36.83/10.37 YES 38.05/10.67 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 38.05/10.67 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 38.05/10.67 38.05/10.67 38.05/10.67 Termination w.r.t. Q of the given QTRS could be proven: 38.05/10.67 38.05/10.67 (0) QTRS 38.05/10.67 (1) DependencyPairsProof [EQUIVALENT, 166 ms] 38.05/10.67 (2) QDP 38.05/10.67 (3) DependencyGraphProof [EQUIVALENT, 1 ms] 38.05/10.67 (4) QDP 38.05/10.67 (5) QDPOrderProof [EQUIVALENT, 258 ms] 38.05/10.67 (6) QDP 38.05/10.67 (7) DependencyGraphProof [EQUIVALENT, 0 ms] 38.05/10.67 (8) QDP 38.05/10.67 (9) UsableRulesProof [EQUIVALENT, 0 ms] 38.05/10.67 (10) QDP 38.05/10.67 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 38.05/10.67 (12) YES 38.05/10.67 38.05/10.67 38.05/10.67 ---------------------------------------- 38.05/10.67 38.05/10.67 (0) 38.05/10.67 Obligation: 38.05/10.67 Q restricted rewrite system: 38.05/10.67 The TRS R consists of the following rules: 38.05/10.67 38.05/10.67 0(1(0(2(x1)))) -> 0(0(3(1(2(x1))))) 38.05/10.67 0(1(3(4(x1)))) -> 0(4(1(0(3(x1))))) 38.05/10.67 0(1(3(4(x1)))) -> 0(4(1(1(3(x1))))) 38.05/10.67 0(1(3(4(x1)))) -> 0(4(1(3(1(x1))))) 38.05/10.67 0(2(1(4(x1)))) -> 0(4(1(2(3(x1))))) 38.05/10.67 0(2(1(4(x1)))) -> 0(4(1(3(2(x1))))) 38.05/10.67 0(2(1(4(x1)))) -> 2(0(4(1(4(x1))))) 38.05/10.67 0(2(1(4(x1)))) -> 5(5(0(4(1(2(x1)))))) 38.05/10.67 0(2(1(5(x1)))) -> 5(0(4(1(2(x1))))) 38.05/10.67 0(2(2(4(x1)))) -> 0(4(2(2(5(x1))))) 38.05/10.67 0(2(2(4(x1)))) -> 0(4(2(5(2(x1))))) 38.05/10.67 3(4(0(2(x1)))) -> 3(0(4(5(2(x1))))) 38.05/10.67 3(4(0(2(x1)))) -> 3(5(0(4(2(x1))))) 38.05/10.67 0(0(1(4(5(x1))))) -> 0(4(1(0(3(5(x1)))))) 38.05/10.67 0(1(0(2(4(x1))))) -> 2(0(0(4(1(1(x1)))))) 38.05/10.67 0(1(2(3(4(x1))))) -> 2(0(4(1(0(3(x1)))))) 38.05/10.67 0(1(3(3(4(x1))))) -> 0(0(3(1(3(4(x1)))))) 38.05/10.67 0(1(4(0(2(x1))))) -> 0(4(1(5(0(2(x1)))))) 38.05/10.67 0(1(4(1(5(x1))))) -> 2(5(0(4(1(1(x1)))))) 38.05/10.67 0(1(4(3(4(x1))))) -> 0(4(0(3(1(4(x1)))))) 38.05/10.67 0(1(4(3(4(x1))))) -> 3(0(4(1(5(4(x1)))))) 38.05/10.67 0(1(4(3(5(x1))))) -> 5(4(5(0(3(1(x1)))))) 38.05/10.67 0(1(5(0(2(x1))))) -> 0(0(4(1(2(5(x1)))))) 38.05/10.67 0(1(5(1(4(x1))))) -> 4(5(0(3(1(1(x1)))))) 38.05/10.67 0(2(1(4(4(x1))))) -> 0(4(1(2(4(3(x1)))))) 38.05/10.67 0(2(1(4(5(x1))))) -> 0(4(1(2(5(2(x1)))))) 38.05/10.67 0(2(1(5(4(x1))))) -> 5(0(2(0(4(1(x1)))))) 38.05/10.67 0(2(4(1(5(x1))))) -> 5(0(4(1(5(2(x1)))))) 38.05/10.67 0(2(4(3(5(x1))))) -> 0(4(5(2(5(3(x1)))))) 38.05/10.67 0(2(5(1(4(x1))))) -> 0(0(5(4(1(2(x1)))))) 38.05/10.67 3(0(1(3(2(x1))))) -> 0(3(1(0(3(2(x1)))))) 38.05/10.67 3(0(2(1(4(x1))))) -> 4(0(4(1(3(2(x1)))))) 38.05/10.67 3(0(2(1(5(x1))))) -> 5(3(2(0(4(1(x1)))))) 38.05/10.67 3(0(4(0(2(x1))))) -> 0(3(4(0(4(2(x1)))))) 38.05/10.67 3(0(4(0(2(x1))))) -> 0(4(1(2(0(3(x1)))))) 38.05/10.67 3(0(5(1(4(x1))))) -> 3(0(4(1(1(5(x1)))))) 38.05/10.67 3(0(5(1(5(x1))))) -> 0(4(1(3(5(5(x1)))))) 38.05/10.67 3(2(4(1(2(x1))))) -> 3(1(2(2(5(4(x1)))))) 38.05/10.67 3(2(4(1(5(x1))))) -> 3(1(4(5(2(5(x1)))))) 38.05/10.67 3(4(0(1(2(x1))))) -> 0(4(2(0(3(1(x1)))))) 38.05/10.67 3(4(0(1(4(x1))))) -> 0(4(1(5(3(4(x1)))))) 38.05/10.67 3(4(0(1(5(x1))))) -> 0(4(1(5(5(3(x1)))))) 38.05/10.67 3(4(0(2(4(x1))))) -> 0(3(4(0(4(2(x1)))))) 38.05/10.67 3(4(1(2(4(x1))))) -> 0(4(1(2(4(3(x1)))))) 38.05/10.67 3(4(1(3(5(x1))))) -> 4(3(0(3(1(5(x1)))))) 38.05/10.67 3(4(3(0(2(x1))))) -> 3(3(0(4(1(2(x1)))))) 38.05/10.67 3(4(5(0(2(x1))))) -> 0(3(0(4(2(5(x1)))))) 38.05/10.67 3(5(0(2(2(x1))))) -> 0(3(2(5(2(5(x1)))))) 38.05/10.67 3(5(2(1(4(x1))))) -> 3(5(1(0(4(2(x1)))))) 38.05/10.67 38.05/10.67 Q is empty. 