115.66/30.43 YES 117.44/30.91 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 117.44/30.91 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 117.44/30.91 117.44/30.91 117.44/30.91 Termination w.r.t. Q of the given QTRS could be proven: 117.44/30.91 117.44/30.91 (0) QTRS 117.44/30.91 (1) DependencyPairsProof [EQUIVALENT, 197 ms] 117.44/30.91 (2) QDP 117.44/30.91 (3) DependencyGraphProof [EQUIVALENT, 6 ms] 117.44/30.91 (4) QDP 117.44/30.91 (5) QDPOrderProof [EQUIVALENT, 273 ms] 117.44/30.91 (6) QDP 117.44/30.91 (7) DependencyGraphProof [EQUIVALENT, 0 ms] 117.44/30.91 (8) QDP 117.44/30.91 (9) QDPOrderProof [EQUIVALENT, 184 ms] 117.44/30.91 (10) QDP 117.44/30.91 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 117.44/30.91 (12) QDP 117.44/30.91 (13) QDPOrderProof [EQUIVALENT, 83 ms] 117.44/30.91 (14) QDP 117.44/30.91 (15) DependencyGraphProof [EQUIVALENT, 0 ms] 117.44/30.91 (16) AND 117.44/30.91 (17) QDP 117.44/30.91 (18) QDPOrderProof [EQUIVALENT, 59 ms] 117.44/30.91 (19) QDP 117.44/30.91 (20) QDPOrderProof [EQUIVALENT, 3259 ms] 117.44/30.91 (21) QDP 117.44/30.91 (22) PisEmptyProof [EQUIVALENT, 0 ms] 117.44/30.91 (23) YES 117.44/30.91 (24) QDP 117.44/30.91 (25) QDPOrderProof [EQUIVALENT, 67 ms] 117.44/30.91 (26) QDP 117.44/30.91 (27) QDPOrderProof [EQUIVALENT, 55 ms] 117.44/30.91 (28) QDP 117.44/30.91 (29) QDPOrderProof [EQUIVALENT, 55 ms] 117.44/30.91 (30) QDP 117.44/30.91 (31) PisEmptyProof [EQUIVALENT, 0 ms] 117.44/30.91 (32) YES 117.44/30.91 117.44/30.91 117.44/30.91 ---------------------------------------- 117.44/30.91 117.44/30.91 (0) 117.44/30.91 Obligation: 117.44/30.91 Q restricted rewrite system: 117.44/30.91 The TRS R consists of the following rules: 117.44/30.91 117.44/30.91 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 117.44/30.91 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 117.44/30.91 0(3(2(x1))) -> 2(1(3(0(x1)))) 117.44/30.91 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 117.44/30.91 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 117.44/30.91 3(2(0(x1))) -> 0(2(1(3(x1)))) 117.44/30.91 3(2(0(x1))) -> 3(0(2(1(x1)))) 117.44/30.91 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 117.44/30.91 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 117.44/30.91 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 117.44/30.91 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 117.44/30.91 3(2(5(x1))) -> 1(3(5(2(x1)))) 117.44/30.91 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 117.44/30.91 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 117.44/30.91 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 117.44/30.91 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 117.44/30.91 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 117.44/30.91 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 117.44/30.91 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 117.44/30.91 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 117.44/30.91 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 117.44/30.91 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 117.44/30.91 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 117.44/30.91 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 117.44/30.91 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 117.44/30.91 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 117.44/30.91 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 117.44/30.91 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 117.44/30.91 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 117.44/30.91 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 117.44/30.91 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 117.44/30.91 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 117.44/30.91 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 117.44/30.91 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 117.44/30.91 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 117.44/30.91 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 117.44/30.91 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 117.44/30.91 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 117.44/30.91 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 117.44/30.91 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 117.44/30.91 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 117.44/30.91 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 117.44/30.91 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 117.44/30.91 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 117.44/30.91 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 117.44/30.91 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 117.44/30.91 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 117.44/30.91 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) 117.44/30.91 117.44/30.91 Q is empty. 117.44/30.91 117.44/30.91 ---------------------------------------- 117.44/30.91 117.44/30.91 (1) DependencyPairsProof (EQUIVALENT) 117.44/30.91 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 117.44/30.91 ---------------------------------------- 117.44/30.91 117.44/30.91 (2) 117.44/30.91 Obligation: 117.44/30.91 Q DP problem: 117.44/30.91 The TRS P consists of the following rules: 117.44/30.91 117.44/30.91 0^1(0(0(x1))) -> 0^1(0(1(0(1(x1))))) 117.44/30.91 0^1(0(0(x1))) -> 0^1(1(0(1(x1)))) 117.44/30.91 0^1(0(0(x1))) -> 0^1(1(x1)) 117.44/30.91 0^1(2(0(x1))) -> 0^1(0(2(1(1(x1))))) 117.44/30.91 0^1(2(0(x1))) -> 0^1(2(1(1(x1)))) 117.44/30.91 0^1(3(2(x1))) -> 3^1(0(x1)) 117.44/30.91 0^1(3(2(x1))) -> 0^1(x1) 117.44/30.91 0^1(4(2(x1))) -> 4^1(1(1(1(0(2(x1)))))) 117.44/30.91 0^1(4(2(x1))) -> 0^1(2(x1)) 117.44/30.91 3^1(0(4(x1))) -> 0^1(1(3(4(1(x1))))) 117.44/30.91 3^1(0(4(x1))) -> 3^1(4(1(x1))) 117.44/30.91 3^1(0(4(x1))) -> 4^1(1(x1)) 117.44/30.91 3^1(2(0(x1))) -> 0^1(2(1(3(x1)))) 117.44/30.91 3^1(2(0(x1))) -> 3^1(x1) 117.44/30.91 3^1(2(0(x1))) -> 3^1(0(2(1(x1)))) 117.44/30.91 3^1(2(0(x1))) -> 0^1(2(1(x1))) 117.44/30.91 3^1(2(0(x1))) -> 0^1(1(2(1(3(x1))))) 117.44/30.91 3^1(2(0(x1))) -> 3^1(0(1(2(1(4(x1)))))) 117.44/30.91 3^1(2(0(x1))) -> 0^1(1(2(1(4(x1))))) 117.44/30.91 3^1(2(0(x1))) -> 4^1(x1) 117.44/30.91 3^1(2(4(x1))) -> 3^1(3(4(x1))) 117.44/30.91 3^1(2(4(x1))) -> 3^1(4(x1)) 117.44/30.91 3^1(2(4(x1))) -> 4^1(1(2(1(1(3(x1)))))) 117.44/30.91 3^1(2(4(x1))) -> 3^1(x1) 117.44/30.91 3^1(2(5(x1))) -> 3^1(5(2(x1))) 117.44/30.91 3^1(2(5(x1))) -> 5^1(2(x1)) 117.44/30.91 3^1(2(5(x1))) -> 3^1(1(4(5(2(x1))))) 117.44/30.91 3^1(2(5(x1))) -> 4^1(5(2(x1))) 117.44/30.91 3^1(2(5(x1))) -> 3^1(3(5(2(x1)))) 117.44/30.91 3^1(2(5(x1))) -> 3^1(4(5(4(2(x1))))) 117.44/30.91 3^1(2(5(x1))) -> 4^1(5(4(2(x1)))) 117.44/30.91 3^1(2(5(x1))) -> 5^1(4(2(x1))) 117.44/30.91 3^1(2(5(x1))) -> 4^1(2(x1)) 117.44/30.91 3^1(5(0(x1))) -> 3^1(1(0(5(1(1(x1)))))) 117.44/30.91 3^1(5(0(x1))) -> 0^1(5(1(1(x1)))) 117.44/30.91 3^1(5(0(x1))) -> 5^1(1(1(x1))) 117.44/30.91 5^1(0(2(x1))) -> 0^1(2(5(x1))) 117.44/30.91 5^1(0(2(x1))) -> 5^1(x1) 117.44/30.91 5^1(0(4(x1))) -> 0^1(5(4(1(1(x1))))) 117.44/30.91 5^1(0(4(x1))) -> 5^1(4(1(1(x1)))) 117.44/30.91 5^1(0(4(x1))) -> 4^1(1(1(x1))) 117.44/30.91 5^1(2(4(x1))) -> 5^1(2(1(1(3(4(x1)))))) 117.44/30.91 5^1(2(4(x1))) -> 3^1(4(x1)) 117.44/30.91 5^1(3(2(x1))) -> 3^1(5(1(2(x1)))) 117.44/30.91 5^1(3(2(x1))) -> 5^1(1(2(x1))) 117.44/30.91 5^1(5(0(x1))) -> 5^1(1(1(5(0(x1))))) 117.44/30.91 0^1(0(2(4(x1)))) -> 0^1(4(0(2(1(1(x1)))))) 117.44/30.91 0^1(0(2(4(x1)))) -> 4^1(0(2(1(1(x1))))) 117.44/30.91 0^1(0(2(4(x1)))) -> 0^1(2(1(1(x1)))) 117.44/30.91 0^1(3(1(2(x1)))) -> 3^1(0(2(x1))) 117.44/30.91 0^1(3(1(2(x1)))) -> 0^1(2(x1)) 117.44/30.91 0^1(3(2(4(x1)))) -> 3^1(0(4(4(x1)))) 117.44/30.91 0^1(3(2(4(x1)))) -> 0^1(4(4(x1))) 117.44/30.91 0^1(3(2(4(x1)))) -> 4^1(4(x1)) 117.44/30.91 0^1(4(3(2(x1)))) -> 3^1(4(0(1(x1)))) 117.44/30.91 0^1(4(3(2(x1)))) -> 4^1(0(1(x1))) 117.44/30.91 0^1(4(3(2(x1)))) -> 0^1(1(x1)) 117.44/30.91 3^1(0(2(4(x1)))) -> 3^1(0(4(2(1(1(x1)))))) 117.44/30.91 3^1(0(2(4(x1)))) -> 0^1(4(2(1(1(x1))))) 117.44/30.91 3^1(0(2(4(x1)))) -> 4^1(2(1(1(x1)))) 117.44/30.91 3^1(0(5(4(x1)))) -> 3^1(4(0(5(1(3(x1)))))) 117.44/30.91 3^1(0(5(4(x1)))) -> 4^1(0(5(1(3(x1))))) 117.44/30.91 3^1(0(5(4(x1)))) -> 0^1(5(1(3(x1)))) 117.44/30.91 3^1(0(5(4(x1)))) -> 5^1(1(3(x1))) 117.44/30.91 3^1(0(5(4(x1)))) -> 3^1(x1) 117.44/30.91 3^1(1(0(0(x1)))) -> 0^1(1(1(3(0(1(x1)))))) 117.44/30.91 3^1(1(0(0(x1)))) -> 3^1(0(1(x1))) 117.44/30.91 3^1(1(0(0(x1)))) -> 0^1(1(x1)) 117.44/30.91 3^1(2(4(0(x1)))) -> 3^1(1(0(4(2(x1))))) 117.44/30.91 3^1(2(4(0(x1)))) -> 0^1(4(2(x1))) 117.44/30.91 3^1(2(4(0(x1)))) -> 4^1(2(x1)) 117.44/30.91 3^1(2(4(5(x1)))) -> 3^1(1(4(2(1(5(x1)))))) 117.44/30.91 3^1(2(4(5(x1)))) -> 4^1(2(1(5(x1)))) 117.44/30.91 3^1(2(5(4(x1)))) -> 3^1(4(3(5(2(x1))))) 117.44/30.91 3^1(2(5(4(x1)))) -> 4^1(3(5(2(x1)))) 117.44/30.91 3^1(2(5(4(x1)))) -> 3^1(5(2(x1))) 117.44/30.91 3^1(2(5(4(x1)))) -> 5^1(2(x1)) 117.44/30.91 3^1(5(3(2(x1)))) -> 3^1(3(0(2(5(x1))))) 117.44/30.91 3^1(5(3(2(x1)))) -> 3^1(0(2(5(x1)))) 117.44/30.91 3^1(5(3(2(x1)))) -> 0^1(2(5(x1))) 117.44/30.91 3^1(5(3(2(x1)))) -> 5^1(x1) 117.44/30.91 4^1(0(4(2(x1)))) -> 4^1(4(2(1(3(0(x1)))))) 117.44/30.91 4^1(0(4(2(x1)))) -> 4^1(2(1(3(0(x1))))) 117.44/30.91 4^1(0(4(2(x1)))) -> 3^1(0(x1)) 117.44/30.91 4^1(0(4(2(x1)))) -> 0^1(x1) 117.44/30.91 4^1(3(2(0(x1)))) -> 0^1(2(1(3(4(x1))))) 117.44/30.91 4^1(3(2(0(x1)))) -> 3^1(4(x1)) 117.44/30.91 4^1(3(2(0(x1)))) -> 4^1(x1) 117.44/30.91 5^1(4(1(0(x1)))) -> 5^1(1(1(4(0(1(x1)))))) 117.44/30.91 5^1(4(1(0(x1)))) -> 4^1(0(1(x1))) 117.44/30.91 5^1(4(1(0(x1)))) -> 0^1(1(x1)) 117.44/30.91 5^1(5(0(0(x1)))) -> 0^1(5(5(0(x1)))) 117.44/30.91 5^1(5(0(0(x1)))) -> 5^1(5(0(x1))) 117.44/30.91 5^1(5(0(0(x1)))) -> 5^1(0(x1)) 117.44/30.91 5^1(5(2(2(x1)))) -> 5^1(5(1(1(2(2(x1)))))) 117.44/30.91 5^1(5(2(2(x1)))) -> 5^1(1(1(2(2(x1))))) 117.44/30.91 0^1(0(1(2(4(x1))))) -> 0^1(0(2(1(1(4(x1)))))) 117.44/30.91 0^1(0(1(2(4(x1))))) -> 0^1(2(1(1(4(x1))))) 117.44/30.91 0^1(5(3(1(2(x1))))) -> 5^1(2(1(3(1(0(x1)))))) 117.44/30.91 0^1(5(3(1(2(x1))))) -> 3^1(1(0(x1))) 117.44/30.91 0^1(5(3(1(2(x1))))) -> 0^1(x1) 117.44/30.91 0^1(5(5(1(0(x1))))) -> 5^1(0(0(5(1(1(x1)))))) 117.44/30.91 0^1(5(5(1(0(x1))))) -> 0^1(0(5(1(1(x1))))) 117.44/30.91 0^1(5(5(1(0(x1))))) -> 0^1(5(1(1(x1)))) 117.44/30.91 0^1(5(5(1(0(x1))))) -> 5^1(1(1(x1))) 117.44/30.91 3^1(0(1(4(5(x1))))) -> 3^1(4(0(5(1(1(x1)))))) 117.44/30.91 3^1(0(1(4(5(x1))))) -> 4^1(0(5(1(1(x1))))) 117.44/30.91 3^1(0(1(4(5(x1))))) -> 0^1(5(1(1(x1)))) 117.44/30.91 3^1(0(1(4(5(x1))))) -> 5^1(1(1(x1))) 117.44/30.91 3^1(0(3(1(2(x1))))) -> 0^1(2(1(1(3(3(x1)))))) 117.44/30.91 3^1(0(3(1(2(x1))))) -> 3^1(3(x1)) 117.44/30.91 3^1(0(3(1(2(x1))))) -> 3^1(x1) 117.44/30.91 3^1(2(2(0(4(x1))))) -> 3^1(2(2(0(1(4(x1)))))) 117.44/30.91 3^1(2(2(0(4(x1))))) -> 0^1(1(4(x1))) 117.44/30.91 3^1(5(1(4(0(x1))))) -> 3^1(3(4(5(0(x1))))) 117.44/30.91 3^1(5(1(4(0(x1))))) -> 3^1(4(5(0(x1)))) 117.44/30.91 3^1(5(1(4(0(x1))))) -> 4^1(5(0(x1))) 117.44/30.91 3^1(5(1(4(0(x1))))) -> 5^1(0(x1)) 117.44/30.91 3^1(5(3(0(4(x1))))) -> 3^1(0(1(5(3(4(x1)))))) 117.44/30.91 3^1(5(3(0(4(x1))))) -> 0^1(1(5(3(4(x1))))) 117.44/30.91 3^1(5(3(0(4(x1))))) -> 5^1(3(4(x1))) 117.44/30.91 3^1(5(3(0(4(x1))))) -> 3^1(4(x1)) 117.44/30.91 5^1(1(0(2(4(x1))))) -> 5^1(4(0(3(x1)))) 117.81/30.91 5^1(1(0(2(4(x1))))) -> 4^1(0(3(x1))) 117.81/30.91 5^1(1(0(2(4(x1))))) -> 0^1(3(x1)) 117.81/30.91 5^1(1(0(2(4(x1))))) -> 3^1(x1) 117.81/30.91 5^1(1(3(2(0(x1))))) -> 3^1(5(0(1(1(x1))))) 117.81/30.91 5^1(1(3(2(0(x1))))) -> 5^1(0(1(1(x1)))) 117.81/30.91 5^1(1(3(2(0(x1))))) -> 0^1(1(1(x1))) 117.81/30.91 5^1(4(2(0(2(x1))))) -> 5^1(0(4(2(1(2(x1)))))) 117.81/30.91 5^1(4(2(0(2(x1))))) -> 0^1(4(2(1(2(x1))))) 117.81/30.91 5^1(4(2(0(2(x1))))) -> 4^1(2(1(2(x1)))) 117.81/30.91 117.81/30.91 The TRS R consists of the following rules: 117.81/30.91 117.81/30.91 