51.09/13.89 YES 51.32/13.95 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 51.32/13.95 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 51.32/13.95 51.32/13.95 51.32/13.95 Termination w.r.t. Q of the given QTRS could be proven: 51.32/13.95 51.32/13.95 (0) QTRS 51.32/13.95 (1) DependencyPairsProof [EQUIVALENT, 374 ms] 51.32/13.95 (2) QDP 51.32/13.95 (3) DependencyGraphProof [EQUIVALENT, 0 ms] 51.32/13.95 (4) QDP 51.32/13.95 (5) QDPOrderProof [EQUIVALENT, 260 ms] 51.32/13.95 (6) QDP 51.32/13.95 (7) DependencyGraphProof [EQUIVALENT, 0 ms] 51.32/13.95 (8) AND 51.32/13.95 (9) QDP 51.32/13.95 (10) QDPOrderProof [EQUIVALENT, 76 ms] 51.32/13.95 (11) QDP 51.32/13.95 (12) PisEmptyProof [EQUIVALENT, 0 ms] 51.32/13.95 (13) YES 51.32/13.95 (14) QDP 51.32/13.95 (15) QDPOrderProof [EQUIVALENT, 72 ms] 51.32/13.95 (16) QDP 51.32/13.95 (17) PisEmptyProof [EQUIVALENT, 0 ms] 51.32/13.95 (18) YES 51.32/13.95 (19) QDP 51.32/13.95 (20) QDPOrderProof [EQUIVALENT, 140 ms] 51.32/13.95 (21) QDP 51.32/13.95 (22) DependencyGraphProof [EQUIVALENT, 0 ms] 51.32/13.95 (23) TRUE 51.32/13.95 51.32/13.95 51.32/13.95 ---------------------------------------- 51.32/13.95 51.32/13.95 (0) 51.32/13.95 Obligation: 51.32/13.95 Q restricted rewrite system: 51.32/13.95 The TRS R consists of the following rules: 51.32/13.95 51.32/13.95 0(1(2(x1))) -> 2(1(0(3(x1)))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(0(x1)))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(0(4(x1))))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(4(0(4(x1)))))) 51.32/13.95 0(0(1(2(x1)))) -> 0(2(1(0(3(x1))))) 51.32/13.95 0(1(1(2(x1)))) -> 1(0(4(1(2(x1))))) 51.32/13.95 0(1(1(2(x1)))) -> 1(2(1(0(3(x1))))) 51.32/13.95 0(1(2(0(x1)))) -> 2(1(0(3(0(3(x1)))))) 51.32/13.95 0(1(2(5(x1)))) -> 2(1(0(3(5(x1))))) 51.32/13.95 0(1(2(5(x1)))) -> 5(1(0(3(2(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 2(1(3(4(0(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 2(4(1(0(3(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 0(2(1(0(3(4(x1)))))) 51.32/13.95 2(1(1(5(x1)))) -> 2(5(1(4(1(x1))))) 51.32/13.95 2(1(1(5(x1)))) -> 5(1(3(2(1(x1))))) 51.32/13.95 2(1(2(0(x1)))) -> 2(2(1(0(3(x1))))) 51.32/13.95 2(3(5(5(x1)))) -> 5(1(0(3(2(5(x1)))))) 51.32/13.95 2(5(3(2(x1)))) -> 5(1(3(2(2(x1))))) 51.32/13.95 5(1(2(0(x1)))) -> 2(5(1(0(3(x1))))) 51.32/13.95 0(1(0(1(2(x1))))) -> 0(2(1(1(3(0(x1)))))) 51.32/13.95 0(1(1(2(5(x1))))) -> 0(1(3(5(1(2(x1)))))) 51.32/13.95 0(1(1(4(2(x1))))) -> 4(1(0(3(2(1(x1)))))) 51.32/13.95 0(1(1(5(0(x1))))) -> 0(1(0(5(4(1(x1)))))) 51.32/13.95 0(1(1(5(4(x1))))) -> 1(4(1(0(3(5(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 0(1(3(4(5(2(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 0(2(5(1(0(3(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 1(0(3(0(2(5(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 5(1(3(0(2(0(x1)))))) 51.32/13.95 0(1(4(0(2(x1))))) -> 4(0(1(3(0(2(x1)))))) 51.32/13.95 0(1(4(0(2(x1))))) -> 4(0(2(1(3(0(x1)))))) 51.32/13.95 0(4(5(3(5(x1))))) -> 5(1(0(3(4(5(x1)))))) 51.32/13.95 2(1(0(1(5(x1))))) -> 1(0(3(2(5(1(x1)))))) 51.32/13.95 2(1(4(3(5(x1))))) -> 1(3(5(4(4(2(x1)))))) 51.32/13.95 2(1(4(3(5(x1))))) -> 5(4(4(1(3(2(x1)))))) 51.32/13.95 2(1(4(5(0(x1))))) -> 2(4(1(0(3(5(x1)))))) 51.32/13.95 2(2(3(5(0(x1))))) -> 5(2(4(2(0(3(x1)))))) 51.32/13.95 2(2(4(3(5(x1))))) -> 5(1(3(4(2(2(x1)))))) 51.32/13.95 2(3(1(1(2(x1))))) -> 1(3(2(1(2(0(x1)))))) 51.32/13.95 2(3(1(1(2(x1))))) -> 4(1(2(2(1(3(x1)))))) 51.32/13.95 2(3(2(0(5(x1))))) -> 2(2(1(0(3(5(x1)))))) 51.32/13.95 2(3(3(1(5(x1))))) -> 1(3(5(1(3(2(x1)))))) 51.32/13.95 2(5(0(3(5(x1))))) -> 5(2(1(3(0(5(x1)))))) 51.32/13.95 4(2(0(1(2(x1))))) -> 4(2(2(1(3(0(x1)))))) 51.32/13.95 5(0(1(2(2(x1))))) -> 4(1(5(2(0(2(x1)))))) 51.32/13.95 5(1(4(2(2(x1))))) -> 5(1(3(2(4(2(x1)))))) 51.32/13.95 5(1(4(3(2(x1))))) -> 4(5(1(3(4(2(x1)))))) 51.32/13.95 5(5(4(3(2(x1))))) -> 5(1(3(4(5(2(x1)))))) 51.32/13.95 51.32/13.95 Q is empty. 51.32/13.95 51.32/13.95 ---------------------------------------- 51.32/13.95 51.32/13.95 (1) DependencyPairsProof (EQUIVALENT) 51.32/13.95 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 51.32/13.95 ---------------------------------------- 51.32/13.95 51.32/13.95 (2) 51.32/13.95 Obligation: 51.32/13.95 Q DP problem: 51.32/13.95 The TRS P consists of the following rules: 51.32/13.95 51.32/13.95 0^1(1(2(x1))) -> 2^1(1(0(3(x1)))) 51.32/13.95 0^1(1(2(x1))) -> 0^1(3(x1)) 51.32/13.95 0^1(1(2(x1))) -> 2^1(1(3(0(x1)))) 51.32/13.95 0^1(1(2(x1))) -> 0^1(x1) 51.32/13.95 0^1(1(2(x1))) -> 2^1(1(3(0(4(x1))))) 51.32/13.95 0^1(1(2(x1))) -> 0^1(4(x1)) 51.32/13.95 0^1(1(2(x1))) -> 4^1(x1) 51.32/13.95 0^1(1(2(x1))) -> 2^1(1(3(4(0(x1))))) 51.32/13.95 0^1(1(2(x1))) -> 4^1(0(x1)) 51.32/13.95 0^1(1(2(x1))) -> 2^1(1(3(4(0(4(x1)))))) 51.32/13.95 0^1(1(2(x1))) -> 4^1(0(4(x1))) 51.32/13.95 0^1(0(1(2(x1)))) -> 0^1(2(1(0(3(x1))))) 51.32/13.95 0^1(0(1(2(x1)))) -> 2^1(1(0(3(x1)))) 51.32/13.95 0^1(0(1(2(x1)))) -> 0^1(3(x1)) 51.32/13.95 0^1(1(1(2(x1)))) -> 0^1(4(1(2(x1)))) 51.32/13.95 0^1(1(1(2(x1)))) -> 4^1(1(2(x1))) 51.32/13.95 0^1(1(1(2(x1)))) -> 2^1(1(0(3(x1)))) 51.32/13.95 0^1(1(1(2(x1)))) -> 0^1(3(x1)) 51.32/13.95 0^1(1(2(0(x1)))) -> 2^1(1(0(3(0(3(x1)))))) 51.32/13.95 0^1(1(2(0(x1)))) -> 0^1(3(0(3(x1)))) 51.32/13.95 0^1(1(2(0(x1)))) -> 0^1(3(x1)) 51.32/13.95 0^1(1(2(5(x1)))) -> 2^1(1(0(3(5(x1))))) 51.32/13.95 0^1(1(2(5(x1)))) -> 0^1(3(5(x1))) 51.32/13.95 0^1(1(2(5(x1)))) -> 5^1(1(0(3(2(x1))))) 51.32/13.95 0^1(1(2(5(x1)))) -> 0^1(3(2(x1))) 51.32/13.95 0^1(1(2(5(x1)))) -> 2^1(x1) 51.32/13.95 0^1(1(4(2(x1)))) -> 2^1(1(3(4(0(x1))))) 51.32/13.95 0^1(1(4(2(x1)))) -> 4^1(0(x1)) 51.32/13.95 0^1(1(4(2(x1)))) -> 0^1(x1) 51.32/13.95 0^1(1(4(2(x1)))) -> 2^1(4(1(0(3(x1))))) 51.32/13.95 0^1(1(4(2(x1)))) -> 4^1(1(0(3(x1)))) 51.32/13.95 0^1(1(4(2(x1)))) -> 0^1(3(x1)) 51.32/13.95 0^1(1(4(2(x1)))) -> 0^1(2(1(0(3(4(x1)))))) 51.32/13.95 0^1(1(4(2(x1)))) -> 2^1(1(0(3(4(x1))))) 51.32/13.95 0^1(1(4(2(x1)))) -> 0^1(3(4(x1))) 51.32/13.95 0^1(1(4(2(x1)))) -> 4^1(x1) 51.32/13.95 2^1(1(1(5(x1)))) -> 2^1(5(1(4(1(x1))))) 51.32/13.95 2^1(1(1(5(x1)))) -> 5^1(1(4(1(x1)))) 51.32/13.95 2^1(1(1(5(x1)))) -> 4^1(1(x1)) 51.32/13.95 2^1(1(1(5(x1)))) -> 5^1(1(3(2(1(x1))))) 51.32/13.95 2^1(1(1(5(x1)))) -> 2^1(1(x1)) 51.32/13.95 2^1(1(2(0(x1)))) -> 2^1(2(1(0(3(x1))))) 51.32/13.95 2^1(1(2(0(x1)))) -> 2^1(1(0(3(x1)))) 51.32/13.95 2^1(1(2(0(x1)))) -> 0^1(3(x1)) 51.32/13.95 2^1(3(5(5(x1)))) -> 5^1(1(0(3(2(5(x1)))))) 51.32/13.95 2^1(3(5(5(x1)))) -> 0^1(3(2(5(x1)))) 51.32/13.95 2^1(3(5(5(x1)))) -> 2^1(5(x1)) 51.32/13.95 2^1(5(3(2(x1)))) -> 5^1(1(3(2(2(x1))))) 51.32/13.95 2^1(5(3(2(x1)))) -> 2^1(2(x1)) 51.32/13.95 5^1(1(2(0(x1)))) -> 2^1(5(1(0(3(x1))))) 51.32/13.95 5^1(1(2(0(x1)))) -> 5^1(1(0(3(x1)))) 51.32/13.95 5^1(1(2(0(x1)))) -> 0^1(3(x1)) 51.32/13.95 0^1(1(0(1(2(x1))))) -> 0^1(2(1(1(3(0(x1)))))) 51.32/13.95 0^1(1(0(1(2(x1))))) -> 