YES proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) DependencyPairsProof [EQUIVALENT, 170 ms] (2) QDP (3) DependencyGraphProof [EQUIVALENT, 0 ms] (4) QDP (5) QDPOrderProof [EQUIVALENT, 185 ms] (6) QDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) TRUE ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 2(5(x1)) -> 1(3(3(0(1(0(x1)))))) 2(5(x1)) -> 2(2(0(5(0(1(x1)))))) 3(5(x1)) -> 1(3(2(0(0(1(x1)))))) 3(5(x1)) -> 3(2(0(5(3(0(x1)))))) 4(5(x1)) -> 2(2(1(3(2(1(x1)))))) 4(5(x1)) -> 3(2(0(5(0(0(x1)))))) 1(2(5(x1))) -> 1(0(5(0(5(4(x1)))))) 1(2(5(x1))) -> 1(2(2(1(0(1(x1)))))) 1(2(5(x1))) -> 2(0(1(3(1(0(x1)))))) 1(4(5(x1))) -> 1(2(4(0(2(1(x1)))))) 2(5(1(x1))) -> 2(2(2(1(2(3(x1)))))) 2(5(2(x1))) -> 4(0(2(2(3(3(x1)))))) 2(5(3(x1))) -> 2(0(4(1(3(3(x1)))))) 2(5(4(x1))) -> 2(0(5(1(0(1(x1)))))) 3(2(5(x1))) -> 3(2(0(1(0(5(x1)))))) 3(4(2(x1))) -> 3(4(0(2(2(2(x1)))))) 3(5(1(x1))) -> 0(4(2(0(0(5(x1)))))) 3(5(1(x1))) -> 0(4(2(2(3(4(x1)))))) 3(5(1(x1))) -> 2(1(4(1(0(1(x1)))))) 3(5(2(x1))) -> 0(4(3(2(2(2(x1)))))) 3(5(2(x1))) -> 2(0(2(2(3(0(x1)))))) 3(5(2(x1))) -> 2(3(3(2(1(2(x1)))))) 3(5(3(x1))) -> 0(2(4(3(3(0(x1)))))) 3(5(3(x1))) -> 0(5(4(3(3(0(x1)))))) 3(5(3(x1))) -> 2(3(4(0(4(2(x1)))))) 3(5(4(x1))) -> 0(2(0(5(0(0(x1)))))) 3(5(4(x1))) -> 0(5(0(0(1(2(x1)))))) 3(5(5(x1))) -> 0(5(4(1(0(5(x1)))))) 4(5(1(x1))) -> 2(1(0(5(3(3(x1)))))) 4(5(2(x1))) -> 0(5(1(0(0(4(x1)))))) 4(5(4(x1))) -> 2(2(1(0(4(2(x1)))))) 4(5(4(x1))) -> 3(2(0(3(2(0(x1)))))) 5(5(3(x1))) -> 5(1(0(1(2(2(x1)))))) 5(5(4(x1))) -> 5(1(0(4(2(2(x1)))))) Q is empty. ---------------------------------------- (1) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (2) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(5(x1)) -> 1^1(3(3(0(1(0(x1)))))) 2^1(5(x1)) -> 3^1(3(0(1(0(x1))))) 2^1(5(x1)) -> 3^1(0(1(0(x1)))) 2^1(5(x1)) -> 1^1(0(x1)) 2^1(5(x1)) -> 2^1(2(0(5(0(1(x1)))))) 2^1(5(x1)) -> 2^1(0(5(0(1(x1))))) 2^1(5(x1)) -> 5^1(0(1(x1))) 