38.05/10.67 38.05/10.67 ---------------------------------------- 38.05/10.67 38.05/10.67 (1) DependencyPairsProof (EQUIVALENT) 38.05/10.67 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 38.05/10.67 ---------------------------------------- 38.05/10.67 38.05/10.67 (2) 38.05/10.67 Obligation: 38.05/10.67 Q DP problem: 38.05/10.67 The TRS P consists of the following rules: 38.05/10.67 38.05/10.67 0^1(1(0(2(x1)))) -> 0^1(0(3(1(2(x1))))) 38.05/10.67 0^1(1(0(2(x1)))) -> 0^1(3(1(2(x1)))) 38.05/10.67 0^1(1(0(2(x1)))) -> 3^1(1(2(x1))) 38.05/10.67 0^1(1(3(4(x1)))) -> 0^1(4(1(0(3(x1))))) 38.05/10.67 0^1(1(3(4(x1)))) -> 0^1(3(x1)) 38.05/10.67 0^1(1(3(4(x1)))) -> 3^1(x1) 38.05/10.67 0^1(1(3(4(x1)))) -> 0^1(4(1(1(3(x1))))) 38.05/10.67 0^1(1(3(4(x1)))) -> 0^1(4(1(3(1(x1))))) 38.05/10.67 0^1(1(3(4(x1)))) -> 3^1(1(x1)) 38.05/10.67 0^1(2(1(4(x1)))) -> 0^1(4(1(2(3(x1))))) 38.05/10.67 0^1(2(1(4(x1)))) -> 3^1(x1) 38.05/10.67 0^1(2(1(4(x1)))) -> 0^1(4(1(3(2(x1))))) 38.05/10.67 0^1(2(1(4(x1)))) -> 3^1(2(x1)) 38.05/10.67 0^1(2(1(4(x1)))) -> 0^1(4(1(4(x1)))) 38.05/10.67 0^1(2(1(4(x1)))) -> 0^1(4(1(2(x1)))) 38.05/10.67 0^1(2(1(5(x1)))) -> 0^1(4(1(2(x1)))) 38.05/10.67 0^1(2(2(4(x1)))) -> 0^1(4(2(2(5(x1))))) 38.05/10.67 0^1(2(2(4(x1)))) -> 0^1(4(2(5(2(x1))))) 38.05/10.67 3^1(4(0(2(x1)))) -> 3^1(0(4(5(2(x1))))) 38.05/10.67 3^1(4(0(2(x1)))) -> 0^1(4(5(2(x1)))) 38.05/10.67 3^1(4(0(2(x1)))) -> 3^1(5(0(4(2(x1))))) 38.05/10.67 3^1(4(0(2(x1)))) -> 0^1(4(2(x1))) 38.05/10.67 0^1(0(1(4(5(x1))))) -> 0^1(4(1(0(3(5(x1)))))) 38.05/10.67 0^1(0(1(4(5(x1))))) -> 0^1(3(5(x1))) 38.05/10.67 0^1(0(1(4(5(x1))))) -> 3^1(5(x1)) 38.05/10.67 0^1(1(0(2(4(x1))))) -> 0^1(0(4(1(1(x1))))) 38.05/10.67 0^1(1(0(2(4(x1))))) -> 0^1(4(1(1(x1)))) 38.05/10.67 0^1(1(2(3(4(x1))))) -> 0^1(4(1(0(3(x1))))) 38.05/10.67 0^1(1(2(3(4(x1))))) -> 0^1(3(x1)) 38.05/10.67 0^1(1(2(3(4(x1))))) -> 3^1(x1) 38.05/10.67 0^1(1(3(3(4(x1))))) -> 0^1(0(3(1(3(4(x1)))))) 38.05/10.67 0^1(1(3(3(4(x1))))) -> 0^1(3(1(3(4(x1))))) 38.05/10.67 0^1(1(3(3(4(x1))))) -> 3^1(1(3(4(x1)))) 38.05/10.67 0^1(1(4(0(2(x1))))) -> 0^1(4(1(5(0(2(x1)))))) 38.05/10.67 0^1(1(4(1(5(x1))))) -> 0^1(4(1(1(x1)))) 38.05/10.67 0^1(1(4(3(4(x1))))) -> 0^1(4(0(3(1(4(x1)))))) 38.05/10.67 0^1(1(4(3(4(x1))))) -> 0^1(3(1(4(x1)))) 38.05/10.67 0^1(1(4(3(4(x1))))) -> 3^1(1(4(x1))) 38.05/10.67 0^1(1(4(3(4(x1))))) -> 3^1(0(4(1(5(4(x1)))))) 38.05/10.67 0^1(1(4(3(4(x1))))) -> 0^1(4(1(5(4(x1))))) 38.05/10.67 0^1(1(4(3(5(x1))))) -> 0^1(3(1(x1))) 38.05/10.67 0^1(1(4(3(5(x1))))) -> 3^1(1(x1)) 38.05/10.67 0^1(1(5(0(2(x1))))) -> 0^1(0(4(1(2(5(x1)))))) 38.05/10.67 0^1(1(5(0(2(x1))))) -> 0^1(4(1(2(5(x1))))) 38.05/10.67 0^1(1(5(1(4(x1))))) -> 0^1(3(1(1(x1)))) 38.05/10.67 0^1(1(5(1(4(x1))))) -> 3^1(1(1(x1))) 38.05/10.67 0^1(2(1(4(4(x1))))) -> 0^1(4(1(2(4(3(x1)))))) 