2^1(1(1(3(0(x1))))) 51.32/13.95 0^1(1(0(1(2(x1))))) -> 0^1(x1) 51.32/13.95 0^1(1(1(2(5(x1))))) -> 0^1(1(3(5(1(2(x1)))))) 51.32/13.95 0^1(1(1(2(5(x1))))) -> 5^1(1(2(x1))) 51.32/13.95 0^1(1(1(2(5(x1))))) -> 2^1(x1) 51.32/13.95 0^1(1(1(4(2(x1))))) -> 4^1(1(0(3(2(1(x1)))))) 51.32/13.95 0^1(1(1(4(2(x1))))) -> 0^1(3(2(1(x1)))) 51.32/13.95 0^1(1(1(4(2(x1))))) -> 2^1(1(x1)) 51.32/13.95 0^1(1(1(5(0(x1))))) -> 0^1(1(0(5(4(1(x1)))))) 51.32/13.95 0^1(1(1(5(0(x1))))) -> 0^1(5(4(1(x1)))) 51.32/13.95 0^1(1(1(5(0(x1))))) -> 5^1(4(1(x1))) 51.32/13.95 0^1(1(1(5(0(x1))))) -> 4^1(1(x1)) 51.32/13.95 0^1(1(1(5(4(x1))))) -> 4^1(1(0(3(5(x1))))) 51.32/13.95 0^1(1(1(5(4(x1))))) -> 0^1(3(5(x1))) 51.32/13.95 0^1(1(1(5(4(x1))))) -> 5^1(x1) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 0^1(1(3(4(5(2(x1)))))) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 4^1(5(2(x1))) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 5^1(2(x1)) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 2^1(x1) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 0^1(2(5(1(0(3(x1)))))) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 2^1(5(1(0(3(x1))))) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 5^1(1(0(3(x1)))) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 0^1(3(x1)) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 0^1(3(0(2(5(x1))))) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 0^1(2(5(x1))) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 2^1(5(x1)) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 5^1(1(3(0(2(0(x1)))))) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 0^1(2(0(x1))) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 2^1(0(x1)) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 0^1(x1) 51.32/13.95 0^1(1(4(0(2(x1))))) -> 4^1(0(1(3(0(2(x1)))))) 51.32/13.95 0^1(1(4(0(2(x1))))) -> 0^1(1(3(0(2(x1))))) 51.32/13.95 0^1(1(4(0(2(x1))))) -> 4^1(0(2(1(3(0(x1)))))) 51.32/13.95 0^1(1(4(0(2(x1))))) -> 0^1(2(1(3(0(x1))))) 51.32/13.95 0^1(1(4(0(2(x1))))) -> 2^1(1(3(0(x1)))) 51.32/13.95 0^1(1(4(0(2(x1))))) -> 0^1(x1) 51.32/13.95 0^1(4(5(3(5(x1))))) -> 5^1(1(0(3(4(5(x1)))))) 51.32/13.95 0^1(4(5(3(5(x1))))) -> 0^1(3(4(5(x1)))) 51.32/13.95 0^1(4(5(3(5(x1))))) -> 4^1(5(x1)) 51.32/13.95 2^1(1(0(1(5(x1))))) -> 0^1(3(2(5(1(x1))))) 51.32/13.95 2^1(1(0(1(5(x1))))) -> 2^1(5(1(x1))) 51.32/13.95 2^1(1(0(1(5(x1))))) -> 5^1(1(x1)) 51.32/13.95 2^1(1(4(3(5(x1))))) -> 5^1(4(4(2(x1)))) 51.32/13.95 2^1(1(4(3(5(x1))))) -> 4^1(4(2(x1))) 51.32/13.95 2^1(1(4(3(5(x1))))) -> 4^1(2(x1)) 51.32/13.95 2^1(1(4(3(5(x1))))) -> 2^1(x1) 51.32/13.95 2^1(1(4(3(5(x1))))) -> 5^1(4(4(1(3(2(x1)))))) 51.32/13.95 2^1(1(4(3(5(x1))))) -> 4^1(4(1(3(2(x1))))) 51.32/13.95 2^1(1(4(3(5(x1))))) -> 4^1(1(3(2(x1)))) 51.32/13.95 2^1(1(4(5(0(x1))))) -> 2^1(4(1(0(3(5(x1)))))) 51.32/13.95 2^1(1(4(5(0(x1))))) -> 4^1(1(0(3(5(x1))))) 51.32/13.95 2^1(1(4(5(0(x1))))) -> 0^1(3(5(x1))) 51.32/13.95 2^1(1(4(5(0(x1))))) -> 5^1(x1) 51.32/13.95 2^1(2(3(5(0(x1))))) -> 5^1(2(4(2(0(3(x1)))))) 51.32/13.95 2^1(2(3(5(0(x1))))) -> 2^1(4(2(0(3(x1))))) 51.32/13.95 2^1(2(3(5(0(x1))))) -> 4^1(2(0(3(x1)))) 51.32/13.95 2^1(2(3(5(0(x1))))) -> 2^1(0(3(x1))) 51.32/13.95 2^1(2(3(5(0(x1))))) -> 0^1(3(x1)) 51.32/13.95 2^1(2(4(3(5(x1))))) -> 5^1(1(3(4(2(2(x1)))))) 51.32/13.95 2^1(2(4(3(5(x1))))) -> 4^1(2(2(x1))) 51.32/13.95 2^1(2(4(3(5(x1))))) -> 2^1(2(x1)) 51.32/13.95 2^1(2(4(3(5(x1))))) -> 2^1(x1) 51.32/13.95 2^1(3(1(1(2(x1))))) -> 2^1(1(2(0(x1)))) 51.32/13.95 2^1(3(1(1(2(x1))))) -> 2^1(0(x1)) 51.32/13.95 2^1(3(1(1(2(x1))))) -> 0^1(x1) 51.32/13.95 2^1(3(1(1(2(x1))))) -> 4^1(1(2(2(1(3(x1)))))) 51.32/13.95 2^1(3(1(1(2(x1))))) -> 2^1(2(1(3(x1)))) 51.32/13.95 2^1(3(1(1(2(x1))))) -> 2^1(1(3(x1))) 51.32/13.95 2^1(3(2(0(5(x1))))) -> 2^1(2(1(0(3(5(x1)))))) 51.32/13.95 2^1(3(2(0(5(x1))))) -> 2^1(1(0(3(5(x1))))) 51.32/13.95 2^1(3(2(0(5(x1))))) -> 0^1(3(5(x1))) 51.32/13.95 2^1(3(3(1(5(x1))))) -> 5^1(1(3(2(x1)))) 51.32/13.95 2^1(3(3(1(5(x1))))) -> 2^1(x1) 51.32/13.95 2^1(5(0(3(5(x1))))) -> 5^1(2(1(3(0(5(x1)))))) 51.32/13.95 2^1(5(0(3(5(x1))))) -> 2^1(1(3(0(5(x1))))) 51.32/13.95 2^1(5(0(3(5(x1))))) -> 0^1(5(x1)) 51.32/13.95 4^1(2(0(1(2(x1))))) -> 4^1(2(2(1(3(0(x1)))))) 51.32/13.95 4^1(2(0(1(2(x1))))) -> 2^1(2(1(3(0(x1))))) 51.32/13.95 4^1(2(0(1(2(x1))))) -> 2^1(1(3(0(x1)))) 51.32/13.95 4^1(2(0(1(2(x1))))) -> 0^1(x1) 51.32/13.95 5^1(0(1(2(2(x1))))) -> 4^1(1(5(2(0(2(x1)))))) 51.32/13.95 5^1(0(1(2(2(x1))))) -> 5^1(2(0(2(x1)))) 51.32/13.95 5^1(0(1(2(2(x1))))) -> 2^1(0(2(x1))) 51.32/13.95 5^1(0(1(2(2(x1))))) -> 0^1(2(x1)) 51.32/13.95 5^1(1(4(2(2(x1))))) -> 5^1(1(3(2(4(2(x1)))))) 51.32/13.95 5^1(1(4(2(2(x1))))) -> 2^1(4(2(x1))) 51.32/13.95 5^1(1(4(2(2(x1))))) -> 4^1(2(x1)) 51.32/13.95 5^1(1(4(3(2(x1))))) -> 4^1(5(1(3(4(2(x1)))))) 51.32/13.95 5^1(1(4(3(2(x1))))) -> 5^1(1(3(4(2(x1))))) 51.32/13.95 5^1(1(4(3(2(x1))))) -> 4^1(2(x1)) 51.32/13.95 5^1(5(4(3(2(x1))))) -> 5^1(1(3(4(5(2(x1)))))) 51.32/13.95 5^1(5(4(3(2(x1))))) -> 4^1(5(2(x1))) 51.32/13.95 5^1(5(4(3(2(x1))))) -> 5^1(2(x1)) 51.32/13.95 51.32/13.95 The TRS R consists of the following rules: 51.32/13.95 51.32/13.95 0(1(2(x1))) -> 2(1(0(3(x1)))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(0(x1)))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(0(4(x1))))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(4(0(4(x1)))))) 51.32/13.95 0(0(1(2(x1)))) -> 0(2(1(0(3(x1))))) 51.32/13.95 0(1(1(2(x1)))) -> 1(0(4(1(2(x1))))) 51.32/13.95 0(1(1(2(x1)))) -> 1(2(1(0(3(x1))))) 51.32/13.95 0(1(2(0(x1)))) -> 2(1(0(3(0(3(x1)))))) 51.32/13.95 0(1(2(5(x1)))) -> 2(1(0(3(5(x1))))) 51.32/13.95 0(1(2(5(x1)))) -> 5(1(0(3(2(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 2(1(3(4(0(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 2(4(1(0(3(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 0(2(1(0(3(4(x1)))))) 51.32/13.95 2(1(1(5(x1)))) -> 2(5(1(4(1(x1))))) 51.32/13.95 2(1(1(5(x1)))) -> 5(1(3(2(1(x1))))) 51.32/13.95 2(1(2(0(x1)))) -> 2(2(1(0(3(x1))))) 51.32/13.95 2(3(5(5(x1)))) -> 5(1(0(3(2(5(x1)))))) 51.32/13.95 2(5(3(2(x1)))) -> 5(1(3(2(2(x1))))) 51.32/13.95 5(1(2(0(x1)))) -> 2(5(1(0(3(x1))))) 51.32/13.95 0(1(0(1(2(x1))))) -> 0(2(1(1(3(0(x1)))))) 51.32/13.95 0(1(1(2(5(x1))))) -> 0(1(3(5(1(2(x1)))))) 51.32/13.95 0(1(1(4(2(x1))))) -> 4(1(0(3(2(1(x1)))))) 51.32/13.95 0(1(1(5(0(x1))))) -> 0(1(0(5(4(1(x1)))))) 51.32/13.95 0(1(1(5(4(x1))))) -> 1(4(1(0(3(5(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 0(1(3(4(5(2(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 0(2(5(1(0(3(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 1(0(3(0(2(5(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 5(1(3(0(2(0(x1)))))) 51.32/13.95 0(1(4(0(2(x1))))) -> 4(0(1(3(0(2(x1)))))) 51.32/13.95 0(1(4(0(2(x1))))) -> 4(0(2(1(3(0(x1)))))) 51.32/13.95 0(4(5(3(5(x1))))) -> 5(1(0(3(4(5(x1)))))) 51.32/13.95 2(1(0(1(5(x1))))) -> 1(0(3(2(5(1(x1)))))) 51.32/13.95 2(1(4(3(5(x1))))) -> 1(3(5(4(4(2(x1)))))) 51.32/13.95 2(1(4(3(5(x1))))) -> 5(4(4(1(3(2(x1)))))) 51.32/13.95 2(1(4(5(0(x1))))) -> 2(4(1(0(3(5(x1)))))) 51.32/13.95 2(2(3(5(0(x1))))) -> 5(2(4(2(0(3(x1)))))) 51.32/13.95 2(2(4(3(5(x1))))) -> 5(1(3(4(2(2(x1)))))) 51.32/13.95 2(3(1(1(2(x1))))) -> 1(3(2(1(2(0(x1)))))) 51.32/13.95 2(3(1(1(2(x1))))) -> 4(1(2(2(1(3(x1)))))) 51.32/13.95 2(3(2(0(5(x1))))) -> 2(2(1(0(3(5(x1)))))) 51.32/13.95 2(3(3(1(5(x1))))) -> 1(3(5(1(3(2(x1)))))) 51.32/13.95 2(5(0(3(5(x1))))) -> 5(2(1(3(0(5(x1)))))) 51.32/13.95 4(2(0(1(2(x1))))) -> 4(2(2(1(3(0(x1)))))) 51.32/13.95 5(0(1(2(2(x1))))) -> 4(1(5(2(0(2(x1)))))) 51.32/13.95 5(1(4(2(2(x1))))) -> 5(1(3(2(4(2(x1)))))) 51.32/13.95 5(1(4(3(2(x1))))) -> 4(5(1(3(4(2(x1)))))) 51.32/13.95 5(5(4(3(2(x1))))) -> 5(1(3(4(5(2(x1)))))) 51.32/13.95 51.32/13.95 Q is empty. 