2^1(5(x1)) -> 1^1(x1) 3^1(5(x1)) -> 1^1(3(2(0(0(1(x1)))))) 3^1(5(x1)) -> 3^1(2(0(0(1(x1))))) 3^1(5(x1)) -> 2^1(0(0(1(x1)))) 3^1(5(x1)) -> 1^1(x1) 3^1(5(x1)) -> 3^1(2(0(5(3(0(x1)))))) 3^1(5(x1)) -> 2^1(0(5(3(0(x1))))) 3^1(5(x1)) -> 5^1(3(0(x1))) 3^1(5(x1)) -> 3^1(0(x1)) 4^1(5(x1)) -> 2^1(2(1(3(2(1(x1)))))) 4^1(5(x1)) -> 2^1(1(3(2(1(x1))))) 4^1(5(x1)) -> 1^1(3(2(1(x1)))) 4^1(5(x1)) -> 3^1(2(1(x1))) 4^1(5(x1)) -> 2^1(1(x1)) 4^1(5(x1)) -> 1^1(x1) 4^1(5(x1)) -> 3^1(2(0(5(0(0(x1)))))) 4^1(5(x1)) -> 2^1(0(5(0(0(x1))))) 4^1(5(x1)) -> 5^1(0(0(x1))) 1^1(2(5(x1))) -> 1^1(0(5(0(5(4(x1)))))) 1^1(2(5(x1))) -> 5^1(0(5(4(x1)))) 1^1(2(5(x1))) -> 5^1(4(x1)) 1^1(2(5(x1))) -> 4^1(x1) 1^1(2(5(x1))) -> 1^1(2(2(1(0(1(x1)))))) 1^1(2(5(x1))) -> 2^1(2(1(0(1(x1))))) 1^1(2(5(x1))) -> 2^1(1(0(1(x1)))) 1^1(2(5(x1))) -> 1^1(0(1(x1))) 1^1(2(5(x1))) -> 1^1(x1) 1^1(2(5(x1))) -> 2^1(0(1(3(1(0(x1)))))) 1^1(2(5(x1))) -> 1^1(3(1(0(x1)))) 1^1(2(5(x1))) -> 3^1(1(0(x1))) 1^1(2(5(x1))) -> 1^1(0(x1)) 1^1(4(5(x1))) -> 1^1(2(4(0(2(1(x1)))))) 1^1(4(5(x1))) -> 2^1(4(0(2(1(x1))))) 1^1(4(5(x1))) -> 4^1(0(2(1(x1)))) 1^1(4(5(x1))) -> 2^1(1(x1)) 1^1(4(5(x1))) -> 1^1(x1) 2^1(5(1(x1))) -> 2^1(2(2(1(2(3(x1)))))) 2^1(5(1(x1))) -> 2^1(2(1(2(3(x1))))) 2^1(5(1(x1))) -> 2^1(1(2(3(x1)))) 2^1(5(1(x1))) -> 1^1(2(3(x1))) 2^1(5(1(x1))) -> 2^1(3(x1)) 2^1(5(1(x1))) -> 3^1(x1) 2^1(5(2(x1))) -> 4^1(0(2(2(3(3(x1)))))) 2^1(5(2(x1))) -> 2^1(2(3(3(x1)))) 2^1(5(2(x1))) -> 2^1(3(3(x1))) 2^1(5(2(x1))) -> 3^1(3(x1)) 2^1(5(2(x1))) -> 3^1(x1) 2^1(5(3(x1))) -> 2^1(0(4(1(3(3(x1)))))) 2^1(5(3(x1))) -> 4^1(1(3(3(x1)))) 2^1(5(3(x1))) -> 1^1(3(3(x1))) 2^1(5(3(x1))) -> 3^1(3(x1)) 2^1(5(4(x1))) -> 2^1(0(5(1(0(1(x1)))))) 2^1(5(4(x1))) -> 5^1(1(0(1(x1)))) 2^1(5(4(x1))) -> 1^1(0(1(x1))) 2^1(5(4(x1))) -> 1^1(x1) 3^1(2(5(x1))) -> 3^1(2(0(1(0(5(x1)))))) 3^1(2(5(x1))) -> 2^1(0(1(0(5(x1))))) 3^1(2(5(x1))) -> 1^1(0(5(x1))) 3^1(4(2(x1))) -> 3^1(4(0(2(2(2(x1)))))) 3^1(4(2(x1))) -> 