38.05/10.67 0^1(2(1(4(4(x1))))) -> 3^1(x1) 38.05/10.67 0^1(2(1(4(5(x1))))) -> 0^1(4(1(2(5(2(x1)))))) 38.05/10.67 0^1(2(1(5(4(x1))))) -> 0^1(2(0(4(1(x1))))) 38.05/10.67 0^1(2(1(5(4(x1))))) -> 0^1(4(1(x1))) 38.05/10.67 0^1(2(4(1(5(x1))))) -> 0^1(4(1(5(2(x1))))) 38.05/10.67 0^1(2(4(3(5(x1))))) -> 0^1(4(5(2(5(3(x1)))))) 38.05/10.67 0^1(2(4(3(5(x1))))) -> 3^1(x1) 38.05/10.67 0^1(2(5(1(4(x1))))) -> 0^1(0(5(4(1(2(x1)))))) 38.05/10.67 0^1(2(5(1(4(x1))))) -> 0^1(5(4(1(2(x1))))) 38.05/10.67 3^1(0(1(3(2(x1))))) -> 0^1(3(1(0(3(2(x1)))))) 38.05/10.67 3^1(0(1(3(2(x1))))) -> 3^1(1(0(3(2(x1))))) 38.05/10.67 3^1(0(1(3(2(x1))))) -> 0^1(3(2(x1))) 38.05/10.67 3^1(0(2(1(4(x1))))) -> 0^1(4(1(3(2(x1))))) 38.05/10.67 3^1(0(2(1(4(x1))))) -> 3^1(2(x1)) 38.05/10.67 3^1(0(2(1(5(x1))))) -> 3^1(2(0(4(1(x1))))) 38.05/10.67 3^1(0(2(1(5(x1))))) -> 0^1(4(1(x1))) 38.05/10.67 3^1(0(4(0(2(x1))))) -> 0^1(3(4(0(4(2(x1)))))) 38.05/10.67 3^1(0(4(0(2(x1))))) -> 3^1(4(0(4(2(x1))))) 38.05/10.67 3^1(0(4(0(2(x1))))) -> 0^1(4(2(x1))) 38.05/10.67 3^1(0(4(0(2(x1))))) -> 0^1(4(1(2(0(3(x1)))))) 38.05/10.67 3^1(0(4(0(2(x1))))) -> 0^1(3(x1)) 38.05/10.67 3^1(0(4(0(2(x1))))) -> 3^1(x1) 38.05/10.67 3^1(0(5(1(4(x1))))) -> 3^1(0(4(1(1(5(x1)))))) 38.05/10.67 3^1(0(5(1(4(x1))))) -> 0^1(4(1(1(5(x1))))) 38.05/10.67 3^1(0(5(1(5(x1))))) -> 0^1(4(1(3(5(5(x1)))))) 38.05/10.67 3^1(0(5(1(5(x1))))) -> 3^1(5(5(x1))) 38.05/10.67 3^1(2(4(1(2(x1))))) -> 3^1(1(2(2(5(4(x1)))))) 38.05/10.67 3^1(2(4(1(5(x1))))) -> 3^1(1(4(5(2(5(x1)))))) 38.05/10.67 3^1(4(0(1(2(x1))))) -> 0^1(4(2(0(3(1(x1)))))) 38.05/10.67 3^1(4(0(1(2(x1))))) -> 0^1(3(1(x1))) 38.05/10.67 3^1(4(0(1(2(x1))))) -> 3^1(1(x1)) 38.05/10.67 3^1(4(0(1(4(x1))))) -> 0^1(4(1(5(3(4(x1)))))) 38.05/10.67 3^1(4(0(1(4(x1))))) -> 3^1(4(x1)) 38.05/10.67 3^1(4(0(1(5(x1))))) -> 0^1(4(1(5(5(3(x1)))))) 38.05/10.67 3^1(4(0(1(5(x1))))) -> 3^1(x1) 38.05/10.67 3^1(4(0(2(4(x1))))) -> 0^1(3(4(0(4(2(x1)))))) 38.05/10.67 3^1(4(0(2(4(x1))))) -> 3^1(4(0(4(2(x1))))) 38.05/10.67 3^1(4(0(2(4(x1))))) -> 0^1(4(2(x1))) 38.05/10.67 3^1(4(1(2(4(x1))))) -> 0^1(4(1(2(4(3(x1)))))) 38.05/10.67 3^1(4(1(2(4(x1))))) -> 3^1(x1) 38.05/10.67 3^1(4(1(3(5(x1))))) -> 3^1(0(3(1(5(x1))))) 38.05/10.67 3^1(4(1(3(5(x1))))) -> 0^1(3(1(5(x1)))) 38.05/10.67 3^1(4(1(3(5(x1))))) -> 3^1(1(5(x1))) 38.05/10.67 3^1(4(3(0(2(x1))))) -> 3^1(3(0(4(1(2(x1)))))) 38.05/10.67 3^1(4(3(0(2(x1))))) -> 3^1(0(4(1(2(x1))))) 38.05/10.67 3^1(4(3(0(2(x1))))) -> 0^1(4(1(2(x1)))) 38.05/10.67 3^1(4(5(0(2(x1))))) -> 0^1(3(0(4(2(5(x1)))))) 38.05/10.67 3^1(4(5(0(2(x1))))) -> 3^1(0(4(2(5(x1))))) 38.05/10.67 3^1(4(5(0(2(x1))))) -> 0^1(4(2(5(x1)))) 38.05/10.67 3^1(5(0(2(2(x1))))) -> 0^1(3(2(5(2(5(x1)))))) 38.05/10.67 3^1(5(0(2(2(x1))))) -> 3^1(2(5(2(5(x1))))) 38.05/10.67 3^1(5(2(1(4(x1))))) -> 3^1(5(1(0(4(2(x1)))))) 38.05/10.67 3^1(5(2(1(4(x1))))) -> 0^1(4(2(x1))) 38.05/10.67 38.05/10.67 The TRS R consists of the following rules: 38.05/10.67 38.05/10.67 0(1(0(2(x1)))) -> 0(0(3(1(2(x1))))) 38.05/10.67 0(1(3(4(x1)))) -> 0(4(1(0(3(x1))))) 38.05/10.67 0(1(3(4(x1)))) -> 0(4(1(1(3(x1))))) 38.05/10.67 0(1(3(4(x1)))) -> 0(4(1(3(1(x1))))) 38.05/10.67 0(2(1(4(x1)))) -> 0(4(1(2(3(x1))))) 38.05/10.67 0(2(1(4(x1)))) -> 