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 117.81/30.91 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 117.81/30.91 0(3(2(x1))) -> 2(1(3(0(x1)))) 117.81/30.91 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 117.81/30.91 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 117.81/30.91 3(2(0(x1))) -> 0(2(1(3(x1)))) 117.81/30.91 3(2(0(x1))) -> 3(0(2(1(x1)))) 117.81/30.91 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 117.81/30.91 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 117.81/30.91 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 117.81/30.91 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 117.81/30.91 3(2(5(x1))) -> 1(3(5(2(x1)))) 117.81/30.91 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 117.81/30.91 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 117.81/30.91 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 117.81/30.91 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 117.81/30.91 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 117.81/30.91 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 117.81/30.91 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 117.81/30.91 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 117.81/30.91 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 117.81/30.91 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 117.81/30.91 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 117.81/30.91 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 117.81/30.91 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 117.81/30.91 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 117.81/30.91 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 117.81/30.91 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 117.81/30.91 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 117.81/30.91 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 117.81/30.91 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 117.81/30.91 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 117.81/30.91 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 117.81/30.91 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 117.81/30.91 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 117.81/30.91 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 117.81/30.91 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 117.81/30.91 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 117.81/30.91 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 117.81/30.91 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 117.81/30.91 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 117.81/30.91 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 117.81/30.91 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 117.81/30.91 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 117.81/30.91 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 117.81/30.91 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 117.81/30.91 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 117.81/30.91 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) 117.81/30.91 117.81/30.91 Q is empty. 117.81/30.91 We have to consider all minimal (P,Q,R)-chains. 117.81/30.91 ---------------------------------------- 117.81/30.91 117.81/30.91 (3) DependencyGraphProof (EQUIVALENT) 117.81/30.91 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 78 less nodes. 117.81/30.91 ---------------------------------------- 117.81/30.91 117.81/30.91 (4) 117.81/30.91 Obligation: 117.81/30.91 Q DP problem: 117.81/30.91 The TRS P consists of the following rules: 117.81/30.91 117.81/30.91 0^1(3(2(x1))) -> 3^1(0(x1)) 117.81/30.91 3^1(2(0(x1))) -> 3^1(x1) 117.81/30.91 3^1(2(0(x1))) -> 4^1(x1) 117.81/30.91 4^1(0(4(2(x1)))) -> 3^1(0(x1)) 117.81/30.91 3^1(2(4(x1))) -> 3^1(3(4(x1))) 117.81/30.91 3^1(2(4(x1))) -> 3^1(4(x1)) 117.81/30.91 3^1(2(4(x1))) -> 3^1(x1) 117.81/30.91 3^1(2(5(x1))) -> 3^1(5(2(x1))) 117.81/30.91 3^1(2(5(x1))) -> 5^1(2(x1)) 117.81/30.91 5^1(2(4(x1))) -> 3^1(4(x1)) 117.81/30.91 3^1(2(5(x1))) -> 4^1(5(2(x1))) 117.81/30.91 4^1(0(4(2(x1)))) -> 0^1(x1) 117.81/30.91 0^1(3(2(x1))) -> 0^1(x1) 117.81/30.91 0^1(3(1(2(x1)))) -> 3^1(0(2(x1))) 117.81/30.91 3^1(2(5(x1))) -> 3^1(3(5(2(x1)))) 117.81/30.91 3^1(2(5(x1))) -> 3^1(4(5(4(2(x1))))) 117.81/30.91 3^1(2(5(x1))) -> 4^1(5(4(2(x1)))) 117.81/30.91 4^1(3(2(0(x1)))) -> 3^1(4(x1)) 117.81/30.91 3^1(2(5(x1))) -> 5^1(4(2(x1))) 117.81/30.91 5^1(4(2(0(2(x1))))) -> 5^1(0(4(2(1(2(x1)))))) 117.81/30.91 5^1(0(2(x1))) -> 5^1(x1) 117.81/30.91 5^1(5(0(0(x1)))) -> 0^1(5(5(0(x1)))) 117.81/30.91 0^1(3(2(4(x1)))) -> 3^1(0(4(4(x1)))) 117.81/30.91 3^1(0(2(4(x1)))) -> 3^1(0(4(2(1(1(x1)))))) 117.81/30.91 3^1(0(5(4(x1)))) -> 3^1(4(0(5(1(3(x1)))))) 117.81/30.91 3^1(0(5(4(x1)))) -> 4^1(0(5(1(3(x1))))) 117.81/30.91 4^1(3(2(0(x1)))) -> 4^1(x1) 117.81/30.91 3^1(0(5(4(x1)))) -> 0^1(5(1(3(x1)))) 117.81/30.91 0^1(3(2(4(x1)))) -> 0^1(4(4(x1))) 117.81/30.91 0^1(3(2(4(x1)))) -> 4^1(4(x1)) 117.81/30.91 0^1(5(3(1(2(x1))))) -> 0^1(x1) 117.81/30.91 3^1(0(5(4(x1)))) -> 5^1(1(3(x1))) 117.81/30.91 5^1(1(0(2(4(x1))))) -> 5^1(4(0(3(x1)))) 117.81/30.91 5^1(5(0(0(x1)))) -> 5^1(5(0(x1))) 117.81/30.91 5^1(5(0(0(x1)))) -> 5^1(0(x1)) 117.81/30.91 5^1(1(0(2(4(x1))))) -> 4^1(0(3(x1))) 117.81/30.91 5^1(1(0(2(4(x1))))) -> 0^1(3(x1)) 117.81/30.91 5^1(1(0(2(4(x1))))) -> 3^1(x1) 117.81/30.91 3^1(0(5(4(x1)))) -> 3^1(x1) 117.81/30.91 3^1(2(5(4(x1)))) -> 3^1(4(3(5(2(x1))))) 117.81/30.91 3^1(2(5(4(x1)))) -> 4^1(3(5(2(x1)))) 117.81/30.91 3^1(2(5(4(x1)))) -> 3^1(5(2(x1))) 117.81/30.91 3^1(2(5(4(x1)))) -> 5^1(2(x1)) 117.81/30.91 3^1(5(3(2(x1)))) -> 3^1(3(0(2(5(x1))))) 117.81/30.91 3^1(5(3(2(x1)))) -> 3^1(0(2(5(x1)))) 117.81/30.91 3^1(5(3(2(x1)))) -> 5^1(x1) 117.81/30.91 3^1(0(3(1(2(x1))))) -> 3^1(3(x1)) 117.81/30.91 3^1(0(3(1(2(x1))))) -> 3^1(x1) 117.81/30.91 3^1(5(1(4(0(x1))))) -> 3^1(3(4(5(0(x1))))) 117.81/30.91 3^1(5(1(4(0(x1))))) -> 3^1(4(5(0(x1)))) 117.81/30.91 3^1(5(1(4(0(x1))))) -> 4^1(5(0(x1))) 117.81/30.91 3^1(5(1(4(0(x1))))) -> 5^1(0(x1)) 117.81/30.91 3^1(5(3(0(4(x1))))) -> 5^1(3(4(x1))) 117.81/30.91 3^1(5(3(0(4(x1))))) -> 3^1(4(x1)) 117.81/30.91 117.81/30.91 The TRS R consists of the following rules: 117.81/30.91 117.81/30.91 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 117.81/30.91 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 117.81/30.91 0(3(2(x1))) -> 2(1(3(0(x1)))) 117.81/30.91 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 117.81/30.91 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 117.81/30.91 3(2(0(x1))) -> 0(2(1(3(x1)))) 117.81/30.91 3(2(0(x1))) -> 3(0(2(1(x1)))) 117.81/30.91 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 117.81/30.91 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 117.81/30.91 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 117.81/30.91 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 117.81/30.91 3(2(5(x1))) -> 1(3(5(2(x1)))) 117.81/30.91 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 117.81/30.91 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 117.81/30.91 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 117.81/30.91 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 117.81/30.91 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 117.81/30.91 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 117.81/30.91 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 117.81/30.91 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 117.81/30.91 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 117.81/30.91 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 117.81/30.91 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 117.81/30.91 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 117.81/30.91 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 117.81/30.91 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 117.81/30.91 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 117.81/30.91 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 117.81/30.91 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 117.81/30.91 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 117.81/30.91 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 117.81/30.91 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 117.81/30.91 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 117.81/30.91 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 117.81/30.91 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 117.81/30.91 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 117.81/30.91 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 117.81/30.91 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 117.81/30.91 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 117.81/30.91 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 117.81/30.91 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 117.81/30.91 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 117.81/30.91 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 117.81/30.91 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 117.81/30.91 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 117.81/30.91 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 117.81/30.91 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 117.81/30.91 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) 117.81/30.91 117.81/30.91 Q is empty. 117.81/30.91 We have to consider all minimal (P,Q,R)-chains. 117.81/30.91 ---------------------------------------- 117.81/30.91 117.81/30.91 (5) QDPOrderProof (EQUIVALENT) 117.81/30.91 We use the reduction pair processor [LPAR04,JAR06]. 117.81/30.91 117.81/30.91 117.81/30.91 The following pairs can be oriented strictly and are deleted. 117.81/30.91 117.81/30.91 5^1(2(4(x1))) -> 3^1(4(x1)) 117.81/30.91 0^1(5(3(1(2(x1))))) -> 0^1(x1) 117.81/30.91 5^1(5(0(0(x1)))) -> 5^1(0(x1)) 117.81/30.91 5^1(1(0(2(4(x1))))) -> 4^1(0(3(x1))) 117.81/30.91 5^1(1(0(2(4(x1))))) -> 0^1(3(x1)) 117.81/30.91 5^1(1(0(2(4(x1))))) -> 3^1(x1) 117.81/30.91 3^1(0(5(4(x1)))) -> 3^1(x1) 117.81/30.91 3^1(5(3(0(4(x1))))) -> 3^1(4(x1)) 117.81/30.91 The remaining pairs can at least be oriented weakly. 