4^1(0(2(2(2(x1))))) 3^1(4(2(x1))) -> 2^1(2(2(x1))) 3^1(4(2(x1))) -> 2^1(2(x1)) 3^1(5(1(x1))) -> 4^1(2(0(0(5(x1))))) 3^1(5(1(x1))) -> 2^1(0(0(5(x1)))) 3^1(5(1(x1))) -> 5^1(x1) 3^1(5(1(x1))) -> 4^1(2(2(3(4(x1))))) 3^1(5(1(x1))) -> 2^1(2(3(4(x1)))) 3^1(5(1(x1))) -> 2^1(3(4(x1))) 3^1(5(1(x1))) -> 3^1(4(x1)) 3^1(5(1(x1))) -> 4^1(x1) 3^1(5(1(x1))) -> 2^1(1(4(1(0(1(x1)))))) 3^1(5(1(x1))) -> 1^1(4(1(0(1(x1))))) 3^1(5(1(x1))) -> 4^1(1(0(1(x1)))) 3^1(5(1(x1))) -> 1^1(0(1(x1))) 3^1(5(2(x1))) -> 4^1(3(2(2(2(x1))))) 3^1(5(2(x1))) -> 3^1(2(2(2(x1)))) 3^1(5(2(x1))) -> 2^1(2(2(x1))) 3^1(5(2(x1))) -> 2^1(2(x1)) 3^1(5(2(x1))) -> 2^1(0(2(2(3(0(x1)))))) 3^1(5(2(x1))) -> 2^1(2(3(0(x1)))) 3^1(5(2(x1))) -> 2^1(3(0(x1))) 3^1(5(2(x1))) -> 3^1(0(x1)) 3^1(5(2(x1))) -> 2^1(3(3(2(1(2(x1)))))) 3^1(5(2(x1))) -> 3^1(3(2(1(2(x1))))) 3^1(5(2(x1))) -> 3^1(2(1(2(x1)))) 3^1(5(2(x1))) -> 2^1(1(2(x1))) 3^1(5(2(x1))) -> 1^1(2(x1)) 3^1(5(3(x1))) -> 2^1(4(3(3(0(x1))))) 3^1(5(3(x1))) -> 4^1(3(3(0(x1)))) 3^1(5(3(x1))) -> 3^1(3(0(x1))) 3^1(5(3(x1))) -> 3^1(0(x1)) 3^1(5(3(x1))) -> 5^1(4(3(3(0(x1))))) 3^1(5(3(x1))) -> 2^1(3(4(0(4(2(x1)))))) 3^1(5(3(x1))) -> 3^1(4(0(4(2(x1))))) 3^1(5(3(x1))) -> 4^1(0(4(2(x1)))) 3^1(5(3(x1))) -> 4^1(2(x1)) 3^1(5(3(x1))) -> 2^1(x1) 3^1(5(4(x1))) -> 2^1(0(5(0(0(x1))))) 3^1(5(4(x1))) -> 5^1(0(0(x1))) 3^1(5(4(x1))) -> 5^1(0(0(1(2(x1))))) 3^1(5(4(x1))) -> 1^1(2(x1)) 3^1(5(4(x1))) -> 2^1(x1) 3^1(5(5(x1))) -> 5^1(4(1(0(5(x1))))) 3^1(5(5(x1))) -> 4^1(1(0(5(x1)))) 3^1(5(5(x1))) -> 1^1(0(5(x1))) 4^1(5(1(x1))) -> 2^1(1(0(5(3(3(x1)))))) 4^1(5(1(x1))) -> 1^1(0(5(3(3(x1))))) 4^1(5(1(x1))) -> 5^1(3(3(x1))) 4^1(5(1(x1))) -> 3^1(3(x1)) 4^1(5(1(x1))) -> 3^1(x1) 4^1(5(2(x1))) -> 5^1(1(0(0(4(x1))))) 4^1(5(2(x1))) -> 1^1(0(0(4(x1)))) 4^1(5(2(x1))) -> 4^1(x1) 4^1(5(4(x1))) -> 2^1(2(1(0(4(2(x1)))))) 4^1(5(4(x1))) -> 2^1(1(0(4(2(x1))))) 4^1(5(4(x1))) -> 1^1(0(4(2(x1)))) 4^1(5(4(x1))) -> 4^1(2(x1)) 4^1(5(4(x1))) -> 2^1(x1) 