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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) DependencyPairsProof [EQUIVALENT, 319 ms] (2) QDP (3) DependencyGraphProof [EQUIVALENT, 0 ms] (4) QDP (5) QDPOrderProof [EQUIVALENT, 334 ms] (6) QDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) QDP (9) QDPOrderProof [EQUIVALENT, 235 ms] (10) QDP (11) DependencyGraphProof [EQUIVALENT, 0 ms] (12) AND (13) QDP (14) QDPOrderProof [EQUIVALENT, 82 ms] (15) QDP (16) QDPOrderProof [EQUIVALENT, 74 ms] (17) QDP (18) QDPOrderProof [EQUIVALENT, 52 ms] (19) QDP (20) QDPOrderProof [EQUIVALENT, 56 ms] (21) QDP (22) QDPOrderProof [EQUIVALENT, 52 ms] (23) QDP (24) PisEmptyProof [EQUIVALENT, 0 ms] (25) YES (26) QDP (27) QDPOrderProof [EQUIVALENT, 78 ms] (28) QDP (29) PisEmptyProof [EQUIVALENT, 0 ms] (30) YES (31) QDP (32) QDPOrderProof [EQUIVALENT, 70 ms] (33) QDP (34) PisEmptyProof [EQUIVALENT, 0 ms] (35) YES (36) QDP (37) QDPOrderProof [EQUIVALENT, 59 ms] (38) QDP (39) PisEmptyProof [EQUIVALENT, 0 ms] (40) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(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(1(2(x1))) -> 0^1(2(1(1(x1)))) 0^1(1(2(x1))) -> 2^1(1(1(x1))) 0^1(3(2(x1))) -> 0^1(2(1(3(x1)))) 0^1(3(2(x1))) -> 2^1(1(3(x1))) 0^1(3(2(x1))) -> 3^1(x1) 2^1(4(1(x1))) -> 4^1(2(1(1(x1)))) 2^1(4(1(x1))) -> 2^1(1(1(x1))) 0^1(0(3(2(x1)))) -> 0^1(0(2(1(3(x1))))) 0^1(0(3(2(x1)))) -> 0^1(2(1(3(x1)))) 0^1(0(3(2(x1)))) -> 2^1(1(3(x1))) 0^1(0(3(2(x1)))) -> 3^1(x1) 0^1(1(0(3(x1)))) -> 0^1(1(1(3(0(x1))))) 0^1(1(0(3(x1)))) -> 3^1(0(x1)) 0^1(1(0(3(x1)))) -> 0^1(x1) 0^1(1(0(5(x1)))) -> 0^1(0(1(1(5(x1))))) 0^1(1(0(5(x1)))) -> 0^1(1(1(5(x1)))) 0^1(1(2(2(x1)))) -> 0^1(2(1(1(2(x1))))) 0^1(1(2(2(x1)))) -> 2^1(1(1(2(x1)))) 0^1(1(2(3(x1)))) -> 0^1(2(1(3(1(x1))))) 0^1(1(2(3(x1)))) -> 2^1(1(3(1(x1)))) 0^1(1(2(3(x1)))) -> 3^1(1(x1)) 0^1(1(2(5(x1)))) -> 2^1(1(1(0(5(x1))))) 0^1(1(2(5(x1)))) -> 0^1(5(x1)) 0^1(1(4(1(x1)))) -> 0^1(4(1(1(1(x1))))) 0^1(1(4(1(x1)))) -> 4^1(1(1(1(x1)))) 0^1(1(4(2(x1)))) -> 0^1(0(2(1(4(x1))))) 0^1(1(4(2(x1)))) -> 0^1(2(1(4(x1)))) 0^1(1(4(2(x1)))) -> 2^1(1(4(x1))) 0^1(1(4(2(x1)))) -> 4^1(x1) 0^1(1(4(2(x1)))) -> 0^1(4(0(2(1(x1))))) 0^1(1(4(2(x1)))) -> 4^1(0(2(1(x1)))) 0^1(1(4(2(x1)))) -> 0^1(2(1(x1))) 0^1(1(4(2(x1)))) -> 2^1(1(x1)) 0^1(1(4(3(x1)))) -> 4^1(0(1(1(3(x1))))) 0^1(1(4(3(x1)))) -> 0^1(1(1(3(x1)))) 0^1(1(4(5(x1)))) -> 0^1(1(1(5(4(x1))))) 0^1(1(4(5(x1)))) -> 5^1(4(x1)) 0^1(1(4(5(x1)))) -> 4^1(x1) 0^1(1(5(2(x1)))) -> 0^1(2(5(1(1(x1))))) 0^1(1(5(2(x1)))) -> 2^1(5(1(1(x1)))) 0^1(1(5(2(x1)))) -> 5^1(1(1(x1))) 0^1(2(0(3(x1)))) -> 0^1(0(2(3(1(x1))))) 0^1(2(0(3(x1)))) -> 0^1(2(3(1(x1)))) 0^1(2(0(3(x1)))) -> 2^1(3(1(x1))) 0^1(2(0(3(x1)))) -> 3^1(1(x1)) 0^1(3(0(1(x1)))) -> 0^1(0(1(3(1(x1))))) 0^1(3(0(1(x1)))) -> 0^1(1(3(1(x1)))) 0^1(3(0(1(x1)))) -> 3^1(1(x1)) 0^1(3(0(2(x1)))) -> 0^1(0(2(3(1(x1))))) 0^1(3(0(2(x1)))) -> 0^1(2(3(1(x1)))) 0^1(3(0(2(x1)))) -> 2^1(3(1(x1))) 0^1(3(0(2(x1)))) -> 3^1(1(x1)) 0^1(3(0(3(x1)))) -> 0^1(0(3(1(3(x1))))) 0^1(3(0(3(x1)))) -> 0^1(3(1(3(x1)))) 0^1(3(0(3(x1)))) -> 3^1(1(3(x1))) 