117.81/30.91 Used ordering: Polynomial interpretation [POLO]: 117.81/30.91 117.81/30.91 POL(0(x_1)) = x_1 117.81/30.91 POL(0^1(x_1)) = x_1 117.81/30.91 POL(1(x_1)) = x_1 117.81/30.91 POL(2(x_1)) = x_1 117.81/30.91 POL(3(x_1)) = x_1 117.81/30.91 POL(3^1(x_1)) = x_1 117.81/30.91 POL(4(x_1)) = x_1 117.81/30.91 POL(4^1(x_1)) = x_1 117.81/30.91 POL(5(x_1)) = 1 + x_1 117.81/30.91 POL(5^1(x_1)) = 1 + x_1 117.81/30.91 117.81/30.91 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 117.81/30.91 117.81/30.91 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 117.81/30.91 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 117.81/30.91 0(3(2(x1))) -> 2(1(3(0(x1)))) 117.81/30.91 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 117.81/30.91 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 117.81/30.91 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 117.81/30.91 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 117.81/30.91 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 117.81/30.91 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 117.81/30.91 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 117.81/30.91 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 117.81/30.91 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 117.81/30.91 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 117.81/30.91 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 117.81/30.91 3(2(0(x1))) -> 0(2(1(3(x1)))) 117.81/30.92 3(2(0(x1))) -> 3(0(2(1(x1)))) 117.81/30.92 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 117.81/30.92 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 117.81/30.92 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 117.81/30.92 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 117.81/30.92 3(2(5(x1))) -> 1(3(5(2(x1)))) 117.81/30.92 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 117.81/30.92 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 117.81/30.92 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 117.81/30.92 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 117.81/30.92 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 117.81/30.92 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 117.81/30.92 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 117.81/30.92 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 117.81/30.92 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 117.81/30.92 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 117.81/30.92 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 117.81/30.92 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 117.81/30.92 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 117.81/30.92 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 117.81/30.92 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 117.81/30.92 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 117.81/30.92 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 117.81/30.92 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) 117.81/30.92 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 117.81/30.92 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 117.81/30.92 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 117.81/30.92 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 117.81/30.92 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 117.81/30.92 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 117.81/30.92 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 117.81/30.92 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 117.81/30.92 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 117.81/30.92 117.81/30.92 117.81/30.92 ---------------------------------------- 117.81/30.92 117.81/30.92 (6) 117.81/30.92 Obligation: 117.81/30.92 Q DP problem: 117.81/30.92 The TRS P consists of the following rules: 117.81/30.92 117.81/30.92 0^1(3(2(x1))) -> 3^1(0(x1)) 117.81/30.92 3^1(2(0(x1))) -> 3^1(x1) 117.81/30.92 3^1(2(0(x1))) -> 4^1(x1) 117.81/30.92 4^1(0(4(2(x1)))) -> 3^1(0(x1)) 117.81/30.92 3^1(2(4(x1))) -> 3^1(3(4(x1))) 117.81/30.92 3^1(2(4(x1))) -> 3^1(4(x1)) 117.81/30.92 3^1(2(4(x1))) -> 3^1(x1) 117.81/30.92 3^1(2(5(x1))) -> 3^1(5(2(x1))) 117.81/30.92 3^1(2(5(x1))) -> 5^1(2(x1)) 117.81/30.92 3^1(2(5(x1))) -> 4^1(5(2(x1))) 117.81/30.92 4^1(0(4(2(x1)))) -> 0^1(x1) 117.81/30.92 0^1(3(2(x1))) -> 0^1(x1) 117.81/30.92 0^1(3(1(2(x1)))) -> 3^1(0(2(x1))) 117.81/30.92 3^1(2(5(x1))) -> 3^1(3(5(2(x1)))) 117.81/30.92 3^1(2(5(x1))) -> 3^1(4(5(4(2(x1))))) 117.81/30.92 3^1(2(5(x1))) -> 4^1(5(4(2(x1)))) 117.81/30.92 4^1(3(2(0(x1)))) -> 3^1(4(x1)) 117.81/30.92 3^1(2(5(x1))) -> 5^1(4(2(x1))) 117.81/30.92 5^1(4(2(0(2(x1))))) -> 5^1(0(4(2(1(2(x1)))))) 117.81/30.92 5^1(0(2(x1))) -> 5^1(x1) 117.81/30.92 5^1(5(0(0(x1)))) -> 0^1(5(5(0(x1)))) 117.81/30.92 0^1(3(2(4(x1)))) -> 3^1(0(4(4(x1)))) 117.81/30.92 3^1(0(2(4(x1)))) -> 3^1(0(4(2(1(1(x1)))))) 117.81/30.92 3^1(0(5(4(x1)))) -> 3^1(4(0(5(1(3(x1)))))) 117.81/30.92 3^1(0(5(4(x1)))) -> 4^1(0(5(1(3(x1))))) 117.81/30.92 4^1(3(2(0(x1)))) -> 4^1(x1) 117.81/30.92 3^1(0(5(4(x1)))) -> 0^1(5(1(3(x1)))) 117.81/30.92 0^1(3(2(4(x1)))) -> 0^1(4(4(x1))) 117.81/30.92 0^1(3(2(4(x1)))) -> 4^1(4(x1)) 117.81/30.92 3^1(0(5(4(x1)))) -> 5^1(1(3(x1))) 117.81/30.92 5^1(1(0(2(4(x1))))) -> 5^1(4(0(3(x1)))) 117.81/30.92 5^1(5(0(0(x1)))) -> 5^1(5(0(x1))) 117.81/30.92 3^1(2(5(4(x1)))) -> 3^1(4(3(5(2(x1))))) 117.81/30.92 3^1(2(5(4(x1)))) -> 4^1(3(5(2(x1)))) 117.81/30.92 3^1(2(5(4(x1)))) -> 3^1(5(2(x1))) 117.81/30.92 3^1(2(5(4(x1)))) -> 5^1(2(x1)) 117.81/30.92 3^1(5(3(2(x1)))) -> 3^1(3(0(2(5(x1))))) 117.81/30.92 3^1(5(3(2(x1)))) -> 3^1(0(2(5(x1)))) 117.81/30.92 3^1(5(3(2(x1)))) -> 5^1(x1) 117.81/30.92 3^1(0(3(1(2(x1))))) -> 3^1(3(x1)) 117.81/30.92 3^1(0(3(1(2(x1))))) -> 3^1(x1) 117.81/30.92 3^1(5(1(4(0(x1))))) -> 3^1(3(4(5(0(x1))))) 117.81/30.92 3^1(5(1(4(0(x1))))) -> 3^1(4(5(0(x1)))) 117.81/30.92 3^1(5(1(4(0(x1))))) -> 4^1(5(0(x1))) 117.81/30.92 3^1(5(1(4(0(x1))))) -> 5^1(0(x1)) 117.81/30.92 3^1(5(3(0(4(x1))))) -> 5^1(3(4(x1))) 117.81/30.92 117.81/30.92 The TRS R consists of the following rules: 117.81/30.92 117.81/30.92 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 117.81/30.92 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 117.81/30.92 0(3(2(x1))) -> 2(1(3(0(x1)))) 117.81/30.92 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 117.81/30.92 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 117.81/30.92 3(2(0(x1))) -> 0(2(1(3(x1)))) 117.81/30.92 3(2(0(x1))) -> 3(0(2(1(x1)))) 117.81/30.92 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 117.81/30.92 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 117.81/30.92 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 117.81/30.92 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 117.81/30.92 3(2(5(x1))) -> 1(3(5(2(x1)))) 117.81/30.92 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 117.81/30.92 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 117.81/30.92 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 117.81/30.92 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 117.81/30.92 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 117.81/30.92 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 117.81/30.92 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 117.81/30.92 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 117.81/30.92 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 117.81/30.92 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 117.81/30.92 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 117.81/30.92 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 117.81/30.92 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 117.81/30.92 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 117.81/30.92 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 117.81/30.92 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 117.81/30.92 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 117.81/30.92 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 117.81/30.92 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 117.81/30.92 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 117.81/30.92 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 117.81/30.92 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 117.81/30.92 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 117.81/30.92 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 117.81/30.92 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 117.81/30.92 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 117.81/30.92 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 117.81/30.92 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 117.81/30.92 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 117.81/30.92 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 117.81/30.92 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 117.81/30.92 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 117.81/30.92 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 117.81/30.92 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 117.81/30.92 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 117.81/30.92 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) 117.81/30.92 117.81/30.92 Q is empty. 117.81/30.92 We have to consider all minimal (P,Q,R)-chains. 117.81/30.92 ---------------------------------------- 117.81/30.92 117.81/30.92 (7) DependencyGraphProof (EQUIVALENT) 117.81/30.92 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 117.81/30.92 ---------------------------------------- 117.81/30.92 117.81/30.92 (8) 117.81/30.92 Obligation: 117.81/30.92 Q DP problem: 117.81/30.92 The TRS P consists of the following rules: 117.81/30.92 117.81/30.92 3^1(2(0(x1))) -> 3^1(x1) 117.81/30.92 3^1(2(0(x1))) -> 4^1(x1) 117.81/30.92 4^1(0(4(2(x1)))) -> 3^1(0(x1)) 117.81/30.92 3^1(2(4(x1))) -> 3^1(3(4(x1))) 117.81/30.92 3^1(2(4(x1))) -> 3^1(4(x1)) 117.81/30.92 3^1(2(4(x1))) -> 3^1(x1) 117.81/30.92 3^1(2(5(x1))) -> 3^1(5(2(x1))) 117.81/30.92 3^1(2(5(x1))) -> 4^1(5(2(x1))) 117.81/30.92 4^1(0(4(2(x1)))) -> 0^1(x1) 117.81/30.92 0^1(3(2(x1))) -> 3^1(0(x1)) 117.81/30.92 3^1(2(5(x1))) -> 3^1(3(5(2(x1)))) 117.81/30.92 3^1(2(5(x1))) -> 3^1(4(5(4(2(x1))))) 117.81/30.92 3^1(2(5(x1))) -> 4^1(5(4(2(x1)))) 117.81/30.92 4^1(3(2(0(x1)))) -> 3^1(4(x1)) 117.81/30.92 3^1(2(5(x1))) -> 5^1(4(2(x1))) 117.81/30.92 5^1(4(2(0(2(x1))))) -> 5^1(0(4(2(1(2(x1)))))) 117.81/30.92 5^1(0(2(x1))) -> 5^1(x1) 117.81/30.92 5^1(5(0(0(x1)))) -> 0^1(5(5(0(x1)))) 117.81/30.92 0^1(3(2(x1))) -> 0^1(x1) 117.81/30.92 0^1(3(1(2(x1)))) -> 3^1(0(2(x1))) 117.81/30.92 3^1(0(2(4(x1)))) -> 3^1(0(4(2(1(1(x1)))))) 117.81/30.92 3^1(0(5(4(x1)))) -> 3^1(4(0(5(1(3(x1)))))) 117.81/30.92 3^1(0(5(4(x1)))) -> 4^1(0(5(1(3(x1))))) 117.81/30.92 4^1(3(2(0(x1)))) -> 4^1(x1) 117.81/30.92 3^1(0(5(4(x1)))) -> 0^1(5(1(3(x1)))) 117.81/30.92 0^1(3(2(4(x1)))) -> 3^1(0(4(4(x1)))) 117.81/30.92 3^1(0(5(4(x1)))) -> 5^1(1(3(x1))) 117.81/30.92 5^1(1(0(2(4(x1))))) -> 5^1(4(0(3(x1)))) 117.81/30.92 5^1(5(0(0(x1)))) -> 5^1(5(0(x1))) 117.81/30.92 3^1(2(5(4(x1)))) -> 3^1(4(3(5(2(x1))))) 117.81/30.92 3^1(2(5(4(x1)))) -> 4^1(3(5(2(x1)))) 117.81/30.92 3^1(2(5(4(x1)))) -> 3^1(5(2(x1))) 117.81/30.92 3^1(5(3(2(x1)))) -> 3^1(3(0(2(5(x1))))) 117.81/30.92 3^1(5(3(2(x1)))) -> 3^1(0(2(5(x1)))) 117.81/30.92 3^1(5(3(2(x1)))) -> 5^1(x1) 117.81/30.92 3^1(0(3(1(2(x1))))) -> 3^1(3(x1)) 117.81/30.92 3^1(0(3(1(2(x1))))) -> 3^1(x1) 117.81/30.92 3^1(5(1(4(0(x1))))) -> 3^1(3(4(5(0(x1))))) 117.81/30.92 3^1(5(1(4(0(x1))))) -> 3^1(4(5(0(x1)))) 117.81/30.92 3^1(5(1(4(0(x1))))) -> 4^1(5(0(x1))) 117.81/30.92 3^1(5(1(4(0(x1))))) -> 5^1(0(x1)) 117.81/30.92 3^1(5(3(0(4(x1))))) -> 5^1(3(4(x1))) 117.81/30.92 0^1(3(2(4(x1)))) -> 0^1(4(4(x1))) 117.81/30.92 0^1(3(2(4(x1)))) -> 4^1(4(x1)) 117.81/30.92 117.81/30.92 The TRS R consists of the following rules: 117.81/30.92 117.81/30.92 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 117.81/30.93 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 117.81/30.93 0(3(2(x1))) -> 2(1(3(0(x1)))) 117.81/30.93 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 117.81/30.93 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 117.81/30.93 3(2(0(x1))) -> 0(2(1(3(x1)))) 117.81/30.93 3(2(0(x1))) -> 3(0(2(1(x1)))) 117.81/30.93 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 117.81/30.93 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 117.81/30.93 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 117.81/30.93 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 117.81/30.93 3(2(5(x1))) -> 1(3(5(2(x1)))) 117.81/30.93 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 117.81/30.93 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 117.81/30.93 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 117.81/30.93 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 117.81/30.93 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 117.81/30.93 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 117.81/30.93 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 117.81/30.93 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 117.81/30.93 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 117.81/30.93 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 117.81/30.93 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 117.81/30.93 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 117.81/30.93 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 117.81/30.93 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 117.81/30.93 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 117.81/30.93 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 117.81/30.94 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 117.81/30.94 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 117.81/30.94 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 117.81/30.94 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 117.81/30.94 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 117.81/30.94 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 117.81/30.94 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 117.81/30.94 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 117.81/30.94 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 117.81/30.94 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 117.81/30.94 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 117.81/30.94 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 117.81/30.94 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 117.81/30.94 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 117.81/30.94 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 117.81/30.94 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 117.81/30.94 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 117.81/30.94 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 117.81/30.94 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 117.81/30.94 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) 117.81/30.94 117.81/30.94 Q is empty. 