YES proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) DependencyPairsProof [EQUIVALENT, 203 ms] (2) QDP (3) DependencyGraphProof [EQUIVALENT, 0 ms] (4) QDP (5) QDPOrderProof [EQUIVALENT, 321 ms] (6) QDP (7) DependencyGraphProof [EQUIVALENT, 1 ms] (8) QDP (9) QDPOrderProof [EQUIVALENT, 196 ms] (10) QDP (11) DependencyGraphProof [EQUIVALENT, 0 ms] (12) QDP (13) QDPOrderProof [EQUIVALENT, 101 ms] (14) QDP (15) DependencyGraphProof [EQUIVALENT, 0 ms] (16) AND (17) QDP (18) QDPOrderProof [EQUIVALENT, 66 ms] (19) QDP (20) QDPOrderProof [EQUIVALENT, 3441 ms] (21) QDP (22) PisEmptyProof [EQUIVALENT, 0 ms] (23) YES (24) QDP (25) QDPOrderProof [EQUIVALENT, 50 ms] (26) QDP (27) QDPOrderProof [EQUIVALENT, 70 ms] (28) QDP (29) QDPOrderProof [EQUIVALENT, 71 ms] (30) QDP (31) PisEmptyProof [EQUIVALENT, 0 ms] (32) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 0(3(2(x1))) -> 2(1(3(0(x1)))) 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 3(2(0(x1))) -> 0(2(1(3(x1)))) 3(2(0(x1))) -> 3(0(2(1(x1)))) 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 3(2(5(x1))) -> 1(3(5(2(x1)))) 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 5(4(2(0(2(x1))))) -> 5(0(4(2(1(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: 0^1(0(0(x1))) -> 0^1(0(1(0(1(x1))))) 0^1(0(0(x1))) -> 0^1(1(0(1(x1)))) 0^1(0(0(x1))) -> 0^1(1(x1)) 0^1(2(0(x1))) -> 0^1(0(2(1(1(x1))))) 0^1(2(0(x1))) -> 0^1(2(1(1(x1)))) 0^1(3(2(x1))) -> 3^1(0(x1)) 0^1(3(2(x1))) -> 0^1(x1) 0^1(4(2(x1))) -> 4^1(1(1(1(0(2(x1)))))) 0^1(4(2(x1))) -> 0^1(2(x1)) 3^1(0(4(x1))) -> 0^1(1(3(4(1(x1))))) 3^1(0(4(x1))) -> 3^1(4(1(x1))) 3^1(0(4(x1))) -> 4^1(1(x1)) 3^1(2(0(x1))) -> 0^1(2(1(3(x1)))) 3^1(2(0(x1))) -> 3^1(x1) 3^1(2(0(x1))) -> 3^1(0(2(1(x1)))) 3^1(2(0(x1))) -> 0^1(2(1(x1))) 3^1(2(0(x1))) -> 0^1(1(2(1(3(x1))))) 3^1(2(0(x1))) -> 3^1(0(1(2(1(4(x1)))))) 3^1(2(0(x1))) -> 0^1(1(2(1(4(x1))))) 3^1(2(0(x1))) -> 4^1(x1) 3^1(2(4(x1))) -> 3^1(3(4(x1))) 3^1(2(4(x1))) -> 3^1(4(x1)) 3^1(2(4(x1))) -> 4^1(1(2(1(1(3(x1)))))) 3^1(2(4(x1))) -> 3^1(x1) 3^1(2(5(x1))) -> 3^1(5(2(x1))) 3^1(2(5(x1))) -> 5^1(2(x1)) 3^1(2(5(x1))) -> 3^1(1(4(5(2(x1))))) 3^1(2(5(x1))) -> 4^1(5(2(x1))) 3^1(2(5(x1))) -> 3^1(3(5(2(x1)))) 3^1(2(5(x1))) -> 3^1(4(5(4(2(x1))))) 3^1(2(5(x1))) -> 4^1(5(4(2(x1)))) 3^1(2(5(x1))) -> 5^1(4(2(x1))) 3^1(2(5(x1))) -> 4^1(2(x1)) 3^1(5(0(x1))) -> 3^1(1(0(5(1(1(x1)))))) 3^1(5(0(x1))) -> 0^1(5(1(1(x1)))) 3^1(5(0(x1))) -> 5^1(1(1(x1))) 5^1(0(2(x1))) -> 0^1(2(5(x1))) 5^1(0(2(x1))) -> 5^1(x1) 5^1(0(4(x1))) -> 0^1(5(4(1(1(x1))))) 5^1(0(4(x1))) -> 5^1(4(1(1(x1)))) 5^1(0(4(x1))) -> 4^1(1(1(x1))) 5^1(2(4(x1))) -> 5^1(2(1(1(3(4(x1)))))) 5^1(2(4(x1))) -> 3^1(4(x1)) 5^1(3(2(x1))) -> 3^1(5(1(2(x1)))) 5^1(3(2(x1))) -> 5^1(1(2(x1))) 5^1(5(0(x1))) -> 5^1(1(1(5(0(x1))))) 0^1(0(2(4(x1)))) -> 0^1(4(0(2(1(1(x1)))))) 0^1(0(2(4(x1)))) -> 4^1(0(2(1(1(x1))))) 0^1(0(2(4(x1)))) -> 0^1(2(1(1(x1)))) 0^1(3(1(2(x1)))) -> 3^1(0(2(x1))) 0^1(3(1(2(x1)))) -> 0^1(2(x1)) 0^1(3(2(4(x1)))) -> 3^1(0(4(4(x1)))) 0^1(3(2(4(x1)))) -> 0^1(4(4(x1))) 0^1(3(2(4(x1)))) -> 4^1(4(x1)) 0^1(4(3(2(x1)))) -> 3^1(4(0(1(x1)))) 0^1(4(3(2(x1)))) -> 4^1(0(1(x1))) 0^1(4(3(2(x1)))) -> 0^1(1(x1)) 3^1(0(2(4(x1)))) -> 