51.32/13.95 We have to consider all minimal (P,Q,R)-chains. 51.32/13.95 ---------------------------------------- 51.32/13.95 51.32/13.95 (3) DependencyGraphProof (EQUIVALENT) 51.32/13.95 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 95 less nodes. 51.32/13.95 ---------------------------------------- 51.32/13.95 51.32/13.95 (4) 51.32/13.95 Obligation: 51.32/13.95 Q DP problem: 51.32/13.95 The TRS P consists of the following rules: 51.32/13.95 51.32/13.95 0^1(1(2(x1))) -> 0^1(4(x1)) 51.32/13.95 0^1(1(2(x1))) -> 0^1(x1) 51.32/13.95 0^1(1(2(x1))) -> 4^1(x1) 51.32/13.95 4^1(2(0(1(2(x1))))) -> 0^1(x1) 51.32/13.95 0^1(1(2(x1))) -> 4^1(0(x1)) 51.32/13.95 0^1(1(2(x1))) -> 4^1(0(4(x1))) 51.32/13.95 0^1(1(2(5(x1)))) -> 2^1(x1) 51.32/13.95 2^1(1(1(5(x1)))) -> 2^1(1(x1)) 51.32/13.95 2^1(1(0(1(5(x1))))) -> 2^1(5(1(x1))) 51.32/13.95 2^1(1(0(1(5(x1))))) -> 5^1(1(x1)) 51.32/13.95 5^1(1(4(2(2(x1))))) -> 2^1(4(2(x1))) 51.32/13.95 2^1(1(4(5(0(x1))))) -> 5^1(x1) 51.32/13.95 5^1(0(1(2(2(x1))))) -> 5^1(2(0(2(x1)))) 51.32/13.95 5^1(0(1(2(2(x1))))) -> 2^1(0(2(x1))) 51.32/13.95 5^1(0(1(2(2(x1))))) -> 0^1(2(x1)) 51.32/13.95 0^1(1(4(2(x1)))) -> 4^1(0(x1)) 51.32/13.95 0^1(1(4(2(x1)))) -> 0^1(x1) 51.32/13.95 0^1(1(4(2(x1)))) -> 4^1(x1) 51.32/13.95 0^1(1(0(1(2(x1))))) -> 0^1(x1) 51.32/13.95 0^1(1(1(2(5(x1))))) -> 5^1(1(2(x1))) 51.32/13.95 5^1(1(4(2(2(x1))))) -> 4^1(2(x1)) 51.32/13.95 0^1(1(1(2(5(x1))))) -> 2^1(x1) 51.32/13.95 2^1(3(5(5(x1)))) -> 2^1(5(x1)) 51.32/13.95 2^1(5(3(2(x1)))) -> 2^1(2(x1)) 51.32/13.95 2^1(2(4(3(5(x1))))) -> 4^1(2(2(x1))) 51.32/13.95 2^1(2(4(3(5(x1))))) -> 2^1(2(x1)) 51.32/13.95 2^1(2(4(3(5(x1))))) -> 2^1(x1) 51.32/13.95 2^1(1(4(3(5(x1))))) -> 5^1(4(4(2(x1)))) 51.32/13.95 2^1(1(4(3(5(x1))))) -> 4^1(4(2(x1))) 51.32/13.95 2^1(1(4(3(5(x1))))) -> 4^1(2(x1)) 51.32/13.95 2^1(1(4(3(5(x1))))) -> 2^1(x1) 51.32/13.95 2^1(3(1(1(2(x1))))) -> 2^1(1(2(0(x1)))) 51.32/13.95 2^1(3(1(1(2(x1))))) -> 2^1(0(x1)) 51.32/13.95 2^1(3(1(1(2(x1))))) -> 0^1(x1) 51.32/13.95 0^1(1(1(4(2(x1))))) -> 2^1(1(x1)) 51.32/13.95 0^1(1(1(5(4(x1))))) -> 5^1(x1) 51.32/13.95 5^1(1(4(3(2(x1))))) -> 4^1(2(x1)) 51.32/13.95 5^1(5(4(3(2(x1))))) -> 4^1(5(2(x1))) 51.32/13.95 5^1(5(4(3(2(x1))))) -> 5^1(2(x1)) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 4^1(5(2(x1))) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 5^1(2(x1)) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 2^1(x1) 51.32/13.95 2^1(3(3(1(5(x1))))) -> 2^1(x1) 51.32/13.95 2^1(5(0(3(5(x1))))) -> 0^1(5(x1)) 51.32/13.95 0^1(1(4(0(2(x1))))) -> 0^1(x1) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 0^1(2(5(x1))) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 2^1(5(x1)) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 0^1(2(0(x1))) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 2^1(0(x1)) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 0^1(x1) 51.32/13.95 0^1(4(5(3(5(x1))))) -> 4^1(5(x1)) 51.32/13.95 51.32/13.95 The TRS R consists of the following rules: 51.32/13.95 51.32/13.95 0(1(2(x1))) -> 2(1(0(3(x1)))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(0(x1)))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(0(4(x1))))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(4(0(4(x1)))))) 51.32/13.95 0(0(1(2(x1)))) -> 0(2(1(0(3(x1))))) 51.32/13.95 0(1(1(2(x1)))) -> 1(0(4(1(2(x1))))) 51.32/13.95 0(1(1(2(x1)))) -> 1(2(1(0(3(x1))))) 51.32/13.95 0(1(2(0(x1)))) -> 2(1(0(3(0(3(x1)))))) 51.32/13.95 0(1(2(5(x1)))) -> 2(1(0(3(5(x1))))) 51.32/13.95 0(1(2(5(x1)))) -> 5(1(0(3(2(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 2(1(3(4(0(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 2(4(1(0(3(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 0(2(1(0(3(4(x1)))))) 51.32/13.95 2(1(1(5(x1)))) -> 2(5(1(4(1(x1))))) 51.32/13.95 2(1(1(5(x1)))) -> 5(1(3(2(1(x1))))) 51.32/13.95 2(1(2(0(x1)))) -> 2(2(1(0(3(x1))))) 51.32/13.95 2(3(5(5(x1)))) -> 5(1(0(3(2(5(x1)))))) 51.32/13.95 2(5(3(2(x1)))) -> 5(1(3(2(2(x1))))) 51.32/13.95 5(1(2(0(x1)))) -> 2(5(1(0(3(x1))))) 51.32/13.95 0(1(0(1(2(x1))))) -> 0(2(1(1(3(0(x1)))))) 51.32/13.95 0(1(1(2(5(x1))))) -> 0(1(3(5(1(2(x1)))))) 51.32/13.95 0(1(1(4(2(x1))))) -> 4(1(0(3(2(1(x1)))))) 51.32/13.95 0(1(1(5(0(x1))))) -> 0(1(0(5(4(1(x1)))))) 51.32/13.95 0(1(1(5(4(x1))))) -> 1(4(1(0(3(5(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 0(1(3(4(5(2(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 0(2(5(1(0(3(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 1(0(3(0(2(5(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 5(1(3(0(2(0(x1)))))) 51.32/13.95 0(1(4(0(2(x1))))) -> 4(0(1(3(0(2(x1)))))) 51.32/13.95 0(1(4(0(2(x1))))) -> 4(0(2(1(3(0(x1)))))) 51.32/13.95 0(4(5(3(5(x1))))) -> 5(1(0(3(4(5(x1)))))) 51.32/13.95 2(1(0(1(5(x1))))) -> 1(0(3(2(5(1(x1)))))) 51.32/13.95 2(1(4(3(5(x1))))) -> 1(3(5(4(4(2(x1)))))) 51.32/13.95 2(1(4(3(5(x1))))) -> 5(4(4(1(3(2(x1)))))) 51.32/13.95 2(1(4(5(0(x1))))) -> 2(4(1(0(3(5(x1)))))) 51.32/13.95 2(2(3(5(0(x1))))) -> 5(2(4(2(0(3(x1)))))) 51.32/13.95 2(2(4(3(5(x1))))) -> 5(1(3(4(2(2(x1)))))) 51.32/13.95 2(3(1(1(2(x1))))) -> 1(3(2(1(2(0(x1)))))) 51.32/13.95 2(3(1(1(2(x1))))) -> 4(1(2(2(1(3(x1)))))) 51.32/13.95 2(3(2(0(5(x1))))) -> 2(2(1(0(3(5(x1)))))) 51.32/13.95 2(3(3(1(5(x1))))) -> 1(3(5(1(3(2(x1)))))) 51.32/13.95 2(5(0(3(5(x1))))) -> 5(2(1(3(0(5(x1)))))) 51.32/13.95 4(2(0(1(2(x1))))) -> 4(2(2(1(3(0(x1)))))) 51.32/13.95 5(0(1(2(2(x1))))) -> 4(1(5(2(0(2(x1)))))) 51.32/13.95 5(1(4(2(2(x1))))) -> 5(1(3(2(4(2(x1)))))) 51.32/13.95 5(1(4(3(2(x1))))) -> 4(5(1(3(4(2(x1)))))) 51.32/13.95 5(5(4(3(2(x1))))) -> 5(1(3(4(5(2(x1)))))) 51.32/13.95 51.32/13.95 Q is empty. 51.32/13.95 We have to consider all minimal (P,Q,R)-chains. 51.32/13.95 ---------------------------------------- 51.32/13.95 51.32/13.95 (5) QDPOrderProof (EQUIVALENT) 51.32/13.95 We use the reduction pair processor [LPAR04,JAR06]. 51.32/13.95 51.32/13.95 51.32/13.95 The following pairs can be oriented strictly and are deleted. 