117.81/30.94 We have to consider all minimal (P,Q,R)-chains. 117.81/30.94 ---------------------------------------- 117.81/30.94 117.81/30.94 (9) QDPOrderProof (EQUIVALENT) 117.81/30.94 We use the reduction pair processor [LPAR04,JAR06]. 117.81/30.94 117.81/30.94 117.81/30.94 The following pairs can be oriented strictly and are deleted. 117.81/30.94 117.81/30.94 3^1(2(0(x1))) -> 3^1(x1) 117.81/30.94 3^1(2(0(x1))) -> 4^1(x1) 117.81/30.94 3^1(2(4(x1))) -> 3^1(3(4(x1))) 117.81/30.94 3^1(2(4(x1))) -> 3^1(4(x1)) 117.81/30.94 3^1(2(4(x1))) -> 3^1(x1) 117.81/30.94 3^1(2(5(x1))) -> 4^1(5(2(x1))) 117.81/30.94 0^1(3(2(x1))) -> 3^1(0(x1)) 117.81/30.94 3^1(2(5(x1))) -> 4^1(5(4(2(x1)))) 117.81/30.94 5^1(0(2(x1))) -> 5^1(x1) 117.81/30.94 0^1(3(2(x1))) -> 0^1(x1) 117.81/30.94 3^1(0(5(4(x1)))) -> 4^1(0(5(1(3(x1))))) 117.81/30.94 4^1(3(2(0(x1)))) -> 4^1(x1) 117.81/30.94 0^1(3(2(4(x1)))) -> 3^1(0(4(4(x1)))) 117.81/30.94 5^1(1(0(2(4(x1))))) -> 5^1(4(0(3(x1)))) 117.81/30.94 3^1(2(5(4(x1)))) -> 4^1(3(5(2(x1)))) 117.81/30.94 3^1(5(3(2(x1)))) -> 5^1(x1) 117.81/30.94 3^1(0(3(1(2(x1))))) -> 3^1(3(x1)) 117.81/30.94 3^1(0(3(1(2(x1))))) -> 3^1(x1) 117.81/30.94 3^1(5(1(4(0(x1))))) -> 4^1(5(0(x1))) 117.81/30.94 0^1(3(2(4(x1)))) -> 0^1(4(4(x1))) 117.81/30.94 0^1(3(2(4(x1)))) -> 4^1(4(x1)) 117.81/30.94 The remaining pairs can at least be oriented weakly. 117.81/30.94 Used ordering: Polynomial interpretation [POLO]: 117.81/30.94 117.81/30.94 POL(0(x_1)) = x_1 117.81/30.94 POL(0^1(x_1)) = 1 + x_1 117.81/30.94 POL(1(x_1)) = x_1 117.81/30.94 POL(2(x_1)) = 1 + x_1 117.81/30.94 POL(3(x_1)) = x_1 117.81/30.94 POL(3^1(x_1)) = 1 + x_1 117.81/30.94 POL(4(x_1)) = x_1 117.81/30.94 POL(4^1(x_1)) = x_1 117.81/30.94 POL(5(x_1)) = x_1 117.81/30.94 POL(5^1(x_1)) = 1 + x_1 117.81/30.94 117.81/30.94 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 117.81/30.94 117.81/30.94 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 117.81/30.94 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 117.81/30.94 0(3(2(x1))) -> 2(1(3(0(x1)))) 117.81/30.94 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 117.81/30.94 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 117.81/30.94 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 117.81/30.94 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 117.81/30.94 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 117.81/30.94 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 117.81/30.94 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 117.81/30.94 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 117.81/30.94 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 117.81/30.94 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 117.81/30.94 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 117.81/30.94 3(2(0(x1))) -> 0(2(1(3(x1)))) 117.81/30.94 3(2(0(x1))) -> 3(0(2(1(x1)))) 117.81/30.94 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 117.81/30.94 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 117.81/30.94 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 117.81/30.94 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 117.81/30.94 3(2(5(x1))) -> 1(3(5(2(x1)))) 117.81/30.94 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 117.81/30.94 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 117.81/30.94 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 117.81/30.94 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 117.81/30.94 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 117.81/30.94 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 117.81/30.94 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 117.81/30.94 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 117.81/30.94 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 117.81/30.94 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 117.81/30.94 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 117.81/30.94 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 117.81/30.94 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 117.81/30.94 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 117.81/30.94 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 117.81/30.94 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 117.81/30.94 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 117.81/30.94 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) 117.81/30.94 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 117.81/30.94 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 117.81/30.94 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 117.81/30.94 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 117.81/30.94 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 117.81/30.94 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 117.81/30.94 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 117.81/30.94 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 117.81/30.94 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 117.81/30.94 117.81/30.94 117.81/30.94 ---------------------------------------- 117.81/30.94 117.81/30.94 (10) 117.81/30.94 Obligation: 117.81/30.94 Q DP problem: 117.81/30.94 The TRS P consists of the following rules: 117.81/30.94 117.81/30.94 4^1(0(4(2(x1)))) -> 3^1(0(x1)) 117.81/30.94 3^1(2(5(x1))) -> 3^1(5(2(x1))) 117.81/30.94 4^1(0(4(2(x1)))) -> 0^1(x1) 117.81/30.94 3^1(2(5(x1))) -> 3^1(3(5(2(x1)))) 117.81/30.94 3^1(2(5(x1))) -> 3^1(4(5(4(2(x1))))) 117.81/30.94 4^1(3(2(0(x1)))) -> 3^1(4(x1)) 117.81/30.94 3^1(2(5(x1))) -> 5^1(4(2(x1))) 117.81/30.94 5^1(4(2(0(2(x1))))) -> 5^1(0(4(2(1(2(x1)))))) 117.81/30.94 5^1(5(0(0(x1)))) -> 0^1(5(5(0(x1)))) 117.81/30.94 0^1(3(1(2(x1)))) -> 3^1(0(2(x1))) 117.81/30.94 3^1(0(2(4(x1)))) -> 3^1(0(4(2(1(1(x1)))))) 117.81/30.94 3^1(0(5(4(x1)))) -> 3^1(4(0(5(1(3(x1)))))) 117.81/30.94 3^1(0(5(4(x1)))) -> 0^1(5(1(3(x1)))) 117.81/30.94 3^1(0(5(4(x1)))) -> 5^1(1(3(x1))) 117.81/30.94 5^1(5(0(0(x1)))) -> 5^1(5(0(x1))) 117.81/30.94 3^1(2(5(4(x1)))) -> 3^1(4(3(5(2(x1))))) 117.81/30.94 3^1(2(5(4(x1)))) -> 3^1(5(2(x1))) 117.81/30.94 3^1(5(3(2(x1)))) -> 3^1(3(0(2(5(x1))))) 117.81/30.94 3^1(5(3(2(x1)))) -> 3^1(0(2(5(x1)))) 117.81/30.94 3^1(5(1(4(0(x1))))) -> 3^1(3(4(5(0(x1))))) 117.81/30.94 3^1(5(1(4(0(x1))))) -> 3^1(4(5(0(x1)))) 117.81/30.94 3^1(5(1(4(0(x1))))) -> 5^1(0(x1)) 117.81/30.94 3^1(5(3(0(4(x1))))) -> 5^1(3(4(x1))) 117.81/30.94 117.81/30.94 The TRS R consists of the following rules: 117.81/30.94 117.81/30.94 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 117.81/30.94 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 117.81/30.94 0(3(2(x1))) -> 2(1(3(0(x1)))) 117.81/30.94 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 117.81/30.94 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 117.81/30.94 3(2(0(x1))) -> 0(2(1(3(x1)))) 117.81/30.94 3(2(0(x1))) -> 3(0(2(1(x1)))) 117.81/30.94 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 117.81/30.94 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 117.81/30.94 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 117.81/30.94 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 117.81/30.94 3(2(5(x1))) -> 1(3(5(2(x1)))) 117.81/30.94 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 117.81/30.94 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 117.81/30.94 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 117.81/30.94 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 117.81/30.94 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 117.81/30.94 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 117.81/30.94 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 117.81/30.94 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 117.81/30.94 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 117.81/30.94 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 117.81/30.94 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 117.81/30.94 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 117.81/30.94 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 117.81/30.94 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 117.81/30.94 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 117.81/30.94 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 117.81/30.94 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 117.81/30.94 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 117.81/30.94 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 117.81/30.94 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 117.81/30.94 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 117.81/30.94 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 117.81/30.94 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 117.81/30.94 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 117.81/30.94 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 117.81/30.94 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 117.81/30.94 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 117.81/30.94 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 117.81/30.94 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 117.81/30.94 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 117.81/30.94 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 117.81/30.94 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 117.81/30.94 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 117.81/30.94 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 117.81/30.94 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 117.81/30.94 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) 117.81/30.94 117.81/30.94 Q is empty. 117.81/30.94 We have to consider all minimal (P,Q,R)-chains. 117.81/30.94 ---------------------------------------- 117.81/30.94 117.81/30.94 (11) DependencyGraphProof (EQUIVALENT) 117.81/30.94 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. 