3^1(0(4(2(1(1(x1)))))) 3^1(0(2(4(x1)))) -> 0^1(4(2(1(1(x1))))) 3^1(0(2(4(x1)))) -> 4^1(2(1(1(x1)))) 3^1(0(5(4(x1)))) -> 3^1(4(0(5(1(3(x1)))))) 3^1(0(5(4(x1)))) -> 4^1(0(5(1(3(x1))))) 3^1(0(5(4(x1)))) -> 0^1(5(1(3(x1)))) 3^1(0(5(4(x1)))) -> 5^1(1(3(x1))) 3^1(0(5(4(x1)))) -> 3^1(x1) 3^1(1(0(0(x1)))) -> 0^1(1(1(3(0(1(x1)))))) 3^1(1(0(0(x1)))) -> 3^1(0(1(x1))) 3^1(1(0(0(x1)))) -> 0^1(1(x1)) 3^1(2(4(0(x1)))) -> 3^1(1(0(4(2(x1))))) 3^1(2(4(0(x1)))) -> 0^1(4(2(x1))) 3^1(2(4(0(x1)))) -> 4^1(2(x1)) 3^1(2(4(5(x1)))) -> 3^1(1(4(2(1(5(x1)))))) 3^1(2(4(5(x1)))) -> 4^1(2(1(5(x1)))) 3^1(2(5(4(x1)))) -> 3^1(4(3(5(2(x1))))) 3^1(2(5(4(x1)))) -> 4^1(3(5(2(x1)))) 3^1(2(5(4(x1)))) -> 3^1(5(2(x1))) 3^1(2(5(4(x1)))) -> 5^1(2(x1)) 3^1(5(3(2(x1)))) -> 3^1(3(0(2(5(x1))))) 3^1(5(3(2(x1)))) -> 3^1(0(2(5(x1)))) 3^1(5(3(2(x1)))) -> 0^1(2(5(x1))) 3^1(5(3(2(x1)))) -> 5^1(x1) 4^1(0(4(2(x1)))) -> 4^1(4(2(1(3(0(x1)))))) 4^1(0(4(2(x1)))) -> 4^1(2(1(3(0(x1))))) 4^1(0(4(2(x1)))) -> 3^1(0(x1)) 4^1(0(4(2(x1)))) -> 0^1(x1) 4^1(3(2(0(x1)))) -> 0^1(2(1(3(4(x1))))) 4^1(3(2(0(x1)))) -> 3^1(4(x1)) 4^1(3(2(0(x1)))) -> 4^1(x1) 5^1(4(1(0(x1)))) -> 5^1(1(1(4(0(1(x1)))))) 5^1(4(1(0(x1)))) -> 4^1(0(1(x1))) 5^1(4(1(0(x1)))) -> 0^1(1(x1)) 5^1(5(0(0(x1)))) -> 0^1(5(5(0(x1)))) 5^1(5(0(0(x1)))) -> 5^1(5(0(x1))) 5^1(5(0(0(x1)))) -> 5^1(0(x1)) 5^1(5(2(2(x1)))) -> 5^1(5(1(1(2(2(x1)))))) 5^1(5(2(2(x1)))) -> 5^1(1(1(2(2(x1))))) 0^1(0(1(2(4(x1))))) -> 0^1(0(2(1(1(4(x1)))))) 0^1(0(1(2(4(x1))))) -> 0^1(2(1(1(4(x1))))) 0^1(5(3(1(2(x1))))) -> 5^1(2(1(3(1(0(x1)))))) 0^1(5(3(1(2(x1))))) -> 3^1(1(0(x1))) 0^1(5(3(1(2(x1))))) -> 0^1(x1) 0^1(5(5(1(0(x1))))) -> 5^1(0(0(5(1(1(x1)))))) 0^1(5(5(1(0(x1))))) -> 0^1(0(5(1(1(x1))))) 0^1(5(5(1(0(x1))))) -> 0^1(5(1(1(x1)))) 0^1(5(5(1(0(x1))))) -> 5^1(1(1(x1))) 3^1(0(1(4(5(x1))))) -> 3^1(4(0(5(1(1(x1)))))) 3^1(0(1(4(5(x1))))) -> 4^1(0(5(1(1(x1))))) 3^1(0(1(4(5(x1))))) -> 0^1(5(1(1(x1)))) 3^1(0(1(4(5(x1))))) -> 5^1(1(1(x1))) 3^1(0(3(1(2(x1))))) -> 0^1(2(1(1(3(3(x1)))))) 3^1(0(3(1(2(x1))))) -> 3^1(3(x1)) 3^1(0(3(1(2(x1))))) -> 3^1(x1) 3^1(2(2(0(4(x1))))) -> 3^1(2(2(0(1(4(x1)))))) 3^1(2(2(0(4(x1))))) -> 0^1(1(4(x1))) 3^1(5(1(4(0(x1))))) -> 3^1(3(4(5(0(x1))))) 3^1(5(1(4(0(x1))))) -> 3^1(4(5(0(x1)))) 3^1(5(1(4(0(x1))))) -> 4^1(5(0(x1))) 3^1(5(1(4(0(x1))))) -> 5^1(0(x1)) 3^1(5(3(0(4(x1))))) -> 3^1(0(1(5(3(4(x1)))))) 3^1(5(3(0(4(x1))))) -> 0^1(1(5(3(4(x1))))) 3^1(5(3(0(4(x1))))) -> 5^1(3(4(x1))) 3^1(5(3(0(4(x1))))) -> 3^1(4(x1)) 5^1(1(0(2(4(x1))))) -> 5^1(4(0(3(x1)))) 5^1(1(0(2(4(x1))))) -> 4^1(0(3(x1))) 5^1(1(0(2(4(x1))))) -> 0^1(3(x1)) 5^1(1(0(2(4(x1))))) -> 3^1(x1) 5^1(1(3(2(0(x1))))) -> 3^1(5(0(1(1(x1))))) 5^1(1(3(2(0(x1))))) -> 5^1(0(1(1(x1)))) 5^1(1(3(2(0(x1))))) -> 0^1(1(1(x1))) 5^1(4(2(0(2(x1))))) -> 5^1(0(4(2(1(2(x1)))))) 5^1(4(2(0(2(x1))))) -> 0^1(4(2(1(2(x1))))) 5^1(4(2(0(2(x1))))) -> 4^1(2(1(2(x1)))) The TRS R consists of the following rules: 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 0(3(2(x1))) -> 2(1(3(0(x1)))) 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 3(2(0(x1))) -> 0(2(1(3(x1)))) 3(2(0(x1))) -> 3(0(2(1(x1)))) 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 3(2(5(x1))) -> 