63.59/17.10 YES 64.53/17.30 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 64.53/17.30 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 64.53/17.30 64.53/17.30 64.53/17.30 Termination w.r.t. Q of the given QTRS could be proven: 64.53/17.30 64.53/17.30 (0) QTRS 64.53/17.30 (1) QTRS Reverse [EQUIVALENT, 0 ms] 64.53/17.30 (2) QTRS 64.53/17.30 (3) DependencyPairsProof [EQUIVALENT, 230 ms] 64.53/17.30 (4) QDP 64.53/17.30 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 64.53/17.30 (6) AND 64.53/17.30 (7) QDP 64.53/17.30 (8) QDPSizeChangeProof [EQUIVALENT, 0 ms] 64.53/17.30 (9) YES 64.53/17.30 (10) QDP 64.53/17.30 (11) QDPOrderProof [EQUIVALENT, 262 ms] 64.53/17.30 (12) QDP 64.53/17.30 (13) QDPOrderProof [EQUIVALENT, 137 ms] 64.53/17.30 (14) QDP 64.53/17.30 (15) DependencyGraphProof [EQUIVALENT, 0 ms] 64.53/17.30 (16) QDP 64.53/17.30 (17) QDPOrderProof [EQUIVALENT, 81 ms] 64.53/17.30 (18) QDP 64.53/17.30 (19) QDPOrderProof [EQUIVALENT, 49 ms] 64.53/17.30 (20) QDP 64.53/17.30 (21) QDPOrderProof [EQUIVALENT, 63 ms] 64.53/17.30 (22) QDP 64.53/17.30 (23) QDPOrderProof [EQUIVALENT, 53 ms] 64.53/17.30 (24) QDP 64.53/17.30 (25) PisEmptyProof [EQUIVALENT, 0 ms] 64.53/17.30 (26) YES 64.53/17.30 (27) QDP 64.53/17.30 (28) UsableRulesProof [EQUIVALENT, 0 ms] 64.53/17.30 (29) QDP 64.53/17.30 (30) QDPSizeChangeProof [EQUIVALENT, 0 ms] 64.53/17.30 (31) YES 64.53/17.30 64.53/17.30 64.53/17.30 ---------------------------------------- 64.53/17.30 64.53/17.30 (0) 64.53/17.30 Obligation: 64.53/17.30 Q restricted rewrite system: 64.53/17.30 The TRS R consists of the following rules: 64.53/17.30 64.53/17.30 0(0(1(2(x1)))) -> 0(2(1(3(3(0(x1)))))) 64.53/17.30 0(1(0(4(x1)))) -> 0(1(3(0(4(x1))))) 64.53/17.30 0(1(0(5(x1)))) -> 0(1(3(5(0(x1))))) 64.53/17.30 0(1(5(0(x1)))) -> 0(1(3(5(0(x1))))) 64.53/17.30 0(3(1(0(x1)))) -> 0(1(3(5(0(x1))))) 64.53/17.30 3(1(0(2(x1)))) -> 1(3(5(0(2(x1))))) 64.53/17.30 3(1(0(2(x1)))) -> 3(2(1(3(0(x1))))) 64.53/17.30 3(1(0(5(x1)))) -> 1(3(5(3(0(x1))))) 64.53/17.30 3(1(2(0(x1)))) -> 2(1(3(3(0(x1))))) 64.53/17.30 3(1(2(0(x1)))) -> 3(0(2(1(4(x1))))) 64.53/17.30 3(1(2(0(x1)))) -> 3(0(3(2(1(x1))))) 64.53/17.30 3(1(2(0(x1)))) -> 3(3(2(1(0(x1))))) 64.53/17.30 3(1(2(0(x1)))) -> 3(5(2(1(0(x1))))) 64.53/17.30 3(1(2(2(x1)))) -> 3(5(2(2(1(x1))))) 64.53/17.30 3(4(2(0(x1)))) -> 3(3(2(4(0(x1))))) 64.53/17.30 4(0(1(0(x1)))) -> 1(3(0(4(0(x1))))) 64.53/17.30 5(1(0(5(x1)))) -> 1(3(3(5(5(0(x1)))))) 64.53/17.30 5(2(5(0(x1)))) -> 5(2(3(5(0(x1))))) 64.53/17.30 5(3(1(0(x1)))) -> 0(1(3(5(5(x1))))) 64.53/17.30 0(0(4(1(0(x1))))) -> 0(4(1(3(0(0(x1)))))) 64.53/17.30 0(1(2(0(5(x1))))) -> 0(3(0(5(1(2(x1)))))) 64.53/17.30 0(1(3(1(2(x1))))) -> 3(0(2(2(1(1(x1)))))) 64.53/17.30 0(1(5(0(4(x1))))) -> 0(4(1(3(5(0(x1)))))) 64.53/17.30 0(1(5(3(5(x1))))) -> 0(1(3(5(3(5(x1)))))) 64.53/17.30 0(2(0(3(4(x1))))) -> 0(4(5(2(3(0(x1)))))) 64.53/17.30 0(2(3(1(5(x1))))) -> 0(1(3(5(5(2(x1)))))) 64.53/17.30 0(2(4(1(2(x1))))) -> 0(2(1(1(4(2(x1)))))) 64.53/17.30 0(2(5(0(2(x1))))) -> 0(0(3(5(2(2(x1)))))) 64.53/17.30 0(2(5(5(0(x1))))) -> 0(0(2(5(1(5(x1)))))) 64.53/17.30 0(3(1(5(0(x1))))) -> 1(3(0(5(3(0(x1)))))) 64.53/17.30 0(5(3(2(0(x1))))) -> 0(0(2(1(3(5(x1)))))) 64.53/17.30 3(1(0(5(2(x1))))) -> 2(4(1(3(5(0(x1)))))) 64.53/17.30 3(1(3(0(2(x1))))) -> 2(1(3(3(3(0(x1)))))) 64.53/17.30 3(1(3(2(0(x1))))) -> 3(1(3(0(5(2(x1)))))) 64.53/17.30 3(1(4(1(2(x1))))) -> 1(4(3(2(2(1(x1)))))) 64.53/17.30 3(1(4(2(0(x1))))) -> 2(1(3(3(0(4(x1)))))) 64.53/17.30 3(1(5(0(2(x1))))) -> 2(5(1(3(0(5(x1)))))) 64.53/17.30 3(3(1(0(0(x1))))) -> 5(1(3(3(0(0(x1)))))) 64.53/17.30 3(4(0(2(0(x1))))) -> 3(0(0(2(1(4(x1)))))) 64.53/17.30 3(4(2(3(2(x1))))) -> 3(3(2(5(4(2(x1)))))) 64.53/17.30 4(3(1(0(2(x1))))) -> 4(2(1(3(0(1(x1)))))) 64.53/17.30 4(3(1(2(0(x1))))) -> 1(3(0(1(4(2(x1)))))) 64.53/17.30 4(5(5(1(2(x1))))) -> 5(4(5(2(1(1(x1)))))) 64.53/17.30 4(5(5(5(0(x1))))) -> 5(5(1(5(0(4(x1)))))) 64.53/17.30 5(1(0(1(2(x1))))) -> 2(1(1(3(5(0(x1)))))) 64.53/17.30 5(1(1(0(4(x1))))) -> 5(1(1(4(3(0(x1)))))) 64.53/17.30 5(1(5(2(0(x1))))) -> 2(1(3(5(0(5(x1)))))) 64.53/17.30 5(4(1(0(5(x1))))) -> 4(1(3(5(5(0(x1)))))) 64.53/17.30 64.53/17.30 Q is empty. 64.53/17.30 64.53/17.30 ---------------------------------------- 64.53/17.30 64.53/17.30 (1) QTRS Reverse (EQUIVALENT) 64.53/17.30 We applied the QTRS Reverse Processor [REVERSE]. 64.53/17.30 ---------------------------------------- 64.53/17.30 64.53/17.30 (2) 64.53/17.30 Obligation: 64.53/17.30 Q restricted rewrite system: 64.53/17.30 The TRS R consists of the following rules: 64.53/17.30 64.53/17.30 2(1(0(0(x1)))) -> 0(3(3(1(2(0(x1)))))) 64.53/17.30 4(0(1(0(x1)))) -> 4(0(3(1(0(x1))))) 64.53/17.30 5(0(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(5(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(1(3(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 2(0(1(3(x1)))) -> 2(0(5(3(1(x1))))) 64.53/17.30 2(0(1(3(x1)))) -> 0(3(1(2(3(x1))))) 64.53/17.30 5(0(1(3(x1)))) -> 0(3(5(3(1(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(3(3(1(2(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 4(1(2(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 1(2(3(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(3(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(5(3(x1))))) 64.53/17.30 2(2(1(3(x1)))) -> 1(2(2(5(3(x1))))) 64.53/17.30 0(2(4(3(x1)))) -> 0(4(2(3(3(x1))))) 64.53/17.30 0(1(0(4(x1)))) -> 0(4(0(3(1(x1))))) 64.53/17.30 5(0(1(5(x1)))) -> 0(5(5(3(3(1(x1)))))) 64.53/17.30 0(5(2(5(x1)))) -> 0(5(3(2(5(x1))))) 64.53/17.30 0(1(3(5(x1)))) -> 5(5(3(1(0(x1))))) 64.53/17.30 0(1(4(0(0(x1))))) -> 0(0(3(1(4(0(x1)))))) 64.53/17.30 5(0(2(1(0(x1))))) -> 2(1(5(0(3(0(x1)))))) 64.53/17.30 2(1(3(1(0(x1))))) -> 1(1(2(2(0(3(x1)))))) 64.53/17.30 4(0(5(1(0(x1))))) -> 0(5(3(1(4(0(x1)))))) 64.53/17.30 5(3(5(1(0(x1))))) -> 5(3(5(3(1(0(x1)))))) 64.53/17.30 4(3(0(2(0(x1))))) -> 0(3(2(5(4(0(x1)))))) 64.53/17.30 5(1(3(2(0(x1))))) -> 2(5(5(3(1(0(x1)))))) 64.53/17.30 2(1(4(2(0(x1))))) -> 2(4(1(1(2(0(x1)))))) 64.53/17.30 2(0(5(2(0(x1))))) -> 2(2(5(3(0(0(x1)))))) 64.53/17.30 0(5(5(2(0(x1))))) -> 5(1(5(2(0(0(x1)))))) 64.53/17.30 0(5(1(3(0(x1))))) -> 0(3(5(0(3(1(x1)))))) 64.53/17.30 0(2(3(5(0(x1))))) -> 5(3(1(2(0(0(x1)))))) 64.53/17.30 2(5(0(1(3(x1))))) -> 0(5(3(1(4(2(x1)))))) 64.53/17.30 2(0(3(1(3(x1))))) -> 0(3(3(3(1(2(x1)))))) 64.53/17.30 0(2(3(1(3(x1))))) -> 2(5(0(3(1(3(x1)))))) 64.53/17.30 2(1(4(1(3(x1))))) -> 1(2(2(3(4(1(x1)))))) 64.53/17.30 0(2(4(1(3(x1))))) -> 4(0(3(3(1(2(x1)))))) 64.53/17.30 2(0(5(1(3(x1))))) -> 5(0(3(1(5(2(x1)))))) 64.53/17.30 0(0(1(3(3(x1))))) -> 0(0(3(3(1(5(x1)))))) 64.53/17.30 0(2(0(4(3(x1))))) -> 4(1(2(0(0(3(x1)))))) 64.53/17.30 2(3(2(4(3(x1))))) -> 2(4(5(2(3(3(x1)))))) 64.53/17.30 2(0(1(3(4(x1))))) -> 1(0(3(1(2(4(x1)))))) 64.53/17.30 0(2(1(3(4(x1))))) -> 2(4(1(0(3(1(x1)))))) 64.53/17.30 2(1(5(5(4(x1))))) -> 1(1(2(5(4(5(x1)))))) 64.53/17.30 0(5(5(5(4(x1))))) -> 4(0(5(1(5(5(x1)))))) 64.53/17.30 2(1(0(1(5(x1))))) -> 0(5(3(1(1(2(x1)))))) 64.53/17.30 4(0(1(1(5(x1))))) -> 0(3(4(1(1(5(x1)))))) 64.53/17.30 0(2(5(1(5(x1))))) -> 5(0(5(3(1(2(x1)))))) 64.53/17.30 5(0(1(4(5(x1))))) -> 0(5(5(3(1(4(x1)))))) 64.53/17.30 64.53/17.30 Q is empty. 64.53/17.30 64.53/17.30 ---------------------------------------- 64.53/17.30 64.53/17.30 (3) DependencyPairsProof (EQUIVALENT) 64.53/17.30 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 64.53/17.30 ---------------------------------------- 64.53/17.30 64.53/17.30 (4) 64.53/17.30 Obligation: 64.53/17.30 Q DP problem: 64.53/17.30 The TRS P consists of the following rules: 64.53/17.30 64.53/17.30 2^1(1(0(0(x1)))) -> 0^1(3(3(1(2(0(x1)))))) 64.53/17.30 2^1(1(0(0(x1)))) -> 2^1(0(x1)) 64.53/17.30 4^1(0(1(0(x1)))) -> 4^1(0(3(1(0(x1))))) 64.53/17.30 4^1(0(1(0(x1)))) -> 0^1(3(1(0(x1)))) 64.53/17.30 5^1(0(1(0(x1)))) -> 0^1(5(3(1(0(x1))))) 64.53/17.30 5^1(0(1(0(x1)))) -> 5^1(3(1(0(x1)))) 64.53/17.30 0^1(5(1(0(x1)))) -> 0^1(5(3(1(0(x1))))) 64.53/17.30 0^1(5(1(0(x1)))) -> 5^1(3(1(0(x1)))) 64.53/17.30 0^1(1(3(0(x1)))) -> 0^1(5(3(1(0(x1))))) 64.53/17.30 0^1(1(3(0(x1)))) -> 5^1(3(1(0(x1)))) 64.53/17.30 2^1(0(1(3(x1)))) -> 2^1(0(5(3(1(x1))))) 64.53/17.30 2^1(0(1(3(x1)))) -> 0^1(5(3(1(x1)))) 64.53/17.30 2^1(0(1(3(x1)))) -> 5^1(3(1(x1))) 64.53/17.30 2^1(0(1(3(x1)))) -> 0^1(3(1(2(3(x1))))) 64.53/17.30 2^1(0(1(3(x1)))) -> 2^1(3(x1)) 64.53/17.30 5^1(0(1(3(x1)))) -> 0^1(3(5(3(1(x1))))) 64.53/17.30 5^1(0(1(3(x1)))) -> 5^1(3(1(x1))) 64.53/17.30 0^1(2(1(3(x1)))) -> 0^1(3(3(1(2(x1))))) 64.53/17.30 0^1(2(1(3(x1)))) -> 2^1(x1) 64.53/17.30 0^1(2(1(3(x1)))) -> 4^1(1(2(0(3(x1))))) 64.53/17.30 0^1(2(1(3(x1)))) -> 2^1(0(3(x1))) 64.53/17.30 0^1(2(1(3(x1)))) -> 0^1(3(x1)) 64.53/17.30 0^1(2(1(3(x1)))) -> 2^1(3(0(3(x1)))) 64.53/17.30 0^1(2(1(3(x1)))) -> 0^1(1(2(3(3(x1))))) 