117.81/30.94 ---------------------------------------- 117.81/30.94 117.81/30.94 (12) 117.81/30.94 Obligation: 117.81/30.94 Q DP problem: 117.81/30.94 The TRS P consists of the following rules: 117.81/30.94 117.81/30.94 3^1(2(5(x1))) -> 3^1(3(5(2(x1)))) 117.81/30.94 3^1(2(5(x1))) -> 3^1(5(2(x1))) 117.81/30.94 3^1(2(5(x1))) -> 3^1(4(5(4(2(x1))))) 117.81/30.94 3^1(2(5(x1))) -> 5^1(4(2(x1))) 117.81/30.94 5^1(4(2(0(2(x1))))) -> 5^1(0(4(2(1(2(x1)))))) 117.81/30.94 5^1(5(0(0(x1)))) -> 0^1(5(5(0(x1)))) 117.81/30.94 0^1(3(1(2(x1)))) -> 3^1(0(2(x1))) 117.81/30.94 3^1(0(2(4(x1)))) -> 3^1(0(4(2(1(1(x1)))))) 117.81/30.94 3^1(0(5(4(x1)))) -> 3^1(4(0(5(1(3(x1)))))) 117.81/30.94 3^1(0(5(4(x1)))) -> 0^1(5(1(3(x1)))) 117.81/30.94 3^1(2(5(4(x1)))) -> 3^1(4(3(5(2(x1))))) 117.81/30.94 3^1(2(5(4(x1)))) -> 3^1(5(2(x1))) 117.81/30.94 3^1(5(3(2(x1)))) -> 3^1(3(0(2(5(x1))))) 117.81/30.94 3^1(5(3(2(x1)))) -> 3^1(0(2(5(x1)))) 117.81/30.94 3^1(5(1(4(0(x1))))) -> 3^1(3(4(5(0(x1))))) 117.81/30.94 3^1(5(1(4(0(x1))))) -> 3^1(4(5(0(x1)))) 117.81/30.94 3^1(5(1(4(0(x1))))) -> 5^1(0(x1)) 117.81/30.94 5^1(5(0(0(x1)))) -> 5^1(5(0(x1))) 117.81/30.94 3^1(5(3(0(4(x1))))) -> 5^1(3(4(x1))) 117.81/30.94 117.81/30.94 The TRS R consists of the following rules: 117.81/30.94 117.81/30.94 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 117.81/30.94 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 117.81/30.94 0(3(2(x1))) -> 2(1(3(0(x1)))) 117.81/30.94 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 117.81/30.94 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 117.81/30.94 3(2(0(x1))) -> 0(2(1(3(x1)))) 117.81/30.94 3(2(0(x1))) -> 3(0(2(1(x1)))) 117.81/30.94 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 117.81/30.94 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 117.81/30.94 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 117.81/30.94 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 117.81/30.94 3(2(5(x1))) -> 1(3(5(2(x1)))) 117.81/30.94 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 117.81/30.94 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 117.81/30.94 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 117.81/30.94 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 117.81/30.94 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 117.81/30.94 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 117.81/30.94 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 117.81/30.94 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 117.81/30.94 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 117.81/30.94 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 117.81/30.94 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 117.81/30.94 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 117.81/30.94 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 117.81/30.94 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 117.81/30.94 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 117.81/30.94 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 117.81/30.94 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 117.81/30.94 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 117.81/30.94 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 117.81/30.94 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 117.81/30.94 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 117.81/30.94 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 117.81/30.94 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 117.81/30.94 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 117.81/30.94 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 117.81/30.94 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 117.81/30.94 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 117.81/30.94 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 117.81/30.94 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 117.81/30.94 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 117.81/30.94 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 117.81/30.94 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 117.81/30.94 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 117.81/30.94 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 117.81/30.94 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 117.81/30.94 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) 117.81/30.94 117.81/30.94 Q is empty. 117.81/30.94 We have to consider all minimal (P,Q,R)-chains. 117.81/30.94 ---------------------------------------- 117.81/30.94 117.81/30.94 (13) QDPOrderProof (EQUIVALENT) 117.81/30.94 We use the reduction pair processor [LPAR04,JAR06]. 117.81/30.94 117.81/30.94 117.81/30.94 The following pairs can be oriented strictly and are deleted. 117.81/30.94 117.81/30.94 0^1(3(1(2(x1)))) -> 3^1(0(2(x1))) 117.81/30.94 The remaining pairs can at least be oriented weakly. 117.81/30.94 Used ordering: Polynomial interpretation [POLO]: 117.81/30.94 117.81/30.94 POL(0(x_1)) = 0 117.81/30.94 POL(0^1(x_1)) = x_1 117.81/30.94 POL(1(x_1)) = 0 117.81/30.94 POL(2(x_1)) = 0 117.81/30.94 POL(3(x_1)) = 1 117.81/30.94 POL(3^1(x_1)) = 0 117.81/30.94 POL(4(x_1)) = 0 117.81/30.94 POL(5(x_1)) = 0 117.81/30.94 POL(5^1(x_1)) = 0 117.81/30.94 117.81/30.94 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 117.81/30.94 117.81/30.94 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 117.81/30.94 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) 117.81/30.94 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 117.81/30.94 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 117.81/30.94 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 117.81/30.94 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 117.81/30.94 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 117.81/30.94 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 117.81/30.94 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 117.81/30.94 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 117.81/30.94 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 117.81/30.94 117.81/30.94 117.81/30.94 ---------------------------------------- 117.81/30.94 117.81/30.94 (14) 117.81/30.94 Obligation: 117.81/30.94 Q DP problem: 117.81/30.94 The TRS P consists of the following rules: 117.81/30.94 117.81/30.94 3^1(2(5(x1))) -> 3^1(3(5(2(x1)))) 117.81/30.94 3^1(2(5(x1))) -> 3^1(5(2(x1))) 117.81/30.94 3^1(2(5(x1))) -> 3^1(4(5(4(2(x1))))) 117.81/30.94 3^1(2(5(x1))) -> 5^1(4(2(x1))) 117.81/30.94 5^1(4(2(0(2(x1))))) -> 5^1(0(4(2(1(2(x1)))))) 117.81/30.94 5^1(5(0(0(x1)))) -> 0^1(5(5(0(x1)))) 117.81/30.94 3^1(0(2(4(x1)))) -> 3^1(0(4(2(1(1(x1)))))) 117.81/30.94 3^1(0(5(4(x1)))) -> 3^1(4(0(5(1(3(x1)))))) 117.81/30.94 3^1(0(5(4(x1)))) -> 0^1(5(1(3(x1)))) 117.81/30.94 3^1(2(5(4(x1)))) -> 3^1(4(3(5(2(x1))))) 117.81/30.94 3^1(2(5(4(x1)))) -> 3^1(5(2(x1))) 117.81/30.94 3^1(5(3(2(x1)))) -> 3^1(3(0(2(5(x1))))) 117.81/30.94 3^1(5(3(2(x1)))) -> 3^1(0(2(5(x1)))) 117.81/30.94 3^1(5(1(4(0(x1))))) -> 3^1(3(4(5(0(x1))))) 117.81/30.94 3^1(5(1(4(0(x1))))) -> 3^1(4(5(0(x1)))) 117.81/30.94 3^1(5(1(4(0(x1))))) -> 5^1(0(x1)) 117.81/30.94 5^1(5(0(0(x1)))) -> 5^1(5(0(x1))) 117.81/30.94 3^1(5(3(0(4(x1))))) -> 5^1(3(4(x1))) 117.81/30.94 117.81/30.94 The TRS R consists of the following rules: 117.81/30.94 117.81/30.94 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 117.81/30.94 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 117.81/30.94 0(3(2(x1))) -> 2(1(3(0(x1)))) 117.81/30.94 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 117.81/30.94 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 117.81/30.94 3(2(0(x1))) -> 0(2(1(3(x1)))) 117.81/30.94 3(2(0(x1))) -> 3(0(2(1(x1)))) 117.81/30.94 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 117.81/30.94 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 117.81/30.94 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 117.81/30.94 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 117.81/30.94 3(2(5(x1))) -> 1(3(5(2(x1)))) 117.81/30.94 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 117.81/30.94 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 117.81/30.94 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 117.81/30.94 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 117.81/30.94 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 117.81/30.94 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 117.81/30.94 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 117.81/30.94 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 117.81/30.94 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 117.81/30.94 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 117.81/30.94 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 117.81/30.94 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 117.81/30.94 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 117.81/30.94 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 117.81/30.94 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 117.81/30.94 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 117.81/30.94 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 117.81/30.94 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 117.81/30.94 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 117.81/30.94 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 117.81/30.94 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 117.81/30.94 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 117.81/30.94 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 117.81/30.94 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 117.81/30.94 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 117.81/30.94 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 117.81/30.94 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 117.81/30.94 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 117.81/30.94 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 117.81/30.94 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 117.81/30.94 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 117.81/30.94 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 117.81/30.94 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 117.81/30.94 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 117.81/30.94 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 117.81/30.94 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) 117.81/30.94 117.81/30.94 Q is empty. 117.81/30.94 We have to consider all minimal (P,Q,R)-chains. 117.81/30.94 ---------------------------------------- 117.81/30.94 117.81/30.94 (15) DependencyGraphProof (EQUIVALENT) 117.81/30.94 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 5 less nodes. 