1(3(5(2(x1)))) 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 5(4(2(0(2(x1))))) -> 5(0(4(2(1(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 78 less nodes. ---------------------------------------- (4) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(3(2(x1))) -> 3^1(0(x1)) 3^1(2(0(x1))) -> 3^1(x1) 3^1(2(0(x1))) -> 4^1(x1) 4^1(0(4(2(x1)))) -> 3^1(0(x1)) 3^1(2(4(x1))) -> 3^1(3(4(x1))) 3^1(2(4(x1))) -> 3^1(4(x1)) 3^1(2(4(x1))) -> 3^1(x1) 3^1(2(5(x1))) -> 3^1(5(2(x1))) 3^1(2(5(x1))) -> 5^1(2(x1)) 5^1(2(4(x1))) -> 3^1(4(x1)) 3^1(2(5(x1))) -> 4^1(5(2(x1))) 4^1(0(4(2(x1)))) -> 0^1(x1) 0^1(3(2(x1))) -> 0^1(x1) 0^1(3(1(2(x1)))) -> 3^1(0(2(x1))) 3^1(2(5(x1))) -> 3^1(3(5(2(x1)))) 3^1(2(5(x1))) -> 3^1(4(5(4(2(x1))))) 3^1(2(5(x1))) -> 4^1(5(4(2(x1)))) 4^1(3(2(0(x1)))) -> 3^1(4(x1)) 3^1(2(5(x1))) -> 5^1(4(2(x1))) 5^1(4(2(0(2(x1))))) -> 5^1(0(4(2(1(2(x1)))))) 5^1(0(2(x1))) -> 5^1(x1) 5^1(5(0(0(x1)))) -> 0^1(5(5(0(x1)))) 0^1(3(2(4(x1)))) -> 3^1(0(4(4(x1)))) 3^1(0(2(4(x1)))) -> 3^1(0(4(2(1(1(x1)))))) 3^1(0(5(4(x1)))) -> 3^1(4(0(5(1(3(x1)))))) 3^1(0(5(4(x1)))) -> 4^1(0(5(1(3(x1))))) 4^1(3(2(0(x1)))) -> 4^1(x1) 3^1(0(5(4(x1)))) -> 0^1(5(1(3(x1)))) 0^1(3(2(4(x1)))) -> 0^1(4(4(x1))) 0^1(3(2(4(x1)))) -> 4^1(4(x1)) 0^1(5(3(1(2(x1))))) -> 0^1(x1) 3^1(0(5(4(x1)))) -> 5^1(1(3(x1))) 5^1(1(0(2(4(x1))))) -> 5^1(4(0(3(x1)))) 5^1(5(0(0(x1)))) -> 5^1(5(0(x1))) 5^1(5(0(0(x1)))) -> 5^1(0(x1)) 5^1(1(0(2(4(x1))))) -> 4^1(0(3(x1))) 5^1(1(0(2(4(x1))))) -> 0^1(3(x1)) 5^1(1(0(2(4(x1))))) -> 3^1(x1) 3^1(0(5(4(x1)))) -> 3^1(x1) 3^1(2(5(4(x1)))) -> 3^1(4(3(5(2(x1))))) 3^1(2(5(4(x1)))) -> 4^1(3(5(2(x1)))) 3^1(2(5(4(x1)))) -> 3^1(5(2(x1))) 3^1(2(5(4(x1)))) -> 5^1(2(x1)) 3^1(5(3(2(x1)))) -> 3^1(3(0(2(5(x1))))) 3^1(5(3(2(x1)))) -> 3^1(0(2(5(x1)))) 3^1(5(3(2(x1)))) -> 5^1(x1) 3^1(0(3(1(2(x1))))) -> 3^1(3(x1)) 3^1(0(3(1(2(x1))))) -> 3^1(x1) 3^1(5(1(4(0(x1))))) -> 3^1(3(4(5(0(x1))))) 3^1(5(1(4(0(x1))))) -> 3^1(4(5(0(x1)))) 3^1(5(1(4(0(x1))))) -> 4^1(5(0(x1))) 3^1(5(1(4(0(x1))))) -> 5^1(0(x1)) 3^1(5(3(0(4(x1))))) -> 5^1(3(4(x1))) 3^1(5(3(0(4(x1))))) -> 3^1(4(x1)) The TRS R consists of the following rules: 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 0(3(2(x1))) -> 2(1(3(0(x1)))) 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 3(2(0(x1))) -> 0(2(1(3(x1)))) 3(2(0(x1))) -> 3(0(2(1(x1)))) 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 3(2(5(x1))) -> 1(3(5(2(x1)))) 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 5(4(2(0(2(x1))))) -> 5(0(4(2(1(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. 5^1(2(4(x1))) -> 3^1(4(x1)) 0^1(5(3(1(2(x1))))) -> 0^1(x1) 5^1(5(0(0(x1)))) -> 5^1(0(x1)) 5^1(1(0(2(4(x1))))) -> 4^1(0(3(x1))) 5^1(1(0(2(4(x1))))) -> 0^1(3(x1)) 5^1(1(0(2(4(x1))))) -> 3^1(x1) 3^1(0(5(4(x1)))) -> 3^1(x1) 3^1(5(3(0(4(x1))))) -> 3^1(4(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(0^1(x_1)) = x_1 POL(1(x_1)) = x_1 POL(2(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)) = 1 + x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 0(3(2(x1))) -> 2(1(3(0(x1)))) 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 3(2(0(x1))) -> 0(2(1(3(x1)))) 3(2(0(x1))) -> 3(0(2(1(x1)))) 