4^1(5(4(x1))) -> 3^1(2(0(3(2(0(x1)))))) 4^1(5(4(x1))) -> 2^1(0(3(2(0(x1))))) 4^1(5(4(x1))) -> 3^1(2(0(x1))) 4^1(5(4(x1))) -> 2^1(0(x1)) 5^1(5(3(x1))) -> 5^1(1(0(1(2(2(x1)))))) 5^1(5(3(x1))) -> 1^1(0(1(2(2(x1))))) 5^1(5(3(x1))) -> 1^1(2(2(x1))) 5^1(5(3(x1))) -> 2^1(2(x1)) 5^1(5(3(x1))) -> 2^1(x1) 5^1(5(4(x1))) -> 5^1(1(0(4(2(2(x1)))))) 5^1(5(4(x1))) -> 1^1(0(4(2(2(x1))))) 5^1(5(4(x1))) -> 4^1(2(2(x1))) 5^1(5(4(x1))) -> 2^1(2(x1)) 5^1(5(4(x1))) -> 2^1(x1) The TRS R consists of the following rules: 2(5(x1)) -> 1(3(3(0(1(0(x1)))))) 2(5(x1)) -> 2(2(0(5(0(1(x1)))))) 3(5(x1)) -> 1(3(2(0(0(1(x1)))))) 3(5(x1)) -> 3(2(0(5(3(0(x1)))))) 4(5(x1)) -> 2(2(1(3(2(1(x1)))))) 4(5(x1)) -> 3(2(0(5(0(0(x1)))))) 1(2(5(x1))) -> 1(0(5(0(5(4(x1)))))) 1(2(5(x1))) -> 1(2(2(1(0(1(x1)))))) 1(2(5(x1))) -> 2(0(1(3(1(0(x1)))))) 1(4(5(x1))) -> 1(2(4(0(2(1(x1)))))) 2(5(1(x1))) -> 2(2(2(1(2(3(x1)))))) 2(5(2(x1))) -> 4(0(2(2(3(3(x1)))))) 2(5(3(x1))) -> 2(0(4(1(3(3(x1)))))) 2(5(4(x1))) -> 2(0(5(1(0(1(x1)))))) 3(2(5(x1))) -> 3(2(0(1(0(5(x1)))))) 3(4(2(x1))) -> 3(4(0(2(2(2(x1)))))) 3(5(1(x1))) -> 0(4(2(0(0(5(x1)))))) 3(5(1(x1))) -> 0(4(2(2(3(4(x1)))))) 3(5(1(x1))) -> 2(1(4(1(0(1(x1)))))) 3(5(2(x1))) -> 0(4(3(2(2(2(x1)))))) 3(5(2(x1))) -> 2(0(2(2(3(0(x1)))))) 3(5(2(x1))) -> 2(3(3(2(1(2(x1)))))) 3(5(3(x1))) -> 0(2(4(3(3(0(x1)))))) 3(5(3(x1))) -> 0(5(4(3(3(0(x1)))))) 3(5(3(x1))) -> 2(3(4(0(4(2(x1)))))) 3(5(4(x1))) -> 0(2(0(5(0(0(x1)))))) 3(5(4(x1))) -> 0(5(0(0(1(2(x1)))))) 3(5(5(x1))) -> 0(5(4(1(0(5(x1)))))) 4(5(1(x1))) -> 2(1(0(5(3(3(x1)))))) 4(5(2(x1))) -> 0(5(1(0(0(4(x1)))))) 4(5(4(x1))) -> 2(2(1(0(4(2(x1)))))) 4(5(4(x1))) -> 3(2(0(3(2(0(x1)))))) 5(5(3(x1))) -> 5(1(0(1(2(2(x1)))))) 5(5(4(x1))) -> 5(1(0(4(2(2(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (3) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 79 less nodes. ---------------------------------------- (4) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(5(x1)) -> 