> 4(0(3(2(1(1(x1)))))) 69.31/18.47 0(1(1(3(4(x1))))) -> 3(4(1(0(2(1(x1)))))) 69.31/18.47 0(5(1(3(4(x1))))) -> 4(2(3(0(1(5(x1)))))) 69.31/18.47 5(0(1(4(4(x1))))) -> 1(4(3(5(0(4(x1)))))) 69.31/18.47 69.31/18.47 Q is empty. 69.31/18.47 69.31/18.47 ---------------------------------------- 69.31/18.47 69.31/18.47 (3) DependencyPairsProof (EQUIVALENT) 69.31/18.47 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 69.31/18.47 ---------------------------------------- 69.31/18.47 69.31/18.47 (4) 69.31/18.47 Obligation: 69.31/18.47 Q DP problem: 69.31/18.47 The TRS P consists of the following rules: 69.31/18.47 69.31/18.47 1^1(0(0(x1))) -> 0^1(2(1(0(x1)))) 69.31/18.47 1^1(0(0(x1))) -> 1^1(0(x1)) 69.31/18.47 1^1(0(0(x1))) -> 0^1(0(2(1(x1)))) 69.31/18.47 1^1(0(0(x1))) -> 0^1(2(1(x1))) 69.31/18.47 1^1(0(0(x1))) -> 1^1(x1) 69.31/18.47 0^1(1(0(x1))) -> 0^1(2(1(3(0(2(x1)))))) 69.31/18.47 0^1(1(0(x1))) -> 1^1(3(0(2(x1)))) 69.31/18.47 0^1(1(0(x1))) -> 0^1(2(x1)) 69.31/18.47 1^1(4(0(x1))) -> 1^1(3(0(4(x1)))) 69.31/18.47 1^1(4(0(x1))) -> 0^1(4(x1)) 69.31/18.47 1^1(4(0(x1))) -> 1^1(2(3(0(4(2(x1)))))) 69.31/18.47 1^1(4(0(x1))) -> 0^1(4(2(x1))) 69.31/18.47 1^1(4(0(x1))) -> 1^1(3(5(0(4(2(x1)))))) 69.31/18.47 1^1(4(0(x1))) -> 5^1(0(4(2(x1)))) 69.31/18.47 1^1(4(0(x1))) -> 0^1(2(1(2(4(x1))))) 69.31/18.47 1^1(4(0(x1))) -> 1^1(2(4(x1))) 69.31/18.47 0^1(1(4(x1))) -> 0^1(2(1(x1))) 69.31/18.47 0^1(1(4(x1))) -> 1^1(x1) 69.31/18.47 1^1(1(0(0(x1)))) -> 1^1(3(0(0(2(1(x1)))))) 69.31/18.47 1^1(1(0(0(x1)))) -> 0^1(0(2(1(x1)))) 69.31/18.47 1^1(1(0(0(x1)))) -> 0^1(2(1(x1))) 69.31/18.47 1^1(1(0(0(x1)))) -> 1^1(x1) 69.31/18.47 1^1(2(0(0(x1)))) -> 1^1(0(3(0(2(2(x1)))))) 69.31/18.47 1^1(2(0(0(x1)))) -> 0^1(3(0(2(2(x1))))) 69.31/18.47 1^1(2(0(0(x1)))) -> 0^1(2(2(x1))) 69.31/18.47 1^1(4(0(0(x1)))) -> 0^1(3(0(4(1(x1))))) 69.31/18.47 1^1(4(0(0(x1)))) -> 0^1(4(1(x1))) 69.31/18.47 1^1(4(0(0(x1)))) -> 1^1(x1) 69.31/18.47 1^1(5(0(0(x1)))) -> 5^1(2(1(3(0(0(x1)))))) 69.31/18.47 1^1(5(0(0(x1)))) -> 1^1(3(0(0(x1)))) 69.31/18.47 0^1(1(1(0(x1)))) -> 1^1(0(0(2(1(x1))))) 69.31/18.47 0^1(1(1(0(x1)))) -> 0^1(0(2(1(x1)))) 69.31/18.47 0^1(1(1(0(x1)))) -> 0^1(2(1(x1))) 69.31/18.47 0^1(1(1(0(x1)))) -> 1^1(x1) 69.31/18.47 1^1(4(1(0(x1)))) -> 1^1(4(2(1(0(x1))))) 69.31/18.47 0^1(5(1(0(x1)))) -> 0^1(0(5(2(1(x1))))) 69.31/18.47 0^1(5(1(0(x1)))) -> 0^1(5(2(1(x1)))) 69.31/18.47 0^1(5(1(0(x1)))) -> 5^1(2(1(x1))) 69.31/18.47 0^1(5(1(0(x1)))) -> 1^1(x1) 69.31/18.47 1^1(4(2(0(x1)))) -> 1^1(3(5(0(4(x1))))) 69.31/18.47 1^1(4(2(0(x1)))) -> 5^1(0(4(x1))) 69.31/18.47 1^1(4(2(0(x1)))) -> 0^1(4(x1)) 69.31/18.47 0^1(1(4(0(x1)))) -> 1^1(0(4(0(x1)))) 69.31/18.47 0^1(1(4(0(x1)))) -> 0^1(4(0(x1))) 69.31/18.47 1^1(2(4(0(x1)))) -> 1^1(2(2(3(0(4(x1)))))) 69.31/18.47 1^1(2(4(0(x1)))) -> 0^1(4(x1)) 69.31/18.47 1^1(2(4(0(x1)))) -> 1^1(5(3(0(4(x1))))) 69.31/18.47 1^1(2(4(0(x1)))) -> 5^1(3(0(4(x1)))) 69.31/18.47 1^1(4(4(0(x1)))) -> 1^1(4(3(0(4(2(x1)))))) 69.31/18.47 1^1(4(4(0(x1)))) -> 0^1(4(2(x1))) 69.31/18.47 1^1(0(5(0(x1)))) -> 1^1(5(3(0(2(0(x1)))))) 69.31/18.47 1^1(0(5(0(x1)))) -> 5^1(3(0(2(0(x1))))) 69.31/18.47 1^1(0(5(0(x1)))) -> 0^1(2(0(x1))) 69.31/18.47 1^1(0(5(0(x1)))) -> 1^1(0(3(0(2(5(x1)))))) 69.31/18.47 1^1(0(5(0(x1)))) -> 0^1(3(0(2(5(x1))))) 69.31/18.47 1^1(0(5(0(x1)))) -> 0^1(2(5(x1))) 69.31/18.47 1^1(0(5(0(x1)))) -> 5^1(x1) 69.31/18.47 0^1(1(5(0(x1)))) -> 5^1(3(0(2(1(0(x1)))))) 69.31/18.47 0^1(1(5(0(x1)))) -> 0^1(2(1(0(x1)))) 69.31/18.47 0^1(1(5(0(x1)))) -> 1^1(0(x1)) 69.31/18.47 0^1(1(5(0(x1)))) -> 0^1(5(3(1(0(2(x1)))))) 69.31/18.47 0^1(1(5(0(x1)))) -> 5^1(3(1(0(2(x1))))) 69.31/18.47 0^1(1(5(0(x1)))) -> 1^1(0(2(x1))) 69.31/18.47 0^1(1(5(0(x1)))) -> 0^1(2(x1)) 69.31/18.47 1^1(5(0(4(x1)))) -> 5^1(0(4(1(x1)))) 69.31/18.47 1^1(5(0(4(x1)))) -> 0^1(4(1(x1))) 69.31/18.47 1^1(5(0(4(x1)))) -> 1^1(x1) 69.31/18.47 0^1(0(1(4(x1)))) -> 0^1(2(1(4(0(x1))))) 69.31/18.47 0^1(0(1(4(x1)))) -> 1^1(4(0(x1))) 69.31/18.47 0^1(0(1(4(x1)))) -> 0^1(x1) 69.31/18.47 1^1(0(1(4(x1)))) -> 0^1(2(1(1(x1)))) 