51.32/13.95 51.32/13.95 0^1(1(2(5(x1)))) -> 2^1(x1) 51.32/13.95 2^1(1(1(5(x1)))) -> 2^1(1(x1)) 51.32/13.95 5^1(1(4(2(2(x1))))) -> 2^1(4(2(x1))) 51.32/13.95 5^1(0(1(2(2(x1))))) -> 2^1(0(2(x1))) 51.32/13.95 5^1(0(1(2(2(x1))))) -> 0^1(2(x1)) 51.32/13.95 5^1(1(4(2(2(x1))))) -> 4^1(2(x1)) 51.32/13.95 0^1(1(1(2(5(x1))))) -> 2^1(x1) 51.32/13.95 2^1(3(5(5(x1)))) -> 2^1(5(x1)) 51.32/13.95 2^1(5(3(2(x1)))) -> 2^1(2(x1)) 51.32/13.95 2^1(2(4(3(5(x1))))) -> 4^1(2(2(x1))) 51.32/13.95 2^1(2(4(3(5(x1))))) -> 2^1(2(x1)) 51.32/13.95 2^1(2(4(3(5(x1))))) -> 2^1(x1) 51.32/13.95 2^1(1(4(3(5(x1))))) -> 4^1(4(2(x1))) 51.32/13.95 2^1(1(4(3(5(x1))))) -> 4^1(2(x1)) 51.32/13.95 2^1(1(4(3(5(x1))))) -> 2^1(x1) 51.32/13.95 5^1(1(4(3(2(x1))))) -> 4^1(2(x1)) 51.32/13.95 5^1(5(4(3(2(x1))))) -> 4^1(5(2(x1))) 51.32/13.95 5^1(5(4(3(2(x1))))) -> 5^1(2(x1)) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 2^1(x1) 51.32/13.95 2^1(3(3(1(5(x1))))) -> 2^1(x1) 51.32/13.95 2^1(5(0(3(5(x1))))) -> 0^1(5(x1)) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 0^1(2(0(x1))) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 2^1(0(x1)) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 0^1(x1) 51.32/13.95 0^1(4(5(3(5(x1))))) -> 4^1(5(x1)) 51.32/13.95 The remaining pairs can at least be oriented weakly. 51.32/13.95 Used ordering: Polynomial interpretation [POLO]: 51.32/13.95 51.32/13.95 POL(0(x_1)) = x_1 51.32/13.95 POL(0^1(x_1)) = x_1 51.32/13.95 POL(1(x_1)) = x_1 51.32/13.95 POL(2(x_1)) = x_1 51.32/13.95 POL(2^1(x_1)) = x_1 51.32/13.95 POL(3(x_1)) = x_1 51.32/13.95 POL(4(x_1)) = x_1 51.32/13.95 POL(4^1(x_1)) = x_1 51.32/13.95 POL(5(x_1)) = 1 + x_1 51.32/13.95 POL(5^1(x_1)) = 1 + x_1 51.32/13.95 51.32/13.95 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 51.32/13.95 51.32/13.95 4(2(0(1(2(x1))))) -> 4(2(2(1(3(0(x1)))))) 51.32/13.95 0(1(2(x1))) -> 2(1(0(3(x1)))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(0(x1)))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(0(4(x1))))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(4(0(4(x1)))))) 51.32/13.95 0(0(1(2(x1)))) -> 0(2(1(0(3(x1))))) 51.32/13.95 0(1(1(2(x1)))) -> 1(0(4(1(2(x1))))) 51.32/13.95 0(1(1(2(x1)))) -> 1(2(1(0(3(x1))))) 51.32/13.95 0(1(2(0(x1)))) -> 2(1(0(3(0(3(x1)))))) 51.32/13.95 0(1(2(5(x1)))) -> 2(1(0(3(5(x1))))) 51.32/13.95 0(1(2(5(x1)))) -> 5(1(0(3(2(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 2(1(3(4(0(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 2(4(1(0(3(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 0(2(1(0(3(4(x1)))))) 51.32/13.95 0(1(0(1(2(x1))))) -> 0(2(1(1(3(0(x1)))))) 51.32/13.95 0(1(1(2(5(x1))))) -> 0(1(3(5(1(2(x1)))))) 51.32/13.95 0(1(1(4(2(x1))))) -> 4(1(0(3(2(1(x1)))))) 51.32/13.95 0(1(1(5(0(x1))))) -> 0(1(0(5(4(1(x1)))))) 51.32/13.95 0(1(1(5(4(x1))))) -> 1(4(1(0(3(5(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 0(1(3(4(5(2(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 0(2(5(1(0(3(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 1(0(3(0(2(5(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 5(1(3(0(2(0(x1)))))) 51.32/13.95 0(1(4(0(2(x1))))) -> 4(0(1(3(0(2(x1)))))) 51.32/13.95 0(1(4(0(2(x1))))) -> 4(0(2(1(3(0(x1)))))) 51.32/13.95 0(4(5(3(5(x1))))) -> 5(1(0(3(4(5(x1)))))) 51.32/13.95 5(1(2(0(x1)))) -> 2(5(1(0(3(x1))))) 51.32/13.95 5(1(4(2(2(x1))))) -> 5(1(3(2(4(2(x1)))))) 51.32/13.95 5(1(4(3(2(x1))))) -> 4(5(1(3(4(2(x1)))))) 51.32/13.95 2(1(1(5(x1)))) -> 2(5(1(4(1(x1))))) 51.32/13.95 2(1(1(5(x1)))) -> 5(1(3(2(1(x1))))) 51.32/13.95 2(1(2(0(x1)))) -> 2(2(1(0(3(x1))))) 51.32/13.95 2(3(5(5(x1)))) -> 5(1(0(3(2(5(x1)))))) 51.32/13.95 2(5(3(2(x1)))) -> 5(1(3(2(2(x1))))) 51.32/13.95 2(1(0(1(5(x1))))) -> 1(0(3(2(5(1(x1)))))) 51.32/13.95 2(1(4(3(5(x1))))) -> 1(3(5(4(4(2(x1)))))) 51.32/13.95 2(1(4(3(5(x1))))) -> 5(4(4(1(3(2(x1)))))) 51.32/13.95 2(1(4(5(0(x1))))) -> 2(4(1(0(3(5(x1)))))) 51.32/13.95 2(2(3(5(0(x1))))) -> 5(2(4(2(0(3(x1)))))) 51.32/13.95 2(2(4(3(5(x1))))) -> 5(1(3(4(2(2(x1)))))) 51.32/13.95 2(3(1(1(2(x1))))) -> 1(3(2(1(2(0(x1)))))) 51.32/13.95 2(3(1(1(2(x1))))) -> 4(1(2(2(1(3(x1)))))) 51.32/13.95 2(3(2(0(5(x1))))) -> 2(2(1(0(3(5(x1)))))) 51.32/13.95 2(3(3(1(5(x1))))) -> 1(3(5(1(3(2(x1)))))) 51.32/13.95 2(5(0(3(5(x1))))) -> 5(2(1(3(0(5(x1)))))) 51.32/13.95 5(0(1(2(2(x1))))) -> 4(1(5(2(0(2(x1)))))) 51.32/13.95 5(5(4(3(2(x1))))) -> 5(1(3(4(5(2(x1)))))) 51.32/13.95 51.32/13.95 51.32/13.95 ---------------------------------------- 51.32/13.95 51.32/13.95 (6) 51.32/13.95 Obligation: 51.32/13.95 Q DP problem: 51.32/13.95 The TRS P consists of the following rules: 51.32/13.95 51.32/13.95 0^1(1(2(x1))) -> 0^1(4(x1)) 51.32/13.95 0^1(1(2(x1))) -> 0^1(x1) 51.32/13.95 0^1(1(2(x1))) -> 4^1(x1) 51.32/13.95 4^1(2(0(1(2(x1))))) -> 0^1(x1) 51.32/13.95 0^1(1(2(x1))) -> 4^1(0(x1)) 51.32/13.95 0^1(1(2(x1))) -> 4^1(0(4(x1))) 51.32/13.95 2^1(1(0(1(5(x1))))) -> 2^1(5(1(x1))) 51.32/13.95 2^1(1(0(1(5(x1))))) -> 5^1(1(x1)) 51.32/13.95 2^1(1(4(5(0(x1))))) -> 5^1(x1) 51.32/13.95 5^1(0(1(2(2(x1))))) -> 5^1(2(0(2(x1)))) 51.32/13.95 0^1(1(4(2(x1)))) -> 4^1(0(x1)) 51.32/13.95 0^1(1(4(2(x1)))) -> 0^1(x1) 51.32/13.95 0^1(1(4(2(x1)))) -> 4^1(x1) 51.32/13.95 0^1(1(0(1(2(x1))))) -> 0^1(x1) 51.32/13.95 0^1(1(1(2(5(x1))))) -> 5^1(1(2(x1))) 51.32/13.95 2^1(1(4(3(5(x1))))) -> 5^1(4(4(2(x1)))) 51.32/13.95 2^1(3(1(1(2(x1))))) -> 2^1(1(2(0(x1)))) 51.32/13.95 2^1(3(1(1(2(x1))))) -> 2^1(0(x1)) 51.32/13.95 2^1(3(1(1(2(x1))))) -> 0^1(x1) 51.32/13.95 0^1(1(1(4(2(x1))))) -> 2^1(1(x1)) 51.32/13.95 0^1(1(1(5(4(x1))))) -> 5^1(x1) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 4^1(5(2(x1))) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 5^1(2(x1)) 51.32/13.95 0^1(1(4(0(2(x1))))) -> 0^1(x1) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 0^1(2(5(x1))) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 2^1(5(x1)) 51.32/13.95 51.32/13.95 The TRS R consists of the following rules: 51.32/13.95 51.32/13.95 0(1(2(x1))) -> 2(1(0(3(x1)))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(0(x1)))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(0(4(x1))))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(4(0(4(x1)))))) 51.32/13.95 0(0(1(2(x1)))) -> 0(2(1(0(3(x1))))) 51.32/13.95 0(1(1(2(x1)))) -> 1(0(4(1(2(x1))))) 51.32/13.95 0(1(1(2(x1)))) -> 1(2(1(0(3(x1))))) 51.32/13.95 0(1(2(0(x1)))) -> 2(1(0(3(0(3(x1)))))) 51.32/13.95 0(1(2(5(x1)))) -> 2(1(0(3(5(x1))))) 51.32/13.95 0(1(2(5(x1)))) -> 5(1(0(3(2(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 2(1(3(4(0(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 2(4(1(0(3(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 0(2(1(0(3(4(x1)))))) 51.32/13.95 2(1(1(5(x1)))) -> 2(5(1(4(1(x1))))) 51.32/13.95 2(1(1(5(x1)))) -> 5(1(3(2(1(x1))))) 51.32/13.95 2(1(2(0(x1)))) -> 2(2(1(0(3(x1))))) 51.32/13.95 2(3(5(5(x1)))) -> 5(1(0(3(2(5(x1)))))) 51.32/13.95 2(5(3(2(x1)))) -> 5(1(3(2(2(x1))))) 51.32/13.95 5(1(2(0(x1)))) -> 2(5(1(0(3(x1))))) 51.32/13.95 0(1(0(1(2(x1))))) -> 