1^1(x1) 1^1(2(5(x1))) -> 5^1(4(x1)) 5^1(5(3(x1))) -> 1^1(2(2(x1))) 1^1(2(5(x1))) -> 4^1(x1) 4^1(5(x1)) -> 2^1(2(1(3(2(1(x1)))))) 2^1(5(1(x1))) -> 2^1(2(2(1(2(3(x1)))))) 2^1(5(1(x1))) -> 2^1(2(1(2(3(x1))))) 2^1(5(1(x1))) -> 2^1(1(2(3(x1)))) 2^1(5(1(x1))) -> 1^1(2(3(x1))) 1^1(2(5(x1))) -> 1^1(x1) 1^1(4(5(x1))) -> 2^1(1(x1)) 2^1(5(1(x1))) -> 2^1(3(x1)) 2^1(5(1(x1))) -> 3^1(x1) 3^1(5(x1)) -> 1^1(x1) 1^1(4(5(x1))) -> 1^1(x1) 3^1(4(2(x1))) -> 2^1(2(2(x1))) 2^1(5(2(x1))) -> 2^1(2(3(3(x1)))) 2^1(5(2(x1))) -> 2^1(3(3(x1))) 2^1(5(2(x1))) -> 3^1(3(x1)) 3^1(4(2(x1))) -> 2^1(2(x1)) 2^1(5(2(x1))) -> 3^1(x1) 3^1(5(1(x1))) -> 5^1(x1) 5^1(5(3(x1))) -> 2^1(2(x1)) 2^1(5(3(x1))) -> 4^1(1(3(3(x1)))) 4^1(5(x1)) -> 2^1(1(3(2(1(x1))))) 2^1(5(3(x1))) -> 1^1(3(3(x1))) 2^1(5(3(x1))) -> 3^1(3(x1)) 3^1(5(1(x1))) -> 4^1(2(2(3(4(x1))))) 4^1(5(x1)) -> 1^1(3(2(1(x1)))) 4^1(5(x1)) -> 3^1(2(1(x1))) 3^1(5(1(x1))) -> 2^1(2(3(4(x1)))) 2^1(5(4(x1))) -> 1^1(x1) 3^1(5(1(x1))) -> 2^1(3(4(x1))) 3^1(5(1(x1))) -> 3^1(4(x1)) 3^1(5(1(x1))) -> 4^1(x1) 4^1(5(x1)) -> 2^1(1(x1)) 4^1(5(x1)) -> 1^1(x1) 4^1(5(1(x1))) -> 5^1(3(3(x1))) 5^1(5(3(x1))) -> 2^1(x1) 5^1(5(4(x1))) -> 4^1(2(2(x1))) 4^1(5(1(x1))) -> 3^1(3(x1)) 3^1(5(2(x1))) -> 4^1(3(2(2(2(x1))))) 4^1(5(1(x1))) -> 3^1(x1) 3^1(5(2(x1))) -> 3^1(2(2(2(x1)))) 3^1(5(2(x1))) -> 2^1(2(2(x1))) 3^1(5(2(x1))) -> 2^1(2(x1)) 3^1(5(2(x1))) -> 2^1(3(3(2(1(2(x1)))))) 3^1(5(2(x1))) -> 3^1(3(2(1(2(x1))))) 3^1(5(2(x1))) -> 3^1(2(1(2(x1)))) 3^1(5(2(x1))) -> 2^1(1(2(x1))) 3^1(5(2(x1))) -> 1^1(2(x1)) 3^1(5(3(x1))) -> 4^1(2(x1)) 4^1(5(2(x1))) -> 4^1(x1) 4^1(5(4(x1))) -> 4^1(2(x1)) 4^1(5(4(x1))) -> 2^1(x1) 3^1(5(3(x1))) -> 2^1(x1) 3^1(5(4(x1))) -> 1^1(2(x1)) 3^1(5(4(x1))) -> 2^1(x1) 5^1(5(4(x1))) -> 2^1(2(x1)) 5^1(5(4(x1))) -> 2^1(x1) The TRS R consists of the following rules: 2(5(x1)) -> 1(3(3(0(1(0(x1)))))) 2(5(x1)) -> 2(2(0(5(0(1(x1)))))) 3(5(x1)) -> 1(3(2(0(0(1(x1)))))) 3(5(x1)) -> 3(2(0(5(3(0(x1)))))) 