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 3(2(5(x1))) -> 1(3(5(2(x1)))) 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(3(2(x1))) -> 3^1(0(x1)) 3^1(2(0(x1))) -> 3^1(x1) 3^1(2(0(x1))) -> 4^1(x1) 4^1(0(4(2(x1)))) -> 3^1(0(x1)) 3^1(2(4(x1))) -> 3^1(3(4(x1))) 3^1(2(4(x1))) -> 3^1(4(x1)) 3^1(2(4(x1))) -> 3^1(x1) 3^1(2(5(x1))) -> 3^1(5(2(x1))) 3^1(2(5(x1))) -> 5^1(2(x1)) 3^1(2(5(x1))) -> 4^1(5(2(x1))) 4^1(0(4(2(x1)))) -> 0^1(x1) 0^1(3(2(x1))) -> 0^1(x1) 0^1(3(1(2(x1)))) -> 3^1(0(2(x1))) 3^1(2(5(x1))) -> 3^1(3(5(2(x1)))) 3^1(2(5(x1))) -> 3^1(4(5(4(2(x1))))) 3^1(2(5(x1))) -> 4^1(5(4(2(x1)))) 4^1(3(2(0(x1)))) -> 3^1(4(x1)) 3^1(2(5(x1))) -> 5^1(4(2(x1))) 5^1(4(2(0(2(x1))))) -> 5^1(0(4(2(1(2(x1)))))) 5^1(0(2(x1))) -> 5^1(x1) 5^1(5(0(0(x1)))) -> 0^1(5(5(0(x1)))) 0^1(3(2(4(x1)))) -> 3^1(0(4(4(x1)))) 3^1(0(2(4(x1)))) -> 3^1(0(4(2(1(1(x1)))))) 3^1(0(5(4(x1)))) -> 3^1(4(0(5(1(3(x1)))))) 3^1(0(5(4(x1)))) -> 4^1(0(5(1(3(x1))))) 4^1(3(2(0(x1)))) -> 4^1(x1) 3^1(0(5(4(x1)))) -> 0^1(5(1(3(x1)))) 0^1(3(2(4(x1)))) -> 0^1(4(4(x1))) 0^1(3(2(4(x1)))) -> 4^1(4(x1)) 3^1(0(5(4(x1)))) -> 5^1(1(3(x1))) 5^1(1(0(2(4(x1))))) -> 5^1(4(0(3(x1)))) 5^1(5(0(0(x1)))) -> 5^1(5(0(x1))) 3^1(2(5(4(x1)))) -> 3^1(4(3(5(2(x1))))) 3^1(2(5(4(x1)))) -> 4^1(3(5(2(x1)))) 3^1(2(5(4(x1)))) -> 3^1(5(2(x1))) 3^1(2(5(4(x1)))) -> 5^1(2(x1)) 3^1(5(3(2(x1)))) -> 3^1(3(0(2(5(x1))))) 3^1(5(3(2(x1)))) -> 3^1(0(2(5(x1)))) 3^1(5(3(2(x1)))) -> 5^1(x1) 3^1(0(3(1(2(x1))))) -> 3^1(3(x1)) 3^1(0(3(1(2(x1))))) -> 3^1(x1) 3^1(5(1(4(0(x1))))) -> 3^1(3(4(5(0(x1))))) 3^1(5(1(4(0(x1))))) -> 3^1(4(5(0(x1)))) 3^1(5(1(4(0(x1))))) -> 4^1(5(0(x1))) 3^1(5(1(4(0(x1))))) -> 5^1(0(x1)) 3^1(5(3(0(4(x1))))) -> 5^1(3(4(x1))) The TRS R consists of the following rules: 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 0(3(2(x1))) -> 2(1(3(0(x1)))) 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 3(2(0(x1))) -> 0(2(1(3(x1)))) 3(2(0(x1))) -> 3(0(2(1(x1)))) 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 3(2(5(x1))) -> 1(3(5(2(x1)))) 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 5(4(2(0(2(x1))))) -> 5(0(4(2(1(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 1 SCC with 2 less nodes. ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(2(0(x1))) -> 3^1(x1) 3^1(2(0(x1))) -> 4^1(x1) 4^1(0(4(2(x1)))) -> 3^1(0(x1)) 3^1(2(4(x1))) -> 3^1(3(4(x1))) 3^1(2(4(x1))) -> 3^1(4(x1)) 3^1(2(4(x1))) -> 3^1(x1) 3^1(2(5(x1))) -> 3^1(5(2(x1))) 3^1(2(5(x1))) -> 4^1(5(2(x1))) 4^1(0(4(2(x1)))) -> 0^1(x1) 0^1(3(2(x1))) -> 3^1(0(x1)) 3^1(2(5(x1))) -> 3^1(3(5(2(x1)))) 3^1(2(5(x1))) -> 3^1(4(5(4(2(x1))))) 3^1(2(5(x1))) -> 4^1(5(4(2(x1)))) 4^1(3(2(0(x1)))) -> 3^1(4(x1)) 3^1(2(5(x1))) -> 5^1(4(2(x1))) 5^1(4(2(0(2(x1))))) -> 5^1(0(4(2(1(2(x1)))))) 5^1(0(2(x1))) -> 5^1(x1) 5^1(5(0(0(x1)))) -> 0^1(5(5(0(x1)))) 0^1(3(2(x1))) -> 0^1(x1) 0^1(3(1(2(x1)))) -> 3^1(0(2(x1))) 3^1(0(2(4(x1)))) -> 3^1(0(4(2(1(1(x1)))))) 3^1(0(5(4(x1)))) -> 3^1(4(0(5(1(3(x1)))))) 3^1(0(5(4(x1)))) -> 4^1(0(5(1(3(x1))))) 4^1(3(2(0(x1)))) -> 4^1(x1) 3^1(0(5(4(x1)))) -> 0^1(5(1(3(x1)))) 0^1(3(2(4(x1)))) -> 3^1(0(4(4(x1)))) 3^1(0(5(4(x1)))) -> 5^1(1(3(x1))) 5^1(1(0(2(4(x1))))) -> 