0(2(1(1(3(0(x1)))))) 51.32/13.95 0(1(1(2(5(x1))))) -> 0(1(3(5(1(2(x1)))))) 51.32/13.95 0(1(1(4(2(x1))))) -> 4(1(0(3(2(1(x1)))))) 51.32/13.95 0(1(1(5(0(x1))))) -> 0(1(0(5(4(1(x1)))))) 51.32/13.95 0(1(1(5(4(x1))))) -> 1(4(1(0(3(5(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 0(1(3(4(5(2(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 0(2(5(1(0(3(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 1(0(3(0(2(5(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 5(1(3(0(2(0(x1)))))) 51.32/13.95 0(1(4(0(2(x1))))) -> 4(0(1(3(0(2(x1)))))) 51.32/13.95 0(1(4(0(2(x1))))) -> 4(0(2(1(3(0(x1)))))) 51.32/13.95 0(4(5(3(5(x1))))) -> 5(1(0(3(4(5(x1)))))) 51.32/13.95 2(1(0(1(5(x1))))) -> 1(0(3(2(5(1(x1)))))) 51.32/13.95 2(1(4(3(5(x1))))) -> 1(3(5(4(4(2(x1)))))) 51.32/13.95 2(1(4(3(5(x1))))) -> 5(4(4(1(3(2(x1)))))) 51.32/13.95 2(1(4(5(0(x1))))) -> 2(4(1(0(3(5(x1)))))) 51.32/13.95 2(2(3(5(0(x1))))) -> 5(2(4(2(0(3(x1)))))) 51.32/13.95 2(2(4(3(5(x1))))) -> 5(1(3(4(2(2(x1)))))) 51.32/13.95 2(3(1(1(2(x1))))) -> 1(3(2(1(2(0(x1)))))) 51.32/13.95 2(3(1(1(2(x1))))) -> 4(1(2(2(1(3(x1)))))) 51.32/13.95 2(3(2(0(5(x1))))) -> 2(2(1(0(3(5(x1)))))) 51.32/13.95 2(3(3(1(5(x1))))) -> 1(3(5(1(3(2(x1)))))) 51.32/13.95 2(5(0(3(5(x1))))) -> 5(2(1(3(0(5(x1)))))) 51.32/13.95 4(2(0(1(2(x1))))) -> 4(2(2(1(3(0(x1)))))) 51.32/13.95 5(0(1(2(2(x1))))) -> 4(1(5(2(0(2(x1)))))) 51.32/13.95 5(1(4(2(2(x1))))) -> 5(1(3(2(4(2(x1)))))) 51.32/13.95 5(1(4(3(2(x1))))) -> 4(5(1(3(4(2(x1)))))) 51.32/13.95 5(5(4(3(2(x1))))) -> 5(1(3(4(5(2(x1)))))) 51.32/13.95 51.32/13.95 Q is empty. 51.32/13.95 We have to consider all minimal (P,Q,R)-chains. 51.32/13.95 ---------------------------------------- 51.32/13.95 51.32/13.95 (7) DependencyGraphProof (EQUIVALENT) 51.32/13.95 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 11 less nodes. 51.32/13.95 ---------------------------------------- 51.32/13.95 51.32/13.95 (8) 51.32/13.95 Complex Obligation (AND) 51.32/13.95 51.32/13.95 ---------------------------------------- 51.32/13.95 51.32/13.95 (9) 51.32/13.95 Obligation: 51.32/13.95 Q DP problem: 51.32/13.95 The TRS P consists of the following rules: 51.32/13.95 51.32/13.95 5^1(0(1(2(2(x1))))) -> 5^1(2(0(2(x1)))) 51.32/13.95 51.32/13.95 The TRS R consists of the following rules: 51.32/13.95 51.32/13.95 0(1(2(x1))) -> 2(1(0(3(x1)))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(0(x1)))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(0(4(x1))))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(4(0(4(x1)))))) 51.32/13.95 0(0(1(2(x1)))) -> 0(2(1(0(3(x1))))) 51.32/13.95 0(1(1(2(x1)))) -> 1(0(4(1(2(x1))))) 51.32/13.95 0(1(1(2(x1)))) -> 1(2(1(0(3(x1))))) 51.32/13.95 0(1(2(0(x1)))) -> 2(1(0(3(0(3(x1)))))) 51.32/13.95 0(1(2(5(x1)))) -> 2(1(0(3(5(x1))))) 51.32/13.95 0(1(2(5(x1)))) -> 5(1(0(3(2(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 2(1(3(4(0(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 2(4(1(0(3(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 0(2(1(0(3(4(x1)))))) 51.32/13.95 2(1(1(5(x1)))) -> 2(5(1(4(1(x1))))) 51.32/13.95 2(1(1(5(x1)))) -> 5(1(3(2(1(x1))))) 51.32/13.95 2(1(2(0(x1)))) -> 2(2(1(0(3(x1))))) 51.32/13.95 2(3(5(5(x1)))) -> 5(1(0(3(2(5(x1)))))) 51.32/13.95 2(5(3(2(x1)))) -> 5(1(3(2(2(x1))))) 51.32/13.95 5(1(2(0(x1)))) -> 2(5(1(0(3(x1))))) 51.32/13.95 0(1(0(1(2(x1))))) -> 0(2(1(1(3(0(x1)))))) 51.32/13.95 0(1(1(2(5(x1))))) -> 0(1(3(5(1(2(x1)))))) 51.32/13.95 0(1(1(4(2(x1))))) -> 4(1(0(3(2(1(x1)))))) 51.32/13.95 0(1(1(5(0(x1))))) -> 0(1(0(5(4(1(x1)))))) 51.32/13.95 0(1(1(5(4(x1))))) -> 1(4(1(0(3(5(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 0(1(3(4(5(2(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 0(2(5(1(0(3(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 1(0(3(0(2(5(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 5(1(3(0(2(0(x1)))))) 51.32/13.95 0(1(4(0(2(x1))))) -> 4(0(1(3(0(2(x1)))))) 51.32/13.95 0(1(4(0(2(x1))))) -> 4(0(2(1(3(0(x1)))))) 51.32/13.95 0(4(5(3(5(x1))))) -> 5(1(0(3(4(5(x1)))))) 51.32/13.95 2(1(0(1(5(x1))))) -> 1(0(3(2(5(1(x1)))))) 51.32/13.95 2(1(4(3(5(x1))))) -> 1(3(5(4(4(2(x1)))))) 51.32/13.95 2(1(4(3(5(x1))))) -> 5(4(4(1(3(2(x1)))))) 51.32/13.95 2(1(4(5(0(x1))))) -> 2(4(1(0(3(5(x1)))))) 51.32/13.95 2(2(3(5(0(x1))))) -> 5(2(4(2(0(3(x1)))))) 51.32/13.95 2(2(4(3(5(x1))))) -> 5(1(3(4(2(2(x1)))))) 51.32/13.95 2(3(1(1(2(x1))))) -> 1(3(2(1(2(0(x1)))))) 51.32/13.95 2(3(1(1(2(x1))))) -> 4(1(2(2(1(3(x1)))))) 51.32/13.95 2(3(2(0(5(x1))))) -> 2(2(1(0(3(5(x1)))))) 51.32/13.95 2(3(3(1(5(x1))))) -> 1(3(5(1(3(2(x1)))))) 51.32/13.95 2(5(0(3(5(x1))))) -> 5(2(1(3(0(5(x1)))))) 51.32/13.95 4(2(0(1(2(x1))))) -> 4(2(2(1(3(0(x1)))))) 51.32/13.95 5(0(1(2(2(x1))))) -> 4(1(5(2(0(2(x1)))))) 51.32/13.95 5(1(4(2(2(x1))))) -> 5(1(3(2(4(2(x1)))))) 51.32/13.95 5(1(4(3(2(x1))))) -> 4(5(1(3(4(2(x1)))))) 51.32/13.95 5(5(4(3(2(x1))))) -> 5(1(3(4(5(2(x1)))))) 51.32/13.95 51.32/13.95 Q is empty. 51.32/13.95 We have to consider all minimal (P,Q,R)-chains. 51.32/13.95 ---------------------------------------- 51.32/13.95 51.32/13.95 (10) QDPOrderProof (EQUIVALENT) 51.32/13.95 We use the reduction pair processor [LPAR04,JAR06]. 51.32/13.95 51.32/13.95 51.32/13.95 The following pairs can be oriented strictly and are deleted. 51.32/13.95 51.32/13.95 5^1(0(1(2(2(x1))))) -> 5^1(2(0(2(x1)))) 51.32/13.95 The remaining pairs can at least be oriented weakly. 51.32/13.95 Used ordering: Polynomial interpretation [POLO]: 51.32/13.95 51.32/13.95 POL(0(x_1)) = 1 + x_1 51.32/13.95 POL(1(x_1)) = 0 51.32/13.95 POL(2(x_1)) = 0 51.32/13.95 POL(3(x_1)) = 1 51.32/13.95 POL(4(x_1)) = x_1 51.32/13.95 POL(5(x_1)) = x_1 51.32/13.95 POL(5^1(x_1)) = x_1 51.32/13.95 51.32/13.95 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 51.32/13.95 51.32/13.95 2(1(1(5(x1)))) -> 2(5(1(4(1(x1))))) 51.32/13.95 2(1(1(5(x1)))) -> 5(1(3(2(1(x1))))) 51.32/13.95 2(1(2(0(x1)))) -> 2(2(1(0(3(x1))))) 51.32/13.95 2(3(5(5(x1)))) -> 5(1(0(3(2(5(x1)))))) 51.32/13.95 2(5(3(2(x1)))) -> 5(1(3(2(2(x1))))) 51.32/13.95 2(1(0(1(5(x1))))) -> 1(0(3(2(5(1(x1)))))) 51.32/13.95 2(1(4(3(5(x1))))) -> 1(3(5(4(4(2(x1)))))) 51.32/13.95 2(1(4(3(5(x1))))) -> 5(4(4(1(3(2(x1)))))) 51.32/13.95 2(1(4(5(0(x1))))) -> 2(4(1(0(3(5(x1)))))) 51.32/13.95 2(2(3(5(0(x1))))) -> 5(2(4(2(0(3(x1)))))) 51.32/13.95 2(2(4(3(5(x1))))) -> 5(1(3(4(2(2(x1)))))) 51.32/13.95 2(3(1(1(2(x1))))) -> 1(3(2(1(2(0(x1)))))) 51.32/13.95 2(3(1(1(2(x1))))) -> 4(1(2(2(1(3(x1)))))) 51.32/13.95 2(3(2(0(5(x1))))) -> 2(2(1(0(3(5(x1)))))) 51.32/13.95 2(3(3(1(5(x1))))) -> 1(3(5(1(3(2(x1)))))) 51.32/13.95 2(5(0(3(5(x1))))) -> 5(2(1(3(0(5(x1)))))) 51.32/13.95 51.32/13.95 51.32/13.95 ---------------------------------------- 51.32/13.95 51.32/13.95 (11) 51.32/13.95 Obligation: 51.32/13.95 Q DP problem: 51.32/13.95 P is empty. 