4(5(x1)) -> 2(2(1(3(2(1(x1)))))) 4(5(x1)) -> 3(2(0(5(0(0(x1)))))) 1(2(5(x1))) -> 1(0(5(0(5(4(x1)))))) 1(2(5(x1))) -> 1(2(2(1(0(1(x1)))))) 1(2(5(x1))) -> 2(0(1(3(1(0(x1)))))) 1(4(5(x1))) -> 1(2(4(0(2(1(x1)))))) 2(5(1(x1))) -> 2(2(2(1(2(3(x1)))))) 2(5(2(x1))) -> 4(0(2(2(3(3(x1)))))) 2(5(3(x1))) -> 2(0(4(1(3(3(x1)))))) 2(5(4(x1))) -> 2(0(5(1(0(1(x1)))))) 3(2(5(x1))) -> 3(2(0(1(0(5(x1)))))) 3(4(2(x1))) -> 3(4(0(2(2(2(x1)))))) 3(5(1(x1))) -> 0(4(2(0(0(5(x1)))))) 3(5(1(x1))) -> 0(4(2(2(3(4(x1)))))) 3(5(1(x1))) -> 2(1(4(1(0(1(x1)))))) 3(5(2(x1))) -> 0(4(3(2(2(2(x1)))))) 3(5(2(x1))) -> 2(0(2(2(3(0(x1)))))) 3(5(2(x1))) -> 2(3(3(2(1(2(x1)))))) 3(5(3(x1))) -> 0(2(4(3(3(0(x1)))))) 3(5(3(x1))) -> 0(5(4(3(3(0(x1)))))) 3(5(3(x1))) -> 2(3(4(0(4(2(x1)))))) 3(5(4(x1))) -> 0(2(0(5(0(0(x1)))))) 3(5(4(x1))) -> 0(5(0(0(1(2(x1)))))) 3(5(5(x1))) -> 0(5(4(1(0(5(x1)))))) 4(5(1(x1))) -> 2(1(0(5(3(3(x1)))))) 4(5(2(x1))) -> 0(5(1(0(0(4(x1)))))) 4(5(4(x1))) -> 2(2(1(0(4(2(x1)))))) 4(5(4(x1))) -> 3(2(0(3(2(0(x1)))))) 5(5(3(x1))) -> 5(1(0(1(2(2(x1)))))) 5(5(4(x1))) -> 5(1(0(4(2(2(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (5) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 2^1(5(x1)) -> 1^1(x1) 1^1(2(5(x1))) -> 5^1(4(x1)) 5^1(5(3(x1))) -> 1^1(2(2(x1))) 1^1(2(5(x1))) -> 4^1(x1) 4^1(5(x1)) -> 2^1(2(1(3(2(1(x1)))))) 2^1(5(1(x1))) -> 2^1(2(2(1(2(3(x1)))))) 2^1(5(1(x1))) -> 2^1(2(1(2(3(x1))))) 2^1(5(1(x1))) -> 2^1(1(2(3(x1)))) 2^1(5(1(x1))) -> 1^1(2(3(x1))) 1^1(2(5(x1))) -> 1^1(x1) 1^1(4(5(x1))) -> 2^1(1(x1)) 2^1(5(1(x1))) -> 2^1(3(x1)) 2^1(5(1(x1))) -> 3^1(x1) 3^1(5(x1)) -> 1^1(x1) 1^1(4(5(x1))) -> 1^1(x1) 2^1(5(2(x1))) -> 2^1(2(3(3(x1)))) 2^1(5(2(x1))) -> 2^1(3(3(x1))) 2^1(5(2(x1))) -> 3^1(3(x1)) 2^1(5(2(x1))) -> 3^1(x1) 3^1(5(1(x1))) -> 5^1(x1) 5^1(5(3(x1))) -> 2^1(2(x1)) 2^1(5(3(x1))) -> 4^1(1(3(3(x1)))) 4^1(5(x1)) -> 2^1(1(3(2(1(x1))))) 2^1(5(3(x1))) -> 1^1(3(3(x1))) 2^1(5(3(x1))) -> 3^1(3(x1)) 