5^1(4(0(3(x1)))) 5^1(5(0(0(x1)))) -> 5^1(5(0(x1))) 3^1(2(5(4(x1)))) -> 3^1(4(3(5(2(x1))))) 3^1(2(5(4(x1)))) -> 4^1(3(5(2(x1)))) 3^1(2(5(4(x1)))) -> 3^1(5(2(x1))) 3^1(5(3(2(x1)))) -> 3^1(3(0(2(5(x1))))) 3^1(5(3(2(x1)))) -> 3^1(0(2(5(x1)))) 3^1(5(3(2(x1)))) -> 5^1(x1) 3^1(0(3(1(2(x1))))) -> 3^1(3(x1)) 3^1(0(3(1(2(x1))))) -> 3^1(x1) 3^1(5(1(4(0(x1))))) -> 3^1(3(4(5(0(x1))))) 3^1(5(1(4(0(x1))))) -> 3^1(4(5(0(x1)))) 3^1(5(1(4(0(x1))))) -> 4^1(5(0(x1))) 3^1(5(1(4(0(x1))))) -> 5^1(0(x1)) 3^1(5(3(0(4(x1))))) -> 5^1(3(4(x1))) 0^1(3(2(4(x1)))) -> 0^1(4(4(x1))) 0^1(3(2(4(x1)))) -> 4^1(4(x1)) The TRS R consists of the following rules: 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 0(3(2(x1))) -> 2(1(3(0(x1)))) 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 3(2(0(x1))) -> 0(2(1(3(x1)))) 3(2(0(x1))) -> 3(0(2(1(x1)))) 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 3(2(5(x1))) -> 1(3(5(2(x1)))) 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 3^1(2(0(x1))) -> 3^1(x1) 3^1(2(0(x1))) -> 4^1(x1) 3^1(2(4(x1))) -> 3^1(3(4(x1))) 3^1(2(4(x1))) -> 3^1(4(x1)) 3^1(2(4(x1))) -> 3^1(x1) 3^1(2(5(x1))) -> 4^1(5(2(x1))) 0^1(3(2(x1))) -> 3^1(0(x1)) 3^1(2(5(x1))) -> 4^1(5(4(2(x1)))) 5^1(0(2(x1))) -> 5^1(x1) 0^1(3(2(x1))) -> 0^1(x1) 3^1(0(5(4(x1)))) -> 4^1(0(5(1(3(x1))))) 4^1(3(2(0(x1)))) -> 4^1(x1) 0^1(3(2(4(x1)))) -> 3^1(0(4(4(x1)))) 5^1(1(0(2(4(x1))))) -> 5^1(4(0(3(x1)))) 3^1(2(5(4(x1)))) -> 4^1(3(5(2(x1)))) 3^1(5(3(2(x1)))) -> 5^1(x1) 3^1(0(3(1(2(x1))))) -> 3^1(3(x1)) 3^1(0(3(1(2(x1))))) -> 3^1(x1) 3^1(5(1(4(0(x1))))) -> 4^1(5(0(x1))) 0^1(3(2(4(x1)))) -> 0^1(4(4(x1))) 0^1(3(2(4(x1)))) -> 4^1(4(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(0^1(x_1)) = 1 + x_1 POL(1(x_1)) = x_1 POL(2(x_1)) = 1 + x_1 POL(3(x_1)) = x_1 POL(3^1(x_1)) = 1 + x_1 POL(4(x_1)) = x_1 POL(4^1(x_1)) = x_1 POL(5(x_1)) = x_1 POL(5^1(x_1)) = 1 + x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 0(3(2(x1))) -> 2(1(3(0(x1)))) 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 3(2(0(x1))) -> 0(2(1(3(x1)))) 3(2(0(x1))) -> 3(0(2(1(x1)))) 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 3(2(5(x1))) -> 1(3(5(2(x1)))) 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: 4^1(0(4(2(x1)))) -> 3^1(0(x1)) 3^1(2(5(x1))) -> 3^1(5(2(x1))) 4^1(0(4(2(x1)))) -> 0^1(x1) 3^1(2(5(x1))) -> 3^1(3(5(2(x1)))) 3^1(2(5(x1))) -> 3^1(4(5(4(2(x1))))) 4^1(3(2(0(x1)))) -> 3^1(4(x1)) 3^1(2(5(x1))) -> 5^1(4(2(x1))) 5^1(4(2(0(2(x1))))) -> 5^1(0(4(2(1(2(x1)))))) 5^1(5(0(0(x1)))) -> 0^1(5(5(0(x1)))) 0^1(3(1(2(x1)))) -> 3^1(0(2(x1))) 3^1(0(2(4(x1)))) -> 3^1(0(4(2(1(1(x1)))))) 3^1(0(5(4(x1)))) -> 3^1(4(0(5(1(3(x1)))))) 3^1(0(5(4(x1)))) -> 0^1(5(1(3(x1)))) 3^1(0(5(4(x1)))) -> 5^1(1(3(x1))) 5^1(5(0(0(x1)))) -> 5^1(5(0(x1))) 3^1(2(5(4(x1)))) -> 3^1(4(3(5(2(x1))))) 3^1(2(5(4(x1)))) -> 3^1(5(2(x1))) 3^1(5(3(2(x1)))) -> 3^1(3(0(2(5(x1))))) 3^1(5(3(2(x1)))) -> 3^1(0(2(5(x1)))) 3^1(5(1(4(0(x1))))) -> 3^1(3(4(5(0(x1))))) 3^1(5(1(4(0(x1))))) -> 3^1(4(5(0(x1)))) 3^1(5(1(4(0(x1))))) -> 5^1(0(x1)) 3^1(5(3(0(4(x1))))) -> 5^1(3(4(x1))) The TRS R consists of the following rules: 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 0(3(2(x1))) -> 2(1(3(0(x1)))) 