0^1(3(2(0(x1)))) -> 0^1(2(1(3(0(x1))))) 0^1(3(2(0(x1)))) -> 2^1(1(3(0(x1)))) 0^1(3(2(0(x1)))) -> 3^1(0(x1)) 0^1(3(5(1(x1)))) -> 0^1(5(3(1(1(x1))))) 0^1(3(5(1(x1)))) -> 5^1(3(1(1(x1)))) 0^1(3(5(1(x1)))) -> 3^1(1(1(x1))) 0^1(4(0(1(x1)))) -> 0^1(0(4(1(1(x1))))) 0^1(4(0(1(x1)))) -> 0^1(4(1(1(x1)))) 0^1(4(0(1(x1)))) -> 4^1(1(1(x1))) 0^1(5(2(0(x1)))) -> 2^1(1(5(0(0(x1))))) 0^1(5(2(0(x1)))) -> 5^1(0(0(x1))) 0^1(5(2(0(x1)))) -> 0^1(0(x1)) 2^1(1(2(3(x1)))) -> 2^1(1(1(3(2(x1))))) 2^1(1(2(3(x1)))) -> 3^1(2(x1)) 2^1(1(2(3(x1)))) -> 2^1(x1) 2^1(1(4(1(x1)))) -> 4^1(2(1(1(1(x1))))) 2^1(1(4(1(x1)))) -> 2^1(1(1(1(x1)))) 2^1(2(0(3(x1)))) -> 2^1(0(2(1(3(x1))))) 2^1(2(0(3(x1)))) -> 0^1(2(1(3(x1)))) 2^1(2(0(3(x1)))) -> 2^1(1(3(x1))) 2^1(2(4(1(x1)))) -> 2^1(4(2(1(1(x1))))) 2^1(2(4(1(x1)))) -> 4^1(2(1(1(x1)))) 2^1(2(4(1(x1)))) -> 2^1(1(1(x1))) 2^1(3(1(4(x1)))) -> 2^1(1(1(3(4(x1))))) 2^1(3(1(4(x1)))) -> 3^1(4(x1)) 2^1(4(1(4(x1)))) -> 4^1(4(2(1(1(x1))))) 2^1(4(1(4(x1)))) -> 4^1(2(1(1(x1)))) 2^1(4(1(4(x1)))) -> 2^1(1(1(x1))) 2^1(4(3(1(x1)))) -> 4^1(2(3(1(1(x1))))) 2^1(4(3(1(x1)))) -> 2^1(3(1(1(x1)))) 2^1(4(3(1(x1)))) -> 3^1(1(1(x1))) 2^1(5(0(1(x1)))) -> 0^1(2(5(1(1(x1))))) 2^1(5(0(1(x1)))) -> 2^1(5(1(1(x1)))) 2^1(5(0(1(x1)))) -> 5^1(1(1(x1))) 2^1(5(0(3(x1)))) -> 5^1(0(2(1(3(x1))))) 2^1(5(0(3(x1)))) -> 0^1(2(1(3(x1)))) 2^1(5(0(3(x1)))) -> 2^1(1(3(x1))) 3^1(0(1(0(x1)))) -> 3^1(1(0(0(0(x1))))) 3^1(0(1(0(x1)))) -> 0^1(0(0(x1))) 3^1(0(1(0(x1)))) -> 0^1(0(x1)) 3^1(0(1(4(x1)))) -> 3^1(1(1(0(4(x1))))) 3^1(0(1(4(x1)))) -> 0^1(4(x1)) 3^1(0(3(0(x1)))) -> 3^1(1(3(0(0(x1))))) 3^1(0(3(0(x1)))) -> 3^1(0(0(x1))) 3^1(0(3(0(x1)))) -> 0^1(0(x1)) 3^1(1(0(3(x1)))) -> 3^1(1(1(3(0(x1))))) 3^1(1(0(3(x1)))) -> 3^1(0(x1)) 3^1(1(0(3(x1)))) -> 0^1(x1) 3^1(1(2(1(x1)))) -> 3^1(2(1(1(x1)))) 3^1(1(2(1(x1)))) -> 2^1(1(1(x1))) 3^1(1(2(1(x1)))) -> 3^1(2(1(1(1(x1))))) 3^1(1(2(1(x1)))) -> 2^1(1(1(1(x1)))) 3^1(2(0(1(x1)))) -> 3^1(0(2(1(x1)))) 3^1(2(0(1(x1)))) -> 0^1(2(1(x1))) 3^1(2(0(1(x1)))) -> 2^1(1(x1)) 3^1(2(0(1(x1)))) -> 3^1(1(0(2(1(x1))))) 3^1(4(1(2(x1)))) -> 3^1(2(1(1(4(x1))))) 3^1(4(1(2(x1)))) -> 2^1(1(1(4(x1)))) 3^1(4(1(2(x1)))) -> 4^1(x1) 3^1(5(1(4(x1)))) -> 3^1(1(1(5(4(x1))))) 3^1(5(1(4(x1)))) -> 5^1(4(x1)) 4^1(1(0(1(x1)))) -> 4^1(0(1(1(1(x1))))) 4^1(1(0(1(x1)))) -> 0^1(1(1(1(x1)))) 4^1(1(2(1(x1)))) -> 2^1(4(1(1(1(x1))))) 4^1(1(2(1(x1)))) -> 4^1(1(1(1(x1)))) 4^1(1(4(3(x1)))) -> 4^1(1(1(3(4(x1))))) 4^1(1(4(3(x1)))) -> 3^1(4(x1)) 4^1(1(4(3(x1)))) -> 4^1(x1) 4^1(2(4(1(x1)))) -> 4^1(4(2(1(1(x1))))) 4^1(2(4(1(x1)))) -> 4^1(2(1(1(x1)))) 4^1(2(4(1(x1)))) -> 2^1(1(1(x1))) 4^1(4(1(2(x1)))) -> 4^1(2(1(1(4(x1))))) 4^1(4(1(2(x1)))) -> 2^1(1(1(4(x1)))) 4^1(4(1(2(x1)))) -> 4^1(x1) 4^1(5(0(1(x1)))) -> 5^1(0(4(1(1(x1))))) 4^1(5(0(1(x1)))) -> 0^1(4(1(1(x1)))) 4^1(5(0(1(x1)))) -> 4^1(1(1(x1))) 5^1(1(0(3(x1)))) -> 5^1(0(1(1(3(x1))))) 5^1(1(0(3(x1)))) -> 0^1(1(1(3(x1)))) 5^1(1(0(3(x1)))) -> 5^1(1(1(3(0(x1))))) 5^1(1(0(3(x1)))) -> 3^1(0(x1)) 5^1(1(0(3(x1)))) -> 0^1(x1) 5^1(1(3(1(x1)))) -> 5^1(3(1(1(1(x1))))) 5^1(1(3(1(x1)))) -> 3^1(1(1(1(x1)))) 5^1(1(4(3(x1)))) -> 