3^1(5(1(x1))) -> 4^1(2(2(3(4(x1))))) 4^1(5(x1)) -> 1^1(3(2(1(x1)))) 4^1(5(x1)) -> 3^1(2(1(x1))) 3^1(5(1(x1))) -> 2^1(2(3(4(x1)))) 2^1(5(4(x1))) -> 1^1(x1) 3^1(5(1(x1))) -> 2^1(3(4(x1))) 3^1(5(1(x1))) -> 3^1(4(x1)) 3^1(5(1(x1))) -> 4^1(x1) 4^1(5(x1)) -> 2^1(1(x1)) 4^1(5(x1)) -> 1^1(x1) 4^1(5(1(x1))) -> 5^1(3(3(x1))) 5^1(5(3(x1))) -> 2^1(x1) 5^1(5(4(x1))) -> 4^1(2(2(x1))) 4^1(5(1(x1))) -> 3^1(3(x1)) 3^1(5(2(x1))) -> 4^1(3(2(2(2(x1))))) 4^1(5(1(x1))) -> 3^1(x1) 3^1(5(2(x1))) -> 3^1(2(2(2(x1)))) 3^1(5(2(x1))) -> 2^1(2(2(x1))) 3^1(5(2(x1))) -> 2^1(2(x1)) 3^1(5(2(x1))) -> 2^1(3(3(2(1(2(x1)))))) 3^1(5(2(x1))) -> 3^1(3(2(1(2(x1))))) 3^1(5(2(x1))) -> 3^1(2(1(2(x1)))) 3^1(5(2(x1))) -> 2^1(1(2(x1))) 3^1(5(2(x1))) -> 1^1(2(x1)) 3^1(5(3(x1))) -> 4^1(2(x1)) 4^1(5(2(x1))) -> 4^1(x1) 4^1(5(4(x1))) -> 4^1(2(x1)) 4^1(5(4(x1))) -> 2^1(x1) 3^1(5(3(x1))) -> 2^1(x1) 3^1(5(4(x1))) -> 1^1(2(x1)) 3^1(5(4(x1))) -> 2^1(x1) 5^1(5(4(x1))) -> 2^1(2(x1)) 5^1(5(4(x1))) -> 2^1(x1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 0 POL(1(x_1)) = x_1 POL(1^1(x_1)) = x_1 POL(2(x_1)) = x_1 POL(2^1(x_1)) = x_1 POL(3(x_1)) = x_1 POL(3^1(x_1)) = x_1 POL(4(x_1)) = x_1 POL(4^1(x_1)) = x_1 POL(5(x_1)) = 1 + x_1 POL(5^1(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 4(5(x1)) -> 2(2(1(3(2(1(x1)))))) 4(5(x1)) -> 3(2(0(5(0(0(x1)))))) 4(5(1(x1))) -> 2(1(0(5(3(3(x1)))))) 4(5(2(x1))) -> 0(5(1(0(0(4(x1)))))) 4(5(4(x1))) -> 2(2(1(0(4(2(x1)))))) 4(5(4(x1))) -> 3(2(0(3(2(0(x1)))))) 2(5(x1)) -> 1(3(3(0(1(0(x1)))))) 2(5(x1)) -> 2(2(0(5(0(1(x1)))))) 2(5(1(x1))) -> 2(2(2(1(2(3(x1)))))) 2(5(2(x1))) -> 4(0(2(2(3(3(x1)))))) 2(5(3(x1))) -> 2(0(4(1(3(3(x1)))))) 2(5(4(x1))) -> 2(0(5(1(0(1(x1)))))) 1(2(5(x1))) -> 1(0(5(0(5(4(x1)))))) 1(2(5(x1))) -> 1(2(2(1(0(1(x1)))))) 1(2(5(x1))) -> 2(0(1(3(1(0(x1)))))) 1(4(5(x1))) -> 1(2(4(0(2(1(x1)))))) 3(5(x1)) -> 1(3(2(0(0(1(x1)))))) 3(5(x1)) -> 3(2(0(5(3(0(x1)))))) 3(2(5(x1))) -> 3(2(0(1(0(5(x1)))))) 