64.53/17.30 0^1(2(1(3(x1)))) -> 2^1(3(3(x1))) 64.53/17.30 0^1(2(1(3(x1)))) -> 0^1(1(2(5(3(x1))))) 64.53/17.30 0^1(2(1(3(x1)))) -> 2^1(5(3(x1))) 64.53/17.30 0^1(2(1(3(x1)))) -> 5^1(3(x1)) 64.53/17.30 2^1(2(1(3(x1)))) -> 2^1(2(5(3(x1)))) 64.53/17.30 2^1(2(1(3(x1)))) -> 2^1(5(3(x1))) 64.53/17.30 2^1(2(1(3(x1)))) -> 5^1(3(x1)) 64.53/17.30 0^1(2(4(3(x1)))) -> 0^1(4(2(3(3(x1))))) 64.53/17.30 0^1(2(4(3(x1)))) -> 4^1(2(3(3(x1)))) 64.53/17.30 0^1(2(4(3(x1)))) -> 2^1(3(3(x1))) 64.53/17.30 0^1(1(0(4(x1)))) -> 0^1(4(0(3(1(x1))))) 64.53/17.30 0^1(1(0(4(x1)))) -> 4^1(0(3(1(x1)))) 64.53/17.30 0^1(1(0(4(x1)))) -> 0^1(3(1(x1))) 64.53/17.30 5^1(0(1(5(x1)))) -> 0^1(5(5(3(3(1(x1)))))) 64.53/17.30 5^1(0(1(5(x1)))) -> 5^1(5(3(3(1(x1))))) 64.53/17.30 5^1(0(1(5(x1)))) -> 5^1(3(3(1(x1)))) 64.53/17.30 0^1(5(2(5(x1)))) -> 0^1(5(3(2(5(x1))))) 64.53/17.30 0^1(5(2(5(x1)))) -> 5^1(3(2(5(x1)))) 64.53/17.30 0^1(1(3(5(x1)))) -> 5^1(5(3(1(0(x1))))) 64.53/17.30 0^1(1(3(5(x1)))) -> 5^1(3(1(0(x1)))) 64.53/17.30 0^1(1(3(5(x1)))) -> 0^1(x1) 64.53/17.30 0^1(1(4(0(0(x1))))) -> 0^1(0(3(1(4(0(x1)))))) 64.53/17.30 0^1(1(4(0(0(x1))))) -> 0^1(3(1(4(0(x1))))) 64.53/17.30 0^1(1(4(0(0(x1))))) -> 4^1(0(x1)) 64.53/17.30 5^1(0(2(1(0(x1))))) -> 2^1(1(5(0(3(0(x1)))))) 64.53/17.30 5^1(0(2(1(0(x1))))) -> 5^1(0(3(0(x1)))) 64.53/17.30 5^1(0(2(1(0(x1))))) -> 0^1(3(0(x1))) 64.53/17.30 2^1(1(3(1(0(x1))))) -> 2^1(2(0(3(x1)))) 64.53/17.30 2^1(1(3(1(0(x1))))) -> 2^1(0(3(x1))) 64.53/17.30 2^1(1(3(1(0(x1))))) -> 0^1(3(x1)) 64.53/17.30 4^1(0(5(1(0(x1))))) -> 0^1(5(3(1(4(0(x1)))))) 64.53/17.30 4^1(0(5(1(0(x1))))) -> 5^1(3(1(4(0(x1))))) 64.53/17.30 4^1(0(5(1(0(x1))))) -> 4^1(0(x1)) 64.53/17.30 5^1(3(5(1(0(x1))))) -> 5^1(3(5(3(1(0(x1)))))) 64.53/17.30 5^1(3(5(1(0(x1))))) -> 5^1(3(1(0(x1)))) 64.53/17.30 4^1(3(0(2(0(x1))))) -> 0^1(3(2(5(4(0(x1)))))) 64.53/17.30 4^1(3(0(2(0(x1))))) -> 2^1(5(4(0(x1)))) 64.53/17.30 4^1(3(0(2(0(x1))))) -> 5^1(4(0(x1))) 64.53/17.30 4^1(3(0(2(0(x1))))) -> 4^1(0(x1)) 64.53/17.30 5^1(1(3(2(0(x1))))) -> 2^1(5(5(3(1(0(x1)))))) 64.53/17.30 5^1(1(3(2(0(x1))))) -> 5^1(5(3(1(0(x1))))) 64.53/17.30 5^1(1(3(2(0(x1))))) -> 5^1(3(1(0(x1)))) 64.53/17.30 2^1(1(4(2(0(x1))))) -> 2^1(4(1(1(2(0(x1)))))) 64.53/17.30 2^1(1(4(2(0(x1))))) -> 4^1(1(1(2(0(x1))))) 64.53/17.30 2^1(0(5(2(0(x1))))) -> 2^1(2(5(3(0(0(x1)))))) 64.53/17.30 2^1(0(5(2(0(x1))))) -> 2^1(5(3(0(0(x1))))) 64.53/17.30 2^1(0(5(2(0(x1))))) -> 5^1(3(0(0(x1)))) 64.53/17.30 2^1(0(5(2(0(x1))))) -> 0^1(0(x1)) 64.53/17.30 0^1(5(5(2(0(x1))))) -> 5^1(1(5(2(0(0(x1)))))) 64.53/17.30 0^1(5(5(2(0(x1))))) -> 5^1(2(0(0(x1)))) 64.53/17.30 0^1(5(5(2(0(x1))))) -> 2^1(0(0(x1))) 64.53/17.30 0^1(5(5(2(0(x1))))) -> 0^1(0(x1)) 64.53/17.30 0^1(5(1(3(0(x1))))) -> 0^1(3(5(0(3(1(x1)))))) 64.53/17.30 0^1(5(1(3(0(x1))))) -> 5^1(0(3(1(x1)))) 64.53/17.30 0^1(5(1(3(0(x1))))) -> 0^1(3(1(x1))) 64.53/17.30 0^1(2(3(5(0(x1))))) -> 5^1(3(1(2(0(0(x1)))))) 64.53/17.30 0^1(2(3(5(0(x1))))) -> 2^1(0(0(x1))) 64.53/17.30 0^1(2(3(5(0(x1))))) -> 0^1(0(x1)) 64.53/17.30 2^1(5(0(1(3(x1))))) -> 0^1(5(3(1(4(2(x1)))))) 64.53/17.30 2^1(5(0(1(3(x1))))) -> 5^1(3(1(4(2(x1))))) 64.53/17.30 2^1(5(0(1(3(x1))))) -> 4^1(2(x1)) 64.53/17.30 2^1(5(0(1(3(x1))))) -> 2^1(x1) 64.53/17.30 2^1(0(3(1(3(x1))))) -> 0^1(3(3(3(1(2(x1)))))) 64.53/17.30 2^1(0(3(1(3(x1))))) -> 2^1(x1) 64.53/17.30 0^1(2(3(1(3(x1))))) -> 2^1(5(0(3(1(3(x1)))))) 64.53/17.30 0^1(2(3(1(3(x1))))) -> 5^1(0(3(1(3(x1))))) 64.53/17.30 0^1(2(3(1(3(x1))))) -> 0^1(3(1(3(x1)))) 64.53/17.30 2^1(1(4(1(3(x1))))) -> 2^1(2(3(4(1(x1))))) 64.53/17.30 2^1(1(4(1(3(x1))))) -> 2^1(3(4(1(x1)))) 64.53/17.30 2^1(1(4(1(3(x1))))) -> 4^1(1(x1)) 64.53/17.30 0^1(2(4(1(3(x1))))) -> 4^1(0(3(3(1(2(x1)))))) 64.53/17.30 0^1(2(4(1(3(x1))))) -> 0^1(3(3(1(2(x1))))) 64.53/17.30 0^1(2(4(1(3(x1))))) -> 2^1(x1) 64.53/17.30 2^1(0(5(1(3(x1))))) -> 5^1(0(3(1(5(2(x1)))))) 64.53/17.30 2^1(0(5(1(3(x1))))) -> 0^1(3(1(5(2(x1))))) 64.53/17.30 2^1(0(5(1(3(x1))))) -> 5^1(2(x1)) 64.53/17.30 2^1(0(5(1(3(x1))))) -> 2^1(x1) 64.53/17.30 0^1(0(1(3(3(x1))))) -> 0^1(0(3(3(1(5(x1)))))) 64.53/17.30 0^1(0(1(3(3(x1))))) -> 0^1(3(3(1(5(x1))))) 64.53/17.30 0^1(0(1(3(3(x1))))) -> 5^1(x1) 64.53/17.30 0^1(2(0(4(3(x1))))) -> 4^1(1(2(0(0(3(x1)))))) 64.53/17.30 0^1(2(0(4(3(x1))))) -> 2^1(0(0(3(x1)))) 64.53/17.30 0^1(2(0(4(3(x1))))) -> 0^1(0(3(x1))) 64.53/17.30 0^1(2(0(4(3(x1))))) -> 0^1(3(x1)) 64.53/17.30 2^1(3(2(4(3(x1))))) -> 2^1(4(5(2(3(3(x1)))))) 64.53/17.30 2^1(3(2(4(3(x1))))) -> 4^1(5(2(3(3(x1))))) 64.53/17.30 2^1(3(2(4(3(x1))))) -> 5^1(2(3(3(x1)))) 64.53/17.30 2^1(3(2(4(3(x1))))) -> 2^1(3(3(x1))) 64.53/17.30 2^1(0(1(3(4(x1))))) -> 0^1(3(1(2(4(x1))))) 64.53/17.30 2^1(0(1(3(4(x1))))) -> 2^1(4(x1)) 64.53/17.30 0^1(2(1(3(4(x1))))) -> 2^1(4(1(0(3(1(x1)))))) 64.53/17.30 0^1(2(1(3(4(x1))))) -> 4^1(1(0(3(1(x1))))) 64.53/17.30 0^1(2(1(3(4(x1))))) -> 0^1(3(1(x1))) 64.53/17.30 2^1(1(5(5(4(x1))))) -> 2^1(5(4(5(x1)))) 64.53/17.30 2^1(1(5(5(4(x1))))) -> 5^1(4(5(x1))) 64.53/17.30 2^1(1(5(5(4(x1))))) -> 4^1(5(x1)) 64.53/17.30 2^1(1(5(5(4(x1))))) -> 5^1(x1) 64.53/17.30 0^1(5(5(5(4(x1))))) -> 4^1(0(5(1(5(5(x1)))))) 64.53/17.30 0^1(5(5(5(4(x1))))) -> 0^1(5(1(5(5(x1))))) 64.53/17.30 0^1(5(5(5(4(x1))))) -> 5^1(1(5(5(x1)))) 64.53/17.30 0^1(5(5(5(4(x1))))) -> 5^1(5(x1)) 64.53/17.30 0^1(5(5(5(4(x1))))) -> 5^1(x1) 64.53/17.30 2^1(1(0(1(5(x1))))) -> 0^1(5(3(1(1(2(x1)))))) 64.53/17.30 2^1(1(0(1(5(x1))))) -> 5^1(3(1(1(2(x1))))) 64.53/17.30 2^1(1(0(1(5(x1))))) -> 2^1(x1) 64.53/17.30 4^1(0(1(1(5(x1))))) -> 0^1(3(4(1(1(5(x1)))))) 64.53/17.30 4^1(0(1(1(5(x1))))) -> 4^1(1(1(5(x1)))) 64.53/17.30 0^1(2(5(1(5(x1))))) -> 5^1(0(5(3(1(2(x1)))))) 64.53/17.30 0^1(2(5(1(5(x1))))) -> 0^1(5(3(1(2(x1))))) 64.53/17.30 0^1(2(5(1(5(x1))))) -> 5^1(3(1(2(x1)))) 64.53/17.30 0^1(2(5(1(5(x1))))) -> 2^1(x1) 64.53/17.30 5^1(0(1(4(5(x1))))) -> 0^1(5(5(3(1(4(x1)))))) 64.53/17.30 5^1(0(1(4(5(x1))))) -> 5^1(5(3(1(4(x1))))) 64.53/17.30 5^1(0(1(4(5(x1))))) -> 5^1(3(1(4(x1)))) 64.53/17.30 5^1(0(1(4(5(x1))))) -> 4^1(x1) 64.53/17.30 64.53/17.30 The TRS R consists of the following rules: 64.53/17.30 64.53/17.30 2(1(0(0(x1)))) -> 0(3(3(1(2(0(x1)))))) 64.53/17.30 4(0(1(0(x1)))) -> 4(0(3(1(0(x1))))) 64.53/17.30 5(0(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(5(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(1(3(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 2(0(1(3(x1)))) -> 2(0(5(3(1(x1))))) 64.53/17.30 2(0(1(3(x1)))) -> 0(3(1(2(3(x1))))) 64.53/17.30 5(0(1(3(x1)))) -> 0(3(5(3(1(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(3(3(1(2(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 4(1(2(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 1(2(3(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(3(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(5(3(x1))))) 64.53/17.30 2(2(1(3(x1)))) -> 1(2(2(5(3(x1))))) 64.53/17.30 0(2(4(3(x1)))) -> 0(4(2(3(3(x1))))) 64.53/17.30 0(1(0(4(x1)))) -> 0(4(0(3(1(x1))))) 64.53/17.30 5(0(1(5(x1)))) -> 0(5(5(3(3(1(x1)))))) 64.53/17.30 0(5(2(5(x1)))) -> 0(5(3(2(5(x1))))) 64.53/17.30 0(1(3(5(x1)))) -> 5(5(3(1(0(x1))))) 64.53/17.30 0(1(4(0(0(x1))))) -> 0(0(3(1(4(0(x1)))))) 64.53/17.30 5(0(2(1(0(x1))))) -> 2(1(5(0(3(0(x1)))))) 64.53/17.30 2(1(3(1(0(x1))))) -> 1(1(2(2(0(3(x1)))))) 64.53/17.30 4(0(5(1(0(x1))))) -> 0(5(3(1(4(0(x1)))))) 64.53/17.30 5(3(5(1(0(x1))))) -> 5(3(5(3(1(0(x1)))))) 64.53/17.30 4(3(0(2(0(x1))))) -> 0(3(2(5(4(0(x1)))))) 64.53/17.30 5(1(3(2(0(x1))))) -> 2(5(5(3(1(0(x1)))))) 64.53/17.30 2(1(4(2(0(x1))))) -> 2(4(1(1(2(0(x1)))))) 64.53/17.30 2(0(5(2(0(x1))))) -> 2(2(5(3(0(0(x1)))))) 64.53/17.30 0(5(5(2(0(x1))))) -> 5(1(5(2(0(0(x1)))))) 64.53/17.30 0(5(1(3(0(x1))))) -> 0(3(5(0(3(1(x1)))))) 64.53/17.30 0(2(3(5(0(x1))))) -> 5(3(1(2(0(0(x1)))))) 64.53/17.30 2(5(0(1(3(x1))))) -> 0(5(3(1(4(2(x1)))))) 64.53/17.30 2(0(3(1(3(x1))))) -> 0(3(3(3(1(2(x1)))))) 64.53/17.30 0(2(3(1(3(x1))))) -> 2(5(0(3(1(3(x1)))))) 64.53/17.30 2(1(4(1(3(x1))))) -> 1(2(2(3(4(1(x1)))))) 64.53/17.30 0(2(4(1(3(x1))))) -> 4(0(3(3(1(2(x1)))))) 64.53/17.30 2(0(5(1(3(x1))))) -> 5(0(3(1(5(2(x1)))))) 64.53/17.30 0(0(1(3(3(x1))))) -> 0(0(3(3(1(5(x1)))))) 64.53/17.30 0(2(0(4(3(x1))))) -> 4(1(2(0(0(3(x1)))))) 64.53/17.30 2(3(2(4(3(x1))))) -> 2(4(5(2(3(3(x1)))))) 64.53/17.30 2(0(1(3(4(x1))))) -> 1(0(3(1(2(4(x1)))))) 64.53/17.30 0(2(1(3(4(x1))))) -> 2(4(1(0(3(1(x1)))))) 64.53/17.30 2(1(5(5(4(x1))))) -> 1(1(2(5(4(5(x1)))))) 64.53/17.30 0(5(5(5(4(x1))))) -> 4(0(5(1(5(5(x1)))))) 64.53/17.30 2(1(0(1(5(x1))))) -> 0(5(3(1(1(2(x1)))))) 64.53/17.30 4(0(1(1(5(x1))))) -> 0(3(4(1(1(5(x1)))))) 64.53/17.30 0(2(5(1(5(x1))))) -> 5(0(5(3(1(2(x1)))))) 64.53/17.30 5(0(1(4(5(x1))))) -> 0(5(5(3(1(4(x1)))))) 64.53/17.30 64.53/17.30 Q is empty. 64.53/17.30 We have to consider all minimal (P,Q,R)-chains. 