4^1(1(1(3(5(x1))))) 5^1(1(4(3(x1)))) -> 3^1(5(x1)) 5^1(1(4(3(x1)))) -> 5^1(x1) 5^1(1(4(4(x1)))) -> 4^1(1(1(5(4(x1))))) 5^1(1(4(4(x1)))) -> 5^1(4(x1)) 5^1(1(5(3(x1)))) -> 5^1(5(1(1(3(x1))))) 5^1(1(5(3(x1)))) -> 5^1(1(1(3(x1)))) 5^1(2(0(1(x1)))) -> 2^1(0(5(1(1(x1))))) 5^1(2(0(1(x1)))) -> 0^1(5(1(1(x1)))) 5^1(2(0(1(x1)))) -> 5^1(1(1(x1))) 5^1(2(0(5(x1)))) -> 5^1(0(2(1(5(x1))))) 5^1(2(0(5(x1)))) -> 0^1(2(1(5(x1)))) 5^1(2(0(5(x1)))) -> 2^1(1(5(x1))) 5^1(3(4(1(x1)))) -> 5^1(4(3(1(1(x1))))) 5^1(3(4(1(x1)))) -> 4^1(3(1(1(x1)))) 5^1(3(4(1(x1)))) -> 3^1(1(1(x1))) 5^1(4(0(1(x1)))) -> 5^1(0(4(1(1(x1))))) 5^1(4(0(1(x1)))) -> 0^1(4(1(1(x1)))) 5^1(4(0(1(x1)))) -> 4^1(1(1(x1))) 5^1(4(1(0(x1)))) -> 4^1(1(1(5(0(x1))))) 5^1(4(1(0(x1)))) -> 5^1(0(x1)) 5^1(4(1(0(x1)))) -> 5^1(1(1(4(0(x1))))) 5^1(4(1(0(x1)))) -> 4^1(0(x1)) The TRS R consists of the following rules: 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(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 82 less nodes. ---------------------------------------- (4) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(3(2(x1))) -> 2^1(1(3(x1))) 2^1(1(2(3(x1)))) -> 3^1(2(x1)) 3^1(0(1(0(x1)))) -> 3^1(1(0(0(0(x1))))) 3^1(1(0(3(x1)))) -> 3^1(0(x1)) 3^1(0(1(0(x1)))) -> 0^1(0(0(x1))) 0^1(3(2(x1))) -> 0^1(2(1(3(x1)))) 0^1(3(2(x1))) -> 3^1(x1) 3^1(0(1(0(x1)))) -> 0^1(0(x1)) 0^1(0(3(2(x1)))) -> 0^1(0(2(1(3(x1))))) 0^1(0(3(2(x1)))) -> 0^1(2(1(3(x1)))) 0^1(0(3(2(x1)))) -> 2^1(1(3(x1))) 2^1(1(2(3(x1)))) -> 2^1(x1) 2^1(2(0(3(x1)))) -> 2^1(0(2(1(3(x1))))) 2^1(2(0(3(x1)))) -> 0^1(2(1(3(x1)))) 0^1(0(3(2(x1)))) -> 3^1(x1) 3^1(0(1(4(x1)))) -> 0^1(4(x1)) 0^1(1(0(3(x1)))) -> 3^1(0(x1)) 3^1(0(3(0(x1)))) -> 3^1(1(3(0(0(x1))))) 3^1(1(0(3(x1)))) -> 0^1(x1) 0^1(1(0(3(x1)))) -> 0^1(x1) 0^1(1(2(3(x1)))) -> 0^1(2(1(3(1(x1))))) 0^1(1(2(3(x1)))) -> 2^1(1(3(1(x1)))) 0^1(1(2(3(x1)))) -> 3^1(1(x1)) 0^1(1(2(5(x1)))) -> 0^1(5(x1)) 0^1(1(4(2(x1)))) -> 0^1(0(2(1(4(x1))))) 0^1(1(4(2(x1)))) -> 0^1(2(1(4(x1)))) 0^1(1(4(2(x1)))) -> 2^1(1(4(x1))) 0^1(1(4(2(x1)))) -> 4^1(x1) 4^1(1(4(3(x1)))) -> 3^1(4(x1)) 3^1(0(3(0(x1)))) -> 3^1(0(0(x1))) 3^1(0(3(0(x1)))) -> 0^1(0(x1)) 0^1(1(4(2(x1)))) -> 0^1(4(0(2(1(x1))))) 0^1(1(4(2(x1)))) -> 4^1(0(2(1(x1)))) 4^1(1(4(3(x1)))) -> 4^1(x1) 4^1(4(1(2(x1)))) -> 4^1(x1) 0^1(1(4(2(x1)))) -> 0^1(2(1(x1))) 0^1(1(4(2(x1)))) -> 2^1(1(x1)) 0^1(1(4(5(x1)))) -> 5^1(4(x1)) 5^1(1(0(3(x1)))) -> 3^1(0(x1)) 3^1(2(0(1(x1)))) -> 3^1(0(2(1(x1)))) 3^1(2(0(1(x1)))) -> 0^1(2(1(x1))) 0^1(1(4(5(x1)))) -> 4^1(x1) 0^1(2(0(3(x1)))) -> 0^1(0(2(3(1(x1))))) 0^1(2(0(3(x1)))) -> 0^1(2(3(1(x1)))) 0^1(2(0(3(x1)))) -> 2^1(3(1(x1))) 2^1(2(0(3(x1)))) -> 2^1(1(3(x1))) 2^1(3(1(4(x1)))) -> 3^1(4(x1)) 3^1(2(0(1(x1)))) -> 2^1(1(x1)) 3^1(2(0(1(x1)))) -> 3^1(1(0(2(1(x1))))) 3^1(4(1(2(x1)))) -> 4^1(x1) 3^1(5(1(4(x1)))) -> 5^1(4(x1)) 5^1(1(0(3(x1)))) -> 0^1(x1) 0^1(2(0(3(x1)))) -> 3^1(1(x1)) 0^1(3(0(1(x1)))) -> 0^1(0(1(3(1(x1))))) 0^1(3(0(1(x1)))) -> 0^1(1(3(1(x1)))) 0^1(3(0(1(x1)))) -> 3^1(1(x1)) 0^1(3(0(2(x1)))) -> 0^1(0(2(3(1(x1))))) 0^1(3(0(2(x1)))) -> 0^1(2(3(1(x1)))) 0^1(3(0(2(x1)))) -> 2^1(3(1(x1))) 2^1(5(0(3(x1)))) -> 