69.31/18.47 1^1(0(1(4(x1)))) -> 1^1(1(x1)) 69.31/18.47 1^1(0(1(4(x1)))) -> 1^1(x1) 69.31/18.47 0^1(5(1(4(x1)))) -> 5^1(0(2(1(x1)))) 69.31/18.47 0^1(5(1(4(x1)))) -> 0^1(2(1(x1))) 69.31/18.47 0^1(5(1(4(x1)))) -> 1^1(x1) 69.31/18.47 0^1(5(1(4(x1)))) -> 5^1(3(0(4(1(x1))))) 69.31/18.47 0^1(5(1(4(x1)))) -> 0^1(4(1(x1))) 69.31/18.47 0^1(1(3(4(x1)))) -> 0^1(2(4(1(x1)))) 69.31/18.47 0^1(1(3(4(x1)))) -> 1^1(x1) 69.31/18.47 0^1(1(3(4(x1)))) -> 1^1(2(0(2(x1)))) 69.31/18.47 0^1(1(3(4(x1)))) -> 0^1(2(x1)) 69.31/18.47 0^1(1(3(4(x1)))) -> 0^1(2(4(1(2(x1))))) 69.31/18.47 0^1(1(3(4(x1)))) -> 1^1(2(x1)) 69.31/18.47 0^1(1(3(4(x1)))) -> 1^1(0(4(2(2(x1))))) 69.31/18.47 0^1(1(3(4(x1)))) -> 0^1(4(2(2(x1)))) 69.31/18.47 0^1(1(0(5(x1)))) -> 0^1(2(1(5(3(0(x1)))))) 69.31/18.47 0^1(1(0(5(x1)))) -> 1^1(5(3(0(x1)))) 69.31/18.47 0^1(1(0(5(x1)))) -> 5^1(3(0(x1))) 69.31/18.47 0^1(1(0(5(x1)))) -> 0^1(x1) 69.31/18.47 0^1(1(0(5(x1)))) -> 0^1(3(0(2(5(1(x1)))))) 69.31/18.47 0^1(1(0(5(x1)))) -> 0^1(2(5(1(x1)))) 69.31/18.47 0^1(1(0(5(x1)))) -> 5^1(1(x1)) 69.31/18.47 0^1(1(0(5(x1)))) -> 1^1(x1) 69.31/18.47 0^1(1(4(5(x1)))) -> 5^1(4(2(1(0(x1))))) 69.31/18.47 0^1(1(4(5(x1)))) -> 1^1(0(x1)) 69.31/18.47 0^1(1(4(5(x1)))) -> 0^1(x1) 69.31/18.47 1^1(5(5(0(0(x1))))) -> 5^1(5(3(0(0(1(x1)))))) 69.31/18.47 1^1(5(5(0(0(x1))))) -> 5^1(3(0(0(1(x1))))) 69.31/18.47 1^1(5(5(0(0(x1))))) -> 0^1(0(1(x1))) 69.31/18.47 1^1(5(5(0(0(x1))))) -> 0^1(1(x1)) 69.31/18.47 1^1(5(5(0(0(x1))))) -> 1^1(x1) 69.31/18.47 0^1(5(0(1(0(x1))))) -> 0^1(0(2(5(1(0(x1)))))) 69.31/18.47 0^1(5(0(1(0(x1))))) -> 0^1(2(5(1(0(x1))))) 69.31/18.47 0^1(5(0(1(0(x1))))) -> 5^1(1(0(x1))) 69.31/18.47 1^1(0(5(2(0(x1))))) -> 1^1(0(5(3(0(2(x1)))))) 69.31/18.47 1^1(0(5(2(0(x1))))) -> 0^1(5(3(0(2(x1))))) 69.31/18.47 1^1(0(5(2(0(x1))))) -> 5^1(3(0(2(x1)))) 69.31/18.47 1^1(0(5(2(0(x1))))) -> 0^1(2(x1)) 69.31/18.47 0^1(0(1(3(0(x1))))) -> 1^1(0(3(0(0(x1))))) 69.31/18.47 0^1(0(1(3(0(x1))))) -> 0^1(3(0(0(x1)))) 69.31/18.47 0^1(0(1(3(0(x1))))) -> 0^1(0(x1)) 69.31/18.47 1^1(4(1(4(0(x1))))) -> 1^1(2(1(0(4(4(x1)))))) 69.31/18.47 1^1(4(1(4(0(x1))))) -> 1^1(0(4(4(x1)))) 69.31/18.47 1^1(4(1(4(0(x1))))) -> 0^1(4(4(x1))) 69.31/18.47 1^1(4(5(5(0(x1))))) -> 5^1(5(0(1(4(x1))))) 69.31/18.47 1^1(4(5(5(0(x1))))) -> 5^1(0(1(4(x1)))) 69.31/18.47 1^1(4(5(5(0(x1))))) -> 0^1(1(4(x1))) 69.31/18.47 1^1(4(5(5(0(x1))))) -> 1^1(4(x1)) 69.31/18.47 0^1(0(5(1(4(x1))))) -> 0^1(5(4(0(2(1(x1)))))) 69.31/18.47 0^1(0(5(1(4(x1))))) -> 5^1(4(0(2(1(x1))))) 69.31/18.47 0^1(0(5(1(4(x1))))) -> 0^1(2(1(x1))) 69.31/18.47 0^1(0(5(1(4(x1))))) -> 1^1(x1) 69.31/18.47 0^1(5(5(1(4(x1))))) -> 5^1(0(3(1(5(x1))))) 69.31/18.47 0^1(5(5(1(4(x1))))) -> 0^1(3(1(5(x1)))) 69.31/18.47 0^1(5(5(1(4(x1))))) -> 1^1(5(x1)) 69.31/18.47 0^1(5(5(1(4(x1))))) -> 5^1(x1) 69.31/18.47 1^1(0(1(3(4(x1))))) -> 0^1(3(2(1(1(x1))))) 69.31/18.47 1^1(0(1(3(4(x1))))) -> 1^1(1(x1)) 69.31/18.47 1^1(0(1(3(4(x1))))) -> 1^1(x1) 69.31/18.47 0^1(1(1(3(4(x1))))) -> 1^1(0(2(1(x1)))) 69.31/18.47 0^1(1(1(3(4(x1))))) -> 0^1(2(1(x1))) 69.31/18.47 0^1(1(1(3(4(x1))))) -> 1^1(x1) 69.31/18.47 0^1(5(1(3(4(x1))))) -> 0^1(1(5(x1))) 69.31/18.47 0^1(5(1(3(4(x1))))) -> 1^1(5(x1)) 69.31/18.47 0^1(5(1(3(4(x1))))) -> 5^1(x1) 69.31/18.47 5^1(0(1(4(4(x1))))) -> 1^1(4(3(5(0(4(x1)))))) 69.31/18.47 5^1(0(1(4(4(x1))))) -> 5^1(0(4(x1))) 69.31/18.47 5^1(0(1(4(4(x1))))) -> 0^1(4(x1)) 69.31/18.47 69.31/18.47 The TRS R consists of the following rules: 69.31/18.47 69.31/18.47 1(0(0(x1))) -> 3(0(2(1(0(x1))))) 69.31/18.47 1(0(0(x1))) -> 2(3(0(2(1(0(x1)))))) 69.31/18.47 1(0(0(x1))) -> 2(3(0(0(2(1(x1)))))) 69.31/18.47 0(1(0(x1))) -> 0(2(1(3(0(2(x1)))))) 69.31/18.47 1(4(0(x1))) -> 1(3(0(4(x1)))) 69.31/18.47 1(4(0(x1))) -> 1(2(3(0(4(2(x1)))))) 69.31/18.47 1(4(0(x1))) -> 1(3(5(0(4(2(x1)))))) 69.31/18.47 1(4(0(x1))) -> 3(0(2(1(2(4(x1)))))) 69.31/18.47 0(1(4(x1))) -> 4(0(2(1(x1)))) 69.31/18.47 1(1(0(0(x1)))) -> 1(3(0(0(2(1(x1)))))) 69.31/18.47 1(2(0(0(x1)))) -> 1(0(3(0(2(2(x1)))))) 69.31/18.47 1(4(0(0(x1)))) -> 3(0(3(0(4(1(x1)))))) 