64.53/17.30 ---------------------------------------- 64.53/17.30 64.53/17.30 (5) DependencyGraphProof (EQUIVALENT) 64.53/17.30 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 108 less nodes. 64.53/17.30 ---------------------------------------- 64.53/17.30 64.53/17.30 (6) 64.53/17.30 Complex Obligation (AND) 64.53/17.30 64.53/17.30 ---------------------------------------- 64.53/17.30 64.53/17.30 (7) 64.53/17.30 Obligation: 64.53/17.30 Q DP problem: 64.53/17.30 The TRS P consists of the following rules: 64.53/17.30 64.53/17.30 4^1(0(5(1(0(x1))))) -> 4^1(0(x1)) 64.53/17.30 64.53/17.30 The TRS R consists of the following rules: 64.53/17.30 64.53/17.30 2(1(0(0(x1)))) -> 0(3(3(1(2(0(x1)))))) 64.53/17.30 4(0(1(0(x1)))) -> 4(0(3(1(0(x1))))) 64.53/17.30 5(0(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(5(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(1(3(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 2(0(1(3(x1)))) -> 2(0(5(3(1(x1))))) 64.53/17.30 2(0(1(3(x1)))) -> 0(3(1(2(3(x1))))) 64.53/17.30 5(0(1(3(x1)))) -> 0(3(5(3(1(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(3(3(1(2(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 4(1(2(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 1(2(3(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(3(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(5(3(x1))))) 64.53/17.30 2(2(1(3(x1)))) -> 1(2(2(5(3(x1))))) 64.53/17.30 0(2(4(3(x1)))) -> 0(4(2(3(3(x1))))) 64.53/17.30 0(1(0(4(x1)))) -> 0(4(0(3(1(x1))))) 64.53/17.30 5(0(1(5(x1)))) -> 0(5(5(3(3(1(x1)))))) 64.53/17.30 0(5(2(5(x1)))) -> 0(5(3(2(5(x1))))) 64.53/17.30 0(1(3(5(x1)))) -> 5(5(3(1(0(x1))))) 64.53/17.30 0(1(4(0(0(x1))))) -> 0(0(3(1(4(0(x1)))))) 64.53/17.30 5(0(2(1(0(x1))))) -> 2(1(5(0(3(0(x1)))))) 64.53/17.30 2(1(3(1(0(x1))))) -> 1(1(2(2(0(3(x1)))))) 64.53/17.30 4(0(5(1(0(x1))))) -> 0(5(3(1(4(0(x1)))))) 64.53/17.30 5(3(5(1(0(x1))))) -> 5(3(5(3(1(0(x1)))))) 64.53/17.30 4(3(0(2(0(x1))))) -> 0(3(2(5(4(0(x1)))))) 64.53/17.30 5(1(3(2(0(x1))))) -> 2(5(5(3(1(0(x1)))))) 64.53/17.30 2(1(4(2(0(x1))))) -> 2(4(1(1(2(0(x1)))))) 64.53/17.30 2(0(5(2(0(x1))))) -> 2(2(5(3(0(0(x1)))))) 64.53/17.30 0(5(5(2(0(x1))))) -> 5(1(5(2(0(0(x1)))))) 64.53/17.30 0(5(1(3(0(x1))))) -> 0(3(5(0(3(1(x1)))))) 64.53/17.30 0(2(3(5(0(x1))))) -> 5(3(1(2(0(0(x1)))))) 64.53/17.30 2(5(0(1(3(x1))))) -> 0(5(3(1(4(2(x1)))))) 64.53/17.30 2(0(3(1(3(x1))))) -> 0(3(3(3(1(2(x1)))))) 64.53/17.30 0(2(3(1(3(x1))))) -> 2(5(0(3(1(3(x1)))))) 64.53/17.30 2(1(4(1(3(x1))))) -> 1(2(2(3(4(1(x1)))))) 64.53/17.30 0(2(4(1(3(x1))))) -> 4(0(3(3(1(2(x1)))))) 64.53/17.30 2(0(5(1(3(x1))))) -> 5(0(3(1(5(2(x1)))))) 64.53/17.30 0(0(1(3(3(x1))))) -> 0(0(3(3(1(5(x1)))))) 64.53/17.30 0(2(0(4(3(x1))))) -> 4(1(2(0(0(3(x1)))))) 64.53/17.30 2(3(2(4(3(x1))))) -> 2(4(5(2(3(3(x1)))))) 64.53/17.30 2(0(1(3(4(x1))))) -> 1(0(3(1(2(4(x1)))))) 64.53/17.30 0(2(1(3(4(x1))))) -> 2(4(1(0(3(1(x1)))))) 64.53/17.30 2(1(5(5(4(x1))))) -> 1(1(2(5(4(5(x1)))))) 64.53/17.30 0(5(5(5(4(x1))))) -> 4(0(5(1(5(5(x1)))))) 64.53/17.30 2(1(0(1(5(x1))))) -> 0(5(3(1(1(2(x1)))))) 64.53/17.30 4(0(1(1(5(x1))))) -> 0(3(4(1(1(5(x1)))))) 64.53/17.30 0(2(5(1(5(x1))))) -> 5(0(5(3(1(2(x1)))))) 64.53/17.30 5(0(1(4(5(x1))))) -> 0(5(5(3(1(4(x1)))))) 64.53/17.30 64.53/17.30 Q is empty. 64.53/17.30 We have to consider all minimal (P,Q,R)-chains. 64.53/17.30 ---------------------------------------- 64.53/17.30 64.53/17.30 (8) QDPSizeChangeProof (EQUIVALENT) 64.53/17.30 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 64.53/17.30 64.53/17.30 From the DPs we obtained the following set of size-change graphs: 64.53/17.30 *4^1(0(5(1(0(x1))))) -> 4^1(0(x1)) 64.53/17.30 The graph contains the following edges 1 > 1 64.53/17.30 64.53/17.30 64.53/17.30 ---------------------------------------- 64.53/17.30 64.53/17.30 (9) 64.53/17.30 YES 64.53/17.30 64.53/17.30 ---------------------------------------- 64.53/17.30 64.53/17.30 (10) 64.53/17.30 Obligation: 64.53/17.30 Q DP problem: 64.53/17.30 The TRS P consists of the following rules: 64.53/17.30 64.53/17.30 2^1(0(5(2(0(x1))))) -> 2^1(2(5(3(0(0(x1)))))) 64.53/17.30 2^1(1(0(0(x1)))) -> 2^1(0(x1)) 64.53/17.30 2^1(0(5(2(0(x1))))) -> 2^1(5(3(0(0(x1))))) 64.53/17.30 2^1(0(5(2(0(x1))))) -> 0^1(0(x1)) 64.53/17.30 0^1(5(2(5(x1)))) -> 0^1(5(3(2(5(x1))))) 64.53/17.30 0^1(5(5(2(0(x1))))) -> 5^1(2(0(0(x1)))) 64.53/17.30 5^1(0(1(4(5(x1))))) -> 4^1(x1) 64.53/17.30 4^1(3(0(2(0(x1))))) -> 2^1(5(4(0(x1)))) 64.53/17.30 2^1(1(5(5(4(x1))))) -> 2^1(5(4(5(x1)))) 64.53/17.30 2^1(1(5(5(4(x1))))) -> 5^1(4(5(x1))) 64.53/17.30 2^1(1(5(5(4(x1))))) -> 5^1(x1) 64.53/17.30 2^1(1(0(1(5(x1))))) -> 2^1(x1) 64.53/17.30 2^1(2(1(3(x1)))) -> 2^1(2(5(3(x1)))) 64.53/17.30 2^1(2(1(3(x1)))) -> 2^1(5(3(x1))) 64.53/17.30 2^1(1(3(1(0(x1))))) -> 2^1(2(0(3(x1)))) 64.53/17.30 2^1(1(3(1(0(x1))))) -> 2^1(0(3(x1))) 64.53/17.30 2^1(0(3(1(3(x1))))) -> 2^1(x1) 64.53/17.30 2^1(5(0(1(3(x1))))) -> 2^1(x1) 64.53/17.30 2^1(0(5(1(3(x1))))) -> 5^1(2(x1)) 64.53/17.30 2^1(0(5(1(3(x1))))) -> 2^1(x1) 64.53/17.30 2^1(0(1(3(4(x1))))) -> 2^1(4(x1)) 64.53/17.30 4^1(3(0(2(0(x1))))) -> 5^1(4(0(x1))) 64.53/17.30 0^1(5(5(2(0(x1))))) -> 2^1(0(0(x1))) 64.53/17.30 0^1(5(5(2(0(x1))))) -> 0^1(0(x1)) 64.53/17.30 0^1(0(1(3(3(x1))))) -> 5^1(x1) 64.53/17.30 0^1(5(5(5(4(x1))))) -> 0^1(5(1(5(5(x1))))) 64.53/17.30 0^1(5(5(5(4(x1))))) -> 5^1(5(x1)) 64.53/17.30 0^1(5(5(5(4(x1))))) -> 5^1(x1) 64.53/17.30 0^1(2(5(1(5(x1))))) -> 2^1(x1) 64.53/17.30 64.53/17.30 The TRS R consists of the following rules: 64.53/17.30 64.53/17.30 2(1(0(0(x1)))) -> 0(3(3(1(2(0(x1)))))) 64.53/17.30 4(0(1(0(x1)))) -> 4(0(3(1(0(x1))))) 64.53/17.30 5(0(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(5(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(1(3(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 2(0(1(3(x1)))) -> 2(0(5(3(1(x1))))) 64.53/17.30 2(0(1(3(x1)))) -> 0(3(1(2(3(x1))))) 64.53/17.30 5(0(1(3(x1)))) -> 0(3(5(3(1(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(3(3(1(2(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 4(1(2(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 1(2(3(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(3(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(5(3(x1))))) 64.53/17.30 2(2(1(3(x1)))) -> 1(2(2(5(3(x1))))) 64.53/17.30 0(2(4(3(x1)))) -> 0(4(2(3(3(x1))))) 64.53/17.30 0(1(0(4(x1)))) -> 0(4(0(3(1(x1))))) 64.53/17.30 5(0(1(5(x1)))) -> 0(5(5(3(3(1(x1)))))) 64.53/17.30 0(5(2(5(x1)))) -> 0(5(3(2(5(x1))))) 64.53/17.30 0(1(3(5(x1)))) -> 5(5(3(1(0(x1))))) 64.53/17.30 0(1(4(0(0(x1))))) -> 0(0(3(1(4(0(x1)))))) 64.53/17.30 5(0(2(1(0(x1))))) -> 2(1(5(0(3(0(x1)))))) 64.53/17.30 2(1(3(1(0(x1))))) -> 1(1(2(2(0(3(x1)))))) 64.53/17.30 4(0(5(1(0(x1))))) -> 0(5(3(1(4(0(x1)))))) 64.53/17.30 5(3(5(1(0(x1))))) -> 5(3(5(3(1(0(x1)))))) 64.53/17.30 4(3(0(2(0(x1))))) -> 0(3(2(5(4(0(x1)))))) 64.53/17.30 5(1(3(2(0(x1))))) -> 2(5(5(3(1(0(x1)))))) 64.53/17.30 2(1(4(2(0(x1))))) -> 2(4(1(1(2(0(x1)))))) 64.53/17.30 2(0(5(2(0(x1))))) -> 2(2(5(3(0(0(x1)))))) 64.53/17.30 0(5(5(2(0(x1))))) -> 5(1(5(2(0(0(x1)))))) 64.53/17.30 0(5(1(3(0(x1))))) -> 0(3(5(0(3(1(x1)))))) 64.53/17.30 0(2(3(5(0(x1))))) -> 5(3(1(2(0(0(x1)))))) 64.53/17.30 2(5(0(1(3(x1))))) -> 0(5(3(1(4(2(x1)))))) 64.53/17.30 2(0(3(1(3(x1))))) -> 0(3(3(3(1(2(x1)))))) 64.53/17.30 0(2(3(1(3(x1))))) -> 2(5(0(3(1(3(x1)))))) 64.53/17.30 2(1(4(1(3(x1))))) -> 1(2(2(3(4(1(x1)))))) 64.53/17.30 0(2(4(1(3(x1))))) -> 4(0(3(3(1(2(x1)))))) 64.53/17.30 2(0(5(1(3(x1))))) -> 5(0(3(1(5(2(x1)))))) 64.53/17.30 0(0(1(3(3(x1))))) -> 0(0(3(3(1(5(x1)))))) 64.53/17.30 0(2(0(4(3(x1))))) -> 4(1(2(0(0(3(x1)))))) 64.53/17.30 2(3(2(4(3(x1))))) -> 2(4(5(2(3(3(x1)))))) 64.53/17.30 2(0(1(3(4(x1))))) -> 1(0(3(1(2(4(x1)))))) 64.53/17.30 0(2(1(3(4(x1))))) -> 2(4(1(0(3(1(x1)))))) 64.53/17.30 2(1(5(5(4(x1))))) -> 1(1(2(5(4(5(x1)))))) 64.53/17.30 0(5(5(5(4(x1))))) -> 4(0(5(1(5(5(x1)))))) 64.53/17.30 2(1(0(1(5(x1))))) -> 0(5(3(1(1(2(x1)))))) 64.53/17.30 4(0(1(1(5(x1))))) -> 0(3(4(1(1(5(x1)))))) 64.53/17.30 0(2(5(1(5(x1))))) -> 5(0(5(3(1(2(x1)))))) 64.53/17.30 5(0(1(4(5(x1))))) -> 0(5(5(3(1(4(x1)))))) 64.53/17.30 64.53/17.30 Q is empty. 64.53/17.30 We have to consider all minimal (P,Q,R)-chains. 64.53/17.30 ---------------------------------------- 64.53/17.30 64.53/17.30 (11) QDPOrderProof (EQUIVALENT) 64.53/17.30 We use the reduction pair processor [LPAR04,JAR06]. 64.53/17.30 64.53/17.30 64.53/17.30 The following pairs can be oriented strictly and are deleted. 64.53/17.30 64.53/17.30 0^1(5(2(5(x1)))) -> 0^1(5(3(2(5(x1))))) 64.53/17.30 The remaining pairs can at least be oriented weakly. 