5^1(0(2(1(3(x1))))) 5^1(1(4(3(x1)))) -> 3^1(5(x1)) 5^1(1(4(3(x1)))) -> 5^1(x1) 5^1(1(4(4(x1)))) -> 5^1(4(x1)) 5^1(2(0(5(x1)))) -> 5^1(0(2(1(5(x1))))) 5^1(2(0(5(x1)))) -> 0^1(2(1(5(x1)))) 0^1(3(0(2(x1)))) -> 3^1(1(x1)) 0^1(3(0(3(x1)))) -> 0^1(0(3(1(3(x1))))) 0^1(3(0(3(x1)))) -> 0^1(3(1(3(x1)))) 0^1(3(0(3(x1)))) -> 3^1(1(3(x1))) 0^1(3(2(0(x1)))) -> 0^1(2(1(3(0(x1))))) 0^1(3(2(0(x1)))) -> 2^1(1(3(0(x1)))) 0^1(3(2(0(x1)))) -> 3^1(0(x1)) 0^1(5(2(0(x1)))) -> 2^1(1(5(0(0(x1))))) 0^1(5(2(0(x1)))) -> 5^1(0(0(x1))) 5^1(2(0(5(x1)))) -> 2^1(1(5(x1))) 5^1(4(1(0(x1)))) -> 5^1(0(x1)) 5^1(4(1(0(x1)))) -> 4^1(0(x1)) 0^1(5(2(0(x1)))) -> 0^1(0(x1)) 2^1(5(0(3(x1)))) -> 0^1(2(1(3(x1)))) 2^1(5(0(3(x1)))) -> 2^1(1(3(x1))) The TRS R consists of the following rules: 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(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. 4^1(1(4(3(x1)))) -> 3^1(4(x1)) 4^1(1(4(3(x1)))) -> 4^1(x1) 4^1(4(1(2(x1)))) -> 4^1(x1) 0^1(1(4(2(x1)))) -> 0^1(2(1(x1))) 0^1(1(4(2(x1)))) -> 2^1(1(x1)) 5^1(1(4(3(x1)))) -> 3^1(5(x1)) 5^1(1(4(3(x1)))) -> 5^1(x1) 5^1(1(4(4(x1)))) -> 5^1(4(x1)) 5^1(4(1(0(x1)))) -> 5^1(0(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(2^1(x_1)) = x_1 POL(3(x_1)) = x_1 POL(3^1(x_1)) = x_1 POL(4(x_1)) = 1 + x_1 POL(4^1(x_1)) = 1 + x_1 POL(5(x_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: 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(3(2(x1))) -> 2^1(1(3(x1))) 2^1(1(2(3(x1)))) -> 3^1(2(x1)) 3^1(0(1(0(x1)))) -> 3^1(1(0(0(0(x1))))) 3^1(1(0(3(x1)))) -> 3^1(0(x1)) 3^1(0(1(0(x1)))) -> 0^1(0(0(x1))) 0^1(3(2(x1))) -> 0^1(2(1(3(x1)))) 0^1(3(2(x1))) -> 3^1(x1) 3^1(0(1(0(x1)))) -> 0^1(0(x1)) 0^1(0(3(2(x1)))) -> 0^1(0(2(1(3(x1))))) 0^1(0(3(2(x1)))) -> 0^1(2(1(3(x1)))) 0^1(0(3(2(x1)))) -> 2^1(1(3(x1))) 2^1(1(2(3(x1)))) -> 2^1(x1) 2^1(2(0(3(x1)))) -> 2^1(0(2(1(3(x1))))) 2^1(2(0(3(x1)))) -> 0^1(2(1(3(x1)))) 0^1(0(3(2(x1)))) -> 3^1(x1) 3^1(0(1(4(x1)))) -> 0^1(4(x1)) 0^1(1(0(3(x1)))) -> 3^1(0(x1)) 3^1(0(3(0(x1)))) -> 3^1(1(3(0(0(x1))))) 3^1(1(0(3(x1)))) -> 0^1(x1) 0^1(1(0(3(x1)))) -> 0^1(x1) 0^1(1(2(3(x1)))) -> 0^1(2(1(3(1(x1))))) 0^1(1(2(3(x1)))) -> 2^1(1(3(1(x1)))) 0^1(1(2(3(x1)))) -> 3^1(1(x1)) 0^1(1(2(5(x1)))) -> 0^1(5(x1)) 0^1(1(4(2(x1)))) -> 0^1(0(2(1(4(x1))))) 0^1(1(4(2(x1)))) -> 0^1(2(1(4(x1)))) 0^1(1(4(2(x1)))) -> 2^1(1(4(x1))) 0^1(1(4(2(x1)))) -> 4^1(x1) 3^1(0(3(0(x1)))) -> 3^1(0(0(x1))) 3^1(0(3(0(x1)))) -> 0^1(0(x1)) 0^1(1(4(2(x1)))) -> 0^1(4(0(2(1(x1))))) 0^1(1(4(2(x1)))) -> 4^1(0(2(1(x1)))) 0^1(1(4(5(x1)))) -> 5^1(4(x1)) 5^1(1(0(3(x1)))) -> 3^1(0(x1)) 3^1(2(0(1(x1)))) -> 3^1(0(2(1(x1)))) 3^1(2(0(1(x1)))) -> 0^1(2(1(x1))) 0^1(1(4(5(x1)))) -> 4^1(x1) 0^1(2(0(3(x1)))) -> 0^1(0(2(3(1(x1))))) 0^1(2(0(3(x1)))) -> 0^1(2(3(1(x1)))) 0^1(2(0(3(x1)))) -> 2^1(3(1(x1))) 2^1(2(0(3(x1)))) -> 2^1(1(3(x1))) 2^1(3(1(4(x1)))) -> 3^1(4(x1)) 3^1(2(0(1(x1)))) -> 2^1(1(x1)) 3^1(2(0(1(x1)))) -> 3^1(1(0(2(1(x1))))) 3^1(4(1(2(x1)))) -> 4^1(x1) 3^1(5(1(4(x1)))) -> 5^1(4(x1)) 5^1(1(0(3(x1)))) -> 0^1(x1) 0^1(2(0(3(x1)))) -> 3^1(1(x1)) 0^1(3(0(1(x1)))) -> 0^1(0(1(3(1(x1))))) 0^1(3(0(1(x1)))) -> 