64.89/17.63 YES 66.11/17.90 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 66.11/17.90 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 66.11/17.90 66.11/17.90 66.11/17.90 Termination w.r.t. Q of the given QTRS could be proven: 66.11/17.90 66.11/17.90 (0) QTRS 66.11/17.90 (1) DependencyPairsProof [EQUIVALENT, 320 ms] 66.11/17.90 (2) QDP 66.11/17.90 (3) DependencyGraphProof [EQUIVALENT, 1 ms] 66.11/17.90 (4) QDP 66.11/17.90 (5) QDPOrderProof [EQUIVALENT, 238 ms] 66.11/17.90 (6) QDP 66.11/17.90 (7) DependencyGraphProof [EQUIVALENT, 0 ms] 66.11/17.90 (8) QDP 66.11/17.90 (9) QDPOrderProof [EQUIVALENT, 254 ms] 66.11/17.90 (10) QDP 66.11/17.90 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 66.11/17.90 (12) AND 66.11/17.90 (13) QDP 66.11/17.90 (14) QDPOrderProof [EQUIVALENT, 0 ms] 66.11/17.90 (15) QDP 66.11/17.90 (16) QDPOrderProof [EQUIVALENT, 51 ms] 66.11/17.90 (17) QDP 66.11/17.90 (18) QDPOrderProof [EQUIVALENT, 36 ms] 66.11/17.90 (19) QDP 66.11/17.90 (20) QDPOrderProof [EQUIVALENT, 40 ms] 66.11/17.90 (21) QDP 66.11/17.90 (22) QDPOrderProof [EQUIVALENT, 31 ms] 66.11/17.90 (23) QDP 66.11/17.90 (24) PisEmptyProof [EQUIVALENT, 0 ms] 66.11/17.90 (25) YES 66.11/17.90 (26) QDP 66.11/17.90 (27) QDPOrderProof [EQUIVALENT, 0 ms] 66.11/17.90 (28) QDP 66.11/17.90 (29) PisEmptyProof [EQUIVALENT, 0 ms] 66.11/17.90 (30) YES 66.11/17.90 (31) QDP 66.11/17.90 (32) QDPOrderProof [EQUIVALENT, 0 ms] 66.11/17.90 (33) QDP 66.11/17.90 (34) PisEmptyProof [EQUIVALENT, 0 ms] 66.11/17.90 (35) YES 66.11/17.90 (36) QDP 66.11/17.90 (37) QDPOrderProof [EQUIVALENT, 44 ms] 66.11/17.90 (38) QDP 66.11/17.90 (39) PisEmptyProof [EQUIVALENT, 0 ms] 66.11/17.90 (40) YES 66.11/17.90 66.11/17.90 66.11/17.90 ---------------------------------------- 66.11/17.90 66.11/17.90 (0) 66.11/17.90 Obligation: 66.11/17.90 Q restricted rewrite system: 66.11/17.90 The TRS R consists of the following rules: 66.11/17.90 66.11/17.90 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.11/17.90 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.11/17.90 2(4(1(x1))) -> 4(2(1(1(x1)))) 66.11/17.90 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.11/17.90 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.11/17.90 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.11/17.90 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.11/17.90 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.11/17.90 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.11/17.90 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.11/17.90 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.11/17.90 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.11/17.90 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.11/17.90 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.11/17.90 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.11/17.90 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.11/17.90 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.11/17.90 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.11/17.90 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.11/17.90 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.11/17.90 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.11/17.90 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.11/17.90 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.11/17.90 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.11/17.90 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.11/17.90 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 66.11/17.90 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 66.11/17.90 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 66.11/17.90 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 66.11/17.90 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 66.11/17.90 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 66.11/17.90 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 66.11/17.90 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 66.11/17.90 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 66.11/17.90 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 66.11/17.90 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 66.11/17.90 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 66.11/17.90 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 66.11/17.90 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 66.11/17.90 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 66.11/17.90 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 66.11/17.90 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 66.32/17.90 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 66.32/17.90 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 66.32/17.90 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 66.32/17.90 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 66.32/17.90 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 66.32/17.90 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.32/17.90 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 66.32/17.90 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 66.32/17.90 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 66.32/17.90 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 66.32/17.90 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 66.32/17.90 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 66.32/17.90 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 66.32/17.90 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 66.32/17.90 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 66.32/17.90 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.32/17.90 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 66.32/17.90 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) 66.32/17.90 66.32/17.90 Q is empty. 66.32/17.90 66.32/17.90 ---------------------------------------- 66.32/17.90 66.32/17.90 (1) DependencyPairsProof (EQUIVALENT) 66.32/17.90 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 66.32/17.90 ---------------------------------------- 66.32/17.90 66.32/17.90 (2) 66.32/17.90 Obligation: 66.32/17.90 Q DP problem: 66.32/17.90 The TRS P consists of the following rules: 66.32/17.90 66.32/17.90 0^1(1(2(x1))) -> 0^1(2(1(1(x1)))) 66.32/17.90 0^1(1(2(x1))) -> 2^1(1(1(x1))) 66.32/17.90 0^1(3(2(x1))) -> 0^1(2(1(3(x1)))) 66.32/17.90 0^1(3(2(x1))) -> 2^1(1(3(x1))) 66.32/17.90 0^1(3(2(x1))) -> 3^1(x1) 66.32/17.90 2^1(4(1(x1))) -> 4^1(2(1(1(x1)))) 66.32/17.90 2^1(4(1(x1))) -> 2^1(1(1(x1))) 66.32/17.90 0^1(0(3(2(x1)))) -> 0^1(0(2(1(3(x1))))) 66.32/17.90 0^1(0(3(2(x1)))) -> 0^1(2(1(3(x1)))) 66.32/17.90 0^1(0(3(2(x1)))) -> 2^1(1(3(x1))) 66.32/17.90 0^1(0(3(2(x1)))) -> 3^1(x1) 66.32/17.90 0^1(1(0(3(x1)))) -> 0^1(1(1(3(0(x1))))) 66.32/17.90 0^1(1(0(3(x1)))) -> 3^1(0(x1)) 66.32/17.90 0^1(1(0(3(x1)))) -> 0^1(x1) 66.32/17.90 0^1(1(0(5(x1)))) -> 0^1(0(1(1(5(x1))))) 66.32/17.90 0^1(1(0(5(x1)))) -> 0^1(1(1(5(x1)))) 66.32/17.90 0^1(1(2(2(x1)))) -> 0^1(2(1(1(2(x1))))) 66.32/17.90 0^1(1(2(2(x1)))) -> 2^1(1(1(2(x1)))) 66.32/17.90 0^1(1(2(3(x1)))) -> 0^1(2(1(3(1(x1))))) 66.32/17.90 0^1(1(2(3(x1)))) -> 2^1(1(3(1(x1)))) 66.32/17.90 0^1(1(2(3(x1)))) -> 3^1(1(x1)) 66.32/17.90 0^1(1(2(5(x1)))) -> 2^1(1(1(0(5(x1))))) 66.32/17.90 0^1(1(2(5(x1)))) -> 0^1(5(x1)) 66.32/17.90 0^1(1(4(1(x1)))) -> 0^1(4(1(1(1(x1))))) 66.32/17.90 0^1(1(4(1(x1)))) -> 4^1(1(1(1(x1)))) 66.32/17.90 0^1(1(4(2(x1)))) -> 0^1(0(2(1(4(x1))))) 66.32/17.90 0^1(1(4(2(x1)))) -> 0^1(2(1(4(x1)))) 66.32/17.90 0^1(1(4(2(x1)))) -> 2^1(1(4(x1))) 66.32/17.90 0^1(1(4(2(x1)))) -> 4^1(x1) 66.32/17.90 0^1(1(4(2(x1)))) -> 0^1(4(0(2(1(x1))))) 66.32/17.90 0^1(1(4(2(x1)))) -> 4^1(0(2(1(x1)))) 66.32/17.90 0^1(1(4(2(x1)))) -> 0^1(2(1(x1))) 66.32/17.90 0^1(1(4(2(x1)))) -> 2^1(1(x1)) 66.32/17.90 0^1(1(4(3(x1)))) -> 4^1(0(1(1(3(x1))))) 66.32/17.90 0^1(1(4(3(x1)))) -> 0^1(1(1(3(x1)))) 66.32/17.90 0^1(1(4(5(x1)))) -> 0^1(1(1(5(4(x1))))) 66.32/17.90 0^1(1(4(5(x1)))) -> 5^1(4(x1)) 66.32/17.90 0^1(1(4(5(x1)))) -> 4^1(x1) 66.32/17.90 0^1(1(5(2(x1)))) -> 0^1(2(5(1(1(x1))))) 66.32/17.90 0^1(1(5(2(x1)))) -> 2^1(5(1(1(x1)))) 66.32/17.90 0^1(1(5(2(x1)))) -> 5^1(1(1(x1))) 66.32/17.90 0^1(2(0(3(x1)))) -> 0^1(0(2(3(1(x1))))) 66.32/17.90 0^1(2(0(3(x1)))) -> 0^1(2(3(1(x1)))) 66.32/17.90 0^1(2(0(3(x1)))) -> 2^1(3(1(x1))) 66.32/17.90 0^1(2(0(3(x1)))) -> 3^1(1(x1)) 66.32/17.90 0^1(3(0(1(x1)))) -> 0^1(0(1(3(1(x1))))) 66.32/17.90 0^1(3(0(1(x1)))) -> 0^1(1(3(1(x1)))) 66.32/17.90 0^1(3(0(1(x1)))) -> 3^1(1(x1)) 66.32/17.90 0^1(3(0(2(x1)))) -> 0^1(0(2(3(1(x1))))) 66.32/17.90 0^1(3(0(2(x1)))) -> 0^1(2(3(1(x1)))) 66.32/17.90 0^1(3(0(2(x1)))) -> 2^1(3(1(x1))) 66.32/17.90 0^1(3(0(2(x1)))) -> 3^1(1(x1)) 66.32/17.90 0^1(3(0(3(x1)))) -> 0^1(0(3(1(3(x1))))) 66.32/17.90 0^1(3(0(3(x1)))) -> 0^1(3(1(3(x1)))) 66.32/17.90 0^1(3(0(3(x1)))) -> 3^1(1(3(x1))) 66.32/17.90 0^1(3(2(0(x1)))) -> 0^1(2(1(3(0(x1))))) 66.32/17.90 0^1(3(2(0(x1)))) -> 2^1(1(3(0(x1)))) 66.32/17.90 0^1(3(2(0(x1)))) -> 3^1(0(x1)) 66.32/17.90 0^1(3(5(1(x1)))) -> 0^1(5(3(1(1(x1))))) 66.32/17.90 0^1(3(5(1(x1)))) -> 5^1(3(1(1(x1)))) 66.32/17.90 0^1(3(5(1(x1)))) -> 3^1(1(1(x1))) 66.32/17.90 0^1(4(0(1(x1)))) -> 0^1(0(4(1(1(x1))))) 66.32/17.90 0^1(4(0(1(x1)))) -> 0^1(4(1(1(x1)))) 66.32/17.90 0^1(4(0(1(x1)))) -> 4^1(1(1(x1))) 66.32/17.90 0^1(5(2(0(x1)))) -> 2^1(1(5(0(0(x1))))) 66.32/17.90 0^1(5(2(0(x1)))) -> 5^1(0(0(x1))) 66.32/17.90 0^1(5(2(0(x1)))) -> 0^1(0(x1)) 66.32/17.90 2^1(1(2(3(x1)))) -> 2^1(1(1(3(2(x1))))) 66.32/17.90 2^1(1(2(3(x1)))) -> 3^1(2(x1)) 66.32/17.90 2^1(1(2(3(x1)))) -> 2^1(x1) 66.32/17.90 2^1(1(4(1(x1)))) -> 4^1(2(1(1(1(x1))))) 66.32/17.90 2^1(1(4(1(x1)))) -> 2^1(1(1(1(x1)))) 66.32/17.90 2^1(2(0(3(x1)))) -> 2^1(0(2(1(3(x1))))) 66.32/17.90 2^1(2(0(3(x1)))) -> 0^1(2(1(3(x1)))) 66.32/17.90 2^1(2(0(3(x1)))) -> 2^1(1(3(x1))) 66.32/17.90 2^1(2(4(1(x1)))) -> 2^1(4(2(1(1(x1))))) 66.32/17.90 2^1(2(4(1(x1)))) -> 4^1(2(1(1(x1)))) 66.32/17.90 2^1(2(4(1(x1)))) -> 2^1(1(1(x1))) 66.32/17.90 2^1(3(1(4(x1)))) -> 2^1(1(1(3(4(x1))))) 66.32/17.90 2^1(3(1(4(x1)))) -> 3^1(4(x1)) 66.32/17.90 2^1(4(1(4(x1)))) -> 4^1(4(2(1(1(x1))))) 66.32/17.90 2^1(4(1(4(x1)))) -> 4^1(2(1(1(x1)))) 66.32/17.90 2^1(4(1(4(x1)))) -> 2^1(1(1(x1))) 66.32/17.90 2^1(4(3(1(x1)))) -> 4^1(2(3(1(1(x1))))) 66.32/17.90 2^1(4(3(1(x1)))) -> 2^1(3(1(1(x1)))) 66.32/17.90 2^1(4(3(1(x1)))) -> 3^1(1(1(x1))) 66.32/17.90 2^1(5(0(1(x1)))) -> 0^1(2(5(1(1(x1))))) 66.32/17.90 2^1(5(0(1(x1)))) -> 2^1(5(1(1(x1)))) 66.32/17.90 2^1(5(0(1(x1)))) -> 5^1(1(1(x1))) 66.32/17.90 2^1(5(0(3(x1)))) -> 5^1(0(2(1(3(x1))))) 66.32/17.90 2^1(5(0(3(x1)))) -> 0^1(2(1(3(x1)))) 66.32/17.90 2^1(5(0(3(x1)))) -> 2^1(1(3(x1))) 66.32/17.90 3^1(0(1(0(x1)))) -> 3^1(1(0(0(0(x1))))) 66.32/17.90 3^1(0(1(0(x1)))) -> 0^1(0(0(x1))) 66.32/17.90 3^1(0(1(0(x1)))) -> 0^1(0(x1)) 66.32/17.90 3^1(0(1(4(x1)))) -> 3^1(1(1(0(4(x1))))) 66.32/17.90 3^1(0(1(4(x1)))) -> 0^1(4(x1)) 66.32/17.90 3^1(0(3(0(x1)))) -> 3^1(1(3(0(0(x1))))) 66.32/17.90 3^1(0(3(0(x1)))) -> 3^1(0(0(x1))) 66.32/17.90 3^1(0(3(0(x1)))) -> 0^1(0(x1)) 66.32/17.90 3^1(1(0(3(x1)))) -> 3^1(1(1(3(0(x1))))) 66.32/17.90 3^1(1(0(3(x1)))) -> 3^1(0(x1)) 66.32/17.90 3^1(1(0(3(x1)))) -> 0^1(x1) 66.32/17.90 3^1(1(2(1(x1)))) -> 3^1(2(1(1(x1)))) 66.32/17.90 3^1(1(2(1(x1)))) -> 2^1(1(1(x1))) 66.32/17.90 3^1(1(2(1(x1)))) -> 3^1(2(1(1(1(x1))))) 66.32/17.90 3^1(1(2(1(x1)))) -> 2^1(1(1(1(x1)))) 66.32/17.92 3^1(2(0(1(x1)))) -> 3^1(0(2(1(x1)))) 66.32/17.92 3^1(2(0(1(x1)))) -> 0^1(2(1(x1))) 66.32/17.92 3^1(2(0(1(x1)))) -> 2^1(1(x1)) 66.32/17.92 3^1(2(0(1(x1)))) -> 3^1(1(0(2(1(x1))))) 66.32/17.92 3^1(4(1(2(x1)))) -> 3^1(2(1(1(4(x1))))) 66.32/17.92 3^1(4(1(2(x1)))) -> 2^1(1(1(4(x1)))) 66.32/17.92 3^1(4(1(2(x1)))) -> 4^1(x1) 66.32/17.92 3^1(5(1(4(x1)))) -> 3^1(1(1(5(4(x1))))) 66.32/17.92 3^1(5(1(4(x1)))) -> 5^1(4(x1)) 66.32/17.92 4^1(1(0(1(x1)))) -> 4^1(0(1(1(1(x1))))) 66.32/17.92 4^1(1(0(1(x1)))) -> 0^1(1(1(1(x1)))) 66.32/17.92 4^1(1(2(1(x1)))) -> 2^1(4(1(1(1(x1))))) 66.32/17.92 4^1(1(2(1(x1)))) -> 4^1(1(1(1(x1)))) 66.32/17.92 4^1(1(4(3(x1)))) -> 4^1(1(1(3(4(x1))))) 66.32/17.92 4^1(1(4(3(x1)))) -> 3^1(4(x1)) 66.32/17.92 4^1(1(4(3(x1)))) -> 4^1(x1) 66.32/17.92 4^1(2(4(1(x1)))) -> 4^1(4(2(1(1(x1))))) 66.32/17.92 4^1(2(4(1(x1)))) -> 4^1(2(1(1(x1)))) 66.32/17.92 4^1(2(4(1(x1)))) -> 2^1(1(1(x1))) 66.32/17.92 4^1(4(1(2(x1)))) -> 4^1(2(1(1(4(x1))))) 66.32/17.92 4^1(4(1(2(x1)))) -> 2^1(1(1(4(x1)))) 66.32/17.92 4^1(4(1(2(x1)))) -> 4^1(x1) 66.32/17.92 4^1(5(0(1(x1)))) -> 5^1(0(4(1(1(x1))))) 66.32/17.92 4^1(5(0(1(x1)))) -> 0^1(4(1(1(x1)))) 66.32/17.92 4^1(5(0(1(x1)))) -> 4^1(1(1(x1))) 66.32/17.92 5^1(1(0(3(x1)))) -> 5^1(0(1(1(3(x1))))) 66.32/17.92 5^1(1(0(3(x1)))) -> 0^1(1(1(3(x1)))) 66.32/17.92 5^1(1(0(3(x1)))) -> 5^1(1(1(3(0(x1))))) 66.32/17.92 5^1(1(0(3(x1)))) -> 3^1(0(x1)) 66.32/17.92 5^1(1(0(3(x1)))) -> 0^1(x1) 66.32/17.92 5^1(1(3(1(x1)))) -> 5^1(3(1(1(1(x1))))) 66.32/17.92 5^1(1(3(1(x1)))) -> 3^1(1(1(1(x1)))) 66.32/17.92 5^1(1(4(3(x1)))) -> 4^1(1(1(3(5(x1))))) 66.32/17.92 5^1(1(4(3(x1)))) -> 3^1(5(x1)) 66.32/17.92 5^1(1(4(3(x1)))) -> 5^1(x1) 66.32/17.92 5^1(1(4(4(x1)))) -> 4^1(1(1(5(4(x1))))) 66.32/17.92 5^1(1(4(4(x1)))) -> 5^1(4(x1)) 66.32/17.92 5^1(1(5(3(x1)))) -> 5^1(5(1(1(3(x1))))) 66.32/17.92 5^1(1(5(3(x1)))) -> 5^1(1(1(3(x1)))) 66.32/17.92 5^1(2(0(1(x1)))) -> 2^1(0(5(1(1(x1))))) 66.32/17.92 5^1(2(0(1(x1)))) -> 0^1(5(1(1(x1)))) 66.32/17.92 5^1(2(0(1(x1)))) -> 5^1(1(1(x1))) 66.32/17.92 5^1(2(0(5(x1)))) -> 5^1(0(2(1(5(x1))))) 66.32/17.92 5^1(2(0(5(x1)))) -> 0^1(2(1(5(x1)))) 66.32/17.92 5^1(2(0(5(x1)))) -> 2^1(1(5(x1))) 66.32/17.92 5^1(3(4(1(x1)))) -> 5^1(4(3(1(1(x1))))) 66.32/17.92 5^1(3(4(1(x1)))) -> 4^1(3(1(1(x1)))) 66.32/17.92 5^1(3(4(1(x1)))) -> 3^1(1(1(x1))) 66.32/17.92 5^1(4(0(1(x1)))) -> 5^1(0(4(1(1(x1))))) 66.32/17.92 5^1(4(0(1(x1)))) -> 0^1(4(1(1(x1)))) 66.32/17.92 5^1(4(0(1(x1)))) -> 4^1(1(1(x1))) 66.32/17.92 5^1(4(1(0(x1)))) -> 4^1(1(1(5(0(x1))))) 66.32/17.92 5^1(4(1(0(x1)))) -> 5^1(0(x1)) 66.32/17.92 5^1(4(1(0(x1)))) -> 5^1(1(1(4(0(x1))))) 66.32/17.92 5^1(4(1(0(x1)))) -> 4^1(0(x1)) 66.32/17.92 66.32/17.92 The TRS R consists of the following rules: 66.32/17.92 66.32/17.92 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.32/17.92 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.32/17.92 2(4(1(x1))) -> 4(2(1(1(x1)))) 66.32/17.92 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.32/17.92 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.32/17.92 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.32/17.92 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.32/17.92 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.32/17.92 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.32/17.92 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.32/17.92 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.32/17.92 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.32/17.92 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.32/17.92 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.32/17.92 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.32/17.92 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.32/17.92 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.32/17.92 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.32/17.92 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.32/17.92 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.32/17.92 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.32/17.92 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.32/17.92 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.32/17.92 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.32/17.92 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.32/17.92 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 66.32/17.92 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 66.32/17.92 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 66.32/17.92 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 66.32/17.92 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 66.32/17.92 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 66.32/17.92 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 66.32/17.92 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 66.32/17.92 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 66.32/17.92 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 66.32/17.92 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 66.32/17.92 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 66.32/17.92 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 66.32/17.92 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 66.32/17.92 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 66.32/17.92 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 66.32/17.92 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 66.32/17.92 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 66.32/17.92 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 66.32/17.92 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 66.32/17.92 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 66.32/17.92 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 66.32/17.92 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.32/17.92 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 66.32/17.92 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 66.32/17.92 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 66.32/17.92 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 66.32/17.92 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 66.32/17.92 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 66.32/17.92 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 66.32/17.92 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 66.32/17.92 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 66.32/17.92 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.32/17.92 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 66.32/17.92 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) 66.32/17.92 66.32/17.92 Q is empty. 