0(4(1(3(2(x1))))) 38.05/10.67 0(2(1(4(x1)))) -> 2(0(4(1(4(x1))))) 38.05/10.67 0(2(1(4(x1)))) -> 5(5(0(4(1(2(x1)))))) 38.05/10.67 0(2(1(5(x1)))) -> 5(0(4(1(2(x1))))) 38.05/10.67 0(2(2(4(x1)))) -> 0(4(2(2(5(x1))))) 38.05/10.67 0(2(2(4(x1)))) -> 0(4(2(5(2(x1))))) 38.05/10.67 3(4(0(2(x1)))) -> 3(0(4(5(2(x1))))) 38.05/10.67 3(4(0(2(x1)))) -> 3(5(0(4(2(x1))))) 38.05/10.67 0(0(1(4(5(x1))))) -> 0(4(1(0(3(5(x1)))))) 38.05/10.67 0(1(0(2(4(x1))))) -> 2(0(0(4(1(1(x1)))))) 38.05/10.67 0(1(2(3(4(x1))))) -> 2(0(4(1(0(3(x1)))))) 38.05/10.67 0(1(3(3(4(x1))))) -> 0(0(3(1(3(4(x1)))))) 38.05/10.67 0(1(4(0(2(x1))))) -> 0(4(1(5(0(2(x1)))))) 38.05/10.67 0(1(4(1(5(x1))))) -> 2(5(0(4(1(1(x1)))))) 38.05/10.67 0(1(4(3(4(x1))))) -> 0(4(0(3(1(4(x1)))))) 38.05/10.67 0(1(4(3(4(x1))))) -> 3(0(4(1(5(4(x1)))))) 38.05/10.67 0(1(4(3(5(x1))))) -> 5(4(5(0(3(1(x1)))))) 38.05/10.67 0(1(5(0(2(x1))))) -> 0(0(4(1(2(5(x1)))))) 38.05/10.67 0(1(5(1(4(x1))))) -> 4(5(0(3(1(1(x1)))))) 38.05/10.67 0(2(1(4(4(x1))))) -> 0(4(1(2(4(3(x1)))))) 38.05/10.67 0(2(1(4(5(x1))))) -> 0(4(1(2(5(2(x1)))))) 38.05/10.67 0(2(1(5(4(x1))))) -> 5(0(2(0(4(1(x1)))))) 38.05/10.67 0(2(4(1(5(x1))))) -> 5(0(4(1(5(2(x1)))))) 38.05/10.67 0(2(4(3(5(x1))))) -> 0(4(5(2(5(3(x1)))))) 38.05/10.67 0(2(5(1(4(x1))))) -> 0(0(5(4(1(2(x1)))))) 38.05/10.67 3(0(1(3(2(x1))))) -> 0(3(1(0(3(2(x1)))))) 38.05/10.67 3(0(2(1(4(x1))))) -> 4(0(4(1(3(2(x1)))))) 38.05/10.67 3(0(2(1(5(x1))))) -> 5(3(2(0(4(1(x1)))))) 38.05/10.67 3(0(4(0(2(x1))))) -> 0(3(4(0(4(2(x1)))))) 38.05/10.67 3(0(4(0(2(x1))))) -> 0(4(1(2(0(3(x1)))))) 38.05/10.67 3(0(5(1(4(x1))))) -> 3(0(4(1(1(5(x1)))))) 38.05/10.67 3(0(5(1(5(x1))))) -> 0(4(1(3(5(5(x1)))))) 38.05/10.67 3(2(4(1(2(x1))))) -> 3(1(2(2(5(4(x1)))))) 38.05/10.67 3(2(4(1(5(x1))))) -> 3(1(4(5(2(5(x1)))))) 38.05/10.67 3(4(0(1(2(x1))))) -> 0(4(2(0(3(1(x1)))))) 38.05/10.67 3(4(0(1(4(x1))))) -> 0(4(1(5(3(4(x1)))))) 38.05/10.67 3(4(0(1(5(x1))))) -> 0(4(1(5(5(3(x1)))))) 38.05/10.67 3(4(0(2(4(x1))))) -> 0(3(4(0(4(2(x1)))))) 38.05/10.67 3(4(1(2(4(x1))))) -> 0(4(1(2(4(3(x1)))))) 38.05/10.67 3(4(1(3(5(x1))))) -> 4(3(0(3(1(5(x1)))))) 38.05/10.67 3(4(3(0(2(x1))))) -> 3(3(0(4(1(2(x1)))))) 38.05/10.67 3(4(5(0(2(x1))))) -> 0(3(0(4(2(5(x1)))))) 38.05/10.67 3(5(0(2(2(x1))))) -> 0(3(2(5(2(5(x1)))))) 38.05/10.67 3(5(2(1(4(x1))))) -> 3(5(1(0(4(2(x1)))))) 38.05/10.67 38.05/10.67 Q is empty. 38.05/10.67 We have to consider all minimal (P,Q,R)-chains. 38.05/10.67 ---------------------------------------- 38.05/10.67 38.05/10.67 (3) DependencyGraphProof (EQUIVALENT) 38.05/10.67 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 93 less nodes. 38.05/10.67 ---------------------------------------- 38.05/10.67 38.05/10.67 (4) 38.05/10.67 Obligation: 38.05/10.67 Q DP problem: 38.05/10.67 The TRS P consists of the following rules: 38.05/10.67 38.05/10.67 0^1(2(4(3(5(x1))))) -> 3^1(x1) 38.05/10.67 3^1(0(1(3(2(x1))))) -> 0^1(3(2(x1))) 38.05/10.67 3^1(0(4(0(2(x1))))) -> 0^1(3(x1)) 38.05/10.67 3^1(0(4(0(2(x1))))) -> 3^1(x1) 38.05/10.67 3^1(4(0(1(4(x1))))) -> 3^1(4(x1)) 38.05/10.67 3^1(4(0(1(5(x1))))) -> 3^1(x1) 38.05/10.67 3^1(4(1(2(4(x1))))) -> 3^1(x1) 38.05/10.67 38.05/10.67 The TRS R consists of the following rules: 