117.81/30.94 ---------------------------------------- 117.81/30.94 117.81/30.94 (16) 117.81/30.94 Complex Obligation (AND) 117.81/30.94 117.81/30.94 ---------------------------------------- 117.81/30.94 117.81/30.94 (17) 117.81/30.94 Obligation: 117.81/30.94 Q DP problem: 117.81/30.94 The TRS P consists of the following rules: 117.81/30.94 117.81/30.94 5^1(5(0(0(x1)))) -> 5^1(5(0(x1))) 117.81/30.94 5^1(4(2(0(2(x1))))) -> 5^1(0(4(2(1(2(x1)))))) 117.81/30.94 117.81/30.94 The TRS R consists of the following rules: 117.81/30.94 117.81/30.94 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 117.81/30.94 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 117.81/30.94 0(3(2(x1))) -> 2(1(3(0(x1)))) 117.81/30.94 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 117.81/30.94 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 117.81/30.94 3(2(0(x1))) -> 0(2(1(3(x1)))) 117.81/30.94 3(2(0(x1))) -> 3(0(2(1(x1)))) 117.81/30.94 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 117.81/30.94 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 117.81/30.94 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 117.81/30.94 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 117.81/30.94 3(2(5(x1))) -> 1(3(5(2(x1)))) 117.81/30.94 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 117.81/30.94 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 117.81/30.94 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 117.81/30.94 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 117.81/30.94 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 117.81/30.94 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 117.81/30.94 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 117.81/30.94 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 117.81/30.94 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 117.81/30.94 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 117.81/30.94 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 117.81/30.94 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 117.81/30.94 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 117.81/30.94 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 117.81/30.94 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 117.81/30.94 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 117.81/30.94 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 117.81/30.94 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 117.81/30.94 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 117.81/30.94 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 117.81/30.94 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 117.81/30.94 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 117.81/30.94 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 117.81/30.94 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 117.81/30.94 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 117.81/30.94 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 117.81/30.94 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 117.81/30.94 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 117.81/30.94 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 117.81/30.94 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 117.81/30.94 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 117.81/30.94 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 117.81/30.94 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 117.81/30.94 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 117.81/30.94 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 117.81/30.94 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) 117.81/30.94 117.81/30.94 Q is empty. 117.81/30.94 We have to consider all minimal (P,Q,R)-chains. 117.81/30.94 ---------------------------------------- 117.81/30.94 117.81/30.94 (18) QDPOrderProof (EQUIVALENT) 117.81/30.94 We use the reduction pair processor [LPAR04,JAR06]. 117.81/30.94 117.81/30.94 117.81/30.94 The following pairs can be oriented strictly and are deleted. 117.81/30.94 117.81/30.94 5^1(4(2(0(2(x1))))) -> 5^1(0(4(2(1(2(x1)))))) 117.81/30.94 The remaining pairs can at least be oriented weakly. 117.81/30.94 Used ordering: Polynomial interpretation [POLO]: 117.81/30.94 117.81/30.94 POL(0(x_1)) = x_1 117.81/30.94 POL(1(x_1)) = 0 117.81/30.94 POL(2(x_1)) = 1 + x_1 117.81/30.94 POL(3(x_1)) = 0 117.81/30.94 POL(4(x_1)) = 1 + x_1 117.81/30.94 POL(5(x_1)) = 1 117.81/30.94 POL(5^1(x_1)) = x_1 117.81/30.94 117.81/30.94 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 117.81/30.94 117.81/30.94 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 117.81/30.94 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 117.81/30.94 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 117.81/30.94 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 117.81/30.94 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 117.81/30.94 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 117.81/30.94 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 117.81/30.94 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 117.81/30.94 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 117.81/30.94 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 117.81/30.94 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 117.81/30.94 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) 117.81/30.94 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 117.81/30.94 117.81/30.94 117.81/30.94 ---------------------------------------- 117.81/30.94 117.81/30.94 (19) 117.81/30.94 Obligation: 117.81/30.94 Q DP problem: 117.81/30.94 The TRS P consists of the following rules: 117.81/30.94 117.81/30.94 5^1(5(0(0(x1)))) -> 5^1(5(0(x1))) 117.81/30.94 117.81/30.94 The TRS R consists of the following rules: 117.81/30.94 117.81/30.94 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 117.81/30.94 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 117.81/30.94 0(3(2(x1))) -> 2(1(3(0(x1)))) 117.81/30.94 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 117.81/30.94 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 117.81/30.94 3(2(0(x1))) -> 0(2(1(3(x1)))) 117.81/30.94 3(2(0(x1))) -> 3(0(2(1(x1)))) 117.81/30.94 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 117.81/30.94 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 117.81/30.94 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 117.81/30.94 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 117.81/30.94 3(2(5(x1))) -> 1(3(5(2(x1)))) 117.81/30.94 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 117.81/30.94 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 117.81/30.94 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 117.81/30.94 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 117.81/30.94 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 117.81/30.94 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 117.81/30.94 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 117.81/30.94 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 117.81/30.94 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 117.81/30.94 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 117.81/30.94 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 117.81/30.94 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 117.81/30.94 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 117.81/30.94 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 117.81/30.94 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 117.81/30.94 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 117.81/30.94 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 117.81/30.94 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 117.81/30.94 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 117.81/30.94 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 117.81/30.94 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 117.81/30.94 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 117.81/30.94 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 117.81/30.94 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 117.81/30.94 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 117.81/30.94 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 117.81/30.94 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 117.81/30.94 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 117.81/30.94 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 117.81/30.94 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 117.81/30.94 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 117.81/30.94 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 117.81/30.94 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 117.81/30.94 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 117.81/30.94 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 117.81/30.94 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) 117.81/30.94 117.81/30.94 Q is empty. 117.81/30.94 We have to consider all minimal (P,Q,R)-chains. 117.81/30.94 ---------------------------------------- 117.81/30.94 117.81/30.94 (20) QDPOrderProof (EQUIVALENT) 117.81/30.94 We use the reduction pair processor [LPAR04,JAR06]. 117.81/30.94 117.81/30.94 117.81/30.94 The following pairs can be oriented strictly and are deleted. 117.81/30.94 117.81/30.94 5^1(5(0(0(x1)))) -> 5^1(5(0(x1))) 117.81/30.94 The remaining pairs can at least be oriented weakly. 117.81/30.94 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 117.81/30.94 117.81/30.94 <<< 117.81/30.94 POL(5^1(x_1)) = [[0A]] + [[-I, 0A, -I]] * x_1 117.81/30.94 >>> 117.81/30.94 117.81/30.94 <<< 117.81/30.94 POL(5(x_1)) = [[1A], [-I], [-I]] + [[0A, 0A, 0A], [-I, -I, 0A], [0A, 0A, -I]] * x_1 117.81/30.94 >>> 117.81/30.94 117.81/30.94 <<< 117.81/30.94 POL(0(x_1)) = [[0A], [1A], [0A]] + [[0A, 0A, 1A], [0A, 0A, 1A], [-I, -I, 1A]] * x_1 117.81/30.94 >>> 117.81/30.94 117.81/30.94 <<< 117.81/30.94 POL(1(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 117.81/30.94 >>> 117.81/30.94 117.81/30.94 <<< 117.81/30.94 POL(2(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, 0A], [-I, 0A, -I], [-I, 0A, -I]] * x_1 117.81/30.94 >>> 117.81/30.94 117.81/30.94 <<< 117.81/30.94 POL(3(x_1)) = [[0A], [-I], [0A]] + [[0A, 0A, 0A], [-I, -I, -I], [0A, 0A, -I]] * x_1 117.81/30.94 >>> 117.81/30.94 117.81/30.94 <<< 117.81/30.94 POL(4(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, -I]] * x_1 117.81/30.94 >>> 117.81/30.94 117.81/30.94 117.81/30.94 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 117.81/30.94 117.81/30.94 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 117.81/30.94 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 117.81/30.94 0(3(2(x1))) -> 2(1(3(0(x1)))) 117.81/30.94 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 117.81/30.94 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 117.81/30.94 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 117.81/30.94 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 117.81/30.94 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 117.81/30.94 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 117.81/30.94 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 117.81/30.94 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 117.81/30.94 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 117.81/30.94 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 117.81/30.94 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 117.81/30.94 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 117.81/30.94 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 117.81/30.94 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 117.81/30.94 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 117.81/30.94 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 117.81/30.94 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 117.81/30.94 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 117.81/30.94 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) 117.81/30.94 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 117.81/30.94 117.81/30.94 117.81/30.94 ---------------------------------------- 117.81/30.94 117.81/30.94 (21) 117.81/30.94 Obligation: 117.81/30.94 Q DP problem: 117.81/30.94 P is empty. 