66.32/17.92 We have to consider all minimal (P,Q,R)-chains. 66.32/17.92 ---------------------------------------- 66.32/17.92 66.32/17.92 (3) DependencyGraphProof (EQUIVALENT) 66.32/17.92 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 82 less nodes. 66.32/17.92 ---------------------------------------- 66.32/17.92 66.32/17.92 (4) 66.32/17.92 Obligation: 66.32/17.92 Q DP problem: 66.32/17.92 The TRS P consists of the following rules: 66.32/17.92 66.32/17.92 0^1(3(2(x1))) -> 2^1(1(3(x1))) 66.32/17.92 2^1(1(2(3(x1)))) -> 3^1(2(x1)) 66.32/17.92 3^1(0(1(0(x1)))) -> 3^1(1(0(0(0(x1))))) 66.32/17.92 3^1(1(0(3(x1)))) -> 3^1(0(x1)) 66.32/17.92 3^1(0(1(0(x1)))) -> 0^1(0(0(x1))) 66.32/17.92 0^1(3(2(x1))) -> 0^1(2(1(3(x1)))) 66.32/17.92 0^1(3(2(x1))) -> 3^1(x1) 66.32/17.92 3^1(0(1(0(x1)))) -> 0^1(0(x1)) 66.32/17.92 0^1(0(3(2(x1)))) -> 0^1(0(2(1(3(x1))))) 66.32/17.92 0^1(0(3(2(x1)))) -> 0^1(2(1(3(x1)))) 66.32/17.92 0^1(0(3(2(x1)))) -> 2^1(1(3(x1))) 66.32/17.92 2^1(1(2(3(x1)))) -> 2^1(x1) 66.32/17.92 2^1(2(0(3(x1)))) -> 2^1(0(2(1(3(x1))))) 66.32/17.92 2^1(2(0(3(x1)))) -> 0^1(2(1(3(x1)))) 66.32/17.92 0^1(0(3(2(x1)))) -> 3^1(x1) 66.32/17.92 3^1(0(1(4(x1)))) -> 0^1(4(x1)) 66.32/17.92 0^1(1(0(3(x1)))) -> 3^1(0(x1)) 66.32/17.92 3^1(0(3(0(x1)))) -> 3^1(1(3(0(0(x1))))) 66.32/17.92 3^1(1(0(3(x1)))) -> 0^1(x1) 66.32/17.92 0^1(1(0(3(x1)))) -> 0^1(x1) 66.32/17.92 0^1(1(2(3(x1)))) -> 0^1(2(1(3(1(x1))))) 66.32/17.92 0^1(1(2(3(x1)))) -> 2^1(1(3(1(x1)))) 66.32/17.92 0^1(1(2(3(x1)))) -> 3^1(1(x1)) 66.32/17.92 0^1(1(2(5(x1)))) -> 0^1(5(x1)) 66.32/17.92 0^1(1(4(2(x1)))) -> 0^1(0(2(1(4(x1))))) 66.32/17.92 0^1(1(4(2(x1)))) -> 0^1(2(1(4(x1)))) 66.32/17.92 0^1(1(4(2(x1)))) -> 2^1(1(4(x1))) 66.32/17.92 0^1(1(4(2(x1)))) -> 4^1(x1) 66.32/17.92 4^1(1(4(3(x1)))) -> 3^1(4(x1)) 66.32/17.92 3^1(0(3(0(x1)))) -> 3^1(0(0(x1))) 66.32/17.92 3^1(0(3(0(x1)))) -> 0^1(0(x1)) 66.32/17.92 0^1(1(4(2(x1)))) -> 0^1(4(0(2(1(x1))))) 66.32/17.92 0^1(1(4(2(x1)))) -> 4^1(0(2(1(x1)))) 66.32/17.92 4^1(1(4(3(x1)))) -> 4^1(x1) 66.32/17.92 4^1(4(1(2(x1)))) -> 4^1(x1) 66.32/17.92 0^1(1(4(2(x1)))) -> 0^1(2(1(x1))) 66.32/17.92 0^1(1(4(2(x1)))) -> 2^1(1(x1)) 66.32/17.92 0^1(1(4(5(x1)))) -> 5^1(4(x1)) 66.32/17.92 5^1(1(0(3(x1)))) -> 3^1(0(x1)) 66.32/17.92 3^1(2(0(1(x1)))) -> 3^1(0(2(1(x1)))) 66.32/17.92 3^1(2(0(1(x1)))) -> 0^1(2(1(x1))) 66.32/17.92 0^1(1(4(5(x1)))) -> 4^1(x1) 66.32/17.92 0^1(2(0(3(x1)))) -> 0^1(0(2(3(1(x1))))) 66.32/17.92 0^1(2(0(3(x1)))) -> 0^1(2(3(1(x1)))) 66.32/17.92 0^1(2(0(3(x1)))) -> 2^1(3(1(x1))) 66.32/17.92 2^1(2(0(3(x1)))) -> 2^1(1(3(x1))) 66.32/17.92 2^1(3(1(4(x1)))) -> 3^1(4(x1)) 66.32/17.92 3^1(2(0(1(x1)))) -> 2^1(1(x1)) 66.32/17.92 3^1(2(0(1(x1)))) -> 3^1(1(0(2(1(x1))))) 66.32/17.92 3^1(4(1(2(x1)))) -> 4^1(x1) 66.32/17.92 3^1(5(1(4(x1)))) -> 5^1(4(x1)) 66.32/17.92 5^1(1(0(3(x1)))) -> 0^1(x1) 66.32/17.92 0^1(2(0(3(x1)))) -> 3^1(1(x1)) 66.32/17.92 0^1(3(0(1(x1)))) -> 0^1(0(1(3(1(x1))))) 66.32/17.92 0^1(3(0(1(x1)))) -> 0^1(1(3(1(x1)))) 66.32/17.92 0^1(3(0(1(x1)))) -> 3^1(1(x1)) 66.32/17.92 0^1(3(0(2(x1)))) -> 0^1(0(2(3(1(x1))))) 66.32/17.92 0^1(3(0(2(x1)))) -> 0^1(2(3(1(x1)))) 66.32/17.92 0^1(3(0(2(x1)))) -> 2^1(3(1(x1))) 66.32/17.92 2^1(5(0(3(x1)))) -> 5^1(0(2(1(3(x1))))) 66.32/17.92 5^1(1(4(3(x1)))) -> 3^1(5(x1)) 66.32/17.92 5^1(1(4(3(x1)))) -> 5^1(x1) 66.32/17.92 5^1(1(4(4(x1)))) -> 5^1(4(x1)) 66.32/17.92 5^1(2(0(5(x1)))) -> 5^1(0(2(1(5(x1))))) 66.32/17.92 5^1(2(0(5(x1)))) -> 0^1(2(1(5(x1)))) 66.32/17.92 0^1(3(0(2(x1)))) -> 3^1(1(x1)) 66.32/17.92 0^1(3(0(3(x1)))) -> 0^1(0(3(1(3(x1))))) 66.32/17.92 0^1(3(0(3(x1)))) -> 0^1(3(1(3(x1)))) 66.32/17.92 0^1(3(0(3(x1)))) -> 3^1(1(3(x1))) 66.32/17.92 0^1(3(2(0(x1)))) -> 0^1(2(1(3(0(x1))))) 66.32/17.92 0^1(3(2(0(x1)))) -> 2^1(1(3(0(x1)))) 66.32/17.92 0^1(3(2(0(x1)))) -> 3^1(0(x1)) 66.32/17.92 0^1(5(2(0(x1)))) -> 2^1(1(5(0(0(x1))))) 66.32/17.92 0^1(5(2(0(x1)))) -> 5^1(0(0(x1))) 66.32/17.92 5^1(2(0(5(x1)))) -> 2^1(1(5(x1))) 66.32/17.92 5^1(4(1(0(x1)))) -> 5^1(0(x1)) 66.32/17.92 5^1(4(1(0(x1)))) -> 4^1(0(x1)) 66.32/17.92 0^1(5(2(0(x1)))) -> 0^1(0(x1)) 66.32/17.92 2^1(5(0(3(x1)))) -> 0^1(2(1(3(x1)))) 66.32/17.92 2^1(5(0(3(x1)))) -> 2^1(1(3(x1))) 66.32/17.92 66.32/17.92 The TRS R consists of the following rules: 66.32/17.92 66.32/17.92 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.32/17.92 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.32/17.92 2(4(1(x1))) -> 4(2(1(1(x1)))) 66.32/17.92 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.32/17.92 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.32/17.92 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.32/17.92 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.32/17.92 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.32/17.92 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.32/17.92 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.32/17.92 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.32/17.92 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.32/17.92 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.32/17.92 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.32/17.92 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.32/17.92 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.32/17.92 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.32/17.92 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.32/17.92 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.32/17.92 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.32/17.92 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.32/17.92 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.32/17.92 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.32/17.92 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.32/17.92 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.32/17.92 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 66.32/17.92 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 66.32/17.92 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 66.32/17.92 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 66.32/17.92 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 66.32/17.92 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 66.32/17.92 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 66.32/17.92 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 66.32/17.92 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 66.32/17.92 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 66.32/17.92 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 66.32/17.92 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 66.32/17.92 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 66.32/17.92 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 66.32/17.92 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 66.32/17.92 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 66.32/17.92 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 66.32/17.92 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 66.32/17.92 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 66.32/17.92 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 66.32/17.92 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 66.32/17.92 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 66.32/17.92 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.32/17.92 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 66.32/17.92 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 66.32/17.92 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 66.32/17.92 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 66.32/17.92 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 66.32/17.92 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 66.32/17.92 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 66.32/17.92 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 66.32/17.92 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 66.32/17.92 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.32/17.92 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 66.32/17.92 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) 66.32/17.92 66.32/17.92 Q is empty. 66.32/17.92 We have to consider all minimal (P,Q,R)-chains. 66.32/17.92 ---------------------------------------- 66.32/17.92 66.32/17.92 (5) QDPOrderProof (EQUIVALENT) 66.32/17.92 We use the reduction pair processor [LPAR04,JAR06]. 66.32/17.92 66.32/17.92 66.32/17.92 The following pairs can be oriented strictly and are deleted. 66.32/17.92 66.32/17.92 4^1(1(4(3(x1)))) -> 3^1(4(x1)) 66.32/17.92 4^1(1(4(3(x1)))) -> 4^1(x1) 66.32/17.92 4^1(4(1(2(x1)))) -> 4^1(x1) 66.32/17.92 0^1(1(4(2(x1)))) -> 0^1(2(1(x1))) 66.32/17.92 0^1(1(4(2(x1)))) -> 2^1(1(x1)) 66.32/17.92 5^1(1(4(3(x1)))) -> 3^1(5(x1)) 66.32/17.92 5^1(1(4(3(x1)))) -> 5^1(x1) 66.32/17.92 5^1(1(4(4(x1)))) -> 5^1(4(x1)) 66.32/17.92 5^1(4(1(0(x1)))) -> 5^1(0(x1)) 66.32/17.92 The remaining pairs can at least be oriented weakly. 66.32/17.92 Used ordering: Polynomial interpretation [POLO]: 66.32/17.92 66.32/17.92 POL(0(x_1)) = x_1 66.32/17.92 POL(0^1(x_1)) = x_1 66.32/17.92 POL(1(x_1)) = x_1 66.32/17.92 POL(2(x_1)) = x_1 66.32/17.92 POL(2^1(x_1)) = x_1 66.32/17.92 POL(3(x_1)) = x_1 66.32/17.92 POL(3^1(x_1)) = x_1 66.32/17.92 POL(4(x_1)) = 1 + x_1 66.32/17.92 POL(4^1(x_1)) = 1 + x_1 66.32/17.92 POL(5(x_1)) = x_1 66.32/17.92 POL(5^1(x_1)) = x_1 66.32/17.92 66.32/17.92 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 66.32/17.92 66.32/17.92 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 66.32/17.92 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 66.32/17.92 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 66.32/17.92 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 66.32/17.92 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 66.32/17.92 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 66.32/17.92 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 66.32/17.92 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 66.32/17.92 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 66.32/17.92 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 66.32/17.92 2(4(1(x1))) -> 4(2(1(1(x1)))) 66.32/17.92 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.32/17.92 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.32/17.92 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 66.32/17.92 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 66.32/17.92 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 66.32/17.92 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 66.32/17.92 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 66.32/17.92 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 66.32/17.92 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 66.32/17.92 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.32/17.92 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.32/17.92 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.32/17.92 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.32/17.92 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.32/17.92 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.32/17.92 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.32/17.92 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.32/17.92 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.32/17.92 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.32/17.92 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.32/17.92 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.32/17.92 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.32/17.92 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.32/17.92 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.32/17.92 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.32/17.92 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.32/17.92 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.32/17.92 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.32/17.92 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.32/17.92 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.32/17.92 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.32/17.92 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 66.32/17.92 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 66.32/17.92 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 66.32/17.92 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 66.32/17.92 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 66.32/17.92 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.32/17.92 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 66.32/17.92 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 66.32/17.92 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 66.32/17.92 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 66.32/17.92 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 66.32/17.92 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 66.32/17.92 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 66.32/17.92 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 66.32/17.92 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 66.32/17.92 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.32/17.92 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 66.32/17.92 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) 66.32/17.92 66.32/17.92 66.32/17.92 ---------------------------------------- 66.32/17.92 66.32/17.92 (6) 66.32/17.92 Obligation: 66.32/17.92 Q DP problem: 66.32/17.92 The TRS P consists of the following rules: 66.32/17.92 66.32/17.92 0^1(3(2(x1))) -> 2^1(1(3(x1))) 66.32/17.92 2^1(1(2(3(x1)))) -> 3^1(2(x1)) 66.32/17.92 3^1(0(1(0(x1)))) -> 3^1(1(0(0(0(x1))))) 66.32/17.92 3^1(1(0(3(x1)))) -> 3^1(0(x1)) 66.32/17.92 3^1(0(1(0(x1)))) -> 0^1(0(0(x1))) 66.32/17.92 0^1(3(2(x1))) -> 0^1(2(1(3(x1)))) 66.32/17.92 0^1(3(2(x1))) -> 3^1(x1) 66.32/17.92 3^1(0(1(0(x1)))) -> 0^1(0(x1)) 66.32/17.92 0^1(0(3(2(x1)))) -> 0^1(0(2(1(3(x1))))) 66.32/17.92 0^1(0(3(2(x1)))) -> 0^1(2(1(3(x1)))) 66.32/17.92 0^1(0(3(2(x1)))) -> 2^1(1(3(x1))) 66.32/17.92 2^1(1(2(3(x1)))) -> 2^1(x1) 66.32/17.92 2^1(2(0(3(x1)))) -> 2^1(0(2(1(3(x1))))) 66.32/17.92 2^1(2(0(3(x1)))) -> 0^1(2(1(3(x1)))) 66.32/17.92 0^1(0(3(2(x1)))) -> 3^1(x1) 66.32/17.92 3^1(0(1(4(x1)))) -> 0^1(4(x1)) 66.32/17.92 0^1(1(0(3(x1)))) -> 3^1(0(x1)) 66.32/17.92 3^1(0(3(0(x1)))) -> 3^1(1(3(0(0(x1))))) 66.32/17.92 3^1(1(0(3(x1)))) -> 0^1(x1) 66.32/17.92 0^1(1(0(3(x1)))) -> 0^1(x1) 66.32/17.92 0^1(1(2(3(x1)))) -> 0^1(2(1(3(1(x1))))) 66.32/17.92 0^1(1(2(3(x1)))) -> 2^1(1(3(1(x1)))) 66.32/17.92 0^1(1(2(3(x1)))) -> 3^1(1(x1)) 66.32/17.92 0^1(1(2(5(x1)))) -> 0^1(5(x1)) 66.32/17.92 0^1(1(4(2(x1)))) -> 0^1(0(2(1(4(x1))))) 66.32/17.92 0^1(1(4(2(x1)))) -> 0^1(2(1(4(x1)))) 66.32/17.92 0^1(1(4(2(x1)))) -> 2^1(1(4(x1))) 66.32/17.92 0^1(1(4(2(x1)))) -> 4^1(x1) 66.32/17.92 3^1(0(3(0(x1)))) -> 3^1(0(0(x1))) 66.32/17.92 3^1(0(3(0(x1)))) -> 0^1(0(x1)) 66.32/17.92 0^1(1(4(2(x1)))) -> 0^1(4(0(2(1(x1))))) 66.32/17.92 0^1(1(4(2(x1)))) -> 4^1(0(2(1(x1)))) 66.32/17.92 0^1(1(4(5(x1)))) -> 5^1(4(x1)) 66.32/17.92 5^1(1(0(3(x1)))) -> 3^1(0(x1)) 66.32/17.92 3^1(2(0(1(x1)))) -> 3^1(0(2(1(x1)))) 66.32/17.92 3^1(2(0(1(x1)))) -> 0^1(2(1(x1))) 66.32/17.92 0^1(1(4(5(x1)))) -> 4^1(x1) 66.32/17.92 0^1(2(0(3(x1)))) -> 0^1(0(2(3(1(x1))))) 66.32/17.92 0^1(2(0(3(x1)))) -> 0^1(2(3(1(x1)))) 66.32/17.92 0^1(2(0(3(x1)))) -> 2^1(3(1(x1))) 66.32/17.92 2^1(2(0(3(x1)))) -> 2^1(1(3(x1))) 66.32/17.92 2^1(3(1(4(x1)))) -> 3^1(4(x1)) 66.32/17.92 3^1(2(0(1(x1)))) -> 2^1(1(x1)) 66.32/17.92 3^1(2(0(1(x1)))) -> 3^1(1(0(2(1(x1))))) 66.32/17.92 3^1(4(1(2(x1)))) -> 4^1(x1) 66.32/17.92 3^1(5(1(4(x1)))) -> 5^1(4(x1)) 66.32/17.92 5^1(1(0(3(x1)))) -> 0^1(x1) 66.32/17.92 0^1(2(0(3(x1)))) -> 3^1(1(x1)) 66.32/17.92 0^1(3(0(1(x1)))) -> 0^1(0(1(3(1(x1))))) 66.32/17.92 0^1(3(0(1(x1)))) -> 0^1(1(3(1(x1)))) 66.32/17.92 0^1(3(0(1(x1)))) -> 3^1(1(x1)) 66.32/17.92 0^1(3(0(2(x1)))) -> 0^1(0(2(3(1(x1))))) 66.32/17.92 0^1(3(0(2(x1)))) -> 0^1(2(3(1(x1)))) 