38.05/10.67 38.05/10.67 0(1(0(2(x1)))) -> 0(0(3(1(2(x1))))) 38.05/10.67 0(1(3(4(x1)))) -> 0(4(1(0(3(x1))))) 38.05/10.67 0(1(3(4(x1)))) -> 0(4(1(1(3(x1))))) 38.05/10.67 0(1(3(4(x1)))) -> 0(4(1(3(1(x1))))) 38.05/10.67 0(2(1(4(x1)))) -> 0(4(1(2(3(x1))))) 38.05/10.67 0(2(1(4(x1)))) -> 0(4(1(3(2(x1))))) 38.05/10.67 0(2(1(4(x1)))) -> 2(0(4(1(4(x1))))) 38.05/10.67 0(2(1(4(x1)))) -> 5(5(0(4(1(2(x1)))))) 38.05/10.67 0(2(1(5(x1)))) -> 5(0(4(1(2(x1))))) 38.05/10.67 0(2(2(4(x1)))) -> 0(4(2(2(5(x1))))) 38.05/10.67 0(2(2(4(x1)))) -> 0(4(2(5(2(x1))))) 38.05/10.67 3(4(0(2(x1)))) -> 3(0(4(5(2(x1))))) 38.05/10.67 3(4(0(2(x1)))) -> 3(5(0(4(2(x1))))) 38.05/10.67 0(0(1(4(5(x1))))) -> 0(4(1(0(3(5(x1)))))) 38.05/10.67 0(1(0(2(4(x1))))) -> 2(0(0(4(1(1(x1)))))) 38.05/10.67 0(1(2(3(4(x1))))) -> 2(0(4(1(0(3(x1)))))) 38.05/10.67 0(1(3(3(4(x1))))) -> 0(0(3(1(3(4(x1)))))) 38.05/10.67 0(1(4(0(2(x1))))) -> 0(4(1(5(0(2(x1)))))) 38.05/10.67 0(1(4(1(5(x1))))) -> 2(5(0(4(1(1(x1)))))) 38.05/10.67 0(1(4(3(4(x1))))) -> 0(4(0(3(1(4(x1)))))) 38.05/10.67 0(1(4(3(4(x1))))) -> 3(0(4(1(5(4(x1)))))) 38.05/10.67 0(1(4(3(5(x1))))) -> 5(4(5(0(3(1(x1)))))) 38.05/10.67 0(1(5(0(2(x1))))) -> 0(0(4(1(2(5(x1)))))) 38.05/10.67 0(1(5(1(4(x1))))) -> 4(5(0(3(1(1(x1)))))) 38.05/10.67 0(2(1(4(4(x1))))) -> 0(4(1(2(4(3(x1)))))) 38.05/10.67 0(2(1(4(5(x1))))) -> 0(4(1(2(5(2(x1)))))) 38.05/10.67 0(2(1(5(4(x1))))) -> 5(0(2(0(4(1(x1)))))) 38.05/10.67 0(2(4(1(5(x1))))) -> 5(0(4(1(5(2(x1)))))) 38.05/10.67 0(2(4(3(5(x1))))) -> 0(4(5(2(5(3(x1)))))) 38.05/10.67 0(2(5(1(4(x1))))) -> 0(0(5(4(1(2(x1)))))) 38.05/10.67 3(0(1(3(2(x1))))) -> 0(3(1(0(3(2(x1)))))) 38.05/10.67 3(0(2(1(4(x1))))) -> 4(0(4(1(3(2(x1)))))) 38.05/10.67 3(0(2(1(5(x1))))) -> 5(3(2(0(4(1(x1)))))) 38.05/10.67 3(0(4(0(2(x1))))) -> 0(3(4(0(4(2(x1)))))) 38.05/10.67 3(0(4(0(2(x1))))) -> 0(4(1(2(0(3(x1)))))) 38.05/10.67 3(0(5(1(4(x1))))) -> 3(0(4(1(1(5(x1)))))) 38.05/10.67 3(0(5(1(5(x1))))) -> 0(4(1(3(5(5(x1)))))) 38.05/10.67 3(2(4(1(2(x1))))) -> 3(1(2(2(5(4(x1)))))) 38.05/10.67 3(2(4(1(5(x1))))) -> 3(1(4(5(2(5(x1)))))) 38.05/10.67 3(4(0(1(2(x1))))) -> 0(4(2(0(3(1(x1)))))) 38.05/10.67 3(4(0(1(4(x1))))) -> 0(4(1(5(3(4(x1)))))) 38.05/10.67 3(4(0(1(5(x1))))) -> 0(4(1(5(5(3(x1)))))) 38.05/10.67 3(4(0(2(4(x1))))) -> 0(3(4(0(4(2(x1)))))) 38.05/10.67 3(4(1(2(4(x1))))) -> 0(4(1(2(4(3(x1)))))) 38.05/10.67 3(4(1(3(5(x1))))) -> 4(3(0(3(1(5(x1)))))) 38.05/10.67 3(4(3(0(2(x1))))) -> 3(3(0(4(1(2(x1)))))) 38.05/10.67 3(4(5(0(2(x1))))) -> 0(3(0(4(2(5(x1)))))) 38.05/10.67 3(5(0(2(2(x1))))) -> 0(3(2(5(2(5(x1)))))) 38.05/10.67 3(5(2(1(4(x1))))) -> 3(5(1(0(4(2(x1)))))) 38.05/10.67 38.05/10.67 Q is empty. 38.05/10.67 We have to consider all minimal (P,Q,R)-chains. 38.05/10.67 ---------------------------------------- 38.05/10.67 38.05/10.67 (5) QDPOrderProof (EQUIVALENT) 38.05/10.67 We use the reduction pair processor [LPAR04,JAR06]. 38.05/10.67 38.05/10.67 38.05/10.67 The following pairs can be oriented strictly and are deleted. 38.05/10.67 38.05/10.67 0^1(2(4(3(5(x1))))) -> 3^1(x1) 38.05/10.67 The remaining pairs can at least be oriented weakly. 