117.81/30.94 The TRS R consists of the following rules: 117.81/30.94 117.81/30.94 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 117.81/30.94 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 117.81/30.94 0(3(2(x1))) -> 2(1(3(0(x1)))) 117.81/30.94 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 117.81/30.94 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 117.81/30.94 3(2(0(x1))) -> 0(2(1(3(x1)))) 117.81/30.94 3(2(0(x1))) -> 3(0(2(1(x1)))) 117.81/30.94 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 117.81/30.94 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 117.81/30.94 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 117.81/30.94 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 117.81/30.94 3(2(5(x1))) -> 1(3(5(2(x1)))) 117.81/30.94 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 117.81/30.94 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 117.81/30.94 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 117.81/30.94 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 117.81/30.94 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 117.81/30.94 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 117.81/30.94 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 117.81/30.94 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 117.81/30.94 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 117.81/30.94 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 117.81/30.94 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 117.81/30.94 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 117.81/30.94 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 117.81/30.94 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 117.81/30.94 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 117.81/30.94 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 117.81/30.94 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 117.81/30.94 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 117.81/30.94 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 117.81/30.94 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 117.81/30.94 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 117.81/30.94 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 117.81/30.94 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 117.81/30.94 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 117.81/30.94 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 117.81/30.94 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 117.81/30.94 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 117.81/30.94 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 117.81/30.94 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 117.81/30.94 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 117.81/30.94 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 117.81/30.94 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 117.81/30.94 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 117.81/30.94 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 117.81/30.94 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 117.81/30.94 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) 117.81/30.94 117.81/30.94 Q is empty. 117.81/30.94 We have to consider all minimal (P,Q,R)-chains. 117.81/30.94 ---------------------------------------- 117.81/30.94 117.81/30.94 (22) PisEmptyProof (EQUIVALENT) 117.81/30.94 The TRS P is empty. Hence, there is no (P,Q,R) chain. 117.81/30.94 ---------------------------------------- 117.81/30.94 117.81/30.94 (23) 117.81/30.94 YES 117.81/30.94 117.81/30.94 ---------------------------------------- 117.81/30.94 117.81/30.94 (24) 117.81/30.94 Obligation: 117.81/30.95 Q DP problem: 117.81/30.95 The TRS P consists of the following rules: 117.81/30.95 117.81/30.95 3^1(2(5(x1))) -> 3^1(5(2(x1))) 117.81/30.95 3^1(2(5(x1))) -> 3^1(3(5(2(x1)))) 117.81/30.95 3^1(2(5(x1))) -> 3^1(4(5(4(2(x1))))) 117.81/30.95 3^1(0(2(4(x1)))) -> 3^1(0(4(2(1(1(x1)))))) 117.81/30.95 3^1(0(5(4(x1)))) -> 3^1(4(0(5(1(3(x1)))))) 117.81/30.95 3^1(2(5(4(x1)))) -> 3^1(4(3(5(2(x1))))) 117.81/30.95 3^1(2(5(4(x1)))) -> 3^1(5(2(x1))) 117.81/30.95 3^1(5(3(2(x1)))) -> 3^1(3(0(2(5(x1))))) 117.81/30.95 3^1(5(3(2(x1)))) -> 3^1(0(2(5(x1)))) 117.81/30.95 3^1(5(1(4(0(x1))))) -> 3^1(3(4(5(0(x1))))) 117.81/30.95 3^1(5(1(4(0(x1))))) -> 3^1(4(5(0(x1)))) 117.81/30.95 117.81/30.95 The TRS R consists of the following rules: 117.81/30.95 117.81/30.95 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 117.81/30.95 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 117.81/30.95 0(3(2(x1))) -> 2(1(3(0(x1)))) 117.81/30.95 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 117.81/30.95 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 117.81/30.95 3(2(0(x1))) -> 0(2(1(3(x1)))) 117.81/30.95 3(2(0(x1))) -> 3(0(2(1(x1)))) 117.81/30.95 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 117.81/30.95 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 117.81/30.95 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 117.81/30.95 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 117.81/30.95 3(2(5(x1))) -> 1(3(5(2(x1)))) 117.81/30.95 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 117.81/30.95 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 117.81/30.95 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 117.81/30.95 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 117.81/30.95 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 117.81/30.95 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 117.81/30.95 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 117.81/30.95 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 117.81/30.95 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 117.81/30.95 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 117.81/30.95 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 117.81/30.95 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 117.81/30.95 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 117.81/30.95 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 117.81/30.95 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 117.81/30.95 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 117.81/30.95 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 117.81/30.95 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 117.81/30.95 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 117.81/30.95 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 117.81/30.95 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 117.81/30.95 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 117.81/30.95 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 117.81/30.95 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 117.81/30.95 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 117.81/30.95 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 117.81/30.95 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 117.81/30.95 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 117.81/30.95 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 117.81/30.95 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 117.81/30.95 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 117.81/30.95 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 117.81/30.95 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 117.81/30.95 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 117.81/30.95 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 117.81/30.95 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) 117.81/30.95 117.81/30.95 Q is empty. 117.81/30.95 We have to consider all minimal (P,Q,R)-chains. 117.81/30.95 ---------------------------------------- 117.81/30.95 117.81/30.95 (25) QDPOrderProof (EQUIVALENT) 117.81/30.95 We use the reduction pair processor [LPAR04,JAR06]. 117.81/30.95 117.81/30.95 117.81/30.95 The following pairs can be oriented strictly and are deleted. 117.81/30.95 117.81/30.95 3^1(2(5(x1))) -> 3^1(5(2(x1))) 117.81/30.95 3^1(2(5(x1))) -> 3^1(3(5(2(x1)))) 117.81/30.95 3^1(2(5(x1))) -> 3^1(4(5(4(2(x1))))) 117.81/30.95 3^1(2(5(4(x1)))) -> 3^1(4(3(5(2(x1))))) 117.81/30.95 3^1(2(5(4(x1)))) -> 3^1(5(2(x1))) 117.81/30.95 The remaining pairs can at least be oriented weakly. 117.81/30.95 Used ordering: Polynomial interpretation [POLO]: 117.81/30.95 117.81/30.95 POL(0(x_1)) = 0 117.81/30.95 POL(1(x_1)) = 0 117.81/30.95 POL(2(x_1)) = 1 117.81/30.95 POL(3(x_1)) = 0 117.81/30.95 POL(3^1(x_1)) = x_1 117.81/30.95 POL(4(x_1)) = 0 117.81/30.95 POL(5(x_1)) = 0 117.81/30.95 117.81/30.95 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 117.81/30.95 117.81/30.95 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 117.81/30.95 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 117.81/30.95 3(2(0(x1))) -> 0(2(1(3(x1)))) 117.81/30.95 3(2(0(x1))) -> 3(0(2(1(x1)))) 117.81/30.95 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 117.81/30.95 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 117.81/30.95 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 117.81/30.95 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 117.81/30.95 3(2(5(x1))) -> 1(3(5(2(x1)))) 117.81/30.95 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 117.81/30.95 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 117.81/30.95 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 117.81/30.95 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 117.81/30.95 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 117.81/30.95 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 117.81/30.95 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 117.81/30.95 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 117.81/30.95 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 117.81/30.95 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 117.81/30.95 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 117.81/30.95 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 117.81/30.95 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 117.81/30.95 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 117.81/30.95 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 117.81/30.95 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 117.81/30.95 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 117.81/30.95 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 117.81/30.95 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 117.81/30.95 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 117.81/30.95 117.81/30.95 117.81/30.95 ---------------------------------------- 117.81/30.95 117.81/30.95 (26) 117.81/30.95 Obligation: 117.81/30.95 Q DP problem: 117.81/30.95 The TRS P consists of the following rules: 117.81/30.95 117.81/30.95 3^1(0(2(4(x1)))) -> 3^1(0(4(2(1(1(x1)))))) 117.81/30.95 3^1(0(5(4(x1)))) -> 3^1(4(0(5(1(3(x1)))))) 117.81/30.95 3^1(5(3(2(x1)))) -> 3^1(3(0(2(5(x1))))) 117.81/30.95 3^1(5(3(2(x1)))) -> 3^1(0(2(5(x1)))) 117.81/30.95 3^1(5(1(4(0(x1))))) -> 3^1(3(4(5(0(x1))))) 117.81/30.95 3^1(5(1(4(0(x1))))) -> 3^1(4(5(0(x1)))) 117.81/30.95 117.81/30.95 The TRS R consists of the following rules: 117.81/30.95 117.81/30.95 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 117.81/30.95 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 117.81/30.95 0(3(2(x1))) -> 2(1(3(0(x1)))) 117.81/30.95 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 117.81/30.95 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 117.81/30.95 3(2(0(x1))) -> 0(2(1(3(x1)))) 117.81/30.95 3(2(0(x1))) -> 3(0(2(1(x1)))) 117.81/30.95 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 117.81/30.95 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 117.81/30.95 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 117.81/30.95 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 117.81/30.95 3(2(5(x1))) -> 1(3(5(2(x1)))) 117.81/30.95 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 117.81/30.95 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 117.81/30.95 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 117.81/30.95 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 117.81/30.95 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 117.81/30.95 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 117.81/30.95 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 117.81/30.95 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 117.81/30.95 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 117.81/30.95 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 117.81/30.95 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 117.81/30.95 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 117.81/30.95 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 117.81/30.95 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 117.81/30.95 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 117.81/30.95 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 117.81/30.95 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 117.81/30.95 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 117.81/30.95 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 117.81/30.95 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 117.81/30.95 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 117.81/30.95 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 117.81/30.95 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 117.81/30.95 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 117.81/30.95 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 117.81/30.95 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 117.81/30.95 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 117.81/30.95 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 117.81/30.95 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 117.81/30.95 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 117.81/30.95 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 117.81/30.95 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 117.81/30.95 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 117.81/30.95 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 117.81/30.95 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 117.81/30.95 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) 117.81/30.95 117.81/30.95 Q is empty. 