66.32/17.92 0^1(3(0(2(x1)))) -> 2^1(3(1(x1))) 66.32/17.92 2^1(5(0(3(x1)))) -> 5^1(0(2(1(3(x1))))) 66.32/17.92 5^1(2(0(5(x1)))) -> 5^1(0(2(1(5(x1))))) 66.32/17.92 5^1(2(0(5(x1)))) -> 0^1(2(1(5(x1)))) 66.32/17.92 0^1(3(0(2(x1)))) -> 3^1(1(x1)) 66.32/17.92 0^1(3(0(3(x1)))) -> 0^1(0(3(1(3(x1))))) 66.32/17.92 0^1(3(0(3(x1)))) -> 0^1(3(1(3(x1)))) 66.32/17.92 0^1(3(0(3(x1)))) -> 3^1(1(3(x1))) 66.32/17.92 0^1(3(2(0(x1)))) -> 0^1(2(1(3(0(x1))))) 66.32/17.92 0^1(3(2(0(x1)))) -> 2^1(1(3(0(x1)))) 66.32/17.92 0^1(3(2(0(x1)))) -> 3^1(0(x1)) 66.32/17.92 0^1(5(2(0(x1)))) -> 2^1(1(5(0(0(x1))))) 66.32/17.92 0^1(5(2(0(x1)))) -> 5^1(0(0(x1))) 66.32/17.92 5^1(2(0(5(x1)))) -> 2^1(1(5(x1))) 66.32/17.92 5^1(4(1(0(x1)))) -> 4^1(0(x1)) 66.32/17.92 0^1(5(2(0(x1)))) -> 0^1(0(x1)) 66.32/17.92 2^1(5(0(3(x1)))) -> 0^1(2(1(3(x1)))) 66.32/17.92 2^1(5(0(3(x1)))) -> 2^1(1(3(x1))) 66.32/17.92 66.32/17.92 The TRS R consists of the following rules: 66.32/17.92 66.32/17.92 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.32/17.92 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.32/17.92 2(4(1(x1))) -> 4(2(1(1(x1)))) 66.32/17.92 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.32/17.92 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.32/17.92 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.32/17.92 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.32/17.92 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.32/17.92 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.32/17.92 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.32/17.92 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.32/17.92 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.32/17.92 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.32/17.92 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.32/17.92 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.32/17.92 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.32/17.92 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.32/17.92 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.32/17.92 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.32/17.92 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.32/17.92 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.32/17.92 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.32/17.92 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.32/17.92 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.32/17.92 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.32/17.92 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 66.32/17.92 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 66.32/17.92 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 66.32/17.92 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 66.32/17.92 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 66.32/17.92 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 66.32/17.92 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 66.32/17.92 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 66.32/17.92 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 66.32/17.92 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 66.32/17.92 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 66.32/17.92 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 66.32/17.92 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 66.32/17.92 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 66.32/17.92 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 66.32/17.92 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 66.32/17.92 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 66.32/17.92 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 66.32/17.92 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 66.32/17.92 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 66.32/17.92 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 66.32/17.92 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 66.32/17.92 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.32/17.92 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 66.32/17.92 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 66.32/17.92 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 66.32/17.92 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 66.32/17.92 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 66.32/17.92 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 66.32/17.92 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 66.32/17.92 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 66.32/17.92 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 66.32/17.92 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.32/17.92 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 66.32/17.92 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) 66.32/17.92 66.32/17.92 Q is empty. 66.32/17.92 We have to consider all minimal (P,Q,R)-chains. 66.32/17.92 ---------------------------------------- 66.32/17.92 66.32/17.92 (7) DependencyGraphProof (EQUIVALENT) 66.32/17.92 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes. 66.32/17.92 ---------------------------------------- 66.32/17.92 66.32/17.92 (8) 66.32/17.92 Obligation: 66.32/17.92 Q DP problem: 66.32/17.92 The TRS P consists of the following rules: 66.32/17.92 66.32/17.92 2^1(1(2(3(x1)))) -> 3^1(2(x1)) 66.32/17.92 3^1(0(1(0(x1)))) -> 3^1(1(0(0(0(x1))))) 66.32/17.92 3^1(1(0(3(x1)))) -> 3^1(0(x1)) 66.32/17.92 3^1(0(1(0(x1)))) -> 0^1(0(0(x1))) 66.32/17.92 0^1(3(2(x1))) -> 0^1(2(1(3(x1)))) 66.32/17.92 0^1(3(2(x1))) -> 2^1(1(3(x1))) 66.32/17.92 2^1(1(2(3(x1)))) -> 2^1(x1) 66.32/17.92 2^1(2(0(3(x1)))) -> 2^1(0(2(1(3(x1))))) 66.32/17.92 2^1(2(0(3(x1)))) -> 0^1(2(1(3(x1)))) 66.32/17.92 0^1(3(2(x1))) -> 3^1(x1) 66.32/17.92 3^1(0(1(0(x1)))) -> 0^1(0(x1)) 66.32/17.92 0^1(0(3(2(x1)))) -> 0^1(0(2(1(3(x1))))) 66.32/17.92 0^1(0(3(2(x1)))) -> 0^1(2(1(3(x1)))) 66.32/17.92 0^1(0(3(2(x1)))) -> 2^1(1(3(x1))) 66.32/17.92 0^1(0(3(2(x1)))) -> 3^1(x1) 66.32/17.92 3^1(0(1(4(x1)))) -> 0^1(4(x1)) 66.32/17.92 0^1(1(0(3(x1)))) -> 3^1(0(x1)) 66.32/17.92 3^1(0(3(0(x1)))) -> 3^1(1(3(0(0(x1))))) 66.32/17.92 3^1(1(0(3(x1)))) -> 0^1(x1) 66.32/17.92 0^1(1(0(3(x1)))) -> 0^1(x1) 66.32/17.92 0^1(1(2(3(x1)))) -> 0^1(2(1(3(1(x1))))) 66.32/17.92 0^1(1(2(3(x1)))) -> 2^1(1(3(1(x1)))) 66.32/17.92 0^1(1(2(3(x1)))) -> 3^1(1(x1)) 66.32/17.92 0^1(1(2(5(x1)))) -> 0^1(5(x1)) 66.32/17.92 0^1(1(4(2(x1)))) -> 0^1(0(2(1(4(x1))))) 66.32/17.92 0^1(1(4(2(x1)))) -> 0^1(2(1(4(x1)))) 66.32/17.92 0^1(1(4(2(x1)))) -> 2^1(1(4(x1))) 66.32/17.92 0^1(1(4(2(x1)))) -> 0^1(4(0(2(1(x1))))) 66.32/17.92 0^1(1(4(5(x1)))) -> 5^1(4(x1)) 66.32/17.92 5^1(1(0(3(x1)))) -> 3^1(0(x1)) 66.32/17.92 3^1(0(3(0(x1)))) -> 3^1(0(0(x1))) 66.32/17.92 3^1(0(3(0(x1)))) -> 0^1(0(x1)) 66.32/17.92 0^1(2(0(3(x1)))) -> 0^1(0(2(3(1(x1))))) 66.32/17.92 0^1(2(0(3(x1)))) -> 0^1(2(3(1(x1)))) 66.32/17.92 0^1(2(0(3(x1)))) -> 2^1(3(1(x1))) 66.32/17.92 2^1(2(0(3(x1)))) -> 2^1(1(3(x1))) 66.32/17.92 2^1(3(1(4(x1)))) -> 3^1(4(x1)) 66.32/17.92 3^1(2(0(1(x1)))) -> 3^1(0(2(1(x1)))) 66.32/17.92 3^1(2(0(1(x1)))) -> 0^1(2(1(x1))) 66.32/17.92 0^1(2(0(3(x1)))) -> 3^1(1(x1)) 66.32/17.92 0^1(3(0(1(x1)))) -> 0^1(0(1(3(1(x1))))) 66.32/17.92 0^1(3(0(1(x1)))) -> 0^1(1(3(1(x1)))) 66.32/17.92 0^1(3(0(1(x1)))) -> 3^1(1(x1)) 66.32/17.92 0^1(3(0(2(x1)))) -> 0^1(0(2(3(1(x1))))) 66.32/17.92 0^1(3(0(2(x1)))) -> 0^1(2(3(1(x1)))) 66.32/17.92 0^1(3(0(2(x1)))) -> 2^1(3(1(x1))) 66.32/17.92 2^1(5(0(3(x1)))) -> 5^1(0(2(1(3(x1))))) 66.32/17.92 5^1(1(0(3(x1)))) -> 0^1(x1) 66.32/17.92 0^1(3(0(2(x1)))) -> 3^1(1(x1)) 66.32/17.92 0^1(3(0(3(x1)))) -> 0^1(0(3(1(3(x1))))) 66.32/17.92 0^1(3(0(3(x1)))) -> 0^1(3(1(3(x1)))) 66.32/17.92 0^1(3(0(3(x1)))) -> 3^1(1(3(x1))) 66.32/17.92 0^1(3(2(0(x1)))) -> 0^1(2(1(3(0(x1))))) 66.32/17.92 0^1(3(2(0(x1)))) -> 2^1(1(3(0(x1)))) 66.32/17.92 0^1(3(2(0(x1)))) -> 3^1(0(x1)) 66.32/17.92 3^1(2(0(1(x1)))) -> 2^1(1(x1)) 66.32/17.92 3^1(2(0(1(x1)))) -> 3^1(1(0(2(1(x1))))) 66.32/17.92 3^1(5(1(4(x1)))) -> 5^1(4(x1)) 66.32/17.92 5^1(2(0(5(x1)))) -> 5^1(0(2(1(5(x1))))) 66.32/17.92 5^1(2(0(5(x1)))) -> 0^1(2(1(5(x1)))) 66.32/17.92 0^1(5(2(0(x1)))) -> 2^1(1(5(0(0(x1))))) 66.32/17.92 0^1(5(2(0(x1)))) -> 5^1(0(0(x1))) 66.32/17.92 5^1(2(0(5(x1)))) -> 2^1(1(5(x1))) 66.32/17.92 0^1(5(2(0(x1)))) -> 0^1(0(x1)) 66.32/17.92 2^1(5(0(3(x1)))) -> 0^1(2(1(3(x1)))) 66.32/17.92 2^1(5(0(3(x1)))) -> 2^1(1(3(x1))) 66.32/17.92 66.32/17.92 The TRS R consists of the following rules: 66.32/17.92 66.32/17.92 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.32/17.92 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.32/17.92 2(4(1(x1))) -> 4(2(1(1(x1)))) 66.32/17.92 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.32/17.92 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.32/17.92 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.32/17.92 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.32/17.92 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.32/17.92 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.32/17.92 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.32/17.92 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.32/17.92 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.32/17.92 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.32/17.92 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.32/17.92 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.32/17.92 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.32/17.92 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.32/17.92 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.32/17.92 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.32/17.92 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.32/17.92 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.32/17.92 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.32/17.92 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.32/17.92 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.32/17.92 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.32/17.92 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 66.32/17.92 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 66.32/17.92 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 66.32/17.92 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 66.32/17.92 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 66.32/17.92 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 66.32/17.92 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 66.32/17.92 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 66.32/17.92 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 66.32/17.92 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 66.32/17.92 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 66.32/17.92 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 66.32/17.92 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 66.32/17.92 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 66.32/17.92 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 66.32/17.92 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 66.32/17.92 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 66.32/17.92 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 66.32/17.92 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 66.32/17.92 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 66.32/17.92 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 66.32/17.92 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 66.32/17.92 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.32/17.92 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 66.32/17.92 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 66.32/17.92 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 66.32/17.92 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 66.32/17.92 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 66.32/17.92 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 66.32/17.92 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 66.32/17.92 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 66.32/17.92 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 66.32/17.92 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.32/17.92 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 66.32/17.92 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) 66.32/17.92 66.32/17.92 Q is empty. 66.32/17.92 We have to consider all minimal (P,Q,R)-chains. 66.32/17.92 ---------------------------------------- 66.32/17.92 66.32/17.92 (9) QDPOrderProof (EQUIVALENT) 66.32/17.92 We use the reduction pair processor [LPAR04,JAR06]. 66.32/17.92 66.32/17.92 66.32/17.92 The following pairs can be oriented strictly and are deleted. 66.32/17.92 66.32/17.92 2^1(1(2(3(x1)))) -> 3^1(2(x1)) 66.32/17.92 3^1(1(0(3(x1)))) -> 3^1(0(x1)) 66.32/17.92 0^1(3(2(x1))) -> 2^1(1(3(x1))) 66.32/17.92 2^1(1(2(3(x1)))) -> 2^1(x1) 66.32/17.92 0^1(3(2(x1))) -> 3^1(x1) 66.32/17.92 0^1(0(3(2(x1)))) -> 2^1(1(3(x1))) 66.32/17.92 0^1(0(3(2(x1)))) -> 3^1(x1) 66.32/17.92 0^1(1(0(3(x1)))) -> 3^1(0(x1)) 66.32/17.92 3^1(1(0(3(x1)))) -> 0^1(x1) 66.32/17.92 0^1(1(0(3(x1)))) -> 0^1(x1) 66.32/17.92 0^1(1(2(3(x1)))) -> 2^1(1(3(1(x1)))) 66.32/17.92 0^1(1(2(3(x1)))) -> 3^1(1(x1)) 66.32/17.92 0^1(1(2(5(x1)))) -> 0^1(5(x1)) 66.51/17.96 0^1(1(4(2(x1)))) -> 2^1(1(4(x1))) 66.51/17.96 0^1(1(4(5(x1)))) -> 5^1(4(x1)) 66.51/17.96 5^1(1(0(3(x1)))) -> 3^1(0(x1)) 66.51/17.96 3^1(0(3(0(x1)))) -> 3^1(0(0(x1))) 66.51/17.96 3^1(0(3(0(x1)))) -> 0^1(0(x1)) 66.51/17.96 0^1(2(0(3(x1)))) -> 2^1(3(1(x1))) 66.51/17.96 2^1(2(0(3(x1)))) -> 2^1(1(3(x1))) 66.51/17.96 2^1(3(1(4(x1)))) -> 3^1(4(x1)) 66.51/17.96 0^1(2(0(3(x1)))) -> 3^1(1(x1)) 66.51/17.96 0^1(3(0(1(x1)))) -> 3^1(1(x1)) 66.51/17.96 0^1(3(0(2(x1)))) -> 2^1(3(1(x1))) 66.51/17.96 5^1(1(0(3(x1)))) -> 0^1(x1) 66.51/17.96 0^1(3(0(2(x1)))) -> 3^1(1(x1)) 66.51/17.96 0^1(3(0(3(x1)))) -> 3^1(1(3(x1))) 66.51/17.96 0^1(3(2(0(x1)))) -> 2^1(1(3(0(x1)))) 66.51/17.96 0^1(3(2(0(x1)))) -> 3^1(0(x1)) 66.51/17.96 3^1(2(0(1(x1)))) -> 2^1(1(x1)) 66.51/17.96 3^1(5(1(4(x1)))) -> 5^1(4(x1)) 66.51/17.96 0^1(5(2(0(x1)))) -> 2^1(1(5(0(0(x1))))) 66.51/17.96 0^1(5(2(0(x1)))) -> 5^1(0(0(x1))) 66.51/17.96 5^1(2(0(5(x1)))) -> 2^1(1(5(x1))) 66.51/17.96 0^1(5(2(0(x1)))) -> 0^1(0(x1)) 66.51/17.96 2^1(5(0(3(x1)))) -> 2^1(1(3(x1))) 66.51/17.96 The remaining pairs can at least be oriented weakly. 66.51/17.96 Used ordering: Polynomial interpretation [POLO]: 66.51/17.96 66.51/17.96 POL(0(x_1)) = x_1 66.51/17.96 POL(0^1(x_1)) = x_1 66.51/17.96 POL(1(x_1)) = x_1 66.51/17.96 POL(2(x_1)) = 1 + x_1 66.51/17.96 POL(2^1(x_1)) = x_1 66.51/17.96 POL(3(x_1)) = 1 + x_1 66.51/17.96 POL(3^1(x_1)) = x_1 66.51/17.96 POL(4(x_1)) = 1 + x_1 66.51/17.96 POL(5(x_1)) = 1 + x_1 66.51/17.96 POL(5^1(x_1)) = x_1 66.51/17.96 66.51/17.96 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 66.51/17.96 66.51/17.96 2(4(1(x1))) -> 4(2(1(1(x1)))) 66.51/17.96 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.51/17.96 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.51/17.96 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 66.51/17.96 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 66.51/17.96 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 66.51/17.96 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 66.51/17.96 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 66.51/17.96 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 66.51/17.96 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 66.51/17.96 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.51/17.96 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.51/17.96 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.51/17.96 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.51/17.96 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.51/17.96 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.51/17.96 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.51/17.96 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.51/17.96 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.51/17.96 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.51/17.96 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.51/17.96 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.51/17.96 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.51/17.96 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.51/17.96 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.51/17.96 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.51/17.96 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.51/17.96 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.51/17.96 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.51/17.96 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.51/17.96 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.51/17.96 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.51/17.96 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 66.51/17.96 