38.05/10.67 Used ordering: Polynomial interpretation [POLO]: 38.05/10.67 38.05/10.67 POL(0(x_1)) = 0 38.05/10.67 POL(0^1(x_1)) = x_1 38.05/10.67 POL(1(x_1)) = 0 38.05/10.67 POL(2(x_1)) = 1 38.05/10.67 POL(3(x_1)) = 0 38.05/10.67 POL(3^1(x_1)) = 0 38.05/10.67 POL(4(x_1)) = 0 38.05/10.67 POL(5(x_1)) = 0 38.05/10.67 38.05/10.67 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 38.05/10.67 38.05/10.67 3(2(4(1(2(x1))))) -> 3(1(2(2(5(4(x1)))))) 38.05/10.67 3(2(4(1(5(x1))))) -> 3(1(4(5(2(5(x1)))))) 38.05/10.67 3(4(0(2(x1)))) -> 3(0(4(5(2(x1))))) 38.05/10.67 3(4(0(2(x1)))) -> 3(5(0(4(2(x1))))) 38.05/10.67 3(0(1(3(2(x1))))) -> 0(3(1(0(3(2(x1)))))) 38.05/10.67 3(0(2(1(4(x1))))) -> 4(0(4(1(3(2(x1)))))) 38.05/10.67 3(0(2(1(5(x1))))) -> 5(3(2(0(4(1(x1)))))) 38.05/10.67 3(0(4(0(2(x1))))) -> 0(3(4(0(4(2(x1)))))) 38.05/10.67 3(0(4(0(2(x1))))) -> 0(4(1(2(0(3(x1)))))) 38.05/10.67 3(0(5(1(4(x1))))) -> 3(0(4(1(1(5(x1)))))) 38.05/10.67 3(0(5(1(5(x1))))) -> 0(4(1(3(5(5(x1)))))) 38.05/10.67 3(4(0(1(2(x1))))) -> 0(4(2(0(3(1(x1)))))) 38.05/10.67 3(4(0(1(4(x1))))) -> 0(4(1(5(3(4(x1)))))) 38.05/10.67 3(4(0(1(5(x1))))) -> 0(4(1(5(5(3(x1)))))) 38.05/10.67 3(4(0(2(4(x1))))) -> 0(3(4(0(4(2(x1)))))) 38.05/10.67 3(4(1(2(4(x1))))) -> 0(4(1(2(4(3(x1)))))) 38.05/10.67 3(4(1(3(5(x1))))) -> 4(3(0(3(1(5(x1)))))) 38.05/10.67 3(4(3(0(2(x1))))) -> 3(3(0(4(1(2(x1)))))) 38.05/10.67 3(4(5(0(2(x1))))) -> 0(3(0(4(2(5(x1)))))) 38.05/10.67 3(5(0(2(2(x1))))) -> 0(3(2(5(2(5(x1)))))) 38.05/10.67 3(5(2(1(4(x1))))) -> 3(5(1(0(4(2(x1)))))) 38.05/10.67 38.05/10.67 38.05/10.67 ---------------------------------------- 38.05/10.67 38.05/10.67 (6) 38.05/10.67 Obligation: 38.05/10.67 Q DP problem: 38.05/10.67 The TRS P consists of the following rules: 38.05/10.67 38.05/10.67 3^1(0(1(3(2(x1))))) -> 0^1(3(2(x1))) 38.05/10.67 3^1(0(4(0(2(x1))))) -> 0^1(3(x1)) 38.05/10.67 3^1(0(4(0(2(x1))))) -> 3^1(x1) 38.05/10.67 3^1(4(0(1(4(x1))))) -> 3^1(4(x1)) 38.05/10.67 3^1(4(0(1(5(x1))))) -> 3^1(x1) 38.05/10.67 3^1(4(1(2(4(x1))))) -> 3^1(x1) 38.05/10.67 38.05/10.67 The TRS R consists of the following rules: 38.05/10.67 38.05/10.67 0(1(0(2(x1)))) -> 0(0(3(1(2(x1))))) 38.05/10.67 0(1(3(4(x1)))) -> 0(4(1(0(3(x1))))) 38.05/10.67 0(1(3(4(x1)))) -> 0(4(1(1(3(x1))))) 38.05/10.67 0(1(3(4(x1)))) -> 0(4(1(3(1(x1))))) 38.05/10.67 0(2(1(4(x1)))) -> 0(4(1(2(3(x1))))) 38.05/10.67 0(2(1(4(x1)))) -> 0(4(1(3(2(x1))))) 38.05/10.67 0(2(1(4(x1)))) -> 2(0(4(1(4(x1))))) 38.05/10.67 0(2(1(4(x1)))) -> 5(5(0(4(1(2(x1)))))) 38.05/10.67 0(2(1(5(x1)))) -> 5(0(4(1(2(x1))))) 38.05/10.67 0(2(2(4(x1)))) -> 0(4(2(2(5(x1))))) 38.05/10.67 0(2(2(4(x1)))) -> 0(4(2(5(2(x1))))) 38.05/10.67 3(4(0(2(x1)))) -> 3(0(4(5(2(x1))))) 38.05/10.67 3(4(0(2(x1)))) -> 3(5(0(4(2(x1))))) 38.05/10.67 0(0(1(4(5(x1))))) -> 0(4(1(0(3(5(x1)))))) 38.05/10.67 0(1(0(2(4(x1))))) -> 2(0(0(4(1(1(x1)))))) 38.05/10.67 0(1(2(3(4(x1))))) -> 2(0(4(1(0(3(x1)))))) 38.05/10.67 0(1(3(3(4(x1))))) -> 0(0(3(1(3(4(x1)))))) 38.05/10.67 0(1(4(0(2(x1))))) -> 0(4(1(5(0(2(x1)))))) 38.05/10.67 0(1(4(1(5(x1))))) -> 2(5(0(4(1(1(x1)))))) 38.05/10.67 0(1(4(3(4(x1))))) -> 0(4(0(3(1(4(x1)))))) 38.05/10.67 0(1(4(3(4(x1))))) -> 3(0(4(1(5(4(x1)))))) 38.05/10.67 0(1(4(3(5(x1))))) -> 5(4(5(0(3(1(x1)))))) 38.05/10.67 0(1(5(0(2(x1))))) -> 0(0(4(1(2(5(x1)))))) 38.05/10.67 