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 3(2(0(x1))) -> 0(2(1(3(x1)))) 3(2(0(x1))) -> 3(0(2(1(x1)))) 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 3(2(5(x1))) -> 1(3(5(2(x1)))) 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(2(5(x1))) -> 3^1(3(5(2(x1)))) 3^1(2(5(x1))) -> 3^1(5(2(x1))) 3^1(2(5(x1))) -> 3^1(4(5(4(2(x1))))) 3^1(2(5(x1))) -> 5^1(4(2(x1))) 5^1(4(2(0(2(x1))))) -> 5^1(0(4(2(1(2(x1)))))) 5^1(5(0(0(x1)))) -> 0^1(5(5(0(x1)))) 0^1(3(1(2(x1)))) -> 3^1(0(2(x1))) 3^1(0(2(4(x1)))) -> 3^1(0(4(2(1(1(x1)))))) 3^1(0(5(4(x1)))) -> 3^1(4(0(5(1(3(x1)))))) 3^1(0(5(4(x1)))) -> 0^1(5(1(3(x1)))) 3^1(2(5(4(x1)))) -> 3^1(4(3(5(2(x1))))) 3^1(2(5(4(x1)))) -> 3^1(5(2(x1))) 3^1(5(3(2(x1)))) -> 3^1(3(0(2(5(x1))))) 3^1(5(3(2(x1)))) -> 3^1(0(2(5(x1)))) 3^1(5(1(4(0(x1))))) -> 3^1(3(4(5(0(x1))))) 3^1(5(1(4(0(x1))))) -> 3^1(4(5(0(x1)))) 3^1(5(1(4(0(x1))))) -> 5^1(0(x1)) 5^1(5(0(0(x1)))) -> 5^1(5(0(x1))) 3^1(5(3(0(4(x1))))) -> 5^1(3(4(x1))) The TRS R consists of the following rules: 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 0(3(2(x1))) -> 2(1(3(0(x1)))) 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 3(2(0(x1))) -> 0(2(1(3(x1)))) 3(2(0(x1))) -> 3(0(2(1(x1)))) 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 3(2(5(x1))) -> 1(3(5(2(x1)))) 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(3(1(2(x1)))) -> 3^1(0(2(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 0 POL(0^1(x_1)) = x_1 POL(1(x_1)) = 0 POL(2(x_1)) = 0 POL(3(x_1)) = 1 POL(3^1(x_1)) = 0 POL(4(x_1)) = 0 POL(5(x_1)) = 0 POL(5^1(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(2(5(x1))) -> 3^1(3(5(2(x1)))) 3^1(2(5(x1))) -> 3^1(5(2(x1))) 3^1(2(5(x1))) -> 3^1(4(5(4(2(x1))))) 3^1(2(5(x1))) -> 5^1(4(2(x1))) 5^1(4(2(0(2(x1))))) -> 5^1(0(4(2(1(2(x1)))))) 5^1(5(0(0(x1)))) -> 0^1(5(5(0(x1)))) 3^1(0(2(4(x1)))) -> 3^1(0(4(2(1(1(x1)))))) 3^1(0(5(4(x1)))) -> 3^1(4(0(5(1(3(x1)))))) 3^1(0(5(4(x1)))) -> 0^1(5(1(3(x1)))) 3^1(2(5(4(x1)))) -> 3^1(4(3(5(2(x1))))) 3^1(2(5(4(x1)))) -> 3^1(5(2(x1))) 3^1(5(3(2(x1)))) -> 3^1(3(0(2(5(x1))))) 3^1(5(3(2(x1)))) -> 3^1(0(2(5(x1)))) 3^1(5(1(4(0(x1))))) -> 3^1(3(4(5(0(x1))))) 3^1(5(1(4(0(x1))))) -> 3^1(4(5(0(x1)))) 3^1(5(1(4(0(x1))))) -> 5^1(0(x1)) 5^1(5(0(0(x1)))) -> 5^1(5(0(x1))) 3^1(5(3(0(4(x1))))) -> 5^1(3(4(x1))) The TRS R consists of the following rules: 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 0(3(2(x1))) -> 2(1(3(0(x1)))) 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 3(2(0(x1))) -> 0(2(1(3(x1)))) 3(2(0(x1))) -> 3(0(2(1(x1)))) 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 3(2(5(x1))) -> 1(3(5(2(x1)))) 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (15) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 5 less nodes. ---------------------------------------- (16) Complex Obligation (AND) ---------------------------------------- (17) Obligation: Q DP problem: The TRS P consists of the following rules: 5^1(5(0(0(x1)))) -> 5^1(5(0(x1))) 5^1(4(2(0(2(x1))))) -> 5^1(0(4(2(1(2(x1)))))) The TRS R consists of the following rules: 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 0(3(2(x1))) -> 2(1(3(0(x1)))) 