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 66.51/17.96 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 66.51/17.96 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 66.51/17.96 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 66.51/17.96 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 66.51/17.96 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 66.51/17.96 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 66.51/17.96 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 66.51/17.96 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 66.51/17.96 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 66.51/17.96 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 66.51/17.96 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 66.51/17.96 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 66.51/17.96 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 66.51/17.96 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.51/17.96 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 66.51/17.96 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 66.51/17.96 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 66.51/17.96 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 66.51/17.96 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 66.51/17.96 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 66.51/17.96 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 66.51/17.96 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 66.51/17.96 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 66.51/17.96 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.51/17.96 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 66.51/17.96 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) 66.51/17.96 66.51/17.96 66.51/17.96 ---------------------------------------- 66.51/17.96 66.51/17.96 (10) 66.51/17.96 Obligation: 66.51/17.96 Q DP problem: 66.51/17.96 The TRS P consists of the following rules: 66.51/17.96 66.51/17.96 3^1(0(1(0(x1)))) -> 3^1(1(0(0(0(x1))))) 66.51/17.96 3^1(0(1(0(x1)))) -> 0^1(0(0(x1))) 66.51/17.96 0^1(3(2(x1))) -> 0^1(2(1(3(x1)))) 66.51/17.96 2^1(2(0(3(x1)))) -> 2^1(0(2(1(3(x1))))) 66.51/17.96 2^1(2(0(3(x1)))) -> 0^1(2(1(3(x1)))) 66.51/17.96 3^1(0(1(0(x1)))) -> 0^1(0(x1)) 66.51/17.96 0^1(0(3(2(x1)))) -> 0^1(0(2(1(3(x1))))) 66.51/17.96 0^1(0(3(2(x1)))) -> 0^1(2(1(3(x1)))) 66.51/17.96 3^1(0(1(4(x1)))) -> 0^1(4(x1)) 66.51/17.96 3^1(0(3(0(x1)))) -> 3^1(1(3(0(0(x1))))) 66.51/17.96 0^1(1(2(3(x1)))) -> 0^1(2(1(3(1(x1))))) 66.51/17.96 0^1(1(4(2(x1)))) -> 0^1(0(2(1(4(x1))))) 66.51/17.96 0^1(1(4(2(x1)))) -> 0^1(2(1(4(x1)))) 66.51/17.96 0^1(1(4(2(x1)))) -> 0^1(4(0(2(1(x1))))) 66.51/17.96 0^1(2(0(3(x1)))) -> 0^1(0(2(3(1(x1))))) 66.51/17.96 0^1(2(0(3(x1)))) -> 0^1(2(3(1(x1)))) 66.51/17.96 3^1(2(0(1(x1)))) -> 3^1(0(2(1(x1)))) 66.51/17.96 3^1(2(0(1(x1)))) -> 0^1(2(1(x1))) 66.51/17.96 0^1(3(0(1(x1)))) -> 0^1(0(1(3(1(x1))))) 66.51/17.96 0^1(3(0(1(x1)))) -> 0^1(1(3(1(x1)))) 66.51/17.96 0^1(3(0(2(x1)))) -> 0^1(0(2(3(1(x1))))) 66.51/17.96 0^1(3(0(2(x1)))) -> 0^1(2(3(1(x1)))) 66.51/17.96 2^1(5(0(3(x1)))) -> 5^1(0(2(1(3(x1))))) 66.51/17.96 0^1(3(0(3(x1)))) -> 0^1(0(3(1(3(x1))))) 66.51/17.96 0^1(3(0(3(x1)))) -> 0^1(3(1(3(x1)))) 66.51/17.96 0^1(3(2(0(x1)))) -> 0^1(2(1(3(0(x1))))) 66.51/17.96 3^1(2(0(1(x1)))) -> 3^1(1(0(2(1(x1))))) 66.51/17.96 5^1(2(0(5(x1)))) -> 5^1(0(2(1(5(x1))))) 66.51/17.96 5^1(2(0(5(x1)))) -> 0^1(2(1(5(x1)))) 66.51/17.96 2^1(5(0(3(x1)))) -> 0^1(2(1(3(x1)))) 66.51/17.96 66.51/17.96 The TRS R consists of the following rules: 66.51/17.96 66.51/17.96 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.51/17.96 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.51/17.96 2(4(1(x1))) -> 4(2(1(1(x1)))) 66.51/17.96 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.51/17.96 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.51/17.96 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.51/17.96 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.51/17.96 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.51/17.96 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.51/17.96 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.51/17.96 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.51/17.96 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.51/17.96 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.51/17.96 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.51/17.96 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.51/17.96 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.51/17.96 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.51/17.96 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.51/17.96 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.51/17.96 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.51/17.96 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.51/17.96 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.51/17.96 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.51/17.96 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.51/17.96 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.51/17.96 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 66.51/17.96 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 66.51/17.96 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 66.51/17.96 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 66.51/17.96 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 66.51/17.96 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 66.51/17.96 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 66.51/17.96 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 66.51/17.96 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 66.51/17.96 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 66.51/17.96 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 66.51/17.96 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 66.51/17.96 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 66.51/17.96 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 66.51/17.96 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 66.51/17.96 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 66.51/17.96 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 66.51/17.96 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 66.51/17.96 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 66.51/17.96 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 66.51/17.96 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 66.51/17.96 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 66.51/17.96 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.51/17.96 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 66.51/17.96 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 66.51/17.96 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 66.51/17.96 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 66.51/17.96 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 66.51/17.96 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 66.51/17.96 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 66.51/17.96 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 66.51/17.96 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 66.51/17.96 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.51/17.96 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 66.51/17.96 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) 66.51/17.96 66.51/17.96 Q is empty. 66.51/17.96 We have to consider all minimal (P,Q,R)-chains. 66.51/17.96 ---------------------------------------- 66.51/17.96 66.51/17.96 (11) DependencyGraphProof (EQUIVALENT) 66.51/17.96 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 11 less nodes. 66.51/17.96 ---------------------------------------- 66.51/17.96 66.51/17.96 (12) 66.51/17.96 Complex Obligation (AND) 66.51/17.96 66.51/17.96 ---------------------------------------- 66.51/17.96 66.51/17.96 (13) 66.51/17.96 Obligation: 66.51/17.96 Q DP problem: 66.51/17.96 The TRS P consists of the following rules: 66.51/17.96 66.51/17.96 0^1(0(3(2(x1)))) -> 0^1(0(2(1(3(x1))))) 66.51/17.96 0^1(3(2(x1))) -> 0^1(2(1(3(x1)))) 66.51/17.96 0^1(0(3(2(x1)))) -> 0^1(2(1(3(x1)))) 66.51/17.96 0^1(1(2(3(x1)))) -> 0^1(2(1(3(1(x1))))) 66.51/17.96 0^1(1(4(2(x1)))) -> 0^1(0(2(1(4(x1))))) 66.51/17.96 0^1(1(4(2(x1)))) -> 0^1(2(1(4(x1)))) 66.51/17.96 0^1(1(4(2(x1)))) -> 0^1(4(0(2(1(x1))))) 66.51/17.96 0^1(2(0(3(x1)))) -> 0^1(0(2(3(1(x1))))) 66.51/17.96 0^1(2(0(3(x1)))) -> 0^1(2(3(1(x1)))) 66.51/17.96 0^1(3(0(1(x1)))) -> 0^1(0(1(3(1(x1))))) 66.51/17.96 0^1(3(0(1(x1)))) -> 0^1(1(3(1(x1)))) 66.51/17.96 0^1(3(0(2(x1)))) -> 0^1(0(2(3(1(x1))))) 66.51/17.96 0^1(3(0(2(x1)))) -> 0^1(2(3(1(x1)))) 66.51/17.96 0^1(3(0(3(x1)))) -> 0^1(0(3(1(3(x1))))) 66.51/17.96 0^1(3(0(3(x1)))) -> 0^1(3(1(3(x1)))) 66.51/17.96 0^1(3(2(0(x1)))) -> 0^1(2(1(3(0(x1))))) 66.51/17.96 66.51/17.96 The TRS R consists of the following rules: 66.51/17.96 66.51/17.96 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.51/17.96 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.51/17.96 2(4(1(x1))) -> 4(2(1(1(x1)))) 66.51/17.96 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.51/17.96 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.51/17.96 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.51/17.96 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.51/17.96 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.51/17.96 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.51/17.96 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.51/17.96 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.51/17.96 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.51/17.96 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.51/17.96 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.51/17.96 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.51/17.96 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.51/17.96 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.51/17.96 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.51/17.96 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.51/17.96 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.51/17.96 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.51/17.96 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.51/17.96 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.51/17.96 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.51/17.96 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.51/17.96 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 66.51/17.96 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 66.51/17.96 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 66.51/17.96 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 66.51/17.96 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 66.51/17.96 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 66.51/17.96 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 66.51/17.96 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 66.51/17.96 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 66.51/17.96 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 66.51/17.96 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 66.51/17.96 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 66.51/17.96 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 66.51/17.96 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 66.51/17.96 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 66.51/17.96 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 66.51/17.96 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 66.51/17.96 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 66.51/17.96 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 66.51/17.96 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 66.51/17.96 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 66.51/17.96 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 66.51/17.96 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.51/17.96 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 66.51/17.96 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 66.51/17.96 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 66.51/17.96 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 66.51/17.96 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 66.51/17.96 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 66.51/17.96 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 66.51/17.96 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 66.51/17.96 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 66.51/17.96 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.51/17.96 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 66.51/17.96 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) 66.51/17.96 66.51/17.96 Q is empty. 66.51/17.96 We have to consider all minimal (P,Q,R)-chains. 66.51/17.96 ---------------------------------------- 66.51/17.96 66.51/17.96 (14) QDPOrderProof (EQUIVALENT) 66.51/17.96 We use the reduction pair processor [LPAR04,JAR06]. 66.51/17.96 66.51/17.96 66.51/17.96 The following pairs can be oriented strictly and are deleted. 66.51/17.96 66.51/17.96 0^1(0(3(2(x1)))) -> 0^1(0(2(1(3(x1))))) 66.51/17.96 0^1(3(2(x1))) -> 0^1(2(1(3(x1)))) 66.51/17.96 0^1(0(3(2(x1)))) -> 0^1(2(1(3(x1)))) 66.51/17.96 0^1(3(0(1(x1)))) -> 0^1(0(1(3(1(x1))))) 66.51/17.96 0^1(3(0(1(x1)))) -> 0^1(1(3(1(x1)))) 66.51/17.96 0^1(3(0(2(x1)))) -> 0^1(0(2(3(1(x1))))) 66.51/17.96 0^1(3(0(2(x1)))) -> 0^1(2(3(1(x1)))) 66.51/17.96 0^1(3(2(0(x1)))) -> 0^1(2(1(3(0(x1))))) 66.51/17.96 The remaining pairs can at least be oriented weakly. 66.51/17.96 Used ordering: Polynomial interpretation [POLO]: 66.51/17.96 66.51/17.96 POL(0(x_1)) = x_1 66.51/17.96 POL(0^1(x_1)) = x_1 66.51/17.96 POL(1(x_1)) = 0 66.51/17.96 POL(2(x_1)) = 0 66.51/17.96 POL(3(x_1)) = 1 66.51/17.96 POL(4(x_1)) = 0 66.51/17.96 POL(5(x_1)) = 0 66.51/17.96 66.51/17.96 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 66.51/17.96 66.51/17.96 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 66.51/17.96 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 66.51/17.96 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 66.51/17.96 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.51/17.96 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.51/17.96 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.51/17.96 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.51/17.96 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.51/17.96 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.51/17.96 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.51/17.96 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.51/17.96 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.51/17.96 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.51/17.96 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.51/17.96 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.51/17.96 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.51/17.96 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.51/17.96 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.51/17.96 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.51/17.96 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.51/17.96 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.51/17.96 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.51/17.96 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.51/17.96 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.51/17.96 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.51/17.96 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.51/17.96 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.51/17.96 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 66.51/17.96 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 66.51/17.96 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 66.51/17.96 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 66.51/17.96 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 66.51/17.96 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.51/17.96 2(4(1(x1))) -> 4(2(1(1(x1)))) 66.51/17.96 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 66.51/17.96 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 66.51/17.96 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 66.51/17.96 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 66.51/17.96 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 66.51/17.96 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 66.51/17.96 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 66.51/17.96 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 66.51/17.96 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 66.51/17.96 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 66.51/17.96 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 66.51/17.96 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 