64.53/17.30 Used ordering: Polynomial interpretation [POLO]: 64.53/17.30 64.53/17.30 POL(0(x_1)) = 1 64.53/17.30 POL(0^1(x_1)) = x_1 64.53/17.30 POL(1(x_1)) = 1 64.53/17.30 POL(2(x_1)) = 1 64.53/17.30 POL(2^1(x_1)) = 1 64.53/17.30 POL(3(x_1)) = 0 64.53/17.30 POL(4(x_1)) = 1 64.53/17.30 POL(4^1(x_1)) = 1 64.53/17.30 POL(5(x_1)) = x_1 64.53/17.30 POL(5^1(x_1)) = 1 64.53/17.30 64.53/17.30 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 64.53/17.30 64.53/17.30 0(5(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(1(3(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(3(3(1(2(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 4(1(2(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 1(2(3(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(3(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(5(3(x1))))) 64.53/17.30 0(2(4(3(x1)))) -> 0(4(2(3(3(x1))))) 64.53/17.30 0(1(0(4(x1)))) -> 0(4(0(3(1(x1))))) 64.53/17.30 0(5(2(5(x1)))) -> 0(5(3(2(5(x1))))) 64.53/17.30 0(1(3(5(x1)))) -> 5(5(3(1(0(x1))))) 64.53/17.30 0(1(4(0(0(x1))))) -> 0(0(3(1(4(0(x1)))))) 64.53/17.30 0(5(5(2(0(x1))))) -> 5(1(5(2(0(0(x1)))))) 64.53/17.30 0(5(1(3(0(x1))))) -> 0(3(5(0(3(1(x1)))))) 64.53/17.30 0(2(3(5(0(x1))))) -> 5(3(1(2(0(0(x1)))))) 64.53/17.30 0(2(3(1(3(x1))))) -> 2(5(0(3(1(3(x1)))))) 64.53/17.30 0(2(4(1(3(x1))))) -> 4(0(3(3(1(2(x1)))))) 64.53/17.30 0(0(1(3(3(x1))))) -> 0(0(3(3(1(5(x1)))))) 64.53/17.30 0(2(0(4(3(x1))))) -> 4(1(2(0(0(3(x1)))))) 64.53/17.30 0(2(1(3(4(x1))))) -> 2(4(1(0(3(1(x1)))))) 64.53/17.30 0(5(5(5(4(x1))))) -> 4(0(5(1(5(5(x1)))))) 64.53/17.30 0(2(5(1(5(x1))))) -> 5(0(5(3(1(2(x1)))))) 64.53/17.30 5(3(5(1(0(x1))))) -> 5(3(5(3(1(0(x1)))))) 64.53/17.30 5(1(3(2(0(x1))))) -> 2(5(5(3(1(0(x1)))))) 64.53/17.30 4(0(1(0(x1)))) -> 4(0(3(1(0(x1))))) 64.53/17.30 4(0(5(1(0(x1))))) -> 0(5(3(1(4(0(x1)))))) 64.53/17.30 4(3(0(2(0(x1))))) -> 0(3(2(5(4(0(x1)))))) 64.53/17.30 4(0(1(1(5(x1))))) -> 0(3(4(1(1(5(x1)))))) 64.53/17.30 64.53/17.30 64.53/17.30 ---------------------------------------- 64.53/17.30 64.53/17.30 (12) 64.53/17.30 Obligation: 64.53/17.30 Q DP problem: 64.53/17.30 The TRS P consists of the following rules: 64.53/17.30 64.53/17.30 2^1(0(5(2(0(x1))))) -> 2^1(2(5(3(0(0(x1)))))) 64.53/17.30 2^1(1(0(0(x1)))) -> 2^1(0(x1)) 64.53/17.30 2^1(0(5(2(0(x1))))) -> 2^1(5(3(0(0(x1))))) 64.53/17.30 2^1(0(5(2(0(x1))))) -> 0^1(0(x1)) 64.53/17.30 0^1(5(5(2(0(x1))))) -> 5^1(2(0(0(x1)))) 64.53/17.30 5^1(0(1(4(5(x1))))) -> 4^1(x1) 64.53/17.30 4^1(3(0(2(0(x1))))) -> 2^1(5(4(0(x1)))) 64.53/17.30 2^1(1(5(5(4(x1))))) -> 2^1(5(4(5(x1)))) 64.53/17.30 2^1(1(5(5(4(x1))))) -> 5^1(4(5(x1))) 64.53/17.30 2^1(1(5(5(4(x1))))) -> 5^1(x1) 64.53/17.30 2^1(1(0(1(5(x1))))) -> 2^1(x1) 64.53/17.30 2^1(2(1(3(x1)))) -> 2^1(2(5(3(x1)))) 64.53/17.30 2^1(2(1(3(x1)))) -> 2^1(5(3(x1))) 64.53/17.30 2^1(1(3(1(0(x1))))) -> 2^1(2(0(3(x1)))) 64.53/17.30 2^1(1(3(1(0(x1))))) -> 2^1(0(3(x1))) 64.53/17.30 2^1(0(3(1(3(x1))))) -> 2^1(x1) 64.53/17.30 2^1(5(0(1(3(x1))))) -> 2^1(x1) 64.53/17.30 2^1(0(5(1(3(x1))))) -> 5^1(2(x1)) 64.53/17.30 2^1(0(5(1(3(x1))))) -> 2^1(x1) 64.53/17.30 2^1(0(1(3(4(x1))))) -> 2^1(4(x1)) 64.53/17.30 4^1(3(0(2(0(x1))))) -> 5^1(4(0(x1))) 64.53/17.30 0^1(5(5(2(0(x1))))) -> 2^1(0(0(x1))) 64.53/17.30 0^1(5(5(2(0(x1))))) -> 0^1(0(x1)) 64.53/17.30 0^1(0(1(3(3(x1))))) -> 5^1(x1) 64.53/17.30 0^1(5(5(5(4(x1))))) -> 0^1(5(1(5(5(x1))))) 64.53/17.30 0^1(5(5(5(4(x1))))) -> 5^1(5(x1)) 64.53/17.30 0^1(5(5(5(4(x1))))) -> 5^1(x1) 64.53/17.30 0^1(2(5(1(5(x1))))) -> 2^1(x1) 64.53/17.30 64.53/17.30 The TRS R consists of the following rules: 64.53/17.30 64.53/17.30 2(1(0(0(x1)))) -> 0(3(3(1(2(0(x1)))))) 64.53/17.30 4(0(1(0(x1)))) -> 4(0(3(1(0(x1))))) 64.53/17.30 5(0(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(5(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(1(3(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 2(0(1(3(x1)))) -> 2(0(5(3(1(x1))))) 64.53/17.30 2(0(1(3(x1)))) -> 0(3(1(2(3(x1))))) 64.53/17.30 5(0(1(3(x1)))) -> 0(3(5(3(1(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(3(3(1(2(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 4(1(2(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 1(2(3(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(3(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(5(3(x1))))) 64.53/17.30 2(2(1(3(x1)))) -> 1(2(2(5(3(x1))))) 64.53/17.30 0(2(4(3(x1)))) -> 0(4(2(3(3(x1))))) 64.53/17.30 0(1(0(4(x1)))) -> 0(4(0(3(1(x1))))) 64.53/17.30 5(0(1(5(x1)))) -> 0(5(5(3(3(1(x1)))))) 64.53/17.30 0(5(2(5(x1)))) -> 0(5(3(2(5(x1))))) 64.53/17.30 0(1(3(5(x1)))) -> 5(5(3(1(0(x1))))) 64.53/17.30 0(1(4(0(0(x1))))) -> 0(0(3(1(4(0(x1)))))) 64.53/17.30 5(0(2(1(0(x1))))) -> 2(1(5(0(3(0(x1)))))) 64.53/17.30 2(1(3(1(0(x1))))) -> 1(1(2(2(0(3(x1)))))) 64.53/17.30 4(0(5(1(0(x1))))) -> 0(5(3(1(4(0(x1)))))) 64.53/17.30 5(3(5(1(0(x1))))) -> 5(3(5(3(1(0(x1)))))) 64.53/17.30 4(3(0(2(0(x1))))) -> 0(3(2(5(4(0(x1)))))) 64.53/17.30 5(1(3(2(0(x1))))) -> 2(5(5(3(1(0(x1)))))) 64.53/17.30 2(1(4(2(0(x1))))) -> 2(4(1(1(2(0(x1)))))) 64.53/17.30 2(0(5(2(0(x1))))) -> 2(2(5(3(0(0(x1)))))) 64.53/17.30 0(5(5(2(0(x1))))) -> 5(1(5(2(0(0(x1)))))) 64.53/17.30 0(5(1(3(0(x1))))) -> 0(3(5(0(3(1(x1)))))) 64.53/17.30 0(2(3(5(0(x1))))) -> 5(3(1(2(0(0(x1)))))) 64.53/17.30 2(5(0(1(3(x1))))) -> 0(5(3(1(4(2(x1)))))) 64.53/17.30 2(0(3(1(3(x1))))) -> 0(3(3(3(1(2(x1)))))) 64.53/17.30 0(2(3(1(3(x1))))) -> 2(5(0(3(1(3(x1)))))) 64.53/17.30 2(1(4(1(3(x1))))) -> 1(2(2(3(4(1(x1)))))) 64.53/17.30 0(2(4(1(3(x1))))) -> 4(0(3(3(1(2(x1)))))) 64.53/17.30 2(0(5(1(3(x1))))) -> 5(0(3(1(5(2(x1)))))) 64.53/17.30 0(0(1(3(3(x1))))) -> 0(0(3(3(1(5(x1)))))) 64.53/17.30 0(2(0(4(3(x1))))) -> 4(1(2(0(0(3(x1)))))) 64.53/17.30 2(3(2(4(3(x1))))) -> 2(4(5(2(3(3(x1)))))) 64.53/17.30 2(0(1(3(4(x1))))) -> 1(0(3(1(2(4(x1)))))) 64.53/17.30 0(2(1(3(4(x1))))) -> 2(4(1(0(3(1(x1)))))) 64.53/17.30 2(1(5(5(4(x1))))) -> 1(1(2(5(4(5(x1)))))) 64.53/17.30 0(5(5(5(4(x1))))) -> 4(0(5(1(5(5(x1)))))) 64.53/17.30 2(1(0(1(5(x1))))) -> 0(5(3(1(1(2(x1)))))) 64.53/17.30 4(0(1(1(5(x1))))) -> 0(3(4(1(1(5(x1)))))) 64.53/17.30 0(2(5(1(5(x1))))) -> 5(0(5(3(1(2(x1)))))) 64.53/17.30 5(0(1(4(5(x1))))) -> 0(5(5(3(1(4(x1)))))) 64.53/17.30 64.53/17.30 Q is empty. 64.53/17.30 We have to consider all minimal (P,Q,R)-chains. 64.53/17.30 ---------------------------------------- 64.53/17.30 64.53/17.30 (13) QDPOrderProof (EQUIVALENT) 64.53/17.30 We use the reduction pair processor [LPAR04,JAR06]. 64.53/17.30 64.53/17.30 64.53/17.30 The following pairs can be oriented strictly and are deleted. 64.53/17.30 64.53/17.30 2^1(1(0(0(x1)))) -> 2^1(0(x1)) 64.53/17.30 5^1(0(1(4(5(x1))))) -> 4^1(x1) 64.53/17.30 4^1(3(0(2(0(x1))))) -> 2^1(5(4(0(x1)))) 64.53/17.30 2^1(1(0(1(5(x1))))) -> 2^1(x1) 64.53/17.30 2^1(0(3(1(3(x1))))) -> 2^1(x1) 64.53/17.30 2^1(5(0(1(3(x1))))) -> 2^1(x1) 64.53/17.30 2^1(0(5(1(3(x1))))) -> 5^1(2(x1)) 64.53/17.30 2^1(0(5(1(3(x1))))) -> 2^1(x1) 64.53/17.30 2^1(0(1(3(4(x1))))) -> 2^1(4(x1)) 64.53/17.30 4^1(3(0(2(0(x1))))) -> 5^1(4(0(x1))) 64.53/17.30 0^1(0(1(3(3(x1))))) -> 5^1(x1) 64.53/17.30 0^1(5(5(5(4(x1))))) -> 5^1(5(x1)) 64.53/17.30 0^1(5(5(5(4(x1))))) -> 5^1(x1) 64.53/17.30 0^1(2(5(1(5(x1))))) -> 2^1(x1) 64.53/17.30 The remaining pairs can at least be oriented weakly. 64.53/17.30 Used ordering: Polynomial interpretation [POLO]: 64.53/17.30 64.53/17.30 POL(0(x_1)) = 1 + x_1 64.53/17.30 POL(0^1(x_1)) = 1 + x_1 64.53/17.30 POL(1(x_1)) = x_1 64.53/17.30 POL(2(x_1)) = x_1 64.53/17.30 POL(2^1(x_1)) = x_1 64.53/17.30 POL(3(x_1)) = x_1 64.53/17.30 POL(4(x_1)) = x_1 64.53/17.30 POL(4^1(x_1)) = x_1 64.53/17.30 POL(5(x_1)) = x_1 64.53/17.30 POL(5^1(x_1)) = x_1 64.53/17.30 64.53/17.30 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 64.53/17.30 64.53/17.30 0(5(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(1(3(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(3(3(1(2(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 4(1(2(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 1(2(3(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(3(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(5(3(x1))))) 64.53/17.30 0(2(4(3(x1)))) -> 0(4(2(3(3(x1))))) 64.53/17.30 0(1(0(4(x1)))) -> 0(4(0(3(1(x1))))) 64.53/17.30 0(5(2(5(x1)))) -> 0(5(3(2(5(x1))))) 64.53/17.30 0(1(3(5(x1)))) -> 5(5(3(1(0(x1))))) 64.53/17.30 0(1(4(0(0(x1))))) -> 0(0(3(1(4(0(x1)))))) 64.53/17.30 0(5(5(2(0(x1))))) -> 5(1(5(2(0(0(x1)))))) 64.53/17.30 0(5(1(3(0(x1))))) -> 0(3(5(0(3(1(x1)))))) 64.53/17.30 0(2(3(5(0(x1))))) -> 5(3(1(2(0(0(x1)))))) 64.53/17.30 0(2(3(1(3(x1))))) -> 2(5(0(3(1(3(x1)))))) 64.53/17.30 0(2(4(1(3(x1))))) -> 4(0(3(3(1(2(x1)))))) 64.53/17.30 0(0(1(3(3(x1))))) -> 0(0(3(3(1(5(x1)))))) 64.53/17.30 0(2(0(4(3(x1))))) -> 4(1(2(0(0(3(x1)))))) 64.53/17.30 0(2(1(3(4(x1))))) -> 2(4(1(0(3(1(x1)))))) 64.53/17.30 0(5(5(5(4(x1))))) -> 4(0(5(1(5(5(x1)))))) 64.53/17.30 0(2(5(1(5(x1))))) -> 5(0(5(3(1(2(x1)))))) 64.53/17.30 5(3(5(1(0(x1))))) -> 5(3(5(3(1(0(x1)))))) 64.53/17.30 