3(4(2(x1))) -> 3(4(0(2(2(2(x1)))))) 3(5(1(x1))) -> 0(4(2(0(0(5(x1)))))) 3(5(1(x1))) -> 0(4(2(2(3(4(x1)))))) 3(5(1(x1))) -> 2(1(4(1(0(1(x1)))))) 3(5(2(x1))) -> 0(4(3(2(2(2(x1)))))) 3(5(2(x1))) -> 2(0(2(2(3(0(x1)))))) 3(5(2(x1))) -> 2(3(3(2(1(2(x1)))))) 3(5(3(x1))) -> 0(2(4(3(3(0(x1)))))) 3(5(3(x1))) -> 0(5(4(3(3(0(x1)))))) 3(5(3(x1))) -> 2(3(4(0(4(2(x1)))))) 3(5(4(x1))) -> 0(2(0(5(0(0(x1)))))) 3(5(4(x1))) -> 0(5(0(0(1(2(x1)))))) 3(5(5(x1))) -> 0(5(4(1(0(5(x1)))))) ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(4(2(x1))) -> 2^1(2(2(x1))) 3^1(4(2(x1))) -> 2^1(2(x1)) The TRS R consists of the following rules: 2(5(x1)) -> 1(3(3(0(1(0(x1)))))) 2(5(x1)) -> 2(2(0(5(0(1(x1)))))) 3(5(x1)) -> 1(3(2(0(0(1(x1)))))) 3(5(x1)) -> 3(2(0(5(3(0(x1)))))) 4(5(x1)) -> 2(2(1(3(2(1(x1)))))) 4(5(x1)) -> 3(2(0(5(0(0(x1)))))) 1(2(5(x1))) -> 1(0(5(0(5(4(x1)))))) 1(2(5(x1))) -> 1(2(2(1(0(1(x1)))))) 1(2(5(x1))) -> 2(0(1(3(1(0(x1)))))) 1(4(5(x1))) -> 1(2(4(0(2(1(x1)))))) 2(5(1(x1))) -> 2(2(2(1(2(3(x1)))))) 2(5(2(x1))) -> 4(0(2(2(3(3(x1)))))) 2(5(3(x1))) -> 2(0(4(1(3(3(x1)))))) 2(5(4(x1))) -> 2(0(5(1(0(1(x1)))))) 3(2(5(x1))) -> 3(2(0(1(0(5(x1)))))) 3(4(2(x1))) -> 3(4(0(2(2(2(x1)))))) 3(5(1(x1))) -> 0(4(2(0(0(5(x1)))))) 3(5(1(x1))) -> 0(4(2(2(3(4(x1)))))) 3(5(1(x1))) -> 2(1(4(1(0(1(x1)))))) 3(5(2(x1))) -> 0(4(3(2(2(2(x1)))))) 3(5(2(x1))) -> 2(0(2(2(3(0(x1)))))) 3(5(2(x1))) -> 2(3(3(2(1(2(x1)))))) 3(5(3(x1))) -> 0(2(4(3(3(0(x1)))))) 3(5(3(x1))) -> 0(5(4(3(3(0(x1)))))) 3(5(3(x1))) -> 2(3(4(0(4(2(x1)))))) 3(5(4(x1))) -> 0(2(0(5(0(0(x1)))))) 3(5(4(x1))) -> 0(5(0(0(1(2(x1)))))) 3(5(5(x1))) -> 0(5(4(1(0(5(x1)))))) 4(5(1(x1))) -> 2(1(0(5(3(3(x1)))))) 4(5(2(x1))) -> 0(5(1(0(0(4(x1)))))) 4(5(4(x1))) -> 2(2(1(0(4(2(x1)))))) 4(5(4(x1))) -> 3(2(0(3(2(0(x1)))))) 5(5(3(x1))) -> 5(1(0(1(2(2(x1)))))) 5(5(4(x1))) -> 5(1(0(4(2(2(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. ---------------------------------------- (8) TRUE