51.32/13.95 The TRS R consists of the following rules: 51.32/13.95 51.32/13.95 0(1(2(x1))) -> 2(1(0(3(x1)))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(0(x1)))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(0(4(x1))))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(4(0(4(x1)))))) 51.32/13.95 0(0(1(2(x1)))) -> 0(2(1(0(3(x1))))) 51.32/13.95 0(1(1(2(x1)))) -> 1(0(4(1(2(x1))))) 51.32/13.95 0(1(1(2(x1)))) -> 1(2(1(0(3(x1))))) 51.32/13.95 0(1(2(0(x1)))) -> 2(1(0(3(0(3(x1)))))) 51.32/13.95 0(1(2(5(x1)))) -> 2(1(0(3(5(x1))))) 51.32/13.95 0(1(2(5(x1)))) -> 5(1(0(3(2(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 2(1(3(4(0(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 2(4(1(0(3(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 0(2(1(0(3(4(x1)))))) 51.32/13.95 2(1(1(5(x1)))) -> 2(5(1(4(1(x1))))) 51.32/13.95 2(1(1(5(x1)))) -> 5(1(3(2(1(x1))))) 51.32/13.95 2(1(2(0(x1)))) -> 2(2(1(0(3(x1))))) 51.32/13.95 2(3(5(5(x1)))) -> 5(1(0(3(2(5(x1)))))) 51.32/13.95 2(5(3(2(x1)))) -> 5(1(3(2(2(x1))))) 51.32/13.95 5(1(2(0(x1)))) -> 2(5(1(0(3(x1))))) 51.32/13.95 0(1(0(1(2(x1))))) -> 0(2(1(1(3(0(x1)))))) 51.32/13.95 0(1(1(2(5(x1))))) -> 0(1(3(5(1(2(x1)))))) 51.32/13.95 0(1(1(4(2(x1))))) -> 4(1(0(3(2(1(x1)))))) 51.32/13.95 0(1(1(5(0(x1))))) -> 0(1(0(5(4(1(x1)))))) 51.32/13.95 0(1(1(5(4(x1))))) -> 1(4(1(0(3(5(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 0(1(3(4(5(2(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 0(2(5(1(0(3(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 1(0(3(0(2(5(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 5(1(3(0(2(0(x1)))))) 51.32/13.95 0(1(4(0(2(x1))))) -> 4(0(1(3(0(2(x1)))))) 51.32/13.95 0(1(4(0(2(x1))))) -> 4(0(2(1(3(0(x1)))))) 51.32/13.95 0(4(5(3(5(x1))))) -> 5(1(0(3(4(5(x1)))))) 51.32/13.95 2(1(0(1(5(x1))))) -> 1(0(3(2(5(1(x1)))))) 51.32/13.95 2(1(4(3(5(x1))))) -> 1(3(5(4(4(2(x1)))))) 51.32/13.95 2(1(4(3(5(x1))))) -> 5(4(4(1(3(2(x1)))))) 51.32/13.95 2(1(4(5(0(x1))))) -> 2(4(1(0(3(5(x1)))))) 51.32/13.95 2(2(3(5(0(x1))))) -> 5(2(4(2(0(3(x1)))))) 51.32/13.95 2(2(4(3(5(x1))))) -> 5(1(3(4(2(2(x1)))))) 51.32/13.95 2(3(1(1(2(x1))))) -> 1(3(2(1(2(0(x1)))))) 51.32/13.95 2(3(1(1(2(x1))))) -> 4(1(2(2(1(3(x1)))))) 51.32/13.95 2(3(2(0(5(x1))))) -> 2(2(1(0(3(5(x1)))))) 51.32/13.95 2(3(3(1(5(x1))))) -> 1(3(5(1(3(2(x1)))))) 51.32/13.95 2(5(0(3(5(x1))))) -> 5(2(1(3(0(5(x1)))))) 51.32/13.95 4(2(0(1(2(x1))))) -> 4(2(2(1(3(0(x1)))))) 51.32/13.95 5(0(1(2(2(x1))))) -> 4(1(5(2(0(2(x1)))))) 51.32/13.95 5(1(4(2(2(x1))))) -> 5(1(3(2(4(2(x1)))))) 51.32/13.95 5(1(4(3(2(x1))))) -> 4(5(1(3(4(2(x1)))))) 51.32/13.95 5(5(4(3(2(x1))))) -> 5(1(3(4(5(2(x1)))))) 51.32/13.95 51.32/13.95 Q is empty. 51.32/13.95 We have to consider all minimal (P,Q,R)-chains. 51.32/13.95 ---------------------------------------- 51.32/13.95 51.32/13.95 (12) PisEmptyProof (EQUIVALENT) 51.32/13.95 The TRS P is empty. Hence, there is no (P,Q,R) chain. 51.32/13.95 ---------------------------------------- 51.32/13.95 51.32/13.95 (13) 51.32/13.95 YES 51.32/13.95 51.32/13.95 ---------------------------------------- 51.32/13.95 51.32/13.95 (14) 51.32/13.95 Obligation: 51.32/13.95 Q DP problem: 51.32/13.95 The TRS P consists of the following rules: 51.32/13.95 51.32/13.95 2^1(1(0(1(5(x1))))) -> 2^1(5(1(x1))) 51.32/13.95 51.32/13.95 The TRS R consists of the following rules: 51.32/13.95 51.32/13.95 0(1(2(x1))) -> 2(1(0(3(x1)))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(0(x1)))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(0(4(x1))))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(4(0(4(x1)))))) 51.32/13.95 0(0(1(2(x1)))) -> 0(2(1(0(3(x1))))) 51.32/13.95 0(1(1(2(x1)))) -> 1(0(4(1(2(x1))))) 51.32/13.95 0(1(1(2(x1)))) -> 1(2(1(0(3(x1))))) 51.32/13.95 0(1(2(0(x1)))) -> 2(1(0(3(0(3(x1)))))) 51.32/13.95 0(1(2(5(x1)))) -> 2(1(0(3(5(x1))))) 51.32/13.95 0(1(2(5(x1)))) -> 5(1(0(3(2(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 2(1(3(4(0(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 2(4(1(0(3(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 0(2(1(0(3(4(x1)))))) 51.32/13.95 2(1(1(5(x1)))) -> 2(5(1(4(1(x1))))) 51.32/13.95 2(1(1(5(x1)))) -> 5(1(3(2(1(x1))))) 51.32/13.95 2(1(2(0(x1)))) -> 2(2(1(0(3(x1))))) 51.32/13.95 2(3(5(5(x1)))) -> 5(1(0(3(2(5(x1)))))) 51.32/13.95 2(5(3(2(x1)))) -> 5(1(3(2(2(x1))))) 51.32/13.95 5(1(2(0(x1)))) -> 2(5(1(0(3(x1))))) 51.32/13.95 0(1(0(1(2(x1))))) -> 0(2(1(1(3(0(x1)))))) 51.32/13.95 0(1(1(2(5(x1))))) -> 0(1(3(5(1(2(x1)))))) 51.32/13.95 0(1(1(4(2(x1))))) -> 4(1(0(3(2(1(x1)))))) 51.32/13.95 0(1(1(5(0(x1))))) -> 0(1(0(5(4(1(x1)))))) 51.32/13.95 0(1(1(5(4(x1))))) -> 1(4(1(0(3(5(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 0(1(3(4(5(2(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 0(2(5(1(0(3(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 1(0(3(0(2(5(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 5(1(3(0(2(0(x1)))))) 51.32/13.95 0(1(4(0(2(x1))))) -> 4(0(1(3(0(2(x1)))))) 51.32/13.95 0(1(4(0(2(x1))))) -> 4(0(2(1(3(0(x1)))))) 51.32/13.95 0(4(5(3(5(x1))))) -> 5(1(0(3(4(5(x1)))))) 51.32/13.95 2(1(0(1(5(x1))))) -> 1(0(3(2(5(1(x1)))))) 51.32/13.95 2(1(4(3(5(x1))))) -> 1(3(5(4(4(2(x1)))))) 51.32/13.95 2(1(4(3(5(x1))))) -> 5(4(4(1(3(2(x1)))))) 51.32/13.95 2(1(4(5(0(x1))))) -> 2(4(1(0(3(5(x1)))))) 51.32/13.95 2(2(3(5(0(x1))))) -> 5(2(4(2(0(3(x1)))))) 51.32/13.95 2(2(4(3(5(x1))))) -> 5(1(3(4(2(2(x1)))))) 51.32/13.95 2(3(1(1(2(x1))))) -> 1(3(2(1(2(0(x1)))))) 51.32/13.95 2(3(1(1(2(x1))))) -> 4(1(2(2(1(3(x1)))))) 51.32/13.95 2(3(2(0(5(x1))))) -> 2(2(1(0(3(5(x1)))))) 51.32/13.95 2(3(3(1(5(x1))))) -> 1(3(5(1(3(2(x1)))))) 51.32/13.95 2(5(0(3(5(x1))))) -> 5(2(1(3(0(5(x1)))))) 51.32/13.95 4(2(0(1(2(x1))))) -> 4(2(2(1(3(0(x1)))))) 51.32/13.95 5(0(1(2(2(x1))))) -> 4(1(5(2(0(2(x1)))))) 51.32/13.95 5(1(4(2(2(x1))))) -> 5(1(3(2(4(2(x1)))))) 51.32/13.95 5(1(4(3(2(x1))))) -> 4(5(1(3(4(2(x1)))))) 51.32/13.95 5(5(4(3(2(x1))))) -> 5(1(3(4(5(2(x1)))))) 51.32/13.95 51.32/13.95 Q is empty. 51.32/13.95 We have to consider all minimal (P,Q,R)-chains. 51.32/13.95 ---------------------------------------- 51.32/13.95 51.32/13.95 (15) QDPOrderProof (EQUIVALENT) 51.32/13.95 We use the reduction pair processor [LPAR04,JAR06]. 51.32/13.95 51.32/13.95 51.32/13.95 The following pairs can be oriented strictly and are deleted. 51.32/13.95 51.32/13.95 2^1(1(0(1(5(x1))))) -> 2^1(5(1(x1))) 51.32/13.95 The remaining pairs can at least be oriented weakly. 51.32/13.95 Used ordering: Polynomial interpretation [POLO]: 51.32/13.95 51.32/13.95 POL(0(x_1)) = 1 + x_1 51.32/13.95 POL(1(x_1)) = 1 + x_1 51.32/13.95 POL(2(x_1)) = 0 51.32/13.95 POL(2^1(x_1)) = x_1 51.32/13.95 POL(3(x_1)) = 0 51.32/13.95 POL(4(x_1)) = x_1 51.32/13.95 POL(5(x_1)) = 1 51.32/13.95 51.32/13.95 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 51.32/13.95 51.32/13.95 5(1(2(0(x1)))) -> 2(5(1(0(3(x1))))) 51.32/13.95 5(1(4(2(2(x1))))) -> 5(1(3(2(4(2(x1)))))) 51.32/13.95 5(1(4(3(2(x1))))) -> 4(5(1(3(4(2(x1)))))) 51.32/13.95 51.32/13.95 51.32/13.95 ---------------------------------------- 51.32/13.95 51.32/13.95 (16) 51.32/13.95 Obligation: 51.32/13.95 Q DP problem: 51.32/13.95 P is empty. 