2(1(0(0(x1)))) -> 0(3(3(1(2(0(x1)))))) 64.53/17.30 2(0(1(3(x1)))) -> 2(0(5(3(1(x1))))) 64.53/17.30 2(0(1(3(x1)))) -> 0(3(1(2(3(x1))))) 64.53/17.30 2(2(1(3(x1)))) -> 1(2(2(5(3(x1))))) 64.53/17.30 2(1(3(1(0(x1))))) -> 1(1(2(2(0(3(x1)))))) 64.53/17.30 2(1(4(2(0(x1))))) -> 2(4(1(1(2(0(x1)))))) 64.53/17.30 2(0(5(2(0(x1))))) -> 2(2(5(3(0(0(x1)))))) 64.53/17.30 2(5(0(1(3(x1))))) -> 0(5(3(1(4(2(x1)))))) 64.53/17.30 2(0(3(1(3(x1))))) -> 0(3(3(3(1(2(x1)))))) 64.53/17.30 2(1(4(1(3(x1))))) -> 1(2(2(3(4(1(x1)))))) 64.53/17.30 2(0(5(1(3(x1))))) -> 5(0(3(1(5(2(x1)))))) 64.53/17.30 2(3(2(4(3(x1))))) -> 2(4(5(2(3(3(x1)))))) 64.53/17.30 2(0(1(3(4(x1))))) -> 1(0(3(1(2(4(x1)))))) 64.53/17.30 2(1(5(5(4(x1))))) -> 1(1(2(5(4(5(x1)))))) 64.53/17.30 2(1(0(1(5(x1))))) -> 0(5(3(1(1(2(x1)))))) 64.53/17.30 4(0(1(0(x1)))) -> 4(0(3(1(0(x1))))) 64.53/17.30 4(0(5(1(0(x1))))) -> 0(5(3(1(4(0(x1)))))) 64.53/17.30 4(3(0(2(0(x1))))) -> 0(3(2(5(4(0(x1)))))) 64.53/17.30 4(0(1(1(5(x1))))) -> 0(3(4(1(1(5(x1)))))) 64.53/17.30 5(0(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 5(0(1(3(x1)))) -> 0(3(5(3(1(x1))))) 64.53/17.30 5(0(1(5(x1)))) -> 0(5(5(3(3(1(x1)))))) 64.53/17.30 5(0(2(1(0(x1))))) -> 2(1(5(0(3(0(x1)))))) 64.53/17.30 5(1(3(2(0(x1))))) -> 2(5(5(3(1(0(x1)))))) 64.53/17.30 5(0(1(4(5(x1))))) -> 0(5(5(3(1(4(x1)))))) 64.53/17.30 64.53/17.30 64.53/17.30 ---------------------------------------- 64.53/17.30 64.53/17.30 (14) 64.53/17.30 Obligation: 64.53/17.30 Q DP problem: 64.53/17.30 The TRS P consists of the following rules: 64.53/17.30 64.53/17.30 2^1(0(5(2(0(x1))))) -> 2^1(2(5(3(0(0(x1)))))) 64.53/17.30 2^1(0(5(2(0(x1))))) -> 2^1(5(3(0(0(x1))))) 64.53/17.30 2^1(0(5(2(0(x1))))) -> 0^1(0(x1)) 64.53/17.30 0^1(5(5(2(0(x1))))) -> 5^1(2(0(0(x1)))) 64.53/17.30 2^1(1(5(5(4(x1))))) -> 2^1(5(4(5(x1)))) 64.53/17.30 2^1(1(5(5(4(x1))))) -> 5^1(4(5(x1))) 64.53/17.30 2^1(1(5(5(4(x1))))) -> 5^1(x1) 64.53/17.30 2^1(2(1(3(x1)))) -> 2^1(2(5(3(x1)))) 64.53/17.30 2^1(2(1(3(x1)))) -> 2^1(5(3(x1))) 64.53/17.30 2^1(1(3(1(0(x1))))) -> 2^1(2(0(3(x1)))) 64.53/17.30 2^1(1(3(1(0(x1))))) -> 2^1(0(3(x1))) 64.53/17.30 0^1(5(5(2(0(x1))))) -> 2^1(0(0(x1))) 64.53/17.30 0^1(5(5(2(0(x1))))) -> 0^1(0(x1)) 64.53/17.30 0^1(5(5(5(4(x1))))) -> 0^1(5(1(5(5(x1))))) 64.53/17.30 64.53/17.30 The TRS R consists of the following rules: 64.53/17.30 64.53/17.30 2(1(0(0(x1)))) -> 0(3(3(1(2(0(x1)))))) 64.53/17.30 4(0(1(0(x1)))) -> 4(0(3(1(0(x1))))) 64.53/17.30 5(0(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(5(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(1(3(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 2(0(1(3(x1)))) -> 2(0(5(3(1(x1))))) 64.53/17.30 2(0(1(3(x1)))) -> 0(3(1(2(3(x1))))) 64.53/17.30 5(0(1(3(x1)))) -> 0(3(5(3(1(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(3(3(1(2(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 4(1(2(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 1(2(3(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(3(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(5(3(x1))))) 64.53/17.30 2(2(1(3(x1)))) -> 1(2(2(5(3(x1))))) 64.53/17.30 0(2(4(3(x1)))) -> 0(4(2(3(3(x1))))) 64.53/17.30 0(1(0(4(x1)))) -> 0(4(0(3(1(x1))))) 64.53/17.30 5(0(1(5(x1)))) -> 0(5(5(3(3(1(x1)))))) 64.53/17.30 0(5(2(5(x1)))) -> 0(5(3(2(5(x1))))) 64.53/17.30 0(1(3(5(x1)))) -> 5(5(3(1(0(x1))))) 64.53/17.30 0(1(4(0(0(x1))))) -> 0(0(3(1(4(0(x1)))))) 64.53/17.30 5(0(2(1(0(x1))))) -> 2(1(5(0(3(0(x1)))))) 64.53/17.30 2(1(3(1(0(x1))))) -> 1(1(2(2(0(3(x1)))))) 64.53/17.30 4(0(5(1(0(x1))))) -> 0(5(3(1(4(0(x1)))))) 64.53/17.30 5(3(5(1(0(x1))))) -> 5(3(5(3(1(0(x1)))))) 64.53/17.30 4(3(0(2(0(x1))))) -> 0(3(2(5(4(0(x1)))))) 64.53/17.30 5(1(3(2(0(x1))))) -> 2(5(5(3(1(0(x1)))))) 64.53/17.30 2(1(4(2(0(x1))))) -> 2(4(1(1(2(0(x1)))))) 64.53/17.30 2(0(5(2(0(x1))))) -> 2(2(5(3(0(0(x1)))))) 64.53/17.30 0(5(5(2(0(x1))))) -> 5(1(5(2(0(0(x1)))))) 64.53/17.30 0(5(1(3(0(x1))))) -> 0(3(5(0(3(1(x1)))))) 64.53/17.30 0(2(3(5(0(x1))))) -> 5(3(1(2(0(0(x1)))))) 64.53/17.30 2(5(0(1(3(x1))))) -> 0(5(3(1(4(2(x1)))))) 64.53/17.30 2(0(3(1(3(x1))))) -> 0(3(3(3(1(2(x1)))))) 64.53/17.30 0(2(3(1(3(x1))))) -> 2(5(0(3(1(3(x1)))))) 64.53/17.30 2(1(4(1(3(x1))))) -> 1(2(2(3(4(1(x1)))))) 64.53/17.30 0(2(4(1(3(x1))))) -> 4(0(3(3(1(2(x1)))))) 64.53/17.30 2(0(5(1(3(x1))))) -> 5(0(3(1(5(2(x1)))))) 64.53/17.30 0(0(1(3(3(x1))))) -> 0(0(3(3(1(5(x1)))))) 64.53/17.30 0(2(0(4(3(x1))))) -> 4(1(2(0(0(3(x1)))))) 64.53/17.30 2(3(2(4(3(x1))))) -> 2(4(5(2(3(3(x1)))))) 64.53/17.30 2(0(1(3(4(x1))))) -> 1(0(3(1(2(4(x1)))))) 64.53/17.30 0(2(1(3(4(x1))))) -> 2(4(1(0(3(1(x1)))))) 64.53/17.30 2(1(5(5(4(x1))))) -> 1(1(2(5(4(5(x1)))))) 64.53/17.30 0(5(5(5(4(x1))))) -> 4(0(5(1(5(5(x1)))))) 64.53/17.30 2(1(0(1(5(x1))))) -> 0(5(3(1(1(2(x1)))))) 64.53/17.30 4(0(1(1(5(x1))))) -> 0(3(4(1(1(5(x1)))))) 64.53/17.30 0(2(5(1(5(x1))))) -> 5(0(5(3(1(2(x1)))))) 64.53/17.30 5(0(1(4(5(x1))))) -> 0(5(5(3(1(4(x1)))))) 64.53/17.30 64.53/17.30 Q is empty. 64.53/17.30 We have to consider all minimal (P,Q,R)-chains. 64.53/17.30 ---------------------------------------- 64.53/17.30 64.53/17.30 (15) DependencyGraphProof (EQUIVALENT) 64.53/17.30 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 7 less nodes. 64.53/17.30 ---------------------------------------- 64.53/17.30 64.53/17.30 (16) 64.53/17.30 Obligation: 64.53/17.30 Q DP problem: 64.53/17.30 The TRS P consists of the following rules: 64.53/17.30 64.53/17.30 2^1(0(5(2(0(x1))))) -> 2^1(5(3(0(0(x1))))) 64.53/17.30 2^1(0(5(2(0(x1))))) -> 2^1(2(5(3(0(0(x1)))))) 64.53/17.30 2^1(0(5(2(0(x1))))) -> 0^1(0(x1)) 64.53/17.30 0^1(5(5(2(0(x1))))) -> 2^1(0(0(x1))) 64.53/17.30 2^1(1(5(5(4(x1))))) -> 2^1(5(4(5(x1)))) 64.53/17.30 0^1(5(5(2(0(x1))))) -> 0^1(0(x1)) 64.53/17.30 0^1(5(5(5(4(x1))))) -> 0^1(5(1(5(5(x1))))) 64.53/17.30 64.53/17.30 The TRS R consists of the following rules: 64.53/17.30 64.53/17.30 2(1(0(0(x1)))) -> 0(3(3(1(2(0(x1)))))) 64.53/17.30 4(0(1(0(x1)))) -> 4(0(3(1(0(x1))))) 64.53/17.30 5(0(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(5(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(1(3(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 2(0(1(3(x1)))) -> 2(0(5(3(1(x1))))) 64.53/17.30 2(0(1(3(x1)))) -> 0(3(1(2(3(x1))))) 64.53/17.30 5(0(1(3(x1)))) -> 0(3(5(3(1(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(3(3(1(2(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 4(1(2(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 1(2(3(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(3(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(5(3(x1))))) 64.53/17.30 2(2(1(3(x1)))) -> 1(2(2(5(3(x1))))) 64.53/17.30 0(2(4(3(x1)))) -> 0(4(2(3(3(x1))))) 64.53/17.30 0(1(0(4(x1)))) -> 0(4(0(3(1(x1))))) 64.53/17.30 5(0(1(5(x1)))) -> 0(5(5(3(3(1(x1)))))) 64.53/17.30 0(5(2(5(x1)))) -> 0(5(3(2(5(x1))))) 64.53/17.30 0(1(3(5(x1)))) -> 5(5(3(1(0(x1))))) 64.53/17.30 0(1(4(0(0(x1))))) -> 0(0(3(1(4(0(x1)))))) 64.53/17.30 5(0(2(1(0(x1))))) -> 2(1(5(0(3(0(x1)))))) 64.53/17.30 2(1(3(1(0(x1))))) -> 1(1(2(2(0(3(x1)))))) 64.53/17.30 4(0(5(1(0(x1))))) -> 0(5(3(1(4(0(x1)))))) 64.53/17.30 5(3(5(1(0(x1))))) -> 5(3(5(3(1(0(x1)))))) 64.53/17.30 4(3(0(2(0(x1))))) -> 0(3(2(5(4(0(x1)))))) 64.53/17.30 5(1(3(2(0(x1))))) -> 2(5(5(3(1(0(x1)))))) 64.53/17.30 2(1(4(2(0(x1))))) -> 2(4(1(1(2(0(x1)))))) 64.53/17.30 2(0(5(2(0(x1))))) -> 2(2(5(3(0(0(x1)))))) 64.53/17.30 0(5(5(2(0(x1))))) -> 5(1(5(2(0(0(x1)))))) 64.53/17.30 0(5(1(3(0(x1))))) -> 0(3(5(0(3(1(x1)))))) 64.53/17.30 0(2(3(5(0(x1))))) -> 5(3(1(2(0(0(x1)))))) 64.53/17.30 2(5(0(1(3(x1))))) -> 0(5(3(1(4(2(x1)))))) 64.53/17.30 2(0(3(1(3(x1))))) -> 0(3(3(3(1(2(x1)))))) 64.53/17.30 0(2(3(1(3(x1))))) -> 2(5(0(3(1(3(x1)))))) 64.53/17.30 2(1(4(1(3(x1))))) -> 1(2(2(3(4(1(x1)))))) 64.53/17.30 0(2(4(1(3(x1))))) -> 4(0(3(3(1(2(x1)))))) 64.53/17.30 2(0(5(1(3(x1))))) -> 5(0(3(1(5(2(x1)))))) 64.53/17.30 0(0(1(3(3(x1))))) -> 0(0(3(3(1(5(x1)))))) 64.53/17.30 0(2(0(4(3(x1))))) -> 4(1(2(0(0(3(x1)))))) 64.53/17.30 2(3(2(4(3(x1))))) -> 2(4(5(2(3(3(x1)))))) 64.53/17.30 2(0(1(3(4(x1))))) -> 1(0(3(1(2(4(x1)))))) 64.53/17.30 0(2(1(3(4(x1))))) -> 2(4(1(0(3(1(x1)))))) 64.53/17.30 2(1(5(5(4(x1))))) -> 1(1(2(5(4(5(x1)))))) 64.53/17.30 0(5(5(5(4(x1))))) -> 4(0(5(1(5(5(x1)))))) 64.53/17.30 2(1(0(1(5(x1))))) -> 0(5(3(1(1(2(x1)))))) 64.53/17.30 4(0(1(1(5(x1))))) -> 0(3(4(1(1(5(x1)))))) 64.53/17.30 0(2(5(1(5(x1))))) -> 5(0(5(3(1(2(x1)))))) 64.53/17.30 5(0(1(4(5(x1))))) -> 0(5(5(3(1(4(x1)))))) 64.53/17.30 64.53/17.30 Q is empty. 64.53/17.30 We have to consider all minimal (P,Q,R)-chains. 64.53/17.30 ---------------------------------------- 64.53/17.30 64.53/17.30 (17) QDPOrderProof (EQUIVALENT) 64.53/17.30 We use the reduction pair processor [LPAR04,JAR06]. 64.53/17.30 64.53/17.30 64.53/17.30 The following pairs can be oriented strictly and are deleted. 