69.31/18.47 1(5(0(0(x1)))) -> 5(2(1(3(0(0(x1)))))) 69.31/18.47 0(1(1(0(x1)))) -> 2(1(0(0(2(1(x1)))))) 69.31/18.47 1(4(1(0(x1)))) -> 2(1(4(2(1(0(x1)))))) 69.31/18.47 0(5(1(0(x1)))) -> 3(0(0(5(2(1(x1)))))) 69.31/18.47 1(4(2(0(x1)))) -> 2(1(3(5(0(4(x1)))))) 69.31/18.47 0(1(4(0(x1)))) -> 2(1(0(4(0(x1))))) 69.31/18.47 1(2(4(0(x1)))) -> 1(2(2(3(0(4(x1)))))) 69.31/18.47 1(2(4(0(x1)))) -> 2(1(5(3(0(4(x1)))))) 69.31/18.47 1(4(4(0(x1)))) -> 1(4(3(0(4(2(x1)))))) 69.31/18.47 1(0(5(0(x1)))) -> 1(5(3(0(2(0(x1)))))) 69.31/18.47 1(0(5(0(x1)))) -> 1(0(3(0(2(5(x1)))))) 69.31/18.47 0(1(5(0(x1)))) -> 5(3(0(2(1(0(x1)))))) 69.31/18.47 0(1(5(0(x1)))) -> 0(5(3(1(0(2(x1)))))) 69.31/18.47 1(5(0(4(x1)))) -> 2(3(5(0(4(1(x1)))))) 69.31/18.47 0(0(1(4(x1)))) -> 0(2(1(4(0(x1))))) 69.31/18.47 1(0(1(4(x1)))) -> 4(0(2(1(1(x1))))) 69.31/18.47 0(5(1(4(x1)))) -> 4(5(0(2(1(x1))))) 69.31/18.47 0(5(1(4(x1)))) -> 5(3(0(4(1(x1))))) 69.31/18.47 0(1(3(4(x1)))) -> 3(0(2(4(1(x1))))) 69.31/18.47 0(1(3(4(x1)))) -> 4(3(1(2(0(2(x1)))))) 69.31/18.47 0(1(3(4(x1)))) -> 3(0(2(4(1(2(x1)))))) 69.31/18.47 0(1(3(4(x1)))) -> 3(1(0(4(2(2(x1)))))) 69.31/18.47 0(1(0(5(x1)))) -> 0(2(1(5(3(0(x1)))))) 69.31/18.47 0(1(0(5(x1)))) -> 0(3(0(2(5(1(x1)))))) 69.31/18.47 0(1(4(5(x1)))) -> 2(5(4(2(1(0(x1)))))) 69.31/18.47 1(5(5(0(0(x1))))) -> 5(5(3(0(0(1(x1)))))) 69.31/18.47 0(5(0(1(0(x1))))) -> 0(0(2(5(1(0(x1)))))) 69.31/18.47 1(0(5(2(0(x1))))) -> 1(0(5(3(0(2(x1)))))) 69.31/18.47 0(0(1(3(0(x1))))) -> 2(1(0(3(0(0(x1)))))) 69.31/18.47 1(4(1(4(0(x1))))) -> 1(2(1(0(4(4(x1)))))) 69.31/18.47 1(4(5(5(0(x1))))) -> 3(5(5(0(1(4(x1)))))) 69.31/18.47 0(0(5(1(4(x1))))) -> 0(5(4(0(2(1(x1)))))) 69.31/18.47 0(5(5(1(4(x1))))) -> 4(5(0(3(1(5(x1)))))) 69.31/18.47 1(0(1(3(4(x1))))) -> 4(0(3(2(1(1(x1)))))) 69.31/18.47 0(1(1(3(4(x1))))) -> 3(4(1(0(2(1(x1)))))) 69.31/18.47 0(5(1(3(4(x1))))) -> 4(2(3(0(1(5(x1)))))) 69.31/18.47 5(0(1(4(4(x1))))) -> 1(4(3(5(0(4(x1)))))) 69.31/18.47 69.31/18.47 Q is empty. 69.31/18.47 We have to consider all minimal (P,Q,R)-chains. 69.31/18.47 ---------------------------------------- 69.31/18.47 69.31/18.47 (5) DependencyGraphProof (EQUIVALENT) 69.31/18.47 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 106 less nodes. 69.31/18.47 ---------------------------------------- 69.31/18.47 69.31/18.47 (6) 69.31/18.47 Obligation: 69.31/18.47 Q DP problem: 69.31/18.47 The TRS P consists of the following rules: 69.31/18.47 69.31/18.47 1^1(0(0(x1))) -> 1^1(x1) 69.31/18.47 1^1(0(0(x1))) -> 1^1(0(x1)) 69.31/18.47 1^1(1(0(0(x1)))) -> 1^1(x1) 69.31/18.47 1^1(4(0(0(x1)))) -> 1^1(x1) 69.31/18.47 1^1(5(0(4(x1)))) -> 1^1(x1) 69.31/18.47 1^1(0(1(4(x1)))) -> 1^1(1(x1)) 69.31/18.47 1^1(0(1(4(x1)))) -> 1^1(x1) 69.31/18.47 1^1(5(5(0(0(x1))))) -> 0^1(0(1(x1))) 69.31/18.47 0^1(1(4(x1))) -> 1^1(x1) 69.31/18.47 1^1(5(5(0(0(x1))))) -> 0^1(1(x1)) 69.31/18.47 0^1(1(1(0(x1)))) -> 1^1(0(0(2(1(x1))))) 69.31/18.47 0^1(1(1(0(x1)))) -> 1^1(x1) 69.31/18.47 1^1(5(5(0(0(x1))))) -> 1^1(x1) 69.31/18.47 1^1(4(5(5(0(x1))))) -> 0^1(1(4(x1))) 69.31/18.47 0^1(5(1(0(x1)))) -> 1^1(x1) 69.31/18.47 1^1(4(5(5(0(x1))))) -> 1^1(4(x1)) 69.31/18.47 1^1(0(1(3(4(x1))))) -> 1^1(1(x1)) 69.31/18.47 1^1(0(1(3(4(x1))))) -> 1^1(x1) 69.31/18.47 0^1(1(5(0(x1)))) -> 1^1(0(x1)) 69.31/18.47 0^1(0(1(4(x1)))) -> 1^1(4(0(x1))) 69.31/18.47 0^1(0(1(4(x1)))) -> 0^1(x1) 69.31/18.47 0^1(5(1(4(x1)))) -> 1^1(x1) 69.31/18.47 0^1(1(3(4(x1)))) -> 1^1(x1) 69.31/18.47 0^1(1(0(5(x1)))) -> 0^1(x1) 69.31/18.47 0^1(1(0(5(x1)))) -> 1^1(x1) 69.31/18.47 0^1(1(4(5(x1)))) -> 1^1(0(x1)) 69.31/18.47 0^1(1(4(5(x1)))) -> 0^1(x1) 69.31/18.47 0^1(0(1(3(0(x1))))) -> 0^1(0(x1)) 69.31/18.47 0^1(0(5(1(4(x1))))) -> 1^1(x1) 69.31/18.47 0^1(5(5(1(4(x1))))) -> 1^1(5(x1)) 69.31/18.47 0^1(1(1(3(4(x1))))) -> 1^1(x1) 69.31/18.47 0^1(5(1(3(4(x1))))) -> 0^1(1(5(x1))) 69.31/18.47 0^1(5(1(3(4(x1))))) -> 1^1(5(x1)) 69.31/18.47 69.31/18.47 The TRS R consists of the following rules: 69.31/18.47 69.31/18.47 1(0(0(x1))) -> 3(0(2(1(0(x1))))) 69.31/18.47 1(0(0(x1))) -> 