0^1(1(3(1(x1)))) 0^1(3(0(1(x1)))) -> 3^1(1(x1)) 0^1(3(0(2(x1)))) -> 0^1(0(2(3(1(x1))))) 0^1(3(0(2(x1)))) -> 0^1(2(3(1(x1)))) 0^1(3(0(2(x1)))) -> 2^1(3(1(x1))) 2^1(5(0(3(x1)))) -> 5^1(0(2(1(3(x1))))) 5^1(2(0(5(x1)))) -> 5^1(0(2(1(5(x1))))) 5^1(2(0(5(x1)))) -> 0^1(2(1(5(x1)))) 0^1(3(0(2(x1)))) -> 3^1(1(x1)) 0^1(3(0(3(x1)))) -> 0^1(0(3(1(3(x1))))) 0^1(3(0(3(x1)))) -> 0^1(3(1(3(x1)))) 0^1(3(0(3(x1)))) -> 3^1(1(3(x1))) 0^1(3(2(0(x1)))) -> 0^1(2(1(3(0(x1))))) 0^1(3(2(0(x1)))) -> 2^1(1(3(0(x1)))) 0^1(3(2(0(x1)))) -> 3^1(0(x1)) 0^1(5(2(0(x1)))) -> 2^1(1(5(0(0(x1))))) 0^1(5(2(0(x1)))) -> 5^1(0(0(x1))) 5^1(2(0(5(x1)))) -> 2^1(1(5(x1))) 5^1(4(1(0(x1)))) -> 4^1(0(x1)) 0^1(5(2(0(x1)))) -> 0^1(0(x1)) 2^1(5(0(3(x1)))) -> 0^1(2(1(3(x1)))) 2^1(5(0(3(x1)))) -> 2^1(1(3(x1))) The TRS R consists of the following rules: 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(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 5 less nodes. ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(1(2(3(x1)))) -> 3^1(2(x1)) 3^1(0(1(0(x1)))) -> 3^1(1(0(0(0(x1))))) 3^1(1(0(3(x1)))) -> 3^1(0(x1)) 3^1(0(1(0(x1)))) -> 0^1(0(0(x1))) 0^1(3(2(x1))) -> 0^1(2(1(3(x1)))) 0^1(3(2(x1))) -> 2^1(1(3(x1))) 2^1(1(2(3(x1)))) -> 2^1(x1) 2^1(2(0(3(x1)))) -> 2^1(0(2(1(3(x1))))) 2^1(2(0(3(x1)))) -> 0^1(2(1(3(x1)))) 0^1(3(2(x1))) -> 3^1(x1) 3^1(0(1(0(x1)))) -> 0^1(0(x1)) 0^1(0(3(2(x1)))) -> 0^1(0(2(1(3(x1))))) 0^1(0(3(2(x1)))) -> 0^1(2(1(3(x1)))) 0^1(0(3(2(x1)))) -> 2^1(1(3(x1))) 0^1(0(3(2(x1)))) -> 3^1(x1) 3^1(0(1(4(x1)))) -> 0^1(4(x1)) 0^1(1(0(3(x1)))) -> 3^1(0(x1)) 3^1(0(3(0(x1)))) -> 3^1(1(3(0(0(x1))))) 3^1(1(0(3(x1)))) -> 0^1(x1) 0^1(1(0(3(x1)))) -> 0^1(x1) 0^1(1(2(3(x1)))) -> 0^1(2(1(3(1(x1))))) 0^1(1(2(3(x1)))) -> 2^1(1(3(1(x1)))) 0^1(1(2(3(x1)))) -> 3^1(1(x1)) 0^1(1(2(5(x1)))) -> 0^1(5(x1)) 0^1(1(4(2(x1)))) -> 0^1(0(2(1(4(x1))))) 0^1(1(4(2(x1)))) -> 0^1(2(1(4(x1)))) 0^1(1(4(2(x1)))) -> 2^1(1(4(x1))) 0^1(1(4(2(x1)))) -> 0^1(4(0(2(1(x1))))) 0^1(1(4(5(x1)))) -> 5^1(4(x1)) 5^1(1(0(3(x1)))) -> 3^1(0(x1)) 3^1(0(3(0(x1)))) -> 3^1(0(0(x1))) 3^1(0(3(0(x1)))) -> 0^1(0(x1)) 0^1(2(0(3(x1)))) -> 0^1(0(2(3(1(x1))))) 0^1(2(0(3(x1)))) -> 0^1(2(3(1(x1)))) 0^1(2(0(3(x1)))) -> 2^1(3(1(x1))) 2^1(2(0(3(x1)))) -> 2^1(1(3(x1))) 2^1(3(1(4(x1)))) -> 3^1(4(x1)) 3^1(2(0(1(x1)))) -> 3^1(0(2(1(x1)))) 3^1(2(0(1(x1)))) -> 0^1(2(1(x1))) 0^1(2(0(3(x1)))) -> 3^1(1(x1)) 0^1(3(0(1(x1)))) -> 0^1(0(1(3(1(x1))))) 0^1(3(0(1(x1)))) -> 0^1(1(3(1(x1)))) 0^1(3(0(1(x1)))) -> 3^1(1(x1)) 0^1(3(0(2(x1)))) -> 0^1(0(2(3(1(x1))))) 0^1(3(0(2(x1)))) -> 0^1(2(3(1(x1)))) 0^1(3(0(2(x1)))) -> 2^1(3(1(x1))) 2^1(5(0(3(x1)))) -> 5^1(0(2(1(3(x1))))) 5^1(1(0(3(x1)))) -> 0^1(x1) 0^1(3(0(2(x1)))) -> 3^1(1(x1)) 0^1(3(0(3(x1)))) -> 0^1(0(3(1(3(x1))))) 0^1(3(0(3(x1)))) -> 0^1(3(1(3(x1)))) 0^1(3(0(3(x1)))) -> 3^1(1(3(x1))) 0^1(3(2(0(x1)))) -> 0^1(2(1(3(0(x1))))) 0^1(3(2(0(x1)))) -> 2^1(1(3(0(x1)))) 0^1(3(2(0(x1)))) -> 3^1(0(x1)) 3^1(2(0(1(x1)))) -> 2^1(1(x1)) 3^1(2(0(1(x1)))) -> 3^1(1(0(2(1(x1))))) 3^1(5(1(4(x1)))) -> 5^1(4(x1)) 5^1(2(0(5(x1)))) -> 5^1(0(2(1(5(x1))))) 5^1(2(0(5(x1)))) -> 0^1(2(1(5(x1)))) 0^1(5(2(0(x1)))) -> 2^1(1(5(0(0(x1))))) 0^1(5(2(0(x1)))) -> 5^1(0(0(x1))) 5^1(2(0(5(x1)))) -> 2^1(1(5(x1))) 0^1(5(2(0(x1)))) -> 0^1(0(x1)) 2^1(5(0(3(x1)))) -> 0^1(2(1(3(x1)))) 2^1(5(0(3(x1)))) -> 2^1(1(3(x1))) The TRS R consists of the following rules: 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(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. 