64.53/17.30 64.53/17.30 2^1(0(5(2(0(x1))))) -> 2^1(5(3(0(0(x1))))) 64.53/17.30 The remaining pairs can at least be oriented weakly. 64.53/17.30 Used ordering: Polynomial interpretation [POLO]: 64.53/17.30 64.53/17.30 POL(0(x_1)) = 1 64.53/17.30 POL(0^1(x_1)) = 1 + x_1 64.53/17.30 POL(1(x_1)) = 1 64.53/17.30 POL(2(x_1)) = 1 64.53/17.30 POL(2^1(x_1)) = 1 + x_1 64.53/17.30 POL(3(x_1)) = 0 64.53/17.30 POL(4(x_1)) = 1 64.53/17.30 POL(5(x_1)) = x_1 64.53/17.30 64.53/17.30 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 64.53/17.30 64.53/17.30 0(5(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(1(3(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(3(3(1(2(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 4(1(2(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 1(2(3(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(3(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(5(3(x1))))) 64.53/17.30 0(2(4(3(x1)))) -> 0(4(2(3(3(x1))))) 64.53/17.30 0(1(0(4(x1)))) -> 0(4(0(3(1(x1))))) 64.53/17.30 0(5(2(5(x1)))) -> 0(5(3(2(5(x1))))) 64.53/17.30 0(1(3(5(x1)))) -> 5(5(3(1(0(x1))))) 64.53/17.30 0(1(4(0(0(x1))))) -> 0(0(3(1(4(0(x1)))))) 64.53/17.30 0(5(5(2(0(x1))))) -> 5(1(5(2(0(0(x1)))))) 64.53/17.30 0(5(1(3(0(x1))))) -> 0(3(5(0(3(1(x1)))))) 64.53/17.30 0(2(3(5(0(x1))))) -> 5(3(1(2(0(0(x1)))))) 64.53/17.30 0(2(3(1(3(x1))))) -> 2(5(0(3(1(3(x1)))))) 64.53/17.30 0(2(4(1(3(x1))))) -> 4(0(3(3(1(2(x1)))))) 64.53/17.30 0(0(1(3(3(x1))))) -> 0(0(3(3(1(5(x1)))))) 64.53/17.30 0(2(0(4(3(x1))))) -> 4(1(2(0(0(3(x1)))))) 64.53/17.30 0(2(1(3(4(x1))))) -> 2(4(1(0(3(1(x1)))))) 64.53/17.30 0(5(5(5(4(x1))))) -> 4(0(5(1(5(5(x1)))))) 64.53/17.30 0(2(5(1(5(x1))))) -> 5(0(5(3(1(2(x1)))))) 64.53/17.30 5(3(5(1(0(x1))))) -> 5(3(5(3(1(0(x1)))))) 64.53/17.30 2(1(0(0(x1)))) -> 0(3(3(1(2(0(x1)))))) 64.53/17.30 2(0(1(3(x1)))) -> 2(0(5(3(1(x1))))) 64.53/17.30 2(0(1(3(x1)))) -> 0(3(1(2(3(x1))))) 64.53/17.30 2(2(1(3(x1)))) -> 1(2(2(5(3(x1))))) 64.53/17.30 2(1(3(1(0(x1))))) -> 1(1(2(2(0(3(x1)))))) 64.53/17.30 2(1(4(2(0(x1))))) -> 2(4(1(1(2(0(x1)))))) 64.53/17.30 2(0(5(2(0(x1))))) -> 2(2(5(3(0(0(x1)))))) 64.53/17.30 2(5(0(1(3(x1))))) -> 0(5(3(1(4(2(x1)))))) 64.53/17.30 2(0(3(1(3(x1))))) -> 0(3(3(3(1(2(x1)))))) 64.53/17.30 2(1(4(1(3(x1))))) -> 1(2(2(3(4(1(x1)))))) 64.53/17.30 2(0(5(1(3(x1))))) -> 5(0(3(1(5(2(x1)))))) 64.53/17.30 2(3(2(4(3(x1))))) -> 2(4(5(2(3(3(x1)))))) 64.53/17.30 2(0(1(3(4(x1))))) -> 1(0(3(1(2(4(x1)))))) 64.53/17.30 2(1(5(5(4(x1))))) -> 1(1(2(5(4(5(x1)))))) 64.53/17.30 2(1(0(1(5(x1))))) -> 0(5(3(1(1(2(x1)))))) 64.53/17.30 5(0(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 5(0(1(3(x1)))) -> 0(3(5(3(1(x1))))) 64.53/17.30 5(0(1(5(x1)))) -> 0(5(5(3(3(1(x1)))))) 64.53/17.30 5(0(2(1(0(x1))))) -> 2(1(5(0(3(0(x1)))))) 64.53/17.30 5(1(3(2(0(x1))))) -> 2(5(5(3(1(0(x1)))))) 64.53/17.30 5(0(1(4(5(x1))))) -> 0(5(5(3(1(4(x1)))))) 64.53/17.30 4(0(1(0(x1)))) -> 4(0(3(1(0(x1))))) 64.53/17.30 4(0(5(1(0(x1))))) -> 0(5(3(1(4(0(x1)))))) 64.53/17.30 4(3(0(2(0(x1))))) -> 0(3(2(5(4(0(x1)))))) 64.53/17.30 4(0(1(1(5(x1))))) -> 0(3(4(1(1(5(x1)))))) 64.53/17.30 64.53/17.30 64.53/17.30 ---------------------------------------- 64.53/17.30 64.53/17.30 (18) 64.53/17.30 Obligation: 64.53/17.30 Q DP problem: 64.53/17.30 The TRS P consists of the following rules: 64.53/17.30 64.53/17.30 2^1(0(5(2(0(x1))))) -> 2^1(2(5(3(0(0(x1)))))) 64.53/17.30 2^1(0(5(2(0(x1))))) -> 0^1(0(x1)) 64.53/17.30 0^1(5(5(2(0(x1))))) -> 2^1(0(0(x1))) 64.53/17.30 2^1(1(5(5(4(x1))))) -> 2^1(5(4(5(x1)))) 64.53/17.30 0^1(5(5(2(0(x1))))) -> 0^1(0(x1)) 64.53/17.30 0^1(5(5(5(4(x1))))) -> 0^1(5(1(5(5(x1))))) 64.53/17.30 64.53/17.30 The TRS R consists of the following rules: 64.53/17.30 64.53/17.30 2(1(0(0(x1)))) -> 0(3(3(1(2(0(x1)))))) 64.53/17.30 4(0(1(0(x1)))) -> 4(0(3(1(0(x1))))) 64.53/17.30 5(0(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(5(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(1(3(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 2(0(1(3(x1)))) -> 2(0(5(3(1(x1))))) 64.53/17.30 2(0(1(3(x1)))) -> 0(3(1(2(3(x1))))) 64.53/17.30 5(0(1(3(x1)))) -> 0(3(5(3(1(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(3(3(1(2(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 4(1(2(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 1(2(3(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(3(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(5(3(x1))))) 64.53/17.30 2(2(1(3(x1)))) -> 1(2(2(5(3(x1))))) 64.53/17.30 0(2(4(3(x1)))) -> 0(4(2(3(3(x1))))) 64.53/17.30 0(1(0(4(x1)))) -> 0(4(0(3(1(x1))))) 64.53/17.30 5(0(1(5(x1)))) -> 0(5(5(3(3(1(x1)))))) 64.53/17.30 0(5(2(5(x1)))) -> 0(5(3(2(5(x1))))) 64.53/17.30 0(1(3(5(x1)))) -> 5(5(3(1(0(x1))))) 64.53/17.30 0(1(4(0(0(x1))))) -> 0(0(3(1(4(0(x1)))))) 64.53/17.30 5(0(2(1(0(x1))))) -> 2(1(5(0(3(0(x1)))))) 64.53/17.30 2(1(3(1(0(x1))))) -> 1(1(2(2(0(3(x1)))))) 64.53/17.30 4(0(5(1(0(x1))))) -> 0(5(3(1(4(0(x1)))))) 64.53/17.30 5(3(5(1(0(x1))))) -> 5(3(5(3(1(0(x1)))))) 64.53/17.30 4(3(0(2(0(x1))))) -> 0(3(2(5(4(0(x1)))))) 64.53/17.30 5(1(3(2(0(x1))))) -> 2(5(5(3(1(0(x1)))))) 64.53/17.30 2(1(4(2(0(x1))))) -> 2(4(1(1(2(0(x1)))))) 64.53/17.30 2(0(5(2(0(x1))))) -> 2(2(5(3(0(0(x1)))))) 64.53/17.30 0(5(5(2(0(x1))))) -> 5(1(5(2(0(0(x1)))))) 64.53/17.30 0(5(1(3(0(x1))))) -> 0(3(5(0(3(1(x1)))))) 64.53/17.30 0(2(3(5(0(x1))))) -> 5(3(1(2(0(0(x1)))))) 64.53/17.30 2(5(0(1(3(x1))))) -> 0(5(3(1(4(2(x1)))))) 64.53/17.30 2(0(3(1(3(x1))))) -> 0(3(3(3(1(2(x1)))))) 64.53/17.30 0(2(3(1(3(x1))))) -> 2(5(0(3(1(3(x1)))))) 64.53/17.30 2(1(4(1(3(x1))))) -> 1(2(2(3(4(1(x1)))))) 64.53/17.30 0(2(4(1(3(x1))))) -> 4(0(3(3(1(2(x1)))))) 64.53/17.30 2(0(5(1(3(x1))))) -> 5(0(3(1(5(2(x1)))))) 64.53/17.30 0(0(1(3(3(x1))))) -> 0(0(3(3(1(5(x1)))))) 64.53/17.30 0(2(0(4(3(x1))))) -> 4(1(2(0(0(3(x1)))))) 64.53/17.30 2(3(2(4(3(x1))))) -> 2(4(5(2(3(3(x1)))))) 64.53/17.30 2(0(1(3(4(x1))))) -> 1(0(3(1(2(4(x1)))))) 64.53/17.30 0(2(1(3(4(x1))))) -> 2(4(1(0(3(1(x1)))))) 64.53/17.30 2(1(5(5(4(x1))))) -> 1(1(2(5(4(5(x1)))))) 64.53/17.30 0(5(5(5(4(x1))))) -> 4(0(5(1(5(5(x1)))))) 64.53/17.30 2(1(0(1(5(x1))))) -> 0(5(3(1(1(2(x1)))))) 64.53/17.30 4(0(1(1(5(x1))))) -> 0(3(4(1(1(5(x1)))))) 64.53/17.30 0(2(5(1(5(x1))))) -> 5(0(5(3(1(2(x1)))))) 64.53/17.30 5(0(1(4(5(x1))))) -> 0(5(5(3(1(4(x1)))))) 64.53/17.30 64.53/17.30 Q is empty. 64.53/17.30 We have to consider all minimal (P,Q,R)-chains. 64.53/17.30 ---------------------------------------- 64.53/17.30 64.53/17.30 (19) QDPOrderProof (EQUIVALENT) 64.53/17.30 We use the reduction pair processor [LPAR04,JAR06]. 64.53/17.30 64.53/17.30 64.53/17.30 The following pairs can be oriented strictly and are deleted. 64.53/17.30 64.53/17.30 0^1(5(5(5(4(x1))))) -> 0^1(5(1(5(5(x1))))) 64.53/17.30 The remaining pairs can at least be oriented weakly. 64.53/17.30 Used ordering: Polynomial interpretation [POLO]: 64.53/17.30 64.53/17.30 POL(0(x_1)) = 1 64.53/17.30 POL(0^1(x_1)) = x_1 64.53/17.30 POL(1(x_1)) = 0 64.53/17.30 POL(2(x_1)) = x_1 64.53/17.30 POL(2^1(x_1)) = 1 64.53/17.30 POL(3(x_1)) = 0 64.53/17.30 POL(4(x_1)) = 1 64.53/17.30 POL(5(x_1)) = x_1 64.53/17.30 64.53/17.30 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 64.53/17.30 64.53/17.30 0(5(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(1(3(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(3(3(1(2(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 4(1(2(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 1(2(3(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(3(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(5(3(x1))))) 64.53/17.30 0(2(4(3(x1)))) -> 0(4(2(3(3(x1))))) 64.53/17.30 0(1(0(4(x1)))) -> 0(4(0(3(1(x1))))) 64.53/17.30 0(5(2(5(x1)))) -> 0(5(3(2(5(x1))))) 64.53/17.30 0(1(3(5(x1)))) -> 5(5(3(1(0(x1))))) 64.53/17.30 0(1(4(0(0(x1))))) -> 0(0(3(1(4(0(x1)))))) 64.53/17.30 0(5(5(2(0(x1))))) -> 5(1(5(2(0(0(x1)))))) 64.53/17.30 0(5(1(3(0(x1))))) -> 0(3(5(0(3(1(x1)))))) 64.53/17.30 0(2(3(5(0(x1))))) -> 5(3(1(2(0(0(x1)))))) 64.53/17.30 0(2(3(1(3(x1))))) -> 2(5(0(3(1(3(x1)))))) 64.53/17.30 0(2(4(1(3(x1))))) -> 4(0(3(3(1(2(x1)))))) 64.53/17.30 0(0(1(3(3(x1))))) -> 0(0(3(3(1(5(x1)))))) 64.53/17.30 0(2(0(4(3(x1))))) -> 4(1(2(0(0(3(x1)))))) 64.53/17.30 0(2(1(3(4(x1))))) -> 2(4(1(0(3(1(x1)))))) 64.53/17.30 0(5(5(5(4(x1))))) -> 4(0(5(1(5(5(x1)))))) 64.53/17.30 0(2(5(1(5(x1))))) -> 5(0(5(3(1(2(x1)))))) 64.53/17.30 5(1(3(2(0(x1))))) -> 2(5(5(3(1(0(x1)))))) 64.53/17.30 4(0(1(0(x1)))) -> 4(0(3(1(0(x1))))) 64.53/17.30 4(0(5(1(0(x1))))) -> 0(5(3(1(4(0(x1)))))) 64.53/17.30 4(3(0(2(0(x1))))) -> 0(3(2(5(4(0(x1)))))) 64.53/17.30 4(0(1(1(5(x1))))) -> 0(3(4(1(1(5(x1)))))) 64.53/17.30 64.53/17.30 64.53/17.30 ---------------------------------------- 64.53/17.30 