2(3(0(2(1(0(x1)))))) 69.31/18.47 1(0(0(x1))) -> 2(3(0(0(2(1(x1)))))) 69.31/18.47 0(1(0(x1))) -> 0(2(1(3(0(2(x1)))))) 69.31/18.47 1(4(0(x1))) -> 1(3(0(4(x1)))) 69.31/18.47 1(4(0(x1))) -> 1(2(3(0(4(2(x1)))))) 69.31/18.47 1(4(0(x1))) -> 1(3(5(0(4(2(x1)))))) 69.31/18.47 1(4(0(x1))) -> 3(0(2(1(2(4(x1)))))) 69.31/18.47 0(1(4(x1))) -> 4(0(2(1(x1)))) 69.31/18.47 1(1(0(0(x1)))) -> 1(3(0(0(2(1(x1)))))) 69.31/18.47 1(2(0(0(x1)))) -> 1(0(3(0(2(2(x1)))))) 69.31/18.47 1(4(0(0(x1)))) -> 3(0(3(0(4(1(x1)))))) 69.31/18.47 1(5(0(0(x1)))) -> 5(2(1(3(0(0(x1)))))) 69.31/18.47 0(1(1(0(x1)))) -> 2(1(0(0(2(1(x1)))))) 69.31/18.47 1(4(1(0(x1)))) -> 2(1(4(2(1(0(x1)))))) 69.31/18.47 0(5(1(0(x1)))) -> 3(0(0(5(2(1(x1)))))) 69.31/18.47 1(4(2(0(x1)))) -> 2(1(3(5(0(4(x1)))))) 69.31/18.47 0(1(4(0(x1)))) -> 2(1(0(4(0(x1))))) 69.31/18.47 1(2(4(0(x1)))) -> 1(2(2(3(0(4(x1)))))) 69.31/18.47 1(2(4(0(x1)))) -> 2(1(5(3(0(4(x1)))))) 69.31/18.47 1(4(4(0(x1)))) -> 1(4(3(0(4(2(x1)))))) 69.31/18.47 1(0(5(0(x1)))) -> 1(5(3(0(2(0(x1)))))) 69.31/18.47 1(0(5(0(x1)))) -> 1(0(3(0(2(5(x1)))))) 69.31/18.47 0(1(5(0(x1)))) -> 5(3(0(2(1(0(x1)))))) 69.31/18.47 0(1(5(0(x1)))) -> 0(5(3(1(0(2(x1)))))) 69.31/18.47 1(5(0(4(x1)))) -> 2(3(5(0(4(1(x1)))))) 69.31/18.47 0(0(1(4(x1)))) -> 0(2(1(4(0(x1))))) 69.31/18.47 1(0(1(4(x1)))) -> 4(0(2(1(1(x1))))) 69.31/18.47 0(5(1(4(x1)))) -> 4(5(0(2(1(x1))))) 69.31/18.47 0(5(1(4(x1)))) -> 5(3(0(4(1(x1))))) 69.31/18.47 0(1(3(4(x1)))) -> 3(0(2(4(1(x1))))) 69.31/18.47 0(1(3(4(x1)))) -> 4(3(1(2(0(2(x1)))))) 69.31/18.47 0(1(3(4(x1)))) -> 3(0(2(4(1(2(x1)))))) 69.31/18.47 0(1(3(4(x1)))) -> 3(1(0(4(2(2(x1)))))) 69.31/18.47 0(1(0(5(x1)))) -> 0(2(1(5(3(0(x1)))))) 69.31/18.47 0(1(0(5(x1)))) -> 0(3(0(2(5(1(x1)))))) 69.31/18.47 0(1(4(5(x1)))) -> 2(5(4(2(1(0(x1)))))) 69.31/18.47 1(5(5(0(0(x1))))) -> 5(5(3(0(0(1(x1)))))) 69.31/18.47 0(5(0(1(0(x1))))) -> 0(0(2(5(1(0(x1)))))) 69.31/18.47 1(0(5(2(0(x1))))) -> 1(0(5(3(0(2(x1)))))) 69.31/18.47 0(0(1(3(0(x1))))) -> 2(1(0(3(0(0(x1)))))) 69.31/18.47 1(4(1(4(0(x1))))) -> 1(2(1(0(4(4(x1)))))) 69.31/18.47 1(4(5(5(0(x1))))) -> 3(5(5(0(1(4(x1)))))) 69.31/18.47 0(0(5(1(4(x1))))) -> 0(5(4(0(2(1(x1)))))) 69.31/18.47 0(5(5(1(4(x1))))) -> 4(5(0(3(1(5(x1)))))) 69.31/18.47 1(0(1(3(4(x1))))) -> 4(0(3(2(1(1(x1)))))) 69.31/18.47 0(1(1(3(4(x1))))) -> 3(4(1(0(2(1(x1)))))) 69.31/18.47 0(5(1(3(4(x1))))) -> 4(2(3(0(1(5(x1)))))) 69.31/18.47 5(0(1(4(4(x1))))) -> 1(4(3(5(0(4(x1)))))) 69.31/18.47 69.31/18.47 Q is empty. 69.31/18.47 We have to consider all minimal (P,Q,R)-chains. 69.31/18.47 ---------------------------------------- 69.31/18.47 69.31/18.47 (7) QDPOrderProof (EQUIVALENT) 69.31/18.47 We use the reduction pair processor [LPAR04,JAR06]. 69.31/18.47 69.31/18.47 69.31/18.47 The following pairs can be oriented strictly and are deleted. 69.31/18.47 69.31/18.47 1^1(0(0(x1))) -> 1^1(x1) 69.31/18.47 1^1(0(0(x1))) -> 1^1(0(x1)) 69.31/18.47 1^1(1(0(0(x1)))) -> 1^1(x1) 69.31/18.47 1^1(4(0(0(x1)))) -> 1^1(x1) 69.31/18.47 1^1(5(0(4(x1)))) -> 1^1(x1) 69.31/18.47 1^1(0(1(4(x1)))) -> 1^1(1(x1)) 69.31/18.47 1^1(0(1(4(x1)))) -> 1^1(x1) 69.31/18.47 0^1(1(4(x1))) -> 1^1(x1) 69.31/18.47 1^1(5(5(0(0(x1))))) -> 0^1(1(x1)) 69.31/18.47 0^1(1(1(0(x1)))) -> 1^1(x1) 69.31/18.47 1^1(5(5(0(0(x1))))) -> 1^1(x1) 69.31/18.47 0^1(5(1(0(x1)))) -> 1^1(x1) 69.31/18.47 1^1(4(5(5(0(x1))))) -> 1^1(4(x1)) 69.31/18.47 1^1(0(1(3(4(x1))))) -> 1^1(1(x1)) 69.31/18.47 1^1(0(1(3(4(x1))))) -> 1^1(x1) 69.31/18.47 0^1(1(5(0(x1)))) -> 1^1(0(x1)) 69.31/18.47 0^1(0(1(4(x1)))) -> 1^1(4(0(x1))) 69.31/18.47 0^1(0(1(4(x1)))) -> 0^1(x1) 69.31/18.47 0^1(5(1(4(x1)))) -> 1^1(x1) 69.31/18.47 0^1(1(3(4(x1)))) -> 1^1(x1) 69.31/18.47 0^1(1(0(5(x1)))) -> 0^1(x1) 69.31/18.47 0^1(1(0(5(x1)))) -> 1^1(x1) 69.31/18.47 0^1(1(4(5(x1)))) -> 0^1(x1) 69.31/18.47 0^1(0(1(3(0(x1))))) -> 0^1(0(x1)) 69.31/18.47 0^1(0(5(1(4(x1))))) -> 1^1(x1) 69.31/18.47 0^1(5(5(1(4(x1))))) -> 1^1(5(x1)) 69.31/18.47 0^1(1(1(3(4(x1))))) -> 1^1(x1) 69.31/18.47 0^1(5(1(3(4(x1))))) -> 1^1(5(x1)) 69.31/18.47 The remaining pairs can at least be oriented weakly. 