2^1(1(2(3(x1)))) -> 3^1(2(x1)) 3^1(1(0(3(x1)))) -> 3^1(0(x1)) 0^1(3(2(x1))) -> 2^1(1(3(x1))) 2^1(1(2(3(x1)))) -> 2^1(x1) 0^1(3(2(x1))) -> 3^1(x1) 0^1(0(3(2(x1)))) -> 2^1(1(3(x1))) 0^1(0(3(2(x1)))) -> 3^1(x1) 0^1(1(0(3(x1)))) -> 3^1(0(x1)) 3^1(1(0(3(x1)))) -> 0^1(x1) 0^1(1(0(3(x1)))) -> 0^1(x1) 0^1(1(2(3(x1)))) -> 2^1(1(3(1(x1)))) 0^1(1(2(3(x1)))) -> 3^1(1(x1)) 0^1(1(2(5(x1)))) -> 0^1(5(x1)) 0^1(1(4(2(x1)))) -> 2^1(1(4(x1))) 0^1(1(4(5(x1)))) -> 5^1(4(x1)) 5^1(1(0(3(x1)))) -> 3^1(0(x1)) 3^1(0(3(0(x1)))) -> 3^1(0(0(x1))) 3^1(0(3(0(x1)))) -> 0^1(0(x1)) 0^1(2(0(3(x1)))) -> 2^1(3(1(x1))) 2^1(2(0(3(x1)))) -> 2^1(1(3(x1))) 2^1(3(1(4(x1)))) -> 3^1(4(x1)) 0^1(2(0(3(x1)))) -> 3^1(1(x1)) 0^1(3(0(1(x1)))) -> 3^1(1(x1)) 0^1(3(0(2(x1)))) -> 2^1(3(1(x1))) 5^1(1(0(3(x1)))) -> 0^1(x1) 0^1(3(0(2(x1)))) -> 3^1(1(x1)) 0^1(3(0(3(x1)))) -> 3^1(1(3(x1))) 0^1(3(2(0(x1)))) -> 2^1(1(3(0(x1)))) 0^1(3(2(0(x1)))) -> 3^1(0(x1)) 3^1(2(0(1(x1)))) -> 2^1(1(x1)) 3^1(5(1(4(x1)))) -> 5^1(4(x1)) 0^1(5(2(0(x1)))) -> 2^1(1(5(0(0(x1))))) 0^1(5(2(0(x1)))) -> 5^1(0(0(x1))) 5^1(2(0(5(x1)))) -> 2^1(1(5(x1))) 0^1(5(2(0(x1)))) -> 0^1(0(x1)) 2^1(5(0(3(x1)))) -> 2^1(1(3(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)) = 1 + x_1 POL(2^1(x_1)) = x_1 POL(3(x_1)) = 1 + x_1 POL(3^1(x_1)) = x_1 POL(4(x_1)) = 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: 2(4(1(x1))) -> 4(2(1(1(x1)))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(0(1(0(x1)))) -> 3^1(1(0(0(0(x1))))) 3^1(0(1(0(x1)))) -> 0^1(0(0(x1))) 0^1(3(2(x1))) -> 0^1(2(1(3(x1)))) 2^1(2(0(3(x1)))) -> 2^1(0(2(1(3(x1))))) 2^1(2(0(3(x1)))) -> 0^1(2(1(3(x1)))) 3^1(0(1(0(x1)))) -> 0^1(0(x1)) 0^1(0(3(2(x1)))) -> 0^1(0(2(1(3(x1))))) 0^1(0(3(2(x1)))) -> 0^1(2(1(3(x1)))) 3^1(0(1(4(x1)))) -> 0^1(4(x1)) 3^1(0(3(0(x1)))) -> 3^1(1(3(0(0(x1))))) 0^1(1(2(3(x1)))) -> 0^1(2(1(3(1(x1))))) 0^1(1(4(2(x1)))) -> 0^1(0(2(1(4(x1))))) 0^1(1(4(2(x1)))) -> 0^1(2(1(4(x1)))) 0^1(1(4(2(x1)))) -> 0^1(4(0(2(1(x1))))) 0^1(2(0(3(x1)))) -> 0^1(0(2(3(1(x1))))) 0^1(2(0(3(x1)))) -> 0^1(2(3(1(x1)))) 3^1(2(0(1(x1)))) -> 3^1(0(2(1(x1)))) 3^1(2(0(1(x1)))) -> 0^1(2(1(x1))) 0^1(3(0(1(x1)))) -> 0^1(0(1(3(1(x1))))) 0^1(3(0(1(x1)))) -> 0^1(1(3(1(x1)))) 0^1(3(0(2(x1)))) -> 0^1(0(2(3(1(x1))))) 0^1(3(0(2(x1)))) -> 0^1(2(3(1(x1)))) 2^1(5(0(3(x1)))) -> 5^1(0(2(1(3(x1))))) 0^1(3(0(3(x1)))) -> 0^1(0(3(1(3(x1))))) 0^1(3(0(3(x1)))) -> 0^1(3(1(3(x1)))) 0^1(3(2(0(x1)))) -> 0^1(2(1(3(0(x1))))) 3^1(2(0(1(x1)))) -> 3^1(1(0(2(1(x1))))) 5^1(2(0(5(x1)))) -> 5^1(0(2(1(5(x1))))) 5^1(2(0(5(x1)))) -> 0^1(2(1(5(x1)))) 2^1(5(0(3(x1)))) -> 0^1(2(1(3(x1)))) The TRS R consists of the following rules: 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(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 