0(1(5(1(4(x1))))) -> 4(5(0(3(1(1(x1)))))) 38.05/10.67 0(2(1(4(4(x1))))) -> 0(4(1(2(4(3(x1)))))) 38.05/10.67 0(2(1(4(5(x1))))) -> 0(4(1(2(5(2(x1)))))) 38.05/10.67 0(2(1(5(4(x1))))) -> 5(0(2(0(4(1(x1)))))) 38.05/10.67 0(2(4(1(5(x1))))) -> 5(0(4(1(5(2(x1)))))) 38.05/10.67 0(2(4(3(5(x1))))) -> 0(4(5(2(5(3(x1)))))) 38.05/10.67 0(2(5(1(4(x1))))) -> 0(0(5(4(1(2(x1)))))) 38.05/10.67 3(0(1(3(2(x1))))) -> 0(3(1(0(3(2(x1)))))) 38.05/10.67 3(0(2(1(4(x1))))) -> 4(0(4(1(3(2(x1)))))) 38.05/10.67 3(0(2(1(5(x1))))) -> 5(3(2(0(4(1(x1)))))) 38.05/10.67 3(0(4(0(2(x1))))) -> 0(3(4(0(4(2(x1)))))) 38.05/10.67 3(0(4(0(2(x1))))) -> 0(4(1(2(0(3(x1)))))) 38.05/10.67 3(0(5(1(4(x1))))) -> 3(0(4(1(1(5(x1)))))) 38.05/10.67 3(0(5(1(5(x1))))) -> 0(4(1(3(5(5(x1)))))) 38.05/10.67 3(2(4(1(2(x1))))) -> 3(1(2(2(5(4(x1)))))) 38.05/10.67 3(2(4(1(5(x1))))) -> 3(1(4(5(2(5(x1)))))) 38.05/10.67 3(4(0(1(2(x1))))) -> 0(4(2(0(3(1(x1)))))) 38.05/10.67 3(4(0(1(4(x1))))) -> 0(4(1(5(3(4(x1)))))) 38.05/10.67 3(4(0(1(5(x1))))) -> 0(4(1(5(5(3(x1)))))) 38.05/10.67 3(4(0(2(4(x1))))) -> 0(3(4(0(4(2(x1)))))) 38.05/10.67 3(4(1(2(4(x1))))) -> 0(4(1(2(4(3(x1)))))) 38.05/10.67 3(4(1(3(5(x1))))) -> 4(3(0(3(1(5(x1)))))) 38.05/10.67 3(4(3(0(2(x1))))) -> 3(3(0(4(1(2(x1)))))) 38.05/10.67 3(4(5(0(2(x1))))) -> 0(3(0(4(2(5(x1)))))) 38.05/10.67 3(5(0(2(2(x1))))) -> 0(3(2(5(2(5(x1)))))) 38.05/10.67 3(5(2(1(4(x1))))) -> 3(5(1(0(4(2(x1)))))) 38.05/10.67 38.05/10.67 Q is empty. 38.05/10.67 We have to consider all minimal (P,Q,R)-chains. 38.05/10.67 ---------------------------------------- 38.05/10.67 38.05/10.67 (7) DependencyGraphProof (EQUIVALENT) 38.05/10.67 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 38.05/10.67 ---------------------------------------- 38.05/10.67 38.05/10.67 (8) 38.05/10.67 Obligation: 38.05/10.67 Q DP problem: 38.05/10.67 The TRS P consists of the following rules: 38.05/10.67 38.05/10.67 3^1(4(0(1(4(x1))))) -> 3^1(4(x1)) 38.05/10.67 3^1(4(0(1(5(x1))))) -> 3^1(x1) 38.05/10.67 3^1(0(4(0(2(x1))))) -> 3^1(x1) 38.05/10.67 3^1(4(1(2(4(x1))))) -> 3^1(x1) 38.05/10.67 38.05/10.67 The TRS R consists of the following rules: 38.05/10.67 38.05/10.67 0(1(0(2(x1)))) -> 0(0(3(1(2(x1))))) 38.05/10.67 0(1(3(4(x1)))) -> 0(4(1(0(3(x1))))) 38.05/10.67 0(1(3(4(x1)))) -> 0(4(1(1(3(x1))))) 38.05/10.67 0(1(3(4(x1)))) -> 0(4(1(3(1(x1))))) 38.05/10.67 0(2(1(4(x1)))) -> 0(4(1(2(3(x1))))) 38.05/10.67 0(2(1(4(x1)))) -> 0(4(1(3(2(x1))))) 38.05/10.67 0(2(1(4(x1)))) -> 2(0(4(1(4(x1))))) 38.05/10.67 0(2(1(4(x1)))) -> 5(5(0(4(1(2(x1)))))) 38.05/10.67 0(2(1(5(x1)))) -> 5(0(4(1(2(x1))))) 38.05/10.67 0(2(2(4(x1)))) -> 0(4(2(2(5(x1))))) 38.05/10.67 0(2(2(4(x1)))) -> 0(4(2(5(2(x1))))) 38.05/10.67 3(4(0(2(x1)))) -> 3(0(4(5(2(x1))))) 38.05/10.67 3(4(0(2(x1)))) -> 3(5(0(4(2(x1))))) 38.05/10.67 0(0(1(4(5(x1))))) -> 0(4(1(0(3(5(x1)))))) 38.05/10.67 0(1(0(2(4(x1))))) -> 2(0(0(4(1(1(x1)))))) 38.05/10.67 0(1(2(3(4(x1))))) -> 2(0(4(1(0(3(x1)))))) 38.05/10.67 0(1(3(3(4(x1))))) -> 0(0(3(1(3(4(x1)))))) 38.05/10.67 0(1(4(0(2(x1))))) -> 0(4(1(5(0(2(x1)))))) 38.05/10.67 0(1(4(1(5(x1))))) -> 2(5(0(4(1(1(x1)))))) 38.05/10.67 0(1(4(3(4(x1))))) -> 0(4(0(3(1(4(x1)))))) 38.05/10.67 0(1(4(3(4(x1))))) -> 3(0(4(1(5(4(x1)))))) 