51.32/13.95 The TRS R consists of the following rules: 51.32/13.95 51.32/13.95 0(1(2(x1))) -> 2(1(0(3(x1)))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(0(x1)))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(0(4(x1))))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(4(0(4(x1)))))) 51.32/13.95 0(0(1(2(x1)))) -> 0(2(1(0(3(x1))))) 51.32/13.95 0(1(1(2(x1)))) -> 1(0(4(1(2(x1))))) 51.32/13.95 0(1(1(2(x1)))) -> 1(2(1(0(3(x1))))) 51.32/13.95 0(1(2(0(x1)))) -> 2(1(0(3(0(3(x1)))))) 51.32/13.95 0(1(2(5(x1)))) -> 2(1(0(3(5(x1))))) 51.32/13.95 0(1(2(5(x1)))) -> 5(1(0(3(2(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 2(1(3(4(0(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 2(4(1(0(3(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 0(2(1(0(3(4(x1)))))) 51.32/13.95 2(1(1(5(x1)))) -> 2(5(1(4(1(x1))))) 51.32/13.95 2(1(1(5(x1)))) -> 5(1(3(2(1(x1))))) 51.32/13.95 2(1(2(0(x1)))) -> 2(2(1(0(3(x1))))) 51.32/13.95 2(3(5(5(x1)))) -> 5(1(0(3(2(5(x1)))))) 51.32/13.95 2(5(3(2(x1)))) -> 5(1(3(2(2(x1))))) 51.32/13.95 5(1(2(0(x1)))) -> 2(5(1(0(3(x1))))) 51.32/13.95 0(1(0(1(2(x1))))) -> 0(2(1(1(3(0(x1)))))) 51.32/13.95 0(1(1(2(5(x1))))) -> 0(1(3(5(1(2(x1)))))) 51.32/13.95 0(1(1(4(2(x1))))) -> 4(1(0(3(2(1(x1)))))) 51.32/13.95 0(1(1(5(0(x1))))) -> 0(1(0(5(4(1(x1)))))) 51.32/13.95 0(1(1(5(4(x1))))) -> 1(4(1(0(3(5(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 0(1(3(4(5(2(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 0(2(5(1(0(3(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 1(0(3(0(2(5(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 5(1(3(0(2(0(x1)))))) 51.32/13.95 0(1(4(0(2(x1))))) -> 4(0(1(3(0(2(x1)))))) 51.32/13.95 0(1(4(0(2(x1))))) -> 4(0(2(1(3(0(x1)))))) 51.32/13.95 0(4(5(3(5(x1))))) -> 5(1(0(3(4(5(x1)))))) 51.32/13.95 2(1(0(1(5(x1))))) -> 1(0(3(2(5(1(x1)))))) 51.32/13.95 2(1(4(3(5(x1))))) -> 1(3(5(4(4(2(x1)))))) 51.32/13.95 2(1(4(3(5(x1))))) -> 5(4(4(1(3(2(x1)))))) 51.32/13.95 2(1(4(5(0(x1))))) -> 2(4(1(0(3(5(x1)))))) 51.32/13.95 2(2(3(5(0(x1))))) -> 5(2(4(2(0(3(x1)))))) 51.32/13.95 2(2(4(3(5(x1))))) -> 5(1(3(4(2(2(x1)))))) 51.32/13.95 2(3(1(1(2(x1))))) -> 1(3(2(1(2(0(x1)))))) 51.32/13.95 2(3(1(1(2(x1))))) -> 4(1(2(2(1(3(x1)))))) 51.32/13.95 2(3(2(0(5(x1))))) -> 2(2(1(0(3(5(x1)))))) 51.32/13.95 2(3(3(1(5(x1))))) -> 1(3(5(1(3(2(x1)))))) 51.32/13.95 2(5(0(3(5(x1))))) -> 5(2(1(3(0(5(x1)))))) 51.32/13.95 4(2(0(1(2(x1))))) -> 4(2(2(1(3(0(x1)))))) 51.32/13.95 5(0(1(2(2(x1))))) -> 4(1(5(2(0(2(x1)))))) 51.32/13.95 5(1(4(2(2(x1))))) -> 5(1(3(2(4(2(x1)))))) 51.32/13.95 5(1(4(3(2(x1))))) -> 4(5(1(3(4(2(x1)))))) 51.32/13.95 5(5(4(3(2(x1))))) -> 5(1(3(4(5(2(x1)))))) 51.32/13.95 51.32/13.95 Q is empty. 51.32/13.95 We have to consider all minimal (P,Q,R)-chains. 51.32/13.95 ---------------------------------------- 51.32/13.95 51.32/13.95 (17) PisEmptyProof (EQUIVALENT) 51.32/13.95 The TRS P is empty. Hence, there is no (P,Q,R) chain. 51.32/13.95 ---------------------------------------- 51.32/13.95 51.32/13.95 (18) 51.32/13.95 YES 51.32/13.95 51.32/13.95 ---------------------------------------- 51.32/13.95 51.32/13.95 (19) 51.32/13.95 Obligation: 51.32/13.95 Q DP problem: 51.32/13.95 The TRS P consists of the following rules: 51.32/13.95 51.32/13.95 0^1(1(2(x1))) -> 0^1(x1) 51.32/13.95 0^1(1(2(x1))) -> 0^1(4(x1)) 51.32/13.95 0^1(1(2(x1))) -> 4^1(x1) 51.32/13.95 4^1(2(0(1(2(x1))))) -> 0^1(x1) 51.32/13.95 0^1(1(2(x1))) -> 4^1(0(x1)) 51.32/13.95 0^1(1(2(x1))) -> 4^1(0(4(x1))) 51.32/13.95 0^1(1(4(2(x1)))) -> 4^1(0(x1)) 51.32/13.95 0^1(1(4(2(x1)))) -> 0^1(x1) 51.32/13.95 0^1(1(4(2(x1)))) -> 4^1(x1) 51.32/13.95 0^1(1(0(1(2(x1))))) -> 0^1(x1) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 4^1(5(2(x1))) 51.32/13.95 0^1(1(2(3(5(x1))))) -> 0^1(2(5(x1))) 51.32/13.95 0^1(1(4(0(2(x1))))) -> 0^1(x1) 51.32/13.95 51.32/13.95 The TRS R consists of the following rules: 51.32/13.95 51.32/13.95 0(1(2(x1))) -> 2(1(0(3(x1)))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(0(x1)))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(0(4(x1))))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 51.32/13.95 0(1(2(x1))) -> 2(1(3(4(0(4(x1)))))) 51.32/13.95 0(0(1(2(x1)))) -> 0(2(1(0(3(x1))))) 51.32/13.95 0(1(1(2(x1)))) -> 1(0(4(1(2(x1))))) 51.32/13.95 0(1(1(2(x1)))) -> 1(2(1(0(3(x1))))) 51.32/13.95 0(1(2(0(x1)))) -> 2(1(0(3(0(3(x1)))))) 51.32/13.95 0(1(2(5(x1)))) -> 2(1(0(3(5(x1))))) 51.32/13.95 0(1(2(5(x1)))) -> 5(1(0(3(2(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 2(1(3(4(0(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 2(4(1(0(3(x1))))) 51.32/13.95 0(1(4(2(x1)))) -> 0(2(1(0(3(4(x1)))))) 51.32/13.95 2(1(1(5(x1)))) -> 2(5(1(4(1(x1))))) 51.32/13.95 2(1(1(5(x1)))) -> 5(1(3(2(1(x1))))) 51.32/13.95 2(1(2(0(x1)))) -> 2(2(1(0(3(x1))))) 51.32/13.95 2(3(5(5(x1)))) -> 5(1(0(3(2(5(x1)))))) 51.32/13.95 2(5(3(2(x1)))) -> 5(1(3(2(2(x1))))) 51.32/13.95 5(1(2(0(x1)))) -> 2(5(1(0(3(x1))))) 51.32/13.95 0(1(0(1(2(x1))))) -> 0(2(1(1(3(0(x1)))))) 51.32/13.95 0(1(1(2(5(x1))))) -> 0(1(3(5(1(2(x1)))))) 51.32/13.95 0(1(1(4(2(x1))))) -> 4(1(0(3(2(1(x1)))))) 51.32/13.95 0(1(1(5(0(x1))))) -> 0(1(0(5(4(1(x1)))))) 51.32/13.95 0(1(1(5(4(x1))))) -> 1(4(1(0(3(5(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 0(1(3(4(5(2(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 0(2(5(1(0(3(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 1(0(3(0(2(5(x1)))))) 51.32/13.95 0(1(2(3(5(x1))))) -> 5(1(3(0(2(0(x1)))))) 51.32/13.96 0(1(4(0(2(x1))))) -> 4(0(1(3(0(2(x1)))))) 51.32/13.96 0(1(4(0(2(x1))))) -> 4(0(2(1(3(0(x1)))))) 51.32/13.96 0(4(5(3(5(x1))))) -> 5(1(0(3(4(5(x1)))))) 51.32/13.96 2(1(0(1(5(x1))))) -> 1(0(3(2(5(1(x1)))))) 51.32/13.96 2(1(4(3(5(x1))))) -> 1(3(5(4(4(2(x1)))))) 51.32/13.96 2(1(4(3(5(x1))))) -> 5(4(4(1(3(2(x1)))))) 51.32/13.96 2(1(4(5(0(x1))))) -> 2(4(1(0(3(5(x1)))))) 51.32/13.96 2(2(3(5(0(x1))))) -> 5(2(4(2(0(3(x1)))))) 51.32/13.96 2(2(4(3(5(x1))))) -> 5(1(3(4(2(2(x1)))))) 51.32/13.96 2(3(1(1(2(x1))))) -> 1(3(2(1(2(0(x1)))))) 51.32/13.96 2(3(1(1(2(x1))))) -> 4(1(2(2(1(3(x1)))))) 51.32/13.96 2(3(2(0(5(x1))))) -> 2(2(1(0(3(5(x1)))))) 51.32/13.96 2(3(3(1(5(x1))))) -> 1(3(5(1(3(2(x1)))))) 51.32/13.96 2(5(0(3(5(x1))))) -> 5(2(1(3(0(5(x1)))))) 51.32/13.96 4(2(0(1(2(x1))))) -> 4(2(2(1(3(0(x1)))))) 51.32/13.96 5(0(1(2(2(x1))))) -> 4(1(5(2(0(2(x1)))))) 51.32/13.96 5(1(4(2(2(x1))))) -> 5(1(3(2(4(2(x1)))))) 51.32/13.96 5(1(4(3(2(x1))))) -> 4(5(1(3(4(2(x1)))))) 51.32/13.96 5(5(4(3(2(x1))))) -> 5(1(3(4(5(2(x1)))))) 51.32/13.96 51.32/13.96 Q is empty. 51.32/13.96 We have to consider all minimal (P,Q,R)-chains. 51.32/13.96 ---------------------------------------- 51.32/13.96 51.32/13.96 (20) QDPOrderProof (EQUIVALENT) 51.32/13.96 We use the reduction pair processor [LPAR04,JAR06]. 