66.51/17.96 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 66.51/17.96 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 66.51/17.96 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) 66.51/17.96 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 66.51/17.96 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 66.51/17.96 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 66.51/17.96 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.51/17.96 66.51/17.96 66.51/17.96 ---------------------------------------- 66.51/17.96 66.51/17.96 (15) 66.51/17.96 Obligation: 66.51/17.96 Q DP problem: 66.51/17.96 The TRS P consists of the following rules: 66.51/17.96 66.51/17.96 0^1(1(2(3(x1)))) -> 0^1(2(1(3(1(x1))))) 66.51/17.96 0^1(1(4(2(x1)))) -> 0^1(0(2(1(4(x1))))) 66.51/17.96 0^1(1(4(2(x1)))) -> 0^1(2(1(4(x1)))) 66.51/17.96 0^1(1(4(2(x1)))) -> 0^1(4(0(2(1(x1))))) 66.51/17.96 0^1(2(0(3(x1)))) -> 0^1(0(2(3(1(x1))))) 66.51/17.96 0^1(2(0(3(x1)))) -> 0^1(2(3(1(x1)))) 66.51/17.96 0^1(3(0(3(x1)))) -> 0^1(0(3(1(3(x1))))) 66.51/17.96 0^1(3(0(3(x1)))) -> 0^1(3(1(3(x1)))) 66.51/17.96 66.51/17.96 The TRS R consists of the following rules: 66.51/17.96 66.51/17.96 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.51/17.96 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.51/17.96 2(4(1(x1))) -> 4(2(1(1(x1)))) 66.51/17.96 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.51/17.96 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.51/17.96 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.51/17.96 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.51/17.96 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.51/17.96 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.51/17.96 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.51/17.96 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.51/17.96 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.51/17.96 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.51/17.96 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.51/17.96 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.51/17.96 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.51/17.96 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.51/17.96 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.51/17.96 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.51/17.96 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.51/17.96 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.51/17.96 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.51/17.96 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.51/17.96 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.51/17.96 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.51/17.96 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 66.51/17.96 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 66.51/17.96 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 66.51/17.96 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 66.51/17.96 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 66.51/17.96 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 66.51/17.96 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 66.51/17.96 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 66.51/17.96 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 66.51/17.96 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 66.51/17.96 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 66.51/17.96 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 66.51/17.96 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 66.51/17.96 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 66.51/17.96 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 66.51/17.96 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 66.51/17.96 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 66.51/17.96 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 66.51/17.96 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 66.51/17.96 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 66.51/17.96 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 66.51/17.96 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 66.51/17.96 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.51/17.96 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 66.51/17.96 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 66.51/17.96 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 66.51/17.96 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 66.51/17.96 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 66.51/17.96 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 66.51/17.96 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 66.51/17.96 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 66.51/17.96 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 66.51/17.96 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.51/17.96 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 66.51/17.96 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) 66.51/17.96 66.51/17.96 Q is empty. 66.51/17.96 We have to consider all minimal (P,Q,R)-chains. 66.51/17.96 ---------------------------------------- 66.51/17.96 66.51/17.96 (16) QDPOrderProof (EQUIVALENT) 66.51/17.96 We use the reduction pair processor [LPAR04,JAR06]. 66.51/17.96 66.51/17.96 66.51/17.96 The following pairs can be oriented strictly and are deleted. 66.51/17.96 66.51/17.96 0^1(1(2(3(x1)))) -> 0^1(2(1(3(1(x1))))) 66.51/17.96 0^1(1(4(2(x1)))) -> 0^1(0(2(1(4(x1))))) 66.51/17.96 0^1(1(4(2(x1)))) -> 0^1(2(1(4(x1)))) 66.51/17.96 0^1(1(4(2(x1)))) -> 0^1(4(0(2(1(x1))))) 66.51/17.96 The remaining pairs can at least be oriented weakly. 66.51/17.96 Used ordering: Polynomial interpretation [POLO]: 66.51/17.96 66.51/17.96 POL(0(x_1)) = x_1 66.51/17.96 POL(0^1(x_1)) = x_1 66.51/17.96 POL(1(x_1)) = 1 66.51/17.96 POL(2(x_1)) = 0 66.51/17.96 POL(3(x_1)) = 1 + x_1 66.51/17.96 POL(4(x_1)) = 0 66.51/17.96 POL(5(x_1)) = 0 66.51/17.96 66.51/17.96 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 66.51/17.96 66.51/17.96 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 66.51/17.96 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 66.51/17.96 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 66.51/17.96 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.51/17.96 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.51/17.96 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 66.51/17.96 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 66.51/17.96 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 66.51/17.96 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 66.51/17.96 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 66.51/17.96 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.51/17.96 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.51/17.96 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.51/17.96 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.51/17.96 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.51/17.96 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.51/17.96 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.51/17.96 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.51/17.96 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.51/17.96 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.51/17.96 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.51/17.96 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.51/17.96 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.51/17.96 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.51/17.96 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.51/17.96 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.51/17.96 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.51/17.96 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.51/17.96 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.51/17.96 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.51/17.96 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.51/17.96 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.51/17.96 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.51/17.96 2(4(1(x1))) -> 4(2(1(1(x1)))) 66.51/17.96 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 66.51/17.96 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 66.51/17.96 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 66.51/17.96 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 66.51/17.96 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 66.51/17.96 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 66.51/17.96 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 66.51/17.96 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 66.51/17.96 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 66.51/17.96 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 66.51/17.96 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 66.51/17.96 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 66.51/17.96 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 66.51/17.96 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 66.51/17.96 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) 66.51/17.96 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 66.51/17.96 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 66.51/17.96 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 66.51/17.96 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.51/17.96 66.51/17.96 66.51/17.96 ---------------------------------------- 66.51/17.96 66.51/17.96 (17) 66.51/17.96 Obligation: 66.51/17.96 Q DP problem: 66.51/17.96 The TRS P consists of the following rules: 66.51/17.96 66.51/17.96 0^1(2(0(3(x1)))) -> 0^1(0(2(3(1(x1))))) 66.51/17.96 0^1(2(0(3(x1)))) -> 0^1(2(3(1(x1)))) 66.51/17.96 0^1(3(0(3(x1)))) -> 0^1(0(3(1(3(x1))))) 66.51/17.96 0^1(3(0(3(x1)))) -> 0^1(3(1(3(x1)))) 66.51/17.96 66.51/17.96 The TRS R consists of the following rules: 66.51/17.96 66.51/17.96 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.51/17.96 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.51/17.96 2(4(1(x1))) -> 4(2(1(1(x1)))) 66.51/17.96 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.51/17.96 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.51/17.96 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.51/17.96 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.51/17.96 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.51/17.96 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.51/17.96 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.51/17.96 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.51/17.96 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.51/17.96 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.51/17.96 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.51/17.96 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.51/17.96 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.51/17.96 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.51/17.96 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.51/17.96 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.51/17.96 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.51/17.96 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.51/17.96 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.51/17.96 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.51/17.96 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.51/17.96 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.51/17.96 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 66.51/17.96 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 66.51/17.96 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 66.51/17.96 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 66.51/17.96 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 66.51/17.96 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 66.51/17.96 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 66.51/17.96 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 66.51/17.96 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 66.51/17.96 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 66.51/17.96 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 66.51/17.96 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 66.51/17.96 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 66.51/17.96 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 66.51/17.96 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 66.51/17.96 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 66.51/17.96 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 66.51/17.96 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 66.51/17.96 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 66.51/17.96 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 66.51/17.96 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 66.51/17.96 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 66.51/17.96 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.51/17.96 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 66.51/17.96 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 66.51/17.96 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 66.51/17.96 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 66.51/17.96 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 66.51/17.96 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 66.51/17.96 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 66.51/17.96 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 66.51/17.96 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 66.51/17.96 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.51/17.96 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 66.51/17.96 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) 66.51/17.96 66.51/17.96 Q is empty. 66.51/17.96 We have to consider all minimal (P,Q,R)-chains. 66.51/17.96 ---------------------------------------- 66.51/17.96 66.51/17.96 (18) QDPOrderProof (EQUIVALENT) 66.51/17.96 We use the reduction pair processor [LPAR04,JAR06]. 66.51/17.96 66.51/17.96 66.51/17.96 The following pairs can be oriented strictly and are deleted. 66.51/17.96 66.51/17.96 0^1(2(0(3(x1)))) -> 0^1(2(3(1(x1)))) 66.51/17.96 0^1(3(0(3(x1)))) -> 0^1(3(1(3(x1)))) 66.51/17.96 The remaining pairs can at least be oriented weakly. 66.51/17.96 Used ordering: Polynomial interpretation [POLO]: 66.51/17.96 66.51/17.96 POL(0(x_1)) = 1 66.51/17.96 POL(0^1(x_1)) = x_1 66.51/17.96 POL(1(x_1)) = 0 66.51/17.96 POL(2(x_1)) = x_1 66.51/17.96 POL(3(x_1)) = x_1 66.51/17.96 POL(4(x_1)) = 0 66.51/17.96 POL(5(x_1)) = 1 66.51/17.96 66.51/17.96 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 66.51/17.96 66.51/17.96 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 66.51/17.96 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 66.51/17.96 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 66.51/17.96 2(4(1(x1))) -> 4(2(1(1(x1)))) 66.51/17.96 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.51/17.96 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.51/17.96 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 66.51/17.96 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 66.51/17.96 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 66.51/17.96 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 66.51/17.96 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 66.51/17.96 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 66.51/17.96 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 66.51/17.96 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.51/17.96 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.51/17.96 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.51/17.96 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.51/17.96 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.51/17.96 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.51/17.96 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.51/17.96 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.51/17.96 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.51/17.96 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.51/17.96 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.51/17.96 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.51/17.96 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.51/17.96 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.51/17.96 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.51/17.96 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.51/17.96 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.51/17.96 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.51/17.96 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.51/17.96 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.51/17.96 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.51/17.96 