4 SCCs with 11 less nodes. ---------------------------------------- (12) Complex Obligation (AND) ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(0(3(2(x1)))) -> 0^1(0(2(1(3(x1))))) 0^1(3(2(x1))) -> 0^1(2(1(3(x1)))) 0^1(0(3(2(x1)))) -> 0^1(2(1(3(x1)))) 0^1(1(2(3(x1)))) -> 0^1(2(1(3(1(x1))))) 0^1(1(4(2(x1)))) -> 0^1(0(2(1(4(x1))))) 0^1(1(4(2(x1)))) -> 0^1(2(1(4(x1)))) 0^1(1(4(2(x1)))) -> 0^1(4(0(2(1(x1))))) 0^1(2(0(3(x1)))) -> 0^1(0(2(3(1(x1))))) 0^1(2(0(3(x1)))) -> 0^1(2(3(1(x1)))) 0^1(3(0(1(x1)))) -> 0^1(0(1(3(1(x1))))) 0^1(3(0(1(x1)))) -> 0^1(1(3(1(x1)))) 0^1(3(0(2(x1)))) -> 0^1(0(2(3(1(x1))))) 0^1(3(0(2(x1)))) -> 0^1(2(3(1(x1)))) 0^1(3(0(3(x1)))) -> 0^1(0(3(1(3(x1))))) 0^1(3(0(3(x1)))) -> 0^1(3(1(3(x1)))) 0^1(3(2(0(x1)))) -> 0^1(2(1(3(0(x1))))) The TRS R consists of the following rules: 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(0(3(2(x1)))) -> 0^1(0(2(1(3(x1))))) 0^1(3(2(x1))) -> 0^1(2(1(3(x1)))) 0^1(0(3(2(x1)))) -> 0^1(2(1(3(x1)))) 0^1(3(0(1(x1)))) -> 0^1(0(1(3(1(x1))))) 0^1(3(0(1(x1)))) -> 0^1(1(3(1(x1)))) 0^1(3(0(2(x1)))) -> 0^1(0(2(3(1(x1))))) 0^1(3(0(2(x1)))) -> 0^1(2(3(1(x1)))) 0^1(3(2(0(x1)))) -> 0^1(2(1(3(0(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)) = 0 POL(2(x_1)) = 0 POL(3(x_1)) = 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: 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(2(3(x1)))) -> 0^1(2(1(3(1(x1))))) 0^1(1(4(2(x1)))) -> 0^1(0(2(1(4(x1))))) 0^1(1(4(2(x1)))) -> 0^1(2(1(4(x1)))) 0^1(1(4(2(x1)))) -> 0^1(4(0(2(1(x1))))) 0^1(2(0(3(x1)))) -> 0^1(0(2(3(1(x1))))) 0^1(2(0(3(x1)))) -> 0^1(2(3(1(x1)))) 0^1(3(0(3(x1)))) -> 0^1(0(3(1(3(x1))))) 0^1(3(0(3(x1)))) -> 0^1(3(1(3(x1)))) The TRS R consists of the following rules: 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(1(2(3(x1)))) -> 0^1(2(1(3(1(x1))))) 0^1(1(4(2(x1)))) -> 0^1(0(2(1(4(x1))))) 0^1(1(4(2(x1)))) -> 0^1(2(1(4(x1)))) 0^1(1(4(2(x1)))) -> 0^1(4(0(2(1(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)) = 1 POL(2(x_1)) = 0 POL(3(x_1)) = 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: 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) ---------------------------------------- (17) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(2(0(3(x1)))) -> 0^1(0(2(3(1(x1))))) 0^1(2(0(3(x1)))) -> 0^1(2(3(1(x1)))) 0^1(3(0(3(x1)))) -> 0^1(0(3(1(3(x1))))) 0^1(3(0(3(x1)))) -> 0^1(3(1(3(x1)))) The TRS R consists of the following rules: 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(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. 0^1(2(0(3(x1)))) -> 0^1(2(3(1(x1)))) 0^1(3(0(3(x1)))) -> 0^1(3(1(3(x1)))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 POL(0^1(x_1)) = x_1 POL(1(x_1)) = 0 POL(2(x_1)) = x_1 POL(3(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: 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) ---------------------------------------- (19) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(2(0(3(x1)))) -> 0^1(0(2(3(1(x1))))) 0^1(3(0(3(x1)))) -> 0^1(0(3(1(3(x1))))) The TRS R consists of the following rules: 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(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. 