69.31/18.47 Used ordering: Polynomial interpretation [POLO]: 69.31/18.47 69.31/18.47 POL(0(x_1)) = 1 + x_1 69.31/18.47 POL(0^1(x_1)) = 1 + x_1 69.31/18.47 POL(1(x_1)) = 1 + x_1 69.31/18.47 POL(1^1(x_1)) = 1 + x_1 69.31/18.47 POL(2(x_1)) = x_1 69.31/18.47 POL(3(x_1)) = x_1 69.31/18.47 POL(4(x_1)) = x_1 69.31/18.47 POL(5(x_1)) = x_1 69.31/18.47 69.31/18.47 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 69.31/18.47 69.31/18.47 0(1(0(x1))) -> 0(2(1(3(0(2(x1)))))) 69.31/18.47 0(1(4(x1))) -> 4(0(2(1(x1)))) 69.31/18.47 0(1(1(0(x1)))) -> 2(1(0(0(2(1(x1)))))) 69.31/18.47 0(5(1(0(x1)))) -> 3(0(0(5(2(1(x1)))))) 69.31/18.47 0(1(4(0(x1)))) -> 2(1(0(4(0(x1))))) 69.31/18.47 0(1(5(0(x1)))) -> 5(3(0(2(1(0(x1)))))) 69.31/18.47 0(1(5(0(x1)))) -> 0(5(3(1(0(2(x1)))))) 69.31/18.47 0(0(1(4(x1)))) -> 0(2(1(4(0(x1))))) 69.31/18.47 0(5(1(4(x1)))) -> 4(5(0(2(1(x1))))) 69.31/18.47 0(5(1(4(x1)))) -> 5(3(0(4(1(x1))))) 69.31/18.47 0(1(3(4(x1)))) -> 3(0(2(4(1(x1))))) 69.31/18.47 0(1(3(4(x1)))) -> 4(3(1(2(0(2(x1)))))) 69.31/18.47 0(1(3(4(x1)))) -> 3(0(2(4(1(2(x1)))))) 69.31/18.47 0(1(3(4(x1)))) -> 3(1(0(4(2(2(x1)))))) 69.31/18.47 0(1(0(5(x1)))) -> 0(2(1(5(3(0(x1)))))) 69.31/18.47 0(1(0(5(x1)))) -> 0(3(0(2(5(1(x1)))))) 69.31/18.47 0(1(4(5(x1)))) -> 2(5(4(2(1(0(x1)))))) 69.31/18.47 0(5(0(1(0(x1))))) -> 0(0(2(5(1(0(x1)))))) 69.31/18.47 0(0(1(3(0(x1))))) -> 2(1(0(3(0(0(x1)))))) 69.31/18.47 0(0(5(1(4(x1))))) -> 0(5(4(0(2(1(x1)))))) 69.31/18.47 0(5(5(1(4(x1))))) -> 4(5(0(3(1(5(x1)))))) 69.31/18.47 0(1(1(3(4(x1))))) -> 3(4(1(0(2(1(x1)))))) 69.31/18.47 0(5(1(3(4(x1))))) -> 4(2(3(0(1(5(x1)))))) 69.31/18.47 1(0(0(x1))) -> 3(0(2(1(0(x1))))) 69.31/18.47 1(0(0(x1))) -> 2(3(0(2(1(0(x1)))))) 69.31/18.47 1(0(0(x1))) -> 2(3(0(0(2(1(x1)))))) 69.31/18.47 1(4(0(x1))) -> 1(3(0(4(x1)))) 69.31/18.47 1(4(0(x1))) -> 1(2(3(0(4(2(x1)))))) 69.31/18.47 1(4(0(x1))) -> 1(3(5(0(4(2(x1)))))) 69.31/18.47 1(4(0(x1))) -> 3(0(2(1(2(4(x1)))))) 69.31/18.47 1(1(0(0(x1)))) -> 1(3(0(0(2(1(x1)))))) 69.31/18.47 1(2(0(0(x1)))) -> 1(0(3(0(2(2(x1)))))) 69.31/18.47 1(4(0(0(x1)))) -> 3(0(3(0(4(1(x1)))))) 69.31/18.47 1(5(0(0(x1)))) -> 5(2(1(3(0(0(x1)))))) 69.31/18.47 1(4(1(0(x1)))) -> 2(1(4(2(1(0(x1)))))) 69.31/18.47 1(4(2(0(x1)))) -> 2(1(3(5(0(4(x1)))))) 69.31/18.48 1(2(4(0(x1)))) -> 1(2(2(3(0(4(x1)))))) 69.31/18.48 1(2(4(0(x1)))) -> 2(1(5(3(0(4(x1)))))) 69.31/18.48 1(4(4(0(x1)))) -> 1(4(3(0(4(2(x1)))))) 69.31/18.48 1(0(5(0(x1)))) -> 1(5(3(0(2(0(x1)))))) 69.31/18.48 1(0(5(0(x1)))) -> 1(0(3(0(2(5(x1)))))) 69.31/18.48 1(5(0(4(x1)))) -> 2(3(5(0(4(1(x1)))))) 69.31/18.48 1(0(1(4(x1)))) -> 4(0(2(1(1(x1))))) 69.31/18.48 1(5(5(0(0(x1))))) -> 5(5(3(0(0(1(x1)))))) 69.31/18.48 1(0(5(2(0(x1))))) -> 1(0(5(3(0(2(x1)))))) 69.31/18.48 1(4(1(4(0(x1))))) -> 1(2(1(0(4(4(x1)))))) 69.31/18.48 1(4(5(5(0(x1))))) -> 3(5(5(0(1(4(x1)))))) 69.31/18.48 1(0(1(3(4(x1))))) -> 4(0(3(2(1(1(x1)))))) 69.31/18.48 5(0(1(4(4(x1))))) -> 1(4(3(5(0(4(x1)))))) 69.31/18.48 69.31/18.48 69.31/18.48 ---------------------------------------- 69.31/18.48 69.31/18.48 (8) 69.31/18.48 Obligation: 69.31/18.48 Q DP problem: 69.31/18.48 The TRS P consists of the following rules: 69.31/18.48 69.31/18.48 1^1(5(5(0(0(x1))))) -> 0^1(0(1(x1))) 69.31/18.48 0^1(1(1(0(x1)))) -> 1^1(0(0(2(1(x1))))) 69.31/18.48 1^1(4(5(5(0(x1))))) -> 0^1(1(4(x1))) 69.31/18.48 0^1(1(4(5(x1)))) -> 1^1(0(x1)) 69.31/18.48 0^1(5(1(3(4(x1))))) -> 0^1(1(5(x1))) 69.31/18.48 69.31/18.48 The TRS R consists of the following rules: 69.31/18.48 69.31/18.48 1(0(0(x1))) -> 3(0(2(1(0(x1))))) 69.31/18.48 1(0(0(x1))) -> 2(3(0(2(1(0(x1)))))) 69.31/18.48 1(0(0(x1))) -> 2(3(0(0(2(1(x1)))))) 69.31/18.48 0(1(0(x1))) -> 0(2(1(3(0(2(x1)))))) 69.31/18.48 1(4(0(x1))) -> 1(3(0(4(x1)))) 69.31/18.48 1(4(0(x1))) -> 1(2(3(0(4(2(x1)))))) 69.31/18.48 1(4(0(x1))) -> 1(3(5(0(4(2(x1)))))) 69.31/18.48 1(4(0(x1))) -> 3(0(2(1(2(4(x1)))))) 69.31/18.48 0(1(4(x1))) -> 4(0(2(1(x1)))) 69.31/18.48 1(1(0(0(x1)))) -> 1(3(0(0(2(1(x1)))))) 69.31/18.48 