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 3(2(0(x1))) -> 0(2(1(3(x1)))) 3(2(0(x1))) -> 3(0(2(1(x1)))) 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 3(2(5(x1))) -> 1(3(5(2(x1)))) 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (18) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 5^1(4(2(0(2(x1))))) -> 5^1(0(4(2(1(2(x1)))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(1(x_1)) = 0 POL(2(x_1)) = 1 + x_1 POL(3(x_1)) = 0 POL(4(x_1)) = 1 + x_1 POL(5(x_1)) = 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: 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) ---------------------------------------- (19) Obligation: Q DP problem: The TRS P consists of the following rules: 5^1(5(0(0(x1)))) -> 5^1(5(0(x1))) The TRS R consists of the following rules: 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 0(3(2(x1))) -> 2(1(3(0(x1)))) 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 3(2(0(x1))) -> 0(2(1(3(x1)))) 3(2(0(x1))) -> 3(0(2(1(x1)))) 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 3(2(5(x1))) -> 1(3(5(2(x1)))) 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (20) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 5^1(5(0(0(x1)))) -> 5^1(5(0(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(5^1(x_1)) = [[0A]] + [[-I, 0A, -I]] * x_1 >>> <<< POL(5(x_1)) = [[1A], [-I], [-I]] + [[0A, 0A, 0A], [-I, -I, 0A], [0A, 0A, -I]] * x_1 >>> <<< POL(0(x_1)) = [[0A], [1A], [0A]] + [[0A, 0A, 1A], [0A, 0A, 1A], [-I, -I, 1A]] * x_1 >>> <<< POL(1(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 >>> <<< POL(2(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, 0A], [-I, 0A, -I], [-I, 0A, -I]] * x_1 >>> <<< POL(3(x_1)) = [[0A], [-I], [0A]] + [[0A, 0A, 0A], [-I, -I, -I], [0A, 0A, -I]] * x_1 >>> <<< POL(4(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 0(3(2(x1))) -> 2(1(3(0(x1)))) 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) ---------------------------------------- (21) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 0(3(2(x1))) -> 2(1(3(0(x1)))) 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 3(2(0(x1))) -> 0(2(1(3(x1)))) 3(2(0(x1))) -> 3(0(2(1(x1)))) 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 3(2(5(x1))) -> 1(3(5(2(x1)))) 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (22) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (23) YES ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(2(5(x1))) -> 3^1(5(2(x1))) 3^1(2(5(x1))) -> 3^1(3(5(2(x1)))) 3^1(2(5(x1))) -> 3^1(4(5(4(2(x1))))) 3^1(0(2(4(x1)))) -> 3^1(0(4(2(1(1(x1)))))) 3^1(0(5(4(x1)))) -> 3^1(4(0(5(1(3(x1)))))) 3^1(2(5(4(x1)))) -> 3^1(4(3(5(2(x1))))) 3^1(2(5(4(x1)))) -> 3^1(5(2(x1))) 3^1(5(3(2(x1)))) -> 3^1(3(0(2(5(x1))))) 3^1(5(3(2(x1)))) -> 3^1(0(2(5(x1)))) 3^1(5(1(4(0(x1))))) -> 3^1(3(4(5(0(x1))))) 3^1(5(1(4(0(x1))))) -> 3^1(4(5(0(x1)))) The TRS R consists of the following rules: 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 0(3(2(x1))) -> 2(1(3(0(x1)))) 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 3(2(0(x1))) -> 0(2(1(3(x1)))) 3(2(0(x1))) -> 3(0(2(1(x1)))) 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 3(2(5(x1))) -> 1(3(5(2(x1)))) 