117.81/30.95 We have to consider all minimal (P,Q,R)-chains. 117.81/30.95 ---------------------------------------- 117.81/30.95 117.81/30.95 (27) QDPOrderProof (EQUIVALENT) 117.81/30.95 We use the reduction pair processor [LPAR04,JAR06]. 117.81/30.95 117.81/30.95 117.81/30.95 The following pairs can be oriented strictly and are deleted. 117.81/30.95 117.81/30.95 3^1(5(3(2(x1)))) -> 3^1(3(0(2(5(x1))))) 117.81/30.95 3^1(5(3(2(x1)))) -> 3^1(0(2(5(x1)))) 117.81/30.95 3^1(5(1(4(0(x1))))) -> 3^1(3(4(5(0(x1))))) 117.81/30.95 3^1(5(1(4(0(x1))))) -> 3^1(4(5(0(x1)))) 117.81/30.95 The remaining pairs can at least be oriented weakly. 117.81/30.95 Used ordering: Polynomial interpretation [POLO]: 117.81/30.95 117.81/30.95 POL(0(x_1)) = 0 117.81/30.95 POL(1(x_1)) = 0 117.81/30.95 POL(2(x_1)) = 0 117.81/30.95 POL(3(x_1)) = 0 117.81/30.95 POL(3^1(x_1)) = x_1 117.81/30.95 POL(4(x_1)) = 0 117.81/30.95 POL(5(x_1)) = 1 117.81/30.95 117.81/30.95 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 117.81/30.95 117.81/30.95 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 117.81/30.95 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 117.81/30.95 3(2(0(x1))) -> 0(2(1(3(x1)))) 117.81/30.95 3(2(0(x1))) -> 3(0(2(1(x1)))) 117.81/30.95 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 117.81/30.95 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 117.81/30.95 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 117.81/30.95 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 117.81/30.95 3(2(5(x1))) -> 1(3(5(2(x1)))) 117.81/30.95 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 117.81/30.95 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 117.81/30.95 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 117.81/30.95 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 117.81/30.95 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 117.81/30.95 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 117.81/30.95 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 117.81/30.95 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 117.81/30.95 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 117.81/30.95 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 117.81/30.95 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 117.81/30.95 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 117.81/30.95 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 117.81/30.95 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 117.81/30.95 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 117.81/30.95 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 117.81/30.95 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 117.81/30.95 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 117.81/30.95 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 117.81/30.95 117.81/30.95 117.81/30.95 ---------------------------------------- 117.81/30.95 117.81/30.95 (28) 117.81/30.95 Obligation: 117.81/30.95 Q DP problem: 117.81/30.95 The TRS P consists of the following rules: 117.81/30.95 117.81/30.95 3^1(0(2(4(x1)))) -> 3^1(0(4(2(1(1(x1)))))) 117.81/30.95 3^1(0(5(4(x1)))) -> 3^1(4(0(5(1(3(x1)))))) 117.81/30.95 117.81/30.95 The TRS R consists of the following rules: 117.81/30.95 117.81/30.95 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 117.81/30.95 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 117.81/30.95 0(3(2(x1))) -> 2(1(3(0(x1)))) 117.81/30.95 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 117.81/30.95 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 117.81/30.95 3(2(0(x1))) -> 0(2(1(3(x1)))) 117.81/30.95 3(2(0(x1))) -> 3(0(2(1(x1)))) 117.81/30.95 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 117.81/30.95 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 117.81/30.95 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 117.81/30.95 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 117.81/30.95 3(2(5(x1))) -> 1(3(5(2(x1)))) 117.81/30.95 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 117.81/30.95 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 117.81/30.95 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 117.81/30.95 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 117.81/30.95 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 117.81/30.95 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 117.81/30.95 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 117.81/30.95 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 117.81/30.95 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 117.81/30.95 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 117.81/30.95 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 117.81/30.95 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 117.81/30.95 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 117.81/30.95 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 117.81/30.95 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 117.81/30.95 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 117.81/30.95 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 117.81/30.95 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 117.81/30.95 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 117.81/30.95 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 117.81/30.95 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 117.81/30.95 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 117.81/30.95 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 117.81/30.95 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 117.81/30.95 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 117.81/30.95 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 117.81/30.95 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 117.81/30.95 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 117.81/30.95 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 117.81/30.95 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 117.81/30.95 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 117.81/30.95 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 117.81/30.95 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 117.81/30.95 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 117.81/30.95 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 117.81/30.95 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) 117.81/30.95 117.81/30.95 Q is empty. 117.81/30.95 We have to consider all minimal (P,Q,R)-chains. 117.81/30.95 ---------------------------------------- 117.81/30.95 117.81/30.95 (29) QDPOrderProof (EQUIVALENT) 117.81/30.95 We use the reduction pair processor [LPAR04,JAR06]. 117.81/30.95 117.81/30.95 117.81/30.95 The following pairs can be oriented strictly and are deleted. 117.81/30.95 117.81/30.95 3^1(0(2(4(x1)))) -> 3^1(0(4(2(1(1(x1)))))) 117.81/30.95 3^1(0(5(4(x1)))) -> 3^1(4(0(5(1(3(x1)))))) 117.81/30.95 The remaining pairs can at least be oriented weakly. 117.81/30.95 Used ordering: Polynomial interpretation [POLO]: 117.81/30.95 117.81/30.95 POL(0(x_1)) = x_1 117.81/30.95 POL(1(x_1)) = 0 117.81/30.95 POL(2(x_1)) = 1 + x_1 117.81/30.95 POL(3(x_1)) = 0 117.81/30.95 POL(3^1(x_1)) = x_1 117.81/30.95 POL(4(x_1)) = 1 117.81/30.95 POL(5(x_1)) = 1 + x_1 117.81/30.95 117.81/30.95 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 117.81/30.95 117.81/30.95 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 117.81/30.95 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 117.81/30.95 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 117.81/30.95 117.81/30.95 117.81/30.95 ---------------------------------------- 117.81/30.95 117.81/30.95 (30) 117.81/30.95 Obligation: 117.81/30.95 Q DP problem: 117.81/30.95 P is empty. 117.81/30.95 The TRS R consists of the following rules: 117.81/30.95 117.81/30.95 0(0(0(x1))) -> 0(0(1(0(1(x1))))) 117.81/30.95 0(2(0(x1))) -> 0(0(2(1(1(x1))))) 117.81/30.95 0(3(2(x1))) -> 2(1(3(0(x1)))) 117.81/30.95 0(4(2(x1))) -> 4(1(1(1(0(2(x1)))))) 117.81/30.95 3(0(4(x1))) -> 0(1(3(4(1(x1))))) 117.81/30.95 3(2(0(x1))) -> 0(2(1(3(x1)))) 117.81/30.95 3(2(0(x1))) -> 3(0(2(1(x1)))) 117.81/30.95 3(2(0(x1))) -> 0(1(2(1(3(x1))))) 117.81/30.95 3(2(0(x1))) -> 3(0(1(2(1(4(x1)))))) 117.81/30.95 3(2(4(x1))) -> 1(2(1(3(3(4(x1)))))) 117.81/30.95 3(2(4(x1))) -> 4(1(2(1(1(3(x1)))))) 117.81/30.95 3(2(5(x1))) -> 1(3(5(2(x1)))) 117.81/30.95 3(2(5(x1))) -> 3(1(4(5(2(x1))))) 117.81/30.95 3(2(5(x1))) -> 1(1(3(3(5(2(x1)))))) 117.81/30.95 3(2(5(x1))) -> 1(3(4(5(4(2(x1)))))) 117.81/30.95 3(5(0(x1))) -> 3(1(0(5(1(1(x1)))))) 117.81/30.95 5(0(2(x1))) -> 1(1(0(2(5(x1))))) 117.81/30.95 5(0(4(x1))) -> 0(5(4(1(1(x1))))) 117.81/30.95 5(2(4(x1))) -> 5(2(1(1(3(4(x1)))))) 117.81/30.95 5(3(2(x1))) -> 1(1(3(5(1(2(x1)))))) 117.81/30.95 5(5(0(x1))) -> 5(1(1(5(0(x1))))) 117.81/30.95 0(0(2(4(x1)))) -> 0(4(0(2(1(1(x1)))))) 117.81/30.95 0(3(1(2(x1)))) -> 1(1(3(0(2(x1))))) 117.81/30.95 0(3(2(4(x1)))) -> 2(1(3(0(4(4(x1)))))) 117.81/30.95 0(4(3(2(x1)))) -> 2(3(4(0(1(x1))))) 117.81/30.95 3(0(2(4(x1)))) -> 3(0(4(2(1(1(x1)))))) 117.81/30.95 3(0(5(4(x1)))) -> 3(4(0(5(1(3(x1)))))) 117.81/30.95 3(1(0(0(x1)))) -> 0(1(1(3(0(1(x1)))))) 117.81/30.95 3(2(4(0(x1)))) -> 3(1(0(4(2(x1))))) 117.81/30.95 3(2(4(5(x1)))) -> 3(1(4(2(1(5(x1)))))) 117.81/30.95 3(2(5(4(x1)))) -> 3(4(3(5(2(x1))))) 117.81/30.95 3(5(3(2(x1)))) -> 3(3(0(2(5(x1))))) 117.81/30.95 4(0(4(2(x1)))) -> 4(4(2(1(3(0(x1)))))) 117.81/30.95 4(3(2(0(x1)))) -> 0(2(1(3(4(x1))))) 117.81/30.95 5(4(1(0(x1)))) -> 5(1(1(4(0(1(x1)))))) 117.81/30.95 5(5(0(0(x1)))) -> 1(1(0(5(5(0(x1)))))) 117.81/30.95 5(5(2(2(x1)))) -> 5(5(1(1(2(2(x1)))))) 117.81/30.95 0(0(1(2(4(x1))))) -> 0(0(2(1(1(4(x1)))))) 117.81/30.95 0(5(3(1(2(x1))))) -> 5(2(1(3(1(0(x1)))))) 117.81/30.95 0(5(5(1(0(x1))))) -> 5(0(0(5(1(1(x1)))))) 117.81/30.95 3(0(1(4(5(x1))))) -> 3(4(0(5(1(1(x1)))))) 117.81/30.95 3(0(3(1(2(x1))))) -> 0(2(1(1(3(3(x1)))))) 117.81/30.95 3(2(2(0(4(x1))))) -> 3(2(2(0(1(4(x1)))))) 117.81/30.95 3(5(1(4(0(x1))))) -> 1(3(3(4(5(0(x1)))))) 117.81/30.95 3(5(3(0(4(x1))))) -> 3(0(1(5(3(4(x1)))))) 117.81/30.95 5(1(0(2(4(x1))))) -> 2(1(5(4(0(3(x1)))))) 117.81/30.95 5(1(3(2(0(x1))))) -> 2(3(5(0(1(1(x1)))))) 117.81/30.95 5(4(2(0(2(x1))))) -> 5(0(4(2(1(2(x1)))))) 117.81/30.95 117.81/30.95 Q is empty. 117.81/30.95 We have to consider all minimal (P,Q,R)-chains. 117.81/30.95 ---------------------------------------- 117.81/30.95 117.81/30.95 (31) PisEmptyProof (EQUIVALENT) 117.81/30.95 The TRS P is empty. Hence, there is no (P,Q,R) chain. 117.81/30.95 ---------------------------------------- 117.81/30.95 117.81/30.95 (32) 117.81/30.95 YES 118.11/31.04 EOF