64.53/17.30 (20) 64.53/17.30 Obligation: 64.53/17.30 Q DP problem: 64.53/17.30 The TRS P consists of the following rules: 64.53/17.30 64.53/17.30 2^1(0(5(2(0(x1))))) -> 2^1(2(5(3(0(0(x1)))))) 64.53/17.30 2^1(0(5(2(0(x1))))) -> 0^1(0(x1)) 64.53/17.30 0^1(5(5(2(0(x1))))) -> 2^1(0(0(x1))) 64.53/17.30 2^1(1(5(5(4(x1))))) -> 2^1(5(4(5(x1)))) 64.53/17.30 0^1(5(5(2(0(x1))))) -> 0^1(0(x1)) 64.53/17.30 64.53/17.30 The TRS R consists of the following rules: 64.53/17.30 64.53/17.30 2(1(0(0(x1)))) -> 0(3(3(1(2(0(x1)))))) 64.53/17.30 4(0(1(0(x1)))) -> 4(0(3(1(0(x1))))) 64.53/17.30 5(0(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(5(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(1(3(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 2(0(1(3(x1)))) -> 2(0(5(3(1(x1))))) 64.53/17.30 2(0(1(3(x1)))) -> 0(3(1(2(3(x1))))) 64.53/17.30 5(0(1(3(x1)))) -> 0(3(5(3(1(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(3(3(1(2(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 4(1(2(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 1(2(3(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(3(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(5(3(x1))))) 64.53/17.30 2(2(1(3(x1)))) -> 1(2(2(5(3(x1))))) 64.53/17.30 0(2(4(3(x1)))) -> 0(4(2(3(3(x1))))) 64.53/17.30 0(1(0(4(x1)))) -> 0(4(0(3(1(x1))))) 64.53/17.30 5(0(1(5(x1)))) -> 0(5(5(3(3(1(x1)))))) 64.53/17.30 0(5(2(5(x1)))) -> 0(5(3(2(5(x1))))) 64.53/17.30 0(1(3(5(x1)))) -> 5(5(3(1(0(x1))))) 64.53/17.30 0(1(4(0(0(x1))))) -> 0(0(3(1(4(0(x1)))))) 64.53/17.30 5(0(2(1(0(x1))))) -> 2(1(5(0(3(0(x1)))))) 64.53/17.30 2(1(3(1(0(x1))))) -> 1(1(2(2(0(3(x1)))))) 64.53/17.30 4(0(5(1(0(x1))))) -> 0(5(3(1(4(0(x1)))))) 64.53/17.30 5(3(5(1(0(x1))))) -> 5(3(5(3(1(0(x1)))))) 64.53/17.30 4(3(0(2(0(x1))))) -> 0(3(2(5(4(0(x1)))))) 64.53/17.30 5(1(3(2(0(x1))))) -> 2(5(5(3(1(0(x1)))))) 64.53/17.30 2(1(4(2(0(x1))))) -> 2(4(1(1(2(0(x1)))))) 64.53/17.30 2(0(5(2(0(x1))))) -> 2(2(5(3(0(0(x1)))))) 64.53/17.30 0(5(5(2(0(x1))))) -> 5(1(5(2(0(0(x1)))))) 64.53/17.30 0(5(1(3(0(x1))))) -> 0(3(5(0(3(1(x1)))))) 64.53/17.30 0(2(3(5(0(x1))))) -> 5(3(1(2(0(0(x1)))))) 64.53/17.30 2(5(0(1(3(x1))))) -> 0(5(3(1(4(2(x1)))))) 64.53/17.30 2(0(3(1(3(x1))))) -> 0(3(3(3(1(2(x1)))))) 64.53/17.30 0(2(3(1(3(x1))))) -> 2(5(0(3(1(3(x1)))))) 64.53/17.30 2(1(4(1(3(x1))))) -> 1(2(2(3(4(1(x1)))))) 64.53/17.30 0(2(4(1(3(x1))))) -> 4(0(3(3(1(2(x1)))))) 64.53/17.30 2(0(5(1(3(x1))))) -> 5(0(3(1(5(2(x1)))))) 64.53/17.30 0(0(1(3(3(x1))))) -> 0(0(3(3(1(5(x1)))))) 64.53/17.30 0(2(0(4(3(x1))))) -> 4(1(2(0(0(3(x1)))))) 64.53/17.30 2(3(2(4(3(x1))))) -> 2(4(5(2(3(3(x1)))))) 64.53/17.30 2(0(1(3(4(x1))))) -> 1(0(3(1(2(4(x1)))))) 64.53/17.30 0(2(1(3(4(x1))))) -> 2(4(1(0(3(1(x1)))))) 64.53/17.30 2(1(5(5(4(x1))))) -> 1(1(2(5(4(5(x1)))))) 64.53/17.30 0(5(5(5(4(x1))))) -> 4(0(5(1(5(5(x1)))))) 64.53/17.30 2(1(0(1(5(x1))))) -> 0(5(3(1(1(2(x1)))))) 64.53/17.30 4(0(1(1(5(x1))))) -> 0(3(4(1(1(5(x1)))))) 64.53/17.30 0(2(5(1(5(x1))))) -> 5(0(5(3(1(2(x1)))))) 64.53/17.30 5(0(1(4(5(x1))))) -> 0(5(5(3(1(4(x1)))))) 64.53/17.30 64.53/17.30 Q is empty. 64.53/17.30 We have to consider all minimal (P,Q,R)-chains. 64.53/17.30 ---------------------------------------- 64.53/17.30 64.53/17.30 (21) QDPOrderProof (EQUIVALENT) 64.53/17.30 We use the reduction pair processor [LPAR04,JAR06]. 64.53/17.30 64.53/17.30 64.53/17.30 The following pairs can be oriented strictly and are deleted. 64.53/17.30 64.53/17.30 2^1(0(5(2(0(x1))))) -> 0^1(0(x1)) 64.53/17.30 0^1(5(5(2(0(x1))))) -> 2^1(0(0(x1))) 64.53/17.30 2^1(1(5(5(4(x1))))) -> 2^1(5(4(5(x1)))) 64.53/17.30 0^1(5(5(2(0(x1))))) -> 0^1(0(x1)) 64.53/17.30 The remaining pairs can at least be oriented weakly. 64.53/17.30 Used ordering: Polynomial interpretation [POLO]: 64.53/17.30 64.53/17.30 POL(0(x_1)) = x_1 64.53/17.30 POL(0^1(x_1)) = 1 + x_1 64.53/17.30 POL(1(x_1)) = 1 64.53/17.30 POL(2(x_1)) = 1 + x_1 64.53/17.30 POL(2^1(x_1)) = 1 + x_1 64.53/17.30 POL(3(x_1)) = 0 64.53/17.30 POL(4(x_1)) = 0 64.53/17.30 POL(5(x_1)) = x_1 64.53/17.30 64.53/17.30 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 64.53/17.30 64.53/17.30 0(5(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(1(3(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(3(3(1(2(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 4(1(2(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 1(2(3(0(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(3(3(x1))))) 64.53/17.30 0(2(1(3(x1)))) -> 0(1(2(5(3(x1))))) 64.53/17.30 0(2(4(3(x1)))) -> 0(4(2(3(3(x1))))) 64.53/17.30 0(1(0(4(x1)))) -> 0(4(0(3(1(x1))))) 64.53/17.30 0(5(2(5(x1)))) -> 0(5(3(2(5(x1))))) 64.53/17.30 0(1(3(5(x1)))) -> 5(5(3(1(0(x1))))) 64.53/17.30 0(1(4(0(0(x1))))) -> 0(0(3(1(4(0(x1)))))) 64.53/17.30 0(5(5(2(0(x1))))) -> 5(1(5(2(0(0(x1)))))) 64.53/17.30 0(5(1(3(0(x1))))) -> 0(3(5(0(3(1(x1)))))) 64.53/17.30 0(2(3(5(0(x1))))) -> 5(3(1(2(0(0(x1)))))) 64.53/17.30 0(2(3(1(3(x1))))) -> 2(5(0(3(1(3(x1)))))) 64.53/17.30 0(2(4(1(3(x1))))) -> 4(0(3(3(1(2(x1)))))) 64.53/17.30 0(0(1(3(3(x1))))) -> 0(0(3(3(1(5(x1)))))) 64.53/17.30 0(2(0(4(3(x1))))) -> 4(1(2(0(0(3(x1)))))) 64.53/17.30 0(2(1(3(4(x1))))) -> 2(4(1(0(3(1(x1)))))) 64.53/17.30 0(5(5(5(4(x1))))) -> 4(0(5(1(5(5(x1)))))) 64.53/17.30 0(2(5(1(5(x1))))) -> 5(0(5(3(1(2(x1)))))) 64.53/17.30 5(3(5(1(0(x1))))) -> 5(3(5(3(1(0(x1)))))) 64.53/17.30 2(1(0(0(x1)))) -> 0(3(3(1(2(0(x1)))))) 64.53/17.30 2(0(1(3(x1)))) -> 2(0(5(3(1(x1))))) 64.53/17.30 2(0(1(3(x1)))) -> 0(3(1(2(3(x1))))) 64.53/17.30 2(2(1(3(x1)))) -> 1(2(2(5(3(x1))))) 64.53/17.30 2(1(3(1(0(x1))))) -> 1(1(2(2(0(3(x1)))))) 64.53/17.30 2(1(4(2(0(x1))))) -> 2(4(1(1(2(0(x1)))))) 64.53/17.30 2(0(5(2(0(x1))))) -> 2(2(5(3(0(0(x1)))))) 64.53/17.30 2(5(0(1(3(x1))))) -> 0(5(3(1(4(2(x1)))))) 64.53/17.30 2(0(3(1(3(x1))))) -> 0(3(3(3(1(2(x1)))))) 64.53/17.30 2(1(4(1(3(x1))))) -> 1(2(2(3(4(1(x1)))))) 64.53/17.30 2(0(5(1(3(x1))))) -> 5(0(3(1(5(2(x1)))))) 64.53/17.30 2(3(2(4(3(x1))))) -> 2(4(5(2(3(3(x1)))))) 64.53/17.30 2(0(1(3(4(x1))))) -> 1(0(3(1(2(4(x1)))))) 64.53/17.30 2(1(5(5(4(x1))))) -> 1(1(2(5(4(5(x1)))))) 64.53/17.31 2(1(0(1(5(x1))))) -> 0(5(3(1(1(2(x1)))))) 64.53/17.31 5(0(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.31 5(0(1(3(x1)))) -> 0(3(5(3(1(x1))))) 64.53/17.31 5(0(1(5(x1)))) -> 0(5(5(3(3(1(x1)))))) 64.53/17.31 5(0(2(1(0(x1))))) -> 2(1(5(0(3(0(x1)))))) 64.53/17.31 5(1(3(2(0(x1))))) -> 2(5(5(3(1(0(x1)))))) 64.53/17.31 5(0(1(4(5(x1))))) -> 0(5(5(3(1(4(x1)))))) 64.53/17.31 4(0(1(0(x1)))) -> 4(0(3(1(0(x1))))) 64.53/17.31 4(0(5(1(0(x1))))) -> 0(5(3(1(4(0(x1)))))) 64.53/17.31 4(3(0(2(0(x1))))) -> 0(3(2(5(4(0(x1)))))) 64.53/17.31 4(0(1(1(5(x1))))) -> 0(3(4(1(1(5(x1)))))) 64.53/17.31 64.53/17.31 64.53/17.31 ---------------------------------------- 64.53/17.31 64.53/17.31 (22) 64.53/17.31 Obligation: 64.53/17.31 Q DP problem: 64.53/17.31 The TRS P consists of the following rules: 64.53/17.31 64.53/17.31 2^1(0(5(2(0(x1))))) -> 2^1(2(5(3(0(0(x1)))))) 64.53/17.31 64.53/17.31 The TRS R consists of the following rules: 64.53/17.31 64.53/17.31 2(1(0(0(x1)))) -> 0(3(3(1(2(0(x1)))))) 64.53/17.31 4(0(1(0(x1)))) -> 4(0(3(1(0(x1))))) 64.53/17.31 5(0(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.31 0(5(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.31 0(1(3(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.31 2(0(1(3(x1)))) -> 2(0(5(3(1(x1))))) 64.53/17.31 2(0(1(3(x1)))) -> 0(3(1(2(3(x1))))) 64.53/17.31 5(0(1(3(x1)))) -> 0(3(5(3(1(x1))))) 64.53/17.31 0(2(1(3(x1)))) -> 0(3(3(1(2(x1))))) 64.53/17.31 0(2(1(3(x1)))) -> 4(1(2(0(3(x1))))) 64.53/17.31 0(2(1(3(x1)))) -> 1(2(3(0(3(x1))))) 64.53/17.31 0(2(1(3(x1)))) -> 0(1(2(3(3(x1))))) 64.53/17.31 0(2(1(3(x1)))) -> 0(1(2(5(3(x1))))) 64.53/17.31 2(2(1(3(x1)))) -> 1(2(2(5(3(x1))))) 64.53/17.31 0(2(4(3(x1)))) -> 0(4(2(3(3(x1))))) 64.53/17.31 0(1(0(4(x1)))) -> 0(4(0(3(1(x1))))) 64.53/17.31 5(0(1(5(x1)))) -> 0(5(5(3(3(1(x1)))))) 64.53/17.31 0(5(2(5(x1)))) -> 0(5(3(2(5(x1))))) 64.53/17.31 0(1(3(5(x1)))) -> 5(5(3(1(0(x1))))) 64.53/17.31 0(1(4(0(0(x1))))) -> 0(0(3(1(4(0(x1)))))) 64.53/17.31 5(0(2(1(0(x1))))) -> 2(1(5(0(3(0(x1)))))) 64.53/17.31 2(1(3(1(0(x1))))) -> 1(1(2(2(0(3(x1)))))) 64.53/17.31 4(0(5(1(0(x1))))) -> 0(5(3(1(4(0(x1)))))) 64.53/17.31 5(3(5(1(0(x1))))) -> 5(3(5(3(1(0(x1)))))) 64.53/17.31 4(3(0(2(0(x1))))) -> 0(3(2(5(4(0(x1)))))) 64.53/17.31 5(1(3(2(0(x1))))) -> 2(5(5(3(1(0(x1)))))) 64.53/17.31 2(1(4(2(0(x1))))) -> 2(4(1(1(2(0(x1)))))) 64.53/17.31 2(0(5(2(0(x1))))) -> 2(2(5(3(0(0(x1)))))) 64.53/17.31 0(5(5(2(0(x1))))) -> 5(1(5(2(0(0(x1)))))) 64.53/17.31 0(5(1(3(0(x1))))) -> 0(3(5(0(3(1(x1)))))) 64.53/17.31 0(2(3(5(0(x1))))) -> 5(3(1(2(0(0(x1)))))) 64.53/17.31 2(5(0(1(3(x1))))) -> 0(5(3(1(4(2(x1)))))) 64.53/17.31 2(0(3(1(3(x1))))) -> 0(3(3(3(1(2(x1)))))) 64.53/17.31 0(2(3(1(3(x1))))) -> 2(5(0(3(1(3(x1)))))) 64.53/17.31 2(1(4(1(3(x1))))) -> 1(2(2(3(4(1(x1)))))) 64.53/17.31 0(2(4(1(3(x1))))) -> 4(0(3(3(1(2(x1)))))) 64.53/17.31 2(0(5(1(3(x1))))) -> 5(0(3(1(5(2(x1)))))) 64.53/17.31 0(0(1(3(3(x1))))) -> 0(0(3(3(1(5(x1)))))) 64.53/17.31 0(2(0(4(3(x1))))) -> 4(1(2(0(0(3(x1)))))) 64.53/17.31 2(3(2(4(3(x1))))) -> 2(4(5(2(3(3(x1)))))) 64.53/17.31 2(0(1(3(4(x1))))) -> 1(0(3(1(2(4(x1)))))) 64.53/17.31 0(2(1(3(4(x1))))) -> 2(4(1(0(3(1(x1)))))) 64.53/17.31 2(1(5(5(4(x1))))) -> 1(1(2(5(4(5(x1)))))) 64.53/17.31 0(5(5(5(4(x1))))) -> 4(0(5(1(5(5(x1)))))) 64.53/17.31 2(1(0(1(5(x1))))) -> 0(5(3(1(1(2(x1)))))) 64.53/17.31 4(0(1(1(5(x1))))) -> 0(3(4(1(1(5(x1)))))) 64.53/17.31 0(2(5(1(5(x1))))) -> 5(0(5(3(1(2(x1)))))) 64.53/17.31 5(0(1(4(5(x1))))) -> 0(5(5(3(1(4(x1)))))) 64.53/17.31 64.53/17.31 Q is empty. 