1(2(0(0(x1)))) -> 1(0(3(0(2(2(x1)))))) 69.31/18.48 1(4(0(0(x1)))) -> 3(0(3(0(4(1(x1)))))) 69.31/18.48 1(5(0(0(x1)))) -> 5(2(1(3(0(0(x1)))))) 69.31/18.48 0(1(1(0(x1)))) -> 2(1(0(0(2(1(x1)))))) 69.31/18.48 1(4(1(0(x1)))) -> 2(1(4(2(1(0(x1)))))) 69.31/18.48 0(5(1(0(x1)))) -> 3(0(0(5(2(1(x1)))))) 69.31/18.48 1(4(2(0(x1)))) -> 2(1(3(5(0(4(x1)))))) 69.31/18.48 0(1(4(0(x1)))) -> 2(1(0(4(0(x1))))) 69.31/18.48 1(2(4(0(x1)))) -> 1(2(2(3(0(4(x1)))))) 69.31/18.48 1(2(4(0(x1)))) -> 2(1(5(3(0(4(x1)))))) 69.31/18.48 1(4(4(0(x1)))) -> 1(4(3(0(4(2(x1)))))) 69.31/18.48 1(0(5(0(x1)))) -> 1(5(3(0(2(0(x1)))))) 69.31/18.48 1(0(5(0(x1)))) -> 1(0(3(0(2(5(x1)))))) 69.31/18.48 0(1(5(0(x1)))) -> 5(3(0(2(1(0(x1)))))) 69.31/18.48 0(1(5(0(x1)))) -> 0(5(3(1(0(2(x1)))))) 69.31/18.48 1(5(0(4(x1)))) -> 2(3(5(0(4(1(x1)))))) 69.31/18.48 0(0(1(4(x1)))) -> 0(2(1(4(0(x1))))) 69.31/18.48 1(0(1(4(x1)))) -> 4(0(2(1(1(x1))))) 69.31/18.48 0(5(1(4(x1)))) -> 4(5(0(2(1(x1))))) 69.31/18.48 0(5(1(4(x1)))) -> 5(3(0(4(1(x1))))) 69.31/18.48 0(1(3(4(x1)))) -> 3(0(2(4(1(x1))))) 69.31/18.48 0(1(3(4(x1)))) -> 4(3(1(2(0(2(x1)))))) 69.31/18.48 0(1(3(4(x1)))) -> 3(0(2(4(1(2(x1)))))) 69.31/18.48 0(1(3(4(x1)))) -> 3(1(0(4(2(2(x1)))))) 69.31/18.48 0(1(0(5(x1)))) -> 0(2(1(5(3(0(x1)))))) 69.31/18.48 0(1(0(5(x1)))) -> 0(3(0(2(5(1(x1)))))) 69.31/18.48 0(1(4(5(x1)))) -> 2(5(4(2(1(0(x1)))))) 69.31/18.48 1(5(5(0(0(x1))))) -> 5(5(3(0(0(1(x1)))))) 69.31/18.48 0(5(0(1(0(x1))))) -> 0(0(2(5(1(0(x1)))))) 69.31/18.48 1(0(5(2(0(x1))))) -> 1(0(5(3(0(2(x1)))))) 69.31/18.48 0(0(1(3(0(x1))))) -> 2(1(0(3(0(0(x1)))))) 69.31/18.48 1(4(1(4(0(x1))))) -> 1(2(1(0(4(4(x1)))))) 69.31/18.48 1(4(5(5(0(x1))))) -> 3(5(5(0(1(4(x1)))))) 69.31/18.48 0(0(5(1(4(x1))))) -> 0(5(4(0(2(1(x1)))))) 69.31/18.48 0(5(5(1(4(x1))))) -> 4(5(0(3(1(5(x1)))))) 69.31/18.48 1(0(1(3(4(x1))))) -> 4(0(3(2(1(1(x1)))))) 69.31/18.48 0(1(1(3(4(x1))))) -> 3(4(1(0(2(1(x1)))))) 69.31/18.48 0(5(1(3(4(x1))))) -> 4(2(3(0(1(5(x1)))))) 69.31/18.48 5(0(1(4(4(x1))))) -> 1(4(3(5(0(4(x1)))))) 69.31/18.48 69.31/18.48 Q is empty. 69.31/18.48 We have to consider all minimal (P,Q,R)-chains. 69.31/18.48 ---------------------------------------- 69.31/18.48 69.31/18.48 (9) DependencyGraphProof (EQUIVALENT) 69.31/18.48 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 69.31/18.48 ---------------------------------------- 69.31/18.48 69.31/18.48 (10) 69.31/18.48 Obligation: 69.31/18.48 Q DP problem: 69.31/18.48 The TRS P consists of the following rules: 69.31/18.48 69.31/18.48 0^1(1(4(5(x1)))) -> 1^1(0(x1)) 69.31/18.48 1^1(5(5(0(0(x1))))) -> 0^1(0(1(x1))) 69.31/18.48 0^1(5(1(3(4(x1))))) -> 0^1(1(5(x1))) 69.31/18.48 1^1(4(5(5(0(x1))))) -> 0^1(1(4(x1))) 69.31/18.48 69.31/18.48 The TRS R consists of the following rules: 69.31/18.48 69.31/18.48 1(0(0(x1))) -> 3(0(2(1(0(x1))))) 69.31/18.48 1(0(0(x1))) -> 2(3(0(2(1(0(x1)))))) 69.31/18.48 1(0(0(x1))) -> 2(3(0(0(2(1(x1)))))) 69.31/18.48 0(1(0(x1))) -> 0(2(1(3(0(2(x1)))))) 69.31/18.48 1(4(0(x1))) -> 1(3(0(4(x1)))) 69.31/18.48 1(4(0(x1))) -> 1(2(3(0(4(2(x1)))))) 69.31/18.48 1(4(0(x1))) -> 1(3(5(0(4(2(x1)))))) 69.31/18.48 1(4(0(x1))) -> 3(0(2(1(2(4(x1)))))) 69.31/18.48 0(1(4(x1))) -> 4(0(2(1(x1)))) 69.31/18.48 1(1(0(0(x1)))) -> 1(3(0(0(2(1(x1)))))) 69.31/18.48 1(2(0(0(x1)))) -> 1(0(3(0(2(2(x1)))))) 69.31/18.48 1(4(0(0(x1)))) -> 3(0(3(0(4(1(x1)))))) 69.31/18.48 1(5(0(0(x1)))) -> 5(2(1(3(0(0(x1)))))) 69.31/18.48 0(1(1(0(x1)))) -> 2(1(0(0(2(1(x1)))))) 69.31/18.48 1(4(1(0(x1)))) -> 2(1(4(2(1(0(x1)))))) 69.31/18.48 0(5(1(0(x1)))) -> 3(0(0(5(2(1(x1)))))) 69.31/18.48 1(4(2(0(x1)))) -> 2(1(3(5(0(4(x1)))))) 69.31/18.48 0(1(4(0(x1)))) -> 2(1(0(4(0(x1))))) 69.31/18.48 1(2(4(0(x1)))) -> 1(2(2(3(0(4(x1)))))) 69.31/18.48 1(2(4(0(x1)))) -> 2(1(5(3(0(4(x1)))))) 69.31/18.48 1(4(4(0(x1)))) -> 1(4(3(0(4(2(x1)))))) 69.31/18.48 1(0(5(0(x1)))) -> 1(5(3(0(2(0(x1)))))) 69.31/18.48 1(0(5(0(x1)))) -> 1(0(3(0(2(5(x1)))))) 69.31/18.48 0(1(5(0(x1)))) -> 5(3(0(2(1(0(x1)))))) 69.31/18.48 0(1(5(0(x1)))) -> 0(5(3(1(0(2(x1)))))) 