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.51/17.96 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 66.51/17.96 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 66.51/17.96 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 66.51/17.96 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 66.51/17.96 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 66.51/17.96 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 66.51/17.96 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 66.51/17.96 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) 66.51/17.96 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 66.51/17.96 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 66.51/17.96 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 66.51/17.96 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.51/17.96 66.51/17.96 66.51/17.96 ---------------------------------------- 66.51/17.96 66.51/17.96 (19) 66.51/17.96 Obligation: 66.51/17.96 Q DP problem: 66.51/17.96 The TRS P consists of the following rules: 66.51/17.96 66.51/17.96 0^1(2(0(3(x1)))) -> 0^1(0(2(3(1(x1))))) 66.51/17.96 0^1(3(0(3(x1)))) -> 0^1(0(3(1(3(x1))))) 66.51/17.96 66.51/17.96 The TRS R consists of the following rules: 66.51/17.96 66.51/17.96 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.51/17.96 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.51/17.96 2(4(1(x1))) -> 4(2(1(1(x1)))) 66.51/17.96 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.51/17.96 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.51/17.96 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.51/17.96 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.51/17.96 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.51/17.96 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.51/17.96 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.51/17.96 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.51/17.96 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.51/17.96 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.51/17.96 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.51/17.96 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.51/17.96 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.51/17.96 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.51/17.96 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.51/17.96 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.51/17.96 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.51/17.96 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.51/17.96 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.51/17.96 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.51/17.96 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.51/17.96 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.51/17.96 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 66.51/17.96 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 66.51/17.96 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 66.51/17.96 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 66.51/17.96 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 66.51/17.96 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 66.51/17.96 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 66.51/17.96 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 66.51/17.96 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 66.51/17.96 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 66.51/17.96 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 66.51/17.96 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 66.51/17.96 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 66.51/17.96 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 66.51/17.96 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 66.51/17.96 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 66.51/17.96 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 66.51/17.96 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 66.51/17.96 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 66.51/17.96 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 66.51/17.96 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 66.51/17.96 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 66.51/17.96 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.51/17.96 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 66.51/17.96 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 66.51/17.96 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 66.51/17.96 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 66.51/17.96 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 66.51/17.96 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 66.51/17.96 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 66.51/17.96 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 66.51/17.96 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 66.51/17.96 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.51/17.96 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 66.51/17.96 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) 66.51/17.96 66.51/17.96 Q is empty. 66.51/17.96 We have to consider all minimal (P,Q,R)-chains. 66.51/17.96 ---------------------------------------- 66.51/17.96 66.51/17.96 (20) QDPOrderProof (EQUIVALENT) 66.51/17.96 We use the reduction pair processor [LPAR04,JAR06]. 66.51/17.96 66.51/17.96 66.51/17.96 The following pairs can be oriented strictly and are deleted. 66.51/17.96 66.51/17.96 0^1(3(0(3(x1)))) -> 0^1(0(3(1(3(x1))))) 66.51/17.96 The remaining pairs can at least be oriented weakly. 66.51/17.96 Used ordering: Polynomial interpretation [POLO]: 66.51/17.96 66.51/17.96 POL(0(x_1)) = 0 66.51/17.96 POL(0^1(x_1)) = x_1 66.51/17.96 POL(1(x_1)) = 0 66.51/17.96 POL(2(x_1)) = 0 66.51/17.96 POL(3(x_1)) = 1 66.51/17.96 POL(4(x_1)) = 0 66.51/17.96 POL(5(x_1)) = 0 66.51/17.96 66.51/17.96 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 66.51/17.96 66.51/17.96 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.51/17.96 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.51/17.96 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.51/17.96 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.51/17.96 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.51/17.96 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.51/17.96 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.51/17.96 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.51/17.96 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.51/17.96 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.51/17.96 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.51/17.96 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.51/17.96 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.51/17.96 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.51/17.96 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.51/17.96 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.51/17.96 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.51/17.96 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.51/17.96 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.51/17.96 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.51/17.96 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.51/17.96 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.51/17.96 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.51/17.96 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.51/17.96 66.51/17.96 66.51/17.96 ---------------------------------------- 66.51/17.96 66.51/17.96 (21) 66.51/17.96 Obligation: 66.51/17.96 Q DP problem: 66.51/17.96 The TRS P consists of the following rules: 66.51/17.96 66.51/17.96 0^1(2(0(3(x1)))) -> 0^1(0(2(3(1(x1))))) 66.51/17.96 66.51/17.96 The TRS R consists of the following rules: 66.51/17.96 66.51/17.96 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.51/17.96 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.51/17.96 2(4(1(x1))) -> 4(2(1(1(x1)))) 66.51/17.96 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.51/17.96 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.51/17.96 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.51/17.96 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.51/17.96 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.51/17.96 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.51/17.96 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.51/17.96 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.51/17.96 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.51/17.96 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.51/17.96 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.51/17.96 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.51/17.96 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.51/17.96 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.51/17.96 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.51/17.96 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.51/17.96 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.51/17.96 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.51/17.96 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.51/17.96 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.51/17.96 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.51/17.96 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.51/17.96 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 66.51/17.96 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 66.51/17.96 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 66.51/17.96 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 66.51/17.96 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 66.51/17.96 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 66.51/17.96 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 66.51/17.96 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 66.51/17.96 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 66.51/17.96 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 66.51/17.96 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 66.51/17.96 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 66.51/17.96 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 66.51/17.96 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 66.51/17.96 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 66.51/17.96 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 66.51/17.96 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 66.51/17.96 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 66.51/17.96 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 66.51/17.96 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 66.51/17.96 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 66.51/17.96 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 66.51/17.96 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.51/17.96 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 66.51/17.96 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 66.51/17.96 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 66.51/17.96 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 66.51/17.96 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 66.51/17.96 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 66.51/17.96 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 66.51/17.96 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 66.51/17.96 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 66.51/17.96 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.51/17.96 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 66.51/17.96 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) 66.51/17.96 66.51/17.96 Q is empty. 66.51/17.96 We have to consider all minimal (P,Q,R)-chains. 66.51/17.96 ---------------------------------------- 66.51/17.96 66.51/17.96 (22) QDPOrderProof (EQUIVALENT) 66.51/17.96 We use the reduction pair processor [LPAR04,JAR06]. 66.51/17.96 66.51/17.96 66.51/17.96 The following pairs can be oriented strictly and are deleted. 66.51/17.96 66.51/17.96 0^1(2(0(3(x1)))) -> 0^1(0(2(3(1(x1))))) 66.51/17.96 The remaining pairs can at least be oriented weakly. 66.51/17.96 Used ordering: Polynomial interpretation [POLO]: 66.51/17.96 66.51/17.96 POL(0(x_1)) = 1 66.51/17.96 POL(0^1(x_1)) = x_1 66.51/17.96 POL(1(x_1)) = 0 66.51/17.96 POL(2(x_1)) = 1 + x_1 66.51/17.96 POL(3(x_1)) = 1 + x_1 66.51/17.96 POL(4(x_1)) = 1 66.51/17.96 POL(5(x_1)) = 0 66.51/17.96 66.51/17.96 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 66.51/17.96 66.51/17.96 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.51/17.96 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.51/17.96 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.51/17.96 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.51/17.96 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.51/17.96 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.51/17.96 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.51/17.96 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.51/17.96 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.51/17.96 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.51/17.96 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.51/17.96 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.51/17.96 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.51/17.96 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.51/17.96 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.51/17.96 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.51/17.96 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.51/17.96 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.51/17.96 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.51/17.96 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.51/17.96 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.51/17.96 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.55/17.97 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.55/17.97 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.55/17.97 66.55/17.97 66.55/17.97 ---------------------------------------- 66.55/17.97 66.55/17.97 (23) 66.55/17.97 Obligation: 66.55/17.97 Q DP problem: 66.55/17.97 P is empty. 66.55/17.97 The TRS R consists of the following rules: 66.55/17.97 66.55/17.97 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.55/17.97 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.55/17.97 2(4(1(x1))) -> 4(2(1(1(x1)))) 66.55/17.97 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.55/17.97 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.55/17.97 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.55/17.97 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.55/17.97 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.55/17.97 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.55/17.97 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.55/17.97 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.55/17.97 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.55/17.97 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.55/17.97 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.55/17.97 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.55/17.97 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.55/17.97 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.55/17.97 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.55/17.97 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.55/17.97 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.55/17.97 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.55/17.97 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.55/17.97 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.55/17.97 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.55/17.97 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.55/17.97 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 66.55/17.97 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 66.55/17.97 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 66.55/17.97 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 66.55/17.97 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 66.55/17.97 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 66.55/17.97 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 66.55/17.97 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 66.55/17.97 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 66.55/17.97 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 66.55/17.97 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 66.55/17.97 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 66.55/17.97 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 66.55/17.97 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 66.55/17.97 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 66.55/17.97 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 66.55/17.97 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 66.55/17.97 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 66.55/17.97 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 66.55/17.97 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 66.55/17.97 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 66.55/17.97 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 66.55/17.97 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.55/17.97 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 66.55/17.97 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 66.55/17.97 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 66.55/17.97 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 66.55/17.97 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 66.55/17.97 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 66.55/17.97 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 66.55/17.97 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 66.55/17.97 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 66.55/17.97 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.55/17.97 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 66.55/17.97 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) 66.55/17.97 66.55/17.97 Q is empty. 