38.05/10.67 0(1(4(3(5(x1))))) -> 5(4(5(0(3(1(x1)))))) 38.05/10.67 0(1(5(0(2(x1))))) -> 0(0(4(1(2(5(x1)))))) 38.05/10.67 0(1(5(1(4(x1))))) -> 4(5(0(3(1(1(x1)))))) 38.05/10.67 0(2(1(4(4(x1))))) -> 0(4(1(2(4(3(x1)))))) 38.05/10.67 0(2(1(4(5(x1))))) -> 0(4(1(2(5(2(x1)))))) 38.05/10.67 0(2(1(5(4(x1))))) -> 5(0(2(0(4(1(x1)))))) 38.05/10.67 0(2(4(1(5(x1))))) -> 5(0(4(1(5(2(x1)))))) 38.05/10.67 0(2(4(3(5(x1))))) -> 0(4(5(2(5(3(x1)))))) 38.05/10.67 0(2(5(1(4(x1))))) -> 0(0(5(4(1(2(x1)))))) 38.05/10.67 3(0(1(3(2(x1))))) -> 0(3(1(0(3(2(x1)))))) 38.05/10.67 3(0(2(1(4(x1))))) -> 4(0(4(1(3(2(x1)))))) 38.05/10.67 3(0(2(1(5(x1))))) -> 5(3(2(0(4(1(x1)))))) 38.05/10.67 3(0(4(0(2(x1))))) -> 0(3(4(0(4(2(x1)))))) 38.05/10.67 3(0(4(0(2(x1))))) -> 0(4(1(2(0(3(x1)))))) 38.05/10.67 3(0(5(1(4(x1))))) -> 3(0(4(1(1(5(x1)))))) 38.05/10.67 3(0(5(1(5(x1))))) -> 0(4(1(3(5(5(x1)))))) 38.05/10.67 3(2(4(1(2(x1))))) -> 3(1(2(2(5(4(x1)))))) 38.05/10.67 3(2(4(1(5(x1))))) -> 3(1(4(5(2(5(x1)))))) 38.05/10.67 3(4(0(1(2(x1))))) -> 0(4(2(0(3(1(x1)))))) 38.05/10.67 3(4(0(1(4(x1))))) -> 0(4(1(5(3(4(x1)))))) 38.05/10.67 3(4(0(1(5(x1))))) -> 0(4(1(5(5(3(x1)))))) 38.05/10.67 3(4(0(2(4(x1))))) -> 0(3(4(0(4(2(x1)))))) 38.05/10.67 3(4(1(2(4(x1))))) -> 0(4(1(2(4(3(x1)))))) 38.05/10.67 3(4(1(3(5(x1))))) -> 4(3(0(3(1(5(x1)))))) 38.05/10.67 3(4(3(0(2(x1))))) -> 3(3(0(4(1(2(x1)))))) 38.05/10.67 3(4(5(0(2(x1))))) -> 0(3(0(4(2(5(x1)))))) 38.05/10.67 3(5(0(2(2(x1))))) -> 0(3(2(5(2(5(x1)))))) 38.05/10.67 3(5(2(1(4(x1))))) -> 3(5(1(0(4(2(x1)))))) 38.05/10.67 38.05/10.67 Q is empty. 38.05/10.67 We have to consider all minimal (P,Q,R)-chains. 38.05/10.67 ---------------------------------------- 38.05/10.67 38.05/10.67 (9) UsableRulesProof (EQUIVALENT) 38.05/10.67 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. 38.05/10.67 ---------------------------------------- 38.05/10.67 38.05/10.67 (10) 38.05/10.67 Obligation: 38.05/10.67 Q DP problem: 38.05/10.67 The TRS P consists of the following rules: 38.05/10.67 38.05/10.67 3^1(4(0(1(4(x1))))) -> 3^1(4(x1)) 38.05/10.67 3^1(4(0(1(5(x1))))) -> 3^1(x1) 38.05/10.67 3^1(0(4(0(2(x1))))) -> 3^1(x1) 38.05/10.67 3^1(4(1(2(4(x1))))) -> 3^1(x1) 38.05/10.67 38.05/10.67 R is empty. 38.05/10.67 Q is empty. 38.05/10.67 We have to consider all minimal (P,Q,R)-chains. 38.05/10.67 ---------------------------------------- 38.05/10.67 38.05/10.67 (11) QDPSizeChangeProof (EQUIVALENT) 38.05/10.67 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. 38.05/10.67 38.05/10.67 From the DPs we obtained the following set of size-change graphs: 38.05/10.67 *3^1(0(4(0(2(x1))))) -> 3^1(x1) 38.05/10.67 The graph contains the following edges 1 > 1 38.05/10.67 38.05/10.67 38.05/10.67 *3^1(4(0(1(4(x1))))) -> 3^1(4(x1)) 38.05/10.67 The graph contains the following edges 1 > 1 38.05/10.67 38.05/10.67 38.05/10.67 *3^1(4(0(1(5(x1))))) -> 3^1(x1) 38.05/10.67 The graph contains the following edges 1 > 1 38.05/10.67 38.05/10.67 38.05/10.67 *3^1(4(1(2(4(x1))))) -> 3^1(x1) 38.05/10.67 The graph contains the following edges 1 > 1 38.05/10.67 38.05/10.67 38.05/10.67 ---------------------------------------- 38.05/10.67 38.05/10.67 (12) 38.05/10.67 YES 38.30/10.76 EOF