69.31/18.48 1(5(0(4(x1)))) -> 2(3(5(0(4(1(x1)))))) 69.31/18.48 0(0(1(4(x1)))) -> 0(2(1(4(0(x1))))) 69.31/18.48 1(0(1(4(x1)))) -> 4(0(2(1(1(x1))))) 69.31/18.48 0(5(1(4(x1)))) -> 4(5(0(2(1(x1))))) 69.31/18.48 0(5(1(4(x1)))) -> 5(3(0(4(1(x1))))) 69.31/18.48 0(1(3(4(x1)))) -> 3(0(2(4(1(x1))))) 69.31/18.48 0(1(3(4(x1)))) -> 4(3(1(2(0(2(x1)))))) 69.31/18.48 0(1(3(4(x1)))) -> 3(0(2(4(1(2(x1)))))) 69.31/18.48 0(1(3(4(x1)))) -> 3(1(0(4(2(2(x1)))))) 69.31/18.48 0(1(0(5(x1)))) -> 0(2(1(5(3(0(x1)))))) 69.31/18.48 0(1(0(5(x1)))) -> 0(3(0(2(5(1(x1)))))) 69.31/18.48 0(1(4(5(x1)))) -> 2(5(4(2(1(0(x1)))))) 69.31/18.48 1(5(5(0(0(x1))))) -> 5(5(3(0(0(1(x1)))))) 69.31/18.48 0(5(0(1(0(x1))))) -> 0(0(2(5(1(0(x1)))))) 69.31/18.48 1(0(5(2(0(x1))))) -> 1(0(5(3(0(2(x1)))))) 69.31/18.48 0(0(1(3(0(x1))))) -> 2(1(0(3(0(0(x1)))))) 69.31/18.48 1(4(1(4(0(x1))))) -> 1(2(1(0(4(4(x1)))))) 69.31/18.48 1(4(5(5(0(x1))))) -> 3(5(5(0(1(4(x1)))))) 69.31/18.48 0(0(5(1(4(x1))))) -> 0(5(4(0(2(1(x1)))))) 69.31/18.48 0(5(5(1(4(x1))))) -> 4(5(0(3(1(5(x1)))))) 69.31/18.48 1(0(1(3(4(x1))))) -> 4(0(3(2(1(1(x1)))))) 69.31/18.48 0(1(1(3(4(x1))))) -> 3(4(1(0(2(1(x1)))))) 69.31/18.48 0(5(1(3(4(x1))))) -> 4(2(3(0(1(5(x1)))))) 69.31/18.48 5(0(1(4(4(x1))))) -> 1(4(3(5(0(4(x1)))))) 69.31/18.48 69.31/18.48 Q is empty. 69.31/18.48 We have to consider all minimal (P,Q,R)-chains. 69.31/18.48 ---------------------------------------- 69.31/18.48 69.31/18.48 (11) QDPOrderProof (EQUIVALENT) 69.31/18.48 We use the reduction pair processor [LPAR04,JAR06]. 69.31/18.48 69.31/18.48 69.31/18.48 The following pairs can be oriented strictly and are deleted. 69.31/18.48 69.31/18.48 0^1(1(4(5(x1)))) -> 1^1(0(x1)) 69.31/18.48 0^1(5(1(3(4(x1))))) -> 0^1(1(5(x1))) 69.31/18.48 The remaining pairs can at least be oriented weakly. 69.31/18.48 Used ordering: Polynomial interpretation [POLO]: 69.31/18.48 69.31/18.48 POL(0(x_1)) = x_1 69.31/18.48 POL(0^1(x_1)) = 1 + x_1 69.31/18.48 POL(1(x_1)) = x_1 69.31/18.48 POL(1^1(x_1)) = 1 + x_1 69.31/18.48 POL(2(x_1)) = 0 69.31/18.48 POL(3(x_1)) = x_1 69.31/18.48 POL(4(x_1)) = 1 + x_1 69.31/18.48 POL(5(x_1)) = x_1 69.31/18.48 69.31/18.48 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 69.31/18.48 69.31/18.48 0(1(0(x1))) -> 0(2(1(3(0(2(x1)))))) 69.31/18.48 0(1(4(x1))) -> 4(0(2(1(x1)))) 69.31/18.48 0(1(1(0(x1)))) -> 2(1(0(0(2(1(x1)))))) 69.31/18.48 0(5(1(0(x1)))) -> 3(0(0(5(2(1(x1)))))) 69.31/18.48 0(1(4(0(x1)))) -> 2(1(0(4(0(x1))))) 69.31/18.48 0(1(5(0(x1)))) -> 5(3(0(2(1(0(x1)))))) 69.31/18.48 0(1(5(0(x1)))) -> 0(5(3(1(0(2(x1)))))) 69.31/18.48 0(0(1(4(x1)))) -> 0(2(1(4(0(x1))))) 69.31/18.48 0(5(1(4(x1)))) -> 4(5(0(2(1(x1))))) 69.31/18.48 0(5(1(4(x1)))) -> 5(3(0(4(1(x1))))) 69.31/18.48 0(1(3(4(x1)))) -> 3(0(2(4(1(x1))))) 69.31/18.48 0(1(3(4(x1)))) -> 4(3(1(2(0(2(x1)))))) 69.31/18.48 0(1(3(4(x1)))) -> 3(0(2(4(1(2(x1)))))) 69.31/18.48 0(1(3(4(x1)))) -> 3(1(0(4(2(2(x1)))))) 69.31/18.48 0(1(0(5(x1)))) -> 0(2(1(5(3(0(x1)))))) 69.31/18.48 0(1(0(5(x1)))) -> 0(3(0(2(5(1(x1)))))) 69.31/18.48 0(1(4(5(x1)))) -> 2(5(4(2(1(0(x1)))))) 69.31/18.48 0(5(0(1(0(x1))))) -> 0(0(2(5(1(0(x1)))))) 69.31/18.48 0(0(1(3(0(x1))))) -> 2(1(0(3(0(0(x1)))))) 69.31/18.48 0(0(5(1(4(x1))))) -> 0(5(4(0(2(1(x1)))))) 69.31/18.48 0(5(5(1(4(x1))))) -> 4(5(0(3(1(5(x1)))))) 69.31/18.48 0(1(1(3(4(x1))))) -> 3(4(1(0(2(1(x1)))))) 69.31/18.48 0(5(1(3(4(x1))))) -> 4(2(3(0(1(5(x1)))))) 69.31/18.48 1(0(0(x1))) -> 3(0(2(1(0(x1))))) 69.31/18.48 1(0(0(x1))) -> 2(3(0(2(1(0(x1)))))) 69.31/18.48 1(0(0(x1))) -> 2(3(0(0(2(1(x1)))))) 69.31/18.48 1(4(0(x1))) -> 1(3(0(4(x1)))) 69.31/18.48 1(4(0(x1))) -> 1(2(3(0(4(2(x1)))))) 69.31/18.48 1(4(0(x1))) -> 1(3(5(0(4(2(x1)))))) 69.31/18.48 1(4(0(x1))) -> 3(0(2(1(2(4(x1)))))) 69.31/18.48 1(1(0(0(x1)))) -> 1(3(0(0(2(1(x1)))))) 69.31/18.48 1(2(0(0(x1)))) -> 1(0(3(0(2(2(x1)))))) 69.31/18.48 1(4(0(0(x1)))) -> 3(0(3(0(4(1(x1)))))) 69.31/18.48 1(5(0(0(x1)))) -> 5(2(1(3(0(0(x1)))))) 69.31/18.48 1(4(1(0(x1)))) -> 2(1(4(2(1(0(x1)))))) 69.31/18.48 1(4(2(0(x1)))) -> 