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 3^1(2(5(x1))) -> 3^1(5(2(x1))) 3^1(2(5(x1))) -> 3^1(3(5(2(x1)))) 3^1(2(5(x1))) -> 3^1(4(5(4(2(x1))))) 3^1(2(5(4(x1)))) -> 3^1(4(3(5(2(x1))))) 3^1(2(5(4(x1)))) -> 3^1(5(2(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 0 POL(1(x_1)) = 0 POL(2(x_1)) = 1 POL(3(x_1)) = 0 POL(3^1(x_1)) = x_1 POL(4(x_1)) = 0 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 3(2(0(x1))) -> 0(2(1(3(x1)))) 3(2(0(x1))) -> 3(0(2(1(x1)))) 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 3(2(5(x1))) -> 1(3(5(2(x1)))) 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 0(2(0(x1))) -> 0(0(2(1(1(x1))))) ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(0(2(4(x1)))) -> 3^1(0(4(2(1(1(x1)))))) 3^1(0(5(4(x1)))) -> 3^1(4(0(5(1(3(x1)))))) 3^1(5(3(2(x1)))) -> 3^1(3(0(2(5(x1))))) 3^1(5(3(2(x1)))) -> 3^1(0(2(5(x1)))) 3^1(5(1(4(0(x1))))) -> 3^1(3(4(5(0(x1))))) 3^1(5(1(4(0(x1))))) -> 3^1(4(5(0(x1)))) The TRS R consists of the following rules: 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 0(3(2(x1))) -> 2(1(3(0(x1)))) 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 3(2(0(x1))) -> 0(2(1(3(x1)))) 3(2(0(x1))) -> 3(0(2(1(x1)))) 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 3(2(5(x1))) -> 1(3(5(2(x1)))) 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 3^1(5(3(2(x1)))) -> 3^1(3(0(2(5(x1))))) 3^1(5(3(2(x1)))) -> 3^1(0(2(5(x1)))) 3^1(5(1(4(0(x1))))) -> 3^1(3(4(5(0(x1))))) 3^1(5(1(4(0(x1))))) -> 3^1(4(5(0(x1)))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 0 POL(1(x_1)) = 0 POL(2(x_1)) = 0 POL(3(x_1)) = 0 POL(3^1(x_1)) = x_1 POL(4(x_1)) = 0 POL(5(x_1)) = 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 3(2(0(x1))) -> 0(2(1(3(x1)))) 3(2(0(x1))) -> 3(0(2(1(x1)))) 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 3(2(5(x1))) -> 1(3(5(2(x1)))) 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) ---------------------------------------- (28) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(0(2(4(x1)))) -> 3^1(0(4(2(1(1(x1)))))) 3^1(0(5(4(x1)))) -> 3^1(4(0(5(1(3(x1)))))) The TRS R consists of the following rules: 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 0(3(2(x1))) -> 2(1(3(0(x1)))) 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 3(2(0(x1))) -> 0(2(1(3(x1)))) 3(2(0(x1))) -> 3(0(2(1(x1)))) 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 3(2(5(x1))) -> 1(3(5(2(x1)))) 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 3^1(0(2(4(x1)))) -> 3^1(0(4(2(1(1(x1)))))) 3^1(0(5(4(x1)))) -> 3^1(4(0(5(1(3(x1)))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(1(x_1)) = 0 POL(2(x_1)) = 1 + x_1 POL(3(x_1)) = 0 POL(3^1(x_1)) = x_1 POL(4(x_1)) = 1 POL(5(x_1)) = 1 + x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) ---------------------------------------- (30) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 0(3(2(x1))) -> 2(1(3(0(x1)))) 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 3(2(0(x1))) -> 0(2(1(3(x1)))) 3(2(0(x1))) -> 3(0(2(1(x1)))) 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 3(2(5(x1))) -> 1(3(5(2(x1)))) 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (32) YES