66.55/17.97 We have to consider all minimal (P,Q,R)-chains. 66.55/17.97 ---------------------------------------- 66.55/17.97 66.55/17.97 (24) PisEmptyProof (EQUIVALENT) 66.55/17.97 The TRS P is empty. Hence, there is no (P,Q,R) chain. 66.55/17.97 ---------------------------------------- 66.55/17.97 66.55/17.97 (25) 66.55/17.97 YES 66.55/17.97 66.55/17.97 ---------------------------------------- 66.55/17.97 66.55/17.97 (26) 66.55/17.97 Obligation: 66.55/17.97 Q DP problem: 66.55/17.97 The TRS P consists of the following rules: 66.55/17.97 66.55/17.97 5^1(2(0(5(x1)))) -> 5^1(0(2(1(5(x1))))) 66.55/17.97 66.55/17.97 The TRS R consists of the following rules: 66.55/17.97 66.55/17.97 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.55/17.97 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.55/17.97 2(4(1(x1))) -> 4(2(1(1(x1)))) 66.55/17.97 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.55/17.97 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.55/17.97 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.55/17.97 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.55/17.97 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.55/17.97 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.55/17.97 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.55/17.97 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.55/17.97 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.55/17.97 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.55/17.97 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.55/17.97 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.55/17.97 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.55/17.97 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.55/17.97 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.55/17.97 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.55/17.97 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.55/17.97 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.55/17.97 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.55/17.97 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.55/17.97 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.55/17.97 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.55/17.97 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 66.55/17.97 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 66.55/17.97 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 66.55/17.97 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 66.55/17.97 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 66.55/17.97 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 66.55/17.97 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 66.55/17.97 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 66.55/17.97 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 66.55/17.97 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 66.55/17.97 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 66.55/17.97 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 66.55/17.97 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 66.55/17.97 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 66.55/17.97 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 66.55/17.97 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 66.55/17.97 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 66.55/17.97 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 66.55/17.97 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 66.55/17.97 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 66.55/17.97 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 66.55/17.97 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 66.55/17.97 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.55/17.97 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 66.55/17.97 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 66.55/17.97 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 66.55/17.97 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 66.55/17.97 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 66.55/17.97 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 66.55/17.97 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 66.55/17.97 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 66.55/17.97 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 66.55/17.97 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.55/17.97 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 66.55/17.97 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) 66.55/17.97 66.55/17.97 Q is empty. 66.55/17.97 We have to consider all minimal (P,Q,R)-chains. 66.55/17.97 ---------------------------------------- 66.55/17.97 66.55/17.97 (27) QDPOrderProof (EQUIVALENT) 66.55/17.97 We use the reduction pair processor [LPAR04,JAR06]. 66.55/17.97 66.55/17.97 66.55/17.97 The following pairs can be oriented strictly and are deleted. 66.55/17.97 66.55/17.97 5^1(2(0(5(x1)))) -> 5^1(0(2(1(5(x1))))) 66.55/17.97 The remaining pairs can at least be oriented weakly. 66.55/17.97 Used ordering: Polynomial interpretation [POLO]: 66.55/17.97 66.55/17.97 POL(0(x_1)) = 1 66.55/17.97 POL(1(x_1)) = 0 66.55/17.97 POL(2(x_1)) = 1 + x_1 66.55/17.97 POL(3(x_1)) = 1 66.55/17.97 POL(4(x_1)) = 0 66.55/17.97 POL(5(x_1)) = 0 66.55/17.97 POL(5^1(x_1)) = x_1 66.55/17.97 66.55/17.97 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 66.55/17.97 66.55/17.97 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.55/17.97 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.55/17.97 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.55/17.97 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.55/17.97 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.55/17.97 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.55/17.97 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.55/17.97 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.55/17.97 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.55/17.97 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.55/17.97 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.55/17.97 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.55/17.97 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.55/17.97 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.55/17.97 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.55/17.97 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.55/17.97 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.55/17.97 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.55/17.97 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.55/17.97 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.55/17.97 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.55/17.97 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.55/17.97 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.55/17.97 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.55/17.97 66.55/17.97 66.55/17.97 ---------------------------------------- 66.55/17.97 66.55/17.97 (28) 66.55/17.97 Obligation: 66.55/17.97 Q DP problem: 66.55/17.97 P is empty. 66.55/17.97 The TRS R consists of the following rules: 66.55/17.97 66.55/17.97 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.55/17.97 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.55/17.97 2(4(1(x1))) -> 4(2(1(1(x1)))) 66.55/17.97 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.55/17.97 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.55/17.97 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.55/17.97 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.55/17.97 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.55/17.97 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.55/17.97 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.55/17.97 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.55/17.97 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.55/17.97 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.55/17.97 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.55/17.97 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.55/17.97 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.55/17.97 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.55/17.97 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.55/17.97 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.55/17.97 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.55/17.97 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.55/17.97 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.55/17.97 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.55/17.97 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.55/17.97 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.55/17.97 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 66.55/17.97 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 66.55/17.97 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 66.55/17.97 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 66.55/17.97 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 66.55/17.97 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 66.55/17.98 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 66.55/17.98 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 66.55/17.98 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 66.55/17.98 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 66.55/17.98 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 66.55/17.98 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 66.55/17.98 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 66.55/17.98 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 66.55/17.98 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 66.55/17.98 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 66.55/17.98 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 66.55/17.98 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 66.55/17.98 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 66.55/17.98 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 66.55/17.98 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 66.55/17.98 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 66.55/17.98 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.55/17.98 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 66.55/17.98 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 66.55/17.98 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 66.55/17.98 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 66.55/17.98 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 66.55/17.98 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 66.55/17.98 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 66.55/17.98 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 66.55/17.98 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 66.55/17.98 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.55/17.98 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 66.55/17.98 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) 66.55/17.98 66.55/17.98 Q is empty. 66.55/17.98 We have to consider all minimal (P,Q,R)-chains. 66.55/17.98 ---------------------------------------- 66.55/17.98 66.55/17.98 (29) PisEmptyProof (EQUIVALENT) 66.55/17.98 The TRS P is empty. Hence, there is no (P,Q,R) chain. 66.55/17.98 ---------------------------------------- 66.55/17.98 66.55/17.98 (30) 66.55/17.98 YES 66.55/17.98 66.55/17.98 ---------------------------------------- 66.55/17.98 66.55/17.98 (31) 66.55/17.98 Obligation: 66.55/17.98 Q DP problem: 66.55/17.98 The TRS P consists of the following rules: 66.55/17.98 66.55/17.98 2^1(2(0(3(x1)))) -> 2^1(0(2(1(3(x1))))) 66.55/17.98 66.55/17.98 The TRS R consists of the following rules: 66.55/17.98 66.55/17.98 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.55/17.98 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.55/17.98 2(4(1(x1))) -> 4(2(1(1(x1)))) 66.55/17.98 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.55/17.98 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.55/17.98 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.55/17.98 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.55/17.98 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.55/17.98 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.55/17.98 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.55/17.98 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.55/17.98 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.55/17.98 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.55/17.98 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.55/17.98 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.55/17.98 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.55/17.98 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.55/17.98 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.55/17.98 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.55/17.98 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.55/17.98 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.55/17.98 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.55/17.98 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.55/17.98 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.55/17.98 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.55/17.98 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 66.55/17.98 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 66.55/17.98 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 66.55/17.98 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 66.55/17.98 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 66.55/17.98 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 66.55/17.98 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 66.55/17.98 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 66.55/17.98 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 66.55/17.98 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 66.55/17.98 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 66.55/17.98 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 66.55/17.98 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 66.55/17.98 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 66.55/17.98 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 66.55/17.98 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 66.55/17.98 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 66.55/17.98 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 66.55/17.98 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 66.55/17.98 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 66.55/17.98 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 66.55/17.98 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 66.55/17.98 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.55/17.98 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 66.55/17.98 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 66.55/17.98 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 66.55/17.98 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 66.55/17.98 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 66.55/17.98 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 66.55/17.98 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 66.55/17.98 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 66.55/17.98 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 66.55/17.98 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.55/17.98 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 66.55/17.98 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) 66.55/17.98 66.55/17.98 Q is empty. 