2(1(3(5(0(4(x1)))))) 69.31/18.48 1(2(4(0(x1)))) -> 1(2(2(3(0(4(x1)))))) 69.31/18.48 1(2(4(0(x1)))) -> 2(1(5(3(0(4(x1)))))) 69.31/18.48 1(4(4(0(x1)))) -> 1(4(3(0(4(2(x1)))))) 69.31/18.48 1(0(5(0(x1)))) -> 1(5(3(0(2(0(x1)))))) 69.31/18.48 1(0(5(0(x1)))) -> 1(0(3(0(2(5(x1)))))) 69.31/18.48 1(5(0(4(x1)))) -> 2(3(5(0(4(1(x1)))))) 69.31/18.48 1(0(1(4(x1)))) -> 4(0(2(1(1(x1))))) 69.31/18.48 1(5(5(0(0(x1))))) -> 5(5(3(0(0(1(x1)))))) 69.31/18.48 1(0(5(2(0(x1))))) -> 1(0(5(3(0(2(x1)))))) 69.31/18.48 1(4(1(4(0(x1))))) -> 1(2(1(0(4(4(x1)))))) 69.31/18.48 1(4(5(5(0(x1))))) -> 3(5(5(0(1(4(x1)))))) 69.31/18.48 1(0(1(3(4(x1))))) -> 4(0(3(2(1(1(x1)))))) 69.31/18.48 5(0(1(4(4(x1))))) -> 1(4(3(5(0(4(x1)))))) 69.31/18.48 69.31/18.48 69.31/18.48 ---------------------------------------- 69.31/18.48 69.31/18.48 (12) 69.31/18.48 Obligation: 69.31/18.48 Q DP problem: 69.31/18.48 The TRS P consists of the following rules: 69.31/18.48 69.31/18.48 1^1(5(5(0(0(x1))))) -> 0^1(0(1(x1))) 69.31/18.48 1^1(4(5(5(0(x1))))) -> 0^1(1(4(x1))) 69.31/18.48 69.31/18.48 The TRS R consists of the following rules: 69.31/18.48 69.31/18.48 1(0(0(x1))) -> 3(0(2(1(0(x1))))) 69.31/18.48 1(0(0(x1))) -> 2(3(0(2(1(0(x1)))))) 69.31/18.48 1(0(0(x1))) -> 2(3(0(0(2(1(x1)))))) 69.31/18.48 0(1(0(x1))) -> 0(2(1(3(0(2(x1)))))) 69.31/18.48 1(4(0(x1))) -> 1(3(0(4(x1)))) 69.31/18.48 1(4(0(x1))) -> 1(2(3(0(4(2(x1)))))) 69.31/18.48 1(4(0(x1))) -> 1(3(5(0(4(2(x1)))))) 69.31/18.48 1(4(0(x1))) -> 3(0(2(1(2(4(x1)))))) 69.31/18.48 0(1(4(x1))) -> 4(0(2(1(x1)))) 69.31/18.48 1(1(0(0(x1)))) -> 1(3(0(0(2(1(x1)))))) 69.31/18.48 1(2(0(0(x1)))) -> 1(0(3(0(2(2(x1)))))) 69.31/18.48 1(4(0(0(x1)))) -> 3(0(3(0(4(1(x1)))))) 69.31/18.48 1(5(0(0(x1)))) -> 5(2(1(3(0(0(x1)))))) 69.31/18.48 0(1(1(0(x1)))) -> 2(1(0(0(2(1(x1)))))) 69.31/18.48 1(4(1(0(x1)))) -> 2(1(4(2(1(0(x1)))))) 69.31/18.48 0(5(1(0(x1)))) -> 3(0(0(5(2(1(x1)))))) 69.31/18.48 1(4(2(0(x1)))) -> 2(1(3(5(0(4(x1)))))) 69.31/18.48 0(1(4(0(x1)))) -> 2(1(0(4(0(x1))))) 69.31/18.48 1(2(4(0(x1)))) -> 1(2(2(3(0(4(x1)))))) 69.31/18.48 1(2(4(0(x1)))) -> 2(1(5(3(0(4(x1)))))) 69.31/18.48 1(4(4(0(x1)))) -> 1(4(3(0(4(2(x1)))))) 69.31/18.48 1(0(5(0(x1)))) -> 1(5(3(0(2(0(x1)))))) 69.31/18.48 1(0(5(0(x1)))) -> 1(0(3(0(2(5(x1)))))) 69.31/18.48 0(1(5(0(x1)))) -> 5(3(0(2(1(0(x1)))))) 69.31/18.48 0(1(5(0(x1)))) -> 0(5(3(1(0(2(x1)))))) 69.31/18.48 1(5(0(4(x1)))) -> 2(3(5(0(4(1(x1)))))) 69.31/18.48 0(0(1(4(x1)))) -> 0(2(1(4(0(x1))))) 69.31/18.48 1(0(1(4(x1)))) -> 4(0(2(1(1(x1))))) 69.31/18.48 0(5(1(4(x1)))) -> 4(5(0(2(1(x1))))) 69.31/18.48 0(5(1(4(x1)))) -> 5(3(0(4(1(x1))))) 69.31/18.48 0(1(3(4(x1)))) -> 3(0(2(4(1(x1))))) 69.31/18.48 0(1(3(4(x1)))) -> 4(3(1(2(0(2(x1)))))) 69.31/18.48 0(1(3(4(x1)))) -> 3(0(2(4(1(2(x1)))))) 69.31/18.48 0(1(3(4(x1)))) -> 3(1(0(4(2(2(x1)))))) 69.31/18.48 0(1(0(5(x1)))) -> 0(2(1(5(3(0(x1)))))) 69.31/18.48 0(1(0(5(x1)))) -> 0(3(0(2(5(1(x1)))))) 69.31/18.48 0(1(4(5(x1)))) -> 2(5(4(2(1(0(x1)))))) 69.31/18.48 1(5(5(0(0(x1))))) -> 5(5(3(0(0(1(x1)))))) 69.31/18.48 0(5(0(1(0(x1))))) -> 0(0(2(5(1(0(x1)))))) 69.31/18.48 1(0(5(2(0(x1))))) -> 1(0(5(3(0(2(x1)))))) 69.31/18.48 0(0(1(3(0(x1))))) -> 2(1(0(3(0(0(x1)))))) 69.31/18.48 1(4(1(4(0(x1))))) -> 1(2(1(0(4(4(x1)))))) 69.31/18.48 1(4(5(5(0(x1))))) -> 3(5(5(0(1(4(x1)))))) 69.31/18.48 0(0(5(1(4(x1))))) -> 0(5(4(0(2(1(x1)))))) 69.31/18.48 0(5(5(1(4(x1))))) -> 4(5(0(3(1(5(x1)))))) 69.31/18.48 1(0(1(3(4(x1))))) -> 4(0(3(2(1(1(x1)))))) 69.31/18.48 0(1(1(3(4(x1))))) -> 3(4(1(0(2(1(x1)))))) 69.31/18.48 0(5(1(3(4(x1))))) -> 4(2(3(0(1(5(x1)))))) 69.31/18.48 5(0(1(4(4(x1))))) -> 1(4(3(5(0(4(x1)))))) 69.31/18.48 69.31/18.48 Q is empty. 69.31/18.48 We have to consider all minimal (P,Q,R)-chains. 69.31/18.48 ---------------------------------------- 69.31/18.48 69.31/18.48 (13) DependencyGraphProof (EQUIVALENT) 69.31/18.48 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 69.31/18.48 ---------------------------------------- 69.31/18.48 69.31/18.48 (14) 69.31/18.48 TRUE 69.60/18.55 EOF