51.32/13.96 51.32/13.96 51.32/13.96 The following pairs can be oriented strictly and are deleted. 51.32/13.96 51.32/13.96 0^1(1(2(x1))) -> 0^1(x1) 51.32/13.96 0^1(1(2(x1))) -> 0^1(4(x1)) 51.32/13.96 0^1(1(2(x1))) -> 4^1(x1) 51.32/13.96 4^1(2(0(1(2(x1))))) -> 0^1(x1) 51.32/13.96 0^1(1(2(x1))) -> 4^1(0(x1)) 51.32/13.96 0^1(1(2(x1))) -> 4^1(0(4(x1))) 51.32/13.96 0^1(1(4(2(x1)))) -> 4^1(0(x1)) 51.32/13.96 0^1(1(4(2(x1)))) -> 0^1(x1) 51.32/13.96 0^1(1(4(2(x1)))) -> 4^1(x1) 51.32/13.96 0^1(1(0(1(2(x1))))) -> 0^1(x1) 51.32/13.96 0^1(1(2(3(5(x1))))) -> 4^1(5(2(x1))) 51.32/13.96 0^1(1(4(0(2(x1))))) -> 0^1(x1) 51.32/13.96 The remaining pairs can at least be oriented weakly. 51.32/13.96 Used ordering: Polynomial interpretation [POLO]: 51.32/13.96 51.32/13.96 POL(0(x_1)) = x_1 51.32/13.96 POL(0^1(x_1)) = 1 + x_1 51.32/13.96 POL(1(x_1)) = x_1 51.32/13.96 POL(2(x_1)) = 1 + x_1 51.32/13.96 POL(3(x_1)) = x_1 51.32/13.96 POL(4(x_1)) = x_1 51.32/13.96 POL(4^1(x_1)) = x_1 51.32/13.96 POL(5(x_1)) = x_1 51.32/13.96 51.32/13.96 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 51.32/13.96 51.32/13.96 4(2(0(1(2(x1))))) -> 4(2(2(1(3(0(x1)))))) 51.32/13.96 0(1(2(x1))) -> 2(1(0(3(x1)))) 51.32/13.96 0(1(2(x1))) -> 2(1(3(0(x1)))) 51.32/13.96 0(1(2(x1))) -> 2(1(3(0(4(x1))))) 51.32/13.96 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 51.32/13.96 0(1(2(x1))) -> 2(1(3(4(0(4(x1)))))) 51.32/13.96 0(0(1(2(x1)))) -> 0(2(1(0(3(x1))))) 51.32/13.96 0(1(1(2(x1)))) -> 1(0(4(1(2(x1))))) 51.32/13.96 0(1(1(2(x1)))) -> 1(2(1(0(3(x1))))) 51.32/13.96 0(1(2(0(x1)))) -> 2(1(0(3(0(3(x1)))))) 51.32/13.96 0(1(2(5(x1)))) -> 2(1(0(3(5(x1))))) 51.32/13.96 0(1(2(5(x1)))) -> 5(1(0(3(2(x1))))) 51.32/13.96 0(1(4(2(x1)))) -> 2(1(3(4(0(x1))))) 51.32/13.96 0(1(4(2(x1)))) -> 2(4(1(0(3(x1))))) 51.32/13.96 0(1(4(2(x1)))) -> 0(2(1(0(3(4(x1)))))) 51.32/13.96 0(1(0(1(2(x1))))) -> 0(2(1(1(3(0(x1)))))) 51.32/13.96 0(1(1(2(5(x1))))) -> 0(1(3(5(1(2(x1)))))) 51.32/13.96 0(1(1(4(2(x1))))) -> 4(1(0(3(2(1(x1)))))) 51.32/13.96 0(1(1(5(0(x1))))) -> 0(1(0(5(4(1(x1)))))) 51.32/13.96 0(1(1(5(4(x1))))) -> 1(4(1(0(3(5(x1)))))) 51.32/13.96 0(1(2(3(5(x1))))) -> 0(1(3(4(5(2(x1)))))) 51.32/13.96 0(1(2(3(5(x1))))) -> 0(2(5(1(0(3(x1)))))) 51.32/13.96 0(1(2(3(5(x1))))) -> 1(0(3(0(2(5(x1)))))) 51.32/13.96 0(1(2(3(5(x1))))) -> 5(1(3(0(2(0(x1)))))) 51.32/13.96 0(1(4(0(2(x1))))) -> 4(0(1(3(0(2(x1)))))) 51.32/13.96 0(1(4(0(2(x1))))) -> 4(0(2(1(3(0(x1)))))) 51.32/13.96 0(4(5(3(5(x1))))) -> 5(1(0(3(4(5(x1)))))) 51.32/13.96 2(1(1(5(x1)))) -> 2(5(1(4(1(x1))))) 51.32/13.96 2(1(1(5(x1)))) -> 5(1(3(2(1(x1))))) 51.32/13.96 2(1(2(0(x1)))) -> 2(2(1(0(3(x1))))) 51.32/13.96 2(3(5(5(x1)))) -> 5(1(0(3(2(5(x1)))))) 51.32/13.96 2(5(3(2(x1)))) -> 5(1(3(2(2(x1))))) 51.32/13.96 2(1(0(1(5(x1))))) -> 1(0(3(2(5(1(x1)))))) 51.32/13.96 2(1(4(3(5(x1))))) -> 1(3(5(4(4(2(x1)))))) 51.32/13.96 2(1(4(3(5(x1))))) -> 5(4(4(1(3(2(x1)))))) 51.32/13.96 2(1(4(5(0(x1))))) -> 2(4(1(0(3(5(x1)))))) 51.32/13.96 2(2(3(5(0(x1))))) -> 5(2(4(2(0(3(x1)))))) 51.32/13.96 2(2(4(3(5(x1))))) -> 5(1(3(4(2(2(x1)))))) 51.32/13.96 2(3(1(1(2(x1))))) -> 1(3(2(1(2(0(x1)))))) 51.32/13.96 2(3(1(1(2(x1))))) -> 4(1(2(2(1(3(x1)))))) 51.32/13.96 2(3(2(0(5(x1))))) -> 2(2(1(0(3(5(x1)))))) 51.32/13.96 2(3(3(1(5(x1))))) -> 1(3(5(1(3(2(x1)))))) 51.32/13.96 2(5(0(3(5(x1))))) -> 5(2(1(3(0(5(x1)))))) 51.32/13.96 5(1(2(0(x1)))) -> 2(5(1(0(3(x1))))) 51.32/13.96 5(0(1(2(2(x1))))) -> 4(1(5(2(0(2(x1)))))) 51.32/13.96 5(1(4(2(2(x1))))) -> 5(1(3(2(4(2(x1)))))) 51.32/13.96 5(1(4(3(2(x1))))) -> 4(5(1(3(4(2(x1)))))) 51.32/13.96 5(5(4(3(2(x1))))) -> 5(1(3(4(5(2(x1)))))) 51.32/13.96 51.32/13.96 51.32/13.96 ---------------------------------------- 51.32/13.96 51.32/13.96 (21) 51.32/13.96 Obligation: 51.32/13.96 Q DP problem: 51.32/13.96 The TRS P consists of the following rules: 51.32/13.96 51.32/13.96 0^1(1(2(3(5(x1))))) -> 0^1(2(5(x1))) 51.32/13.96 51.32/13.96 The TRS R consists of the following rules: 51.32/13.96 51.32/13.96 0(1(2(x1))) -> 2(1(0(3(x1)))) 51.32/13.96 0(1(2(x1))) -> 2(1(3(0(x1)))) 51.32/13.96 0(1(2(x1))) -> 2(1(3(0(4(x1))))) 51.32/13.96 0(1(2(x1))) -> 2(1(3(4(0(x1))))) 51.32/13.96 0(1(2(x1))) -> 2(1(3(4(0(4(x1)))))) 51.32/13.96 0(0(1(2(x1)))) -> 0(2(1(0(3(x1))))) 51.32/13.96 0(1(1(2(x1)))) -> 1(0(4(1(2(x1))))) 51.32/13.96 0(1(1(2(x1)))) -> 1(2(1(0(3(x1))))) 51.32/13.96 0(1(2(0(x1)))) -> 2(1(0(3(0(3(x1)))))) 51.32/13.96 0(1(2(5(x1)))) -> 2(1(0(3(5(x1))))) 51.32/13.96 0(1(2(5(x1)))) -> 5(1(0(3(2(x1))))) 51.32/13.96 0(1(4(2(x1)))) -> 2(1(3(4(0(x1))))) 51.32/13.96 0(1(4(2(x1)))) -> 2(4(1(0(3(x1))))) 51.32/13.96 0(1(4(2(x1)))) -> 0(2(1(0(3(4(x1)))))) 51.32/13.96 2(1(1(5(x1)))) -> 2(5(1(4(1(x1))))) 51.32/13.96 2(1(1(5(x1)))) -> 5(1(3(2(1(x1))))) 51.32/13.96 2(1(2(0(x1)))) -> 2(2(1(0(3(x1))))) 51.32/13.96 2(3(5(5(x1)))) -> 5(1(0(3(2(5(x1)))))) 51.32/13.96 2(5(3(2(x1)))) -> 5(1(3(2(2(x1))))) 51.32/13.96 5(1(2(0(x1)))) -> 2(5(1(0(3(x1))))) 51.32/13.96 0(1(0(1(2(x1))))) -> 0(2(1(1(3(0(x1)))))) 51.32/13.96 0(1(1(2(5(x1))))) -> 0(1(3(5(1(2(x1)))))) 51.32/13.96 0(1(1(4(2(x1))))) -> 4(1(0(3(2(1(x1)))))) 51.32/13.96 0(1(1(5(0(x1))))) -> 0(1(0(5(4(1(x1)))))) 51.32/13.96 0(1(1(5(4(x1))))) -> 1(4(1(0(3(5(x1)))))) 51.32/13.96 0(1(2(3(5(x1))))) -> 0(1(3(4(5(2(x1)))))) 51.32/13.96 0(1(2(3(5(x1))))) -> 0(2(5(1(0(3(x1)))))) 51.32/13.96 0(1(2(3(5(x1))))) -> 1(0(3(0(2(5(x1)))))) 51.32/13.96 0(1(2(3(5(x1))))) -> 5(1(3(0(2(0(x1)))))) 51.32/13.96 0(1(4(0(2(x1))))) -> 4(0(1(3(0(2(x1)))))) 51.32/13.96 0(1(4(0(2(x1))))) -> 4(0(2(1(3(0(x1)))))) 51.32/13.96 0(4(5(3(5(x1))))) -> 5(1(0(3(4(5(x1)))))) 51.32/13.96 2(1(0(1(5(x1))))) -> 1(0(3(2(5(1(x1)))))) 51.32/13.96 2(1(4(3(5(x1))))) -> 1(3(5(4(4(2(x1)))))) 51.32/13.96 2(1(4(3(5(x1))))) -> 5(4(4(1(3(2(x1)))))) 51.32/13.96 2(1(4(5(0(x1))))) -> 2(4(1(0(3(5(x1)))))) 51.32/13.96 2(2(3(5(0(x1))))) -> 5(2(4(2(0(3(x1)))))) 51.32/13.96 2(2(4(3(5(x1))))) -> 5(1(3(4(2(2(x1)))))) 51.32/13.96 2(3(1(1(2(x1))))) -> 1(3(2(1(2(0(x1)))))) 51.32/13.96 2(3(1(1(2(x1))))) -> 4(1(2(2(1(3(x1)))))) 51.32/13.96 2(3(2(0(5(x1))))) -> 2(2(1(0(3(5(x1)))))) 51.32/13.96 2(3(3(1(5(x1))))) -> 1(3(5(1(3(2(x1)))))) 51.32/13.96 2(5(0(3(5(x1))))) -> 5(2(1(3(0(5(x1)))))) 51.32/13.96 4(2(0(1(2(x1))))) -> 4(2(2(1(3(0(x1)))))) 51.32/13.96 5(0(1(2(2(x1))))) -> 4(1(5(2(0(2(x1)))))) 51.32/13.96 5(1(4(2(2(x1))))) -> 5(1(3(2(4(2(x1)))))) 51.32/13.96 5(1(4(3(2(x1))))) -> 4(5(1(3(4(2(x1)))))) 51.32/13.96 5(5(4(3(2(x1))))) -> 5(1(3(4(5(2(x1)))))) 51.32/13.96 51.32/13.96 Q is empty. 51.32/13.96 We have to consider all minimal (P,Q,R)-chains. 51.32/13.96 ---------------------------------------- 51.32/13.96 51.32/13.96 (22) DependencyGraphProof (EQUIVALENT) 51.32/13.96 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 51.32/13.96 ---------------------------------------- 51.32/13.96 51.32/13.96 (23) 51.32/13.96 TRUE 51.53/14.00 EOF