66.55/17.98 We have to consider all minimal (P,Q,R)-chains. 66.55/17.98 ---------------------------------------- 66.55/17.98 66.55/17.98 (32) QDPOrderProof (EQUIVALENT) 66.55/17.98 We use the reduction pair processor [LPAR04,JAR06]. 66.55/17.98 66.55/17.98 66.55/17.98 The following pairs can be oriented strictly and are deleted. 66.55/17.98 66.55/17.98 2^1(2(0(3(x1)))) -> 2^1(0(2(1(3(x1))))) 66.55/17.98 The remaining pairs can at least be oriented weakly. 66.55/17.98 Used ordering: Polynomial interpretation [POLO]: 66.55/17.98 66.55/17.98 POL(0(x_1)) = 1 66.55/17.98 POL(1(x_1)) = 0 66.55/17.98 POL(2(x_1)) = 1 + x_1 66.55/17.98 POL(2^1(x_1)) = x_1 66.55/17.98 POL(3(x_1)) = 0 66.55/17.98 POL(4(x_1)) = 0 66.55/17.98 POL(5(x_1)) = 0 66.55/17.98 66.55/17.98 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 66.55/17.98 66.55/17.98 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.55/17.98 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.55/17.98 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.55/17.98 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.55/17.98 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.55/17.98 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.55/17.98 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.55/17.98 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.55/17.98 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.55/17.98 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.55/17.98 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.55/17.98 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.55/17.98 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.55/17.98 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.55/17.98 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.55/17.98 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.55/17.98 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.55/17.98 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.55/17.98 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.55/17.98 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.55/17.98 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.55/17.98 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.55/17.98 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.55/17.98 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.55/17.98 66.55/17.98 66.55/17.98 ---------------------------------------- 66.55/17.98 66.55/17.98 (33) 66.55/17.98 Obligation: 66.55/17.98 Q DP problem: 66.55/17.98 P is empty. 66.55/17.98 The TRS R consists of the following rules: 66.55/17.98 66.55/17.98 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.55/17.98 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.55/17.98 2(4(1(x1))) -> 4(2(1(1(x1)))) 66.55/17.98 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.55/17.98 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.55/17.98 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.55/17.98 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.55/17.98 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.55/17.98 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.55/17.98 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.55/17.98 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.55/17.98 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.55/17.98 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.55/17.98 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.55/17.98 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.55/17.98 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.55/17.98 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.55/17.98 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.55/17.98 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.55/17.98 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.55/17.98 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.55/17.98 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.55/17.98 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.55/17.98 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.55/17.98 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.55/17.98 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 66.55/17.98 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 66.55/17.98 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 66.55/17.98 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 66.55/17.98 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 66.55/17.98 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 66.55/17.98 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 66.55/17.98 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 66.55/17.98 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 66.55/17.98 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 66.55/17.98 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 66.55/17.98 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 66.55/17.98 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 66.55/17.98 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 66.55/17.98 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 66.55/17.98 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 66.55/17.98 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 66.55/17.98 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 66.55/17.98 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 66.55/17.98 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 66.55/17.98 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 66.55/17.98 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 66.55/17.98 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.55/17.98 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 66.55/17.98 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 66.55/17.98 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 66.55/17.98 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 66.55/17.98 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 66.55/17.98 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 66.55/17.98 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 66.55/17.98 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 66.55/17.98 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 66.55/17.98 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.55/17.98 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 66.55/17.98 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) 66.55/17.98 66.55/17.98 Q is empty. 66.55/17.98 We have to consider all minimal (P,Q,R)-chains. 66.55/17.98 ---------------------------------------- 66.55/17.98 66.55/17.98 (34) PisEmptyProof (EQUIVALENT) 66.55/17.98 The TRS P is empty. Hence, there is no (P,Q,R) chain. 66.55/17.98 ---------------------------------------- 66.55/17.98 66.55/17.98 (35) 66.55/17.98 YES 66.55/17.98 66.55/17.98 ---------------------------------------- 66.55/17.98 66.55/17.98 (36) 66.55/17.98 Obligation: 66.55/17.98 Q DP problem: 66.55/17.98 The TRS P consists of the following rules: 66.55/17.98 66.55/17.98 3^1(2(0(1(x1)))) -> 3^1(0(2(1(x1)))) 66.55/17.98 66.55/17.98 The TRS R consists of the following rules: 66.55/17.98 66.55/17.98 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.55/17.98 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.55/17.98 2(4(1(x1))) -> 4(2(1(1(x1)))) 66.55/17.98 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.55/17.98 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.55/17.98 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.55/17.98 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.55/17.98 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.55/17.98 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.55/17.98 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.55/17.98 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.55/17.98 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.55/17.98 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.55/17.98 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.55/17.98 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.55/17.98 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.55/17.98 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.55/17.98 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.55/17.98 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.55/17.98 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.55/17.98 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.55/17.98 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.55/17.98 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.55/17.98 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.55/17.98 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.55/17.98 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 66.55/17.98 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 66.55/17.98 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 66.55/17.98 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 66.55/17.98 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 66.55/17.98 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 66.55/17.98 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 66.55/17.98 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 66.55/17.98 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 66.55/17.98 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 66.55/17.98 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 66.55/17.98 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 66.55/17.98 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 66.55/17.98 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 66.55/17.98 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 66.55/17.98 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 66.55/17.98 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 66.55/17.98 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 66.55/17.98 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 66.55/17.98 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 66.55/17.98 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 66.55/17.98 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 66.55/17.98 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.55/17.98 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 66.55/17.98 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 66.55/17.98 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 66.55/17.98 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 66.55/17.98 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 66.55/17.98 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 66.55/17.98 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 66.55/17.98 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 66.55/17.98 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 66.55/17.98 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.55/17.98 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 66.55/17.98 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) 66.55/17.98 66.55/17.98 Q is empty. 66.55/17.98 We have to consider all minimal (P,Q,R)-chains. 66.55/17.98 ---------------------------------------- 66.55/17.98 66.55/17.98 (37) QDPOrderProof (EQUIVALENT) 66.55/17.98 We use the reduction pair processor [LPAR04,JAR06]. 66.55/17.98 66.55/17.98 66.55/17.98 The following pairs can be oriented strictly and are deleted. 66.55/17.98 66.55/17.98 3^1(2(0(1(x1)))) -> 3^1(0(2(1(x1)))) 66.55/17.98 The remaining pairs can at least be oriented weakly. 66.55/17.98 Used ordering: Polynomial interpretation [POLO]: 66.55/17.98 66.55/17.98 POL(0(x_1)) = 1 66.55/17.98 POL(1(x_1)) = 0 66.55/17.98 POL(2(x_1)) = 1 + x_1 66.55/17.98 POL(3(x_1)) = 1 + x_1 66.55/17.98 POL(3^1(x_1)) = x_1 66.55/17.98 POL(4(x_1)) = 1 66.55/17.98 POL(5(x_1)) = 0 66.55/17.98 66.55/17.98 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 66.55/17.98 66.55/17.98 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.55/17.98 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.55/17.98 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.55/17.98 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.55/17.98 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.55/17.98 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.55/17.98 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.55/17.98 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.55/17.98 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.55/17.98 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.55/17.98 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.55/17.98 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.55/17.98 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.55/17.98 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.55/17.98 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.55/17.98 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.55/17.98 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.55/17.98 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.55/17.98 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.55/17.98 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.55/17.98 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.55/17.98 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.55/17.98 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.55/17.98 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.55/17.98 66.55/17.98 66.55/17.98 ---------------------------------------- 66.55/17.98 66.55/17.98 (38) 66.55/17.98 Obligation: 66.55/17.98 Q DP problem: 66.55/17.98 P is empty. 66.55/17.98 The TRS R consists of the following rules: 66.55/17.98 66.55/17.98 0(1(2(x1))) -> 0(2(1(1(x1)))) 66.55/17.98 0(3(2(x1))) -> 0(2(1(3(x1)))) 66.55/17.98 2(4(1(x1))) -> 4(2(1(1(x1)))) 66.55/17.98 0(0(3(2(x1)))) -> 0(0(2(1(3(x1))))) 66.55/17.98 0(1(0(3(x1)))) -> 0(1(1(3(0(x1))))) 66.55/17.98 0(1(0(5(x1)))) -> 0(0(1(1(5(x1))))) 66.55/17.98 0(1(2(2(x1)))) -> 0(2(1(1(2(x1))))) 66.55/17.98 0(1(2(3(x1)))) -> 0(2(1(3(1(x1))))) 66.55/17.98 0(1(2(5(x1)))) -> 2(1(1(0(5(x1))))) 66.55/17.98 0(1(4(1(x1)))) -> 0(4(1(1(1(x1))))) 66.55/17.98 0(1(4(2(x1)))) -> 0(0(2(1(4(x1))))) 66.55/17.98 0(1(4(2(x1)))) -> 0(4(0(2(1(x1))))) 66.55/17.98 0(1(4(3(x1)))) -> 4(0(1(1(3(x1))))) 66.55/17.98 0(1(4(5(x1)))) -> 0(1(1(5(4(x1))))) 66.55/17.98 0(1(5(2(x1)))) -> 0(2(5(1(1(x1))))) 66.55/17.98 0(2(0(3(x1)))) -> 0(0(2(3(1(x1))))) 66.55/17.98 0(3(0(1(x1)))) -> 0(0(1(3(1(x1))))) 66.55/17.98 0(3(0(2(x1)))) -> 0(0(2(3(1(x1))))) 66.55/17.98 0(3(0(3(x1)))) -> 0(0(3(1(3(x1))))) 66.55/17.98 0(3(2(0(x1)))) -> 0(2(1(3(0(x1))))) 66.55/17.98 0(3(5(1(x1)))) -> 0(5(3(1(1(x1))))) 66.55/17.98 0(4(0(1(x1)))) -> 0(0(4(1(1(x1))))) 66.55/17.98 0(5(2(0(x1)))) -> 2(1(5(0(0(x1))))) 66.55/17.98 2(1(2(3(x1)))) -> 2(1(1(3(2(x1))))) 66.55/17.98 2(1(4(1(x1)))) -> 4(2(1(1(1(x1))))) 66.55/17.98 2(2(0(3(x1)))) -> 2(0(2(1(3(x1))))) 66.55/17.98 2(2(4(1(x1)))) -> 2(4(2(1(1(x1))))) 66.55/17.98 2(3(1(4(x1)))) -> 2(1(1(3(4(x1))))) 66.55/17.98 2(4(1(4(x1)))) -> 4(4(2(1(1(x1))))) 66.55/17.98 2(4(3(1(x1)))) -> 4(2(3(1(1(x1))))) 66.55/17.98 2(5(0(1(x1)))) -> 0(2(5(1(1(x1))))) 66.55/17.98 2(5(0(3(x1)))) -> 5(0(2(1(3(x1))))) 66.55/17.98 3(0(1(0(x1)))) -> 3(1(0(0(0(x1))))) 66.55/17.98 3(0(1(4(x1)))) -> 3(1(1(0(4(x1))))) 66.55/17.98 3(0(3(0(x1)))) -> 3(1(3(0(0(x1))))) 66.55/17.98 3(1(0(3(x1)))) -> 3(1(1(3(0(x1))))) 66.55/17.98 3(1(2(1(x1)))) -> 1(3(2(1(1(x1))))) 66.55/17.98 3(1(2(1(x1)))) -> 3(2(1(1(1(x1))))) 66.55/17.98 3(2(0(1(x1)))) -> 1(3(0(2(1(x1))))) 66.55/17.98 3(2(0(1(x1)))) -> 3(1(0(2(1(x1))))) 66.55/17.98 3(4(1(2(x1)))) -> 3(2(1(1(4(x1))))) 66.55/17.98 3(5(1(4(x1)))) -> 3(1(1(5(4(x1))))) 66.55/17.98 4(1(0(1(x1)))) -> 4(0(1(1(1(x1))))) 66.55/17.98 4(1(2(1(x1)))) -> 2(4(1(1(1(x1))))) 66.55/17.98 4(1(4(3(x1)))) -> 4(1(1(3(4(x1))))) 66.55/17.98 4(2(4(1(x1)))) -> 4(4(2(1(1(x1))))) 66.55/17.98 4(4(1(2(x1)))) -> 4(2(1(1(4(x1))))) 66.55/17.98 4(5(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.55/17.98 5(1(0(3(x1)))) -> 5(0(1(1(3(x1))))) 66.55/17.98 5(1(0(3(x1)))) -> 5(1(1(3(0(x1))))) 66.55/17.98 5(1(3(1(x1)))) -> 5(3(1(1(1(x1))))) 66.55/17.98 5(1(4(3(x1)))) -> 4(1(1(3(5(x1))))) 66.55/17.98 5(1(4(4(x1)))) -> 4(1(1(5(4(x1))))) 66.55/17.98 5(1(5(3(x1)))) -> 5(5(1(1(3(x1))))) 66.55/17.98 5(2(0(1(x1)))) -> 2(0(5(1(1(x1))))) 66.55/17.98 5(2(0(5(x1)))) -> 5(0(2(1(5(x1))))) 66.55/17.98 5(3(4(1(x1)))) -> 5(4(3(1(1(x1))))) 66.55/17.98 5(4(0(1(x1)))) -> 5(0(4(1(1(x1))))) 66.55/17.98 5(4(1(0(x1)))) -> 4(1(1(5(0(x1))))) 66.55/17.98 5(4(1(0(x1)))) -> 5(1(1(4(0(x1))))) 66.55/17.98 66.55/17.98 Q is empty. 66.55/17.98 We have to consider all minimal (P,Q,R)-chains. 66.55/17.98 ---------------------------------------- 66.55/17.98 66.55/17.98 (39) PisEmptyProof (EQUIVALENT) 66.55/17.98 The TRS P is empty. Hence, there is no (P,Q,R) chain. 66.55/17.98 ---------------------------------------- 66.55/17.98 66.55/17.98 (40) 66.55/17.98 YES 67.00/18.22 EOF