0^1(3(0(3(x1)))) -> 0^1(0(3(1(3(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(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: 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) ---------------------------------------- (21) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(2(0(3(x1)))) -> 0^1(0(2(3(1(x1))))) The TRS R consists of the following rules: 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (22) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(2(0(3(x1)))) -> 0^1(0(2(3(1(x1))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 POL(0^1(x_1)) = x_1 POL(1(x_1)) = 0 POL(2(x_1)) = 1 + x_1 POL(3(x_1)) = 1 + x_1 POL(4(x_1)) = 1 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) ---------------------------------------- (23) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (24) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (25) YES ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: 5^1(2(0(5(x1)))) -> 5^1(0(2(1(5(x1))))) The TRS R consists of the following rules: 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(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. 5^1(2(0(5(x1)))) -> 5^1(0(2(1(5(x1))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 POL(1(x_1)) = 0 POL(2(x_1)) = 1 + x_1 POL(3(x_1)) = 1 POL(4(x_1)) = 0 POL(5(x_1)) = 0 POL(5^1(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) ---------------------------------------- (28) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (30) YES ---------------------------------------- (31) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(2(0(3(x1)))) -> 2^1(0(2(1(3(x1))))) The TRS R consists of the following rules: 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (32) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 2^1(2(0(3(x1)))) -> 2^1(0(2(1(3(x1))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 POL(1(x_1)) = 0 POL(2(x_1)) = 1 + x_1 POL(2^1(x_1)) = x_1 POL(3(x_1)) = 0 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: 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) ---------------------------------------- (33) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (34) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (35) YES ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(2(0(1(x1)))) -> 3^1(0(2(1(x1)))) The TRS R consists of the following rules: 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (37) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 3^1(2(0(1(x1)))) -> 3^1(0(2(1(x1)))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 POL(1(x_1)) = 0 POL(2(x_1)) = 1 + x_1 POL(3(x_1)) = 1 + x_1 POL(3^1(x_1)) = x_1 POL(4(x_1)) = 1 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) ---------------------------------------- (38) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: 0(1(2(x1))) -> 0(2(1(1(x1)))) 0(3(2(x1))) -> 0(2(1(3(x1)))) 2(4(1(x1))) -> 4(2(1(1(x1)))) 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (39) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (40) YES