64.53/17.31 We have to consider all minimal (P,Q,R)-chains. 64.53/17.31 ---------------------------------------- 64.53/17.31 64.53/17.31 (23) QDPOrderProof (EQUIVALENT) 64.53/17.31 We use the reduction pair processor [LPAR04,JAR06]. 64.53/17.31 64.53/17.31 64.53/17.31 The following pairs can be oriented strictly and are deleted. 64.53/17.31 64.53/17.31 2^1(0(5(2(0(x1))))) -> 2^1(2(5(3(0(0(x1)))))) 64.53/17.31 The remaining pairs can at least be oriented weakly. 64.53/17.31 Used ordering: Polynomial interpretation [POLO]: 64.53/17.31 64.53/17.31 POL(0(x_1)) = 1 + x_1 64.53/17.31 POL(1(x_1)) = 1 64.53/17.31 POL(2(x_1)) = x_1 64.53/17.31 POL(2^1(x_1)) = x_1 64.53/17.31 POL(3(x_1)) = 0 64.53/17.31 POL(4(x_1)) = 0 64.53/17.31 POL(5(x_1)) = x_1 64.53/17.31 64.53/17.31 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 64.53/17.31 64.53/17.31 5(3(5(1(0(x1))))) -> 5(3(5(3(1(0(x1)))))) 64.53/17.31 2(1(0(0(x1)))) -> 0(3(3(1(2(0(x1)))))) 64.53/17.31 2(0(1(3(x1)))) -> 2(0(5(3(1(x1))))) 64.53/17.31 2(0(1(3(x1)))) -> 0(3(1(2(3(x1))))) 64.53/17.31 2(2(1(3(x1)))) -> 1(2(2(5(3(x1))))) 64.53/17.31 2(1(3(1(0(x1))))) -> 1(1(2(2(0(3(x1)))))) 64.53/17.31 2(1(4(2(0(x1))))) -> 2(4(1(1(2(0(x1)))))) 64.53/17.31 2(0(5(2(0(x1))))) -> 2(2(5(3(0(0(x1)))))) 64.53/17.31 2(5(0(1(3(x1))))) -> 0(5(3(1(4(2(x1)))))) 64.53/17.31 2(0(3(1(3(x1))))) -> 0(3(3(3(1(2(x1)))))) 64.53/17.31 2(1(4(1(3(x1))))) -> 1(2(2(3(4(1(x1)))))) 64.53/17.31 2(0(5(1(3(x1))))) -> 5(0(3(1(5(2(x1)))))) 64.53/17.31 2(3(2(4(3(x1))))) -> 2(4(5(2(3(3(x1)))))) 64.53/17.31 2(0(1(3(4(x1))))) -> 1(0(3(1(2(4(x1)))))) 64.53/17.31 2(1(5(5(4(x1))))) -> 1(1(2(5(4(5(x1)))))) 64.53/17.31 2(1(0(1(5(x1))))) -> 0(5(3(1(1(2(x1)))))) 64.53/17.31 64.53/17.31 64.53/17.31 ---------------------------------------- 64.53/17.31 64.53/17.31 (24) 64.53/17.31 Obligation: 64.53/17.31 Q DP problem: 64.53/17.31 P is empty. 64.53/17.31 The TRS R consists of the following rules: 64.53/17.31 64.53/17.31 2(1(0(0(x1)))) -> 0(3(3(1(2(0(x1)))))) 64.53/17.31 4(0(1(0(x1)))) -> 4(0(3(1(0(x1))))) 64.53/17.31 5(0(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.31 0(5(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.31 0(1(3(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.31 2(0(1(3(x1)))) -> 2(0(5(3(1(x1))))) 64.53/17.31 2(0(1(3(x1)))) -> 0(3(1(2(3(x1))))) 64.53/17.31 5(0(1(3(x1)))) -> 0(3(5(3(1(x1))))) 64.53/17.31 0(2(1(3(x1)))) -> 0(3(3(1(2(x1))))) 64.53/17.31 0(2(1(3(x1)))) -> 4(1(2(0(3(x1))))) 64.53/17.31 0(2(1(3(x1)))) -> 1(2(3(0(3(x1))))) 64.53/17.31 0(2(1(3(x1)))) -> 0(1(2(3(3(x1))))) 64.53/17.31 0(2(1(3(x1)))) -> 0(1(2(5(3(x1))))) 64.53/17.31 2(2(1(3(x1)))) -> 1(2(2(5(3(x1))))) 64.53/17.31 0(2(4(3(x1)))) -> 0(4(2(3(3(x1))))) 64.53/17.31 0(1(0(4(x1)))) -> 0(4(0(3(1(x1))))) 64.53/17.31 5(0(1(5(x1)))) -> 0(5(5(3(3(1(x1)))))) 64.53/17.31 0(5(2(5(x1)))) -> 0(5(3(2(5(x1))))) 64.53/17.31 0(1(3(5(x1)))) -> 5(5(3(1(0(x1))))) 64.53/17.31 0(1(4(0(0(x1))))) -> 0(0(3(1(4(0(x1)))))) 64.53/17.31 5(0(2(1(0(x1))))) -> 2(1(5(0(3(0(x1)))))) 64.53/17.31 2(1(3(1(0(x1))))) -> 1(1(2(2(0(3(x1)))))) 64.53/17.31 4(0(5(1(0(x1))))) -> 0(5(3(1(4(0(x1)))))) 64.53/17.31 5(3(5(1(0(x1))))) -> 5(3(5(3(1(0(x1)))))) 64.53/17.31 4(3(0(2(0(x1))))) -> 0(3(2(5(4(0(x1)))))) 64.53/17.31 5(1(3(2(0(x1))))) -> 2(5(5(3(1(0(x1)))))) 64.53/17.31 2(1(4(2(0(x1))))) -> 2(4(1(1(2(0(x1)))))) 64.53/17.31 2(0(5(2(0(x1))))) -> 2(2(5(3(0(0(x1)))))) 64.53/17.31 0(5(5(2(0(x1))))) -> 5(1(5(2(0(0(x1)))))) 64.53/17.31 0(5(1(3(0(x1))))) -> 0(3(5(0(3(1(x1)))))) 64.53/17.31 0(2(3(5(0(x1))))) -> 5(3(1(2(0(0(x1)))))) 64.53/17.31 2(5(0(1(3(x1))))) -> 0(5(3(1(4(2(x1)))))) 64.53/17.31 2(0(3(1(3(x1))))) -> 0(3(3(3(1(2(x1)))))) 64.53/17.31 0(2(3(1(3(x1))))) -> 2(5(0(3(1(3(x1)))))) 64.53/17.31 2(1(4(1(3(x1))))) -> 1(2(2(3(4(1(x1)))))) 64.53/17.31 0(2(4(1(3(x1))))) -> 4(0(3(3(1(2(x1)))))) 64.53/17.31 2(0(5(1(3(x1))))) -> 5(0(3(1(5(2(x1)))))) 64.53/17.31 0(0(1(3(3(x1))))) -> 0(0(3(3(1(5(x1)))))) 64.53/17.31 0(2(0(4(3(x1))))) -> 4(1(2(0(0(3(x1)))))) 64.53/17.31 2(3(2(4(3(x1))))) -> 2(4(5(2(3(3(x1)))))) 64.53/17.31 2(0(1(3(4(x1))))) -> 1(0(3(1(2(4(x1)))))) 64.53/17.31 0(2(1(3(4(x1))))) -> 2(4(1(0(3(1(x1)))))) 64.53/17.31 2(1(5(5(4(x1))))) -> 1(1(2(5(4(5(x1)))))) 64.53/17.31 0(5(5(5(4(x1))))) -> 4(0(5(1(5(5(x1)))))) 64.53/17.31 2(1(0(1(5(x1))))) -> 0(5(3(1(1(2(x1)))))) 64.53/17.31 4(0(1(1(5(x1))))) -> 0(3(4(1(1(5(x1)))))) 64.53/17.31 0(2(5(1(5(x1))))) -> 5(0(5(3(1(2(x1)))))) 64.53/17.31 5(0(1(4(5(x1))))) -> 0(5(5(3(1(4(x1)))))) 64.53/17.31 64.53/17.31 Q is empty. 64.53/17.31 We have to consider all minimal (P,Q,R)-chains. 64.53/17.31 ---------------------------------------- 64.53/17.31 64.53/17.31 (25) PisEmptyProof (EQUIVALENT) 64.53/17.31 The TRS P is empty. Hence, there is no (P,Q,R) chain. 64.53/17.31 ---------------------------------------- 64.53/17.31 64.53/17.31 (26) 64.53/17.31 YES 64.53/17.31 64.53/17.31 ---------------------------------------- 64.53/17.31 64.53/17.31 (27) 64.53/17.31 Obligation: 64.53/17.31 Q DP problem: 64.53/17.31 The TRS P consists of the following rules: 64.53/17.31 64.53/17.31 0^1(1(3(5(x1)))) -> 0^1(x1) 64.53/17.31 64.53/17.31 The TRS R consists of the following rules: 64.53/17.31 64.53/17.31 2(1(0(0(x1)))) -> 0(3(3(1(2(0(x1)))))) 64.53/17.31 4(0(1(0(x1)))) -> 4(0(3(1(0(x1))))) 64.53/17.31 5(0(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.31 0(5(1(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.31 0(1(3(0(x1)))) -> 0(5(3(1(0(x1))))) 64.53/17.31 2(0(1(3(x1)))) -> 2(0(5(3(1(x1))))) 64.53/17.31 2(0(1(3(x1)))) -> 0(3(1(2(3(x1))))) 64.53/17.31 5(0(1(3(x1)))) -> 0(3(5(3(1(x1))))) 64.53/17.31 0(2(1(3(x1)))) -> 0(3(3(1(2(x1))))) 64.53/17.31 0(2(1(3(x1)))) -> 4(1(2(0(3(x1))))) 64.53/17.31 0(2(1(3(x1)))) -> 1(2(3(0(3(x1))))) 64.53/17.31 0(2(1(3(x1)))) -> 0(1(2(3(3(x1))))) 64.53/17.31 0(2(1(3(x1)))) -> 0(1(2(5(3(x1))))) 64.53/17.31 2(2(1(3(x1)))) -> 1(2(2(5(3(x1))))) 64.53/17.31 0(2(4(3(x1)))) -> 0(4(2(3(3(x1))))) 64.53/17.31 0(1(0(4(x1)))) -> 0(4(0(3(1(x1))))) 64.53/17.31 5(0(1(5(x1)))) -> 0(5(5(3(3(1(x1)))))) 64.53/17.31 0(5(2(5(x1)))) -> 0(5(3(2(5(x1))))) 64.53/17.31 0(1(3(5(x1)))) -> 5(5(3(1(0(x1))))) 64.53/17.31 0(1(4(0(0(x1))))) -> 0(0(3(1(4(0(x1)))))) 64.53/17.31 5(0(2(1(0(x1))))) -> 2(1(5(0(3(0(x1)))))) 64.53/17.31 2(1(3(1(0(x1))))) -> 1(1(2(2(0(3(x1)))))) 64.53/17.31 4(0(5(1(0(x1))))) -> 0(5(3(1(4(0(x1)))))) 64.53/17.31 5(3(5(1(0(x1))))) -> 5(3(5(3(1(0(x1)))))) 64.53/17.31 4(3(0(2(0(x1))))) -> 0(3(2(5(4(0(x1)))))) 64.53/17.31 5(1(3(2(0(x1))))) -> 2(5(5(3(1(0(x1)))))) 64.53/17.31 2(1(4(2(0(x1))))) -> 2(4(1(1(2(0(x1)))))) 64.53/17.31 2(0(5(2(0(x1))))) -> 2(2(5(3(0(0(x1)))))) 64.53/17.31 0(5(5(2(0(x1))))) -> 5(1(5(2(0(0(x1)))))) 64.53/17.31 0(5(1(3(0(x1))))) -> 0(3(5(0(3(1(x1)))))) 64.53/17.31 0(2(3(5(0(x1))))) -> 5(3(1(2(0(0(x1)))))) 64.53/17.31 2(5(0(1(3(x1))))) -> 0(5(3(1(4(2(x1)))))) 64.53/17.31 2(0(3(1(3(x1))))) -> 0(3(3(3(1(2(x1)))))) 64.53/17.31 0(2(3(1(3(x1))))) -> 2(5(0(3(1(3(x1)))))) 64.53/17.31 2(1(4(1(3(x1))))) -> 1(2(2(3(4(1(x1)))))) 64.53/17.31 0(2(4(1(3(x1))))) -> 4(0(3(3(1(2(x1)))))) 64.53/17.31 2(0(5(1(3(x1))))) -> 5(0(3(1(5(2(x1)))))) 64.53/17.31 0(0(1(3(3(x1))))) -> 0(0(3(3(1(5(x1)))))) 64.53/17.31 0(2(0(4(3(x1))))) -> 4(1(2(0(0(3(x1)))))) 64.53/17.31 2(3(2(4(3(x1))))) -> 2(4(5(2(3(3(x1)))))) 64.53/17.31 2(0(1(3(4(x1))))) -> 1(0(3(1(2(4(x1)))))) 64.53/17.31 0(2(1(3(4(x1))))) -> 2(4(1(0(3(1(x1)))))) 64.53/17.31 2(1(5(5(4(x1))))) -> 1(1(2(5(4(5(x1)))))) 64.53/17.31 0(5(5(5(4(x1))))) -> 4(0(5(1(5(5(x1)))))) 64.53/17.31 2(1(0(1(5(x1))))) -> 0(5(3(1(1(2(x1)))))) 64.53/17.31 4(0(1(1(5(x1))))) -> 0(3(4(1(1(5(x1)))))) 64.53/17.31 0(2(5(1(5(x1))))) -> 5(0(5(3(1(2(x1)))))) 64.53/17.31 5(0(1(4(5(x1))))) -> 0(5(5(3(1(4(x1)))))) 64.53/17.31 64.53/17.31 Q is empty. 64.53/17.31 We have to consider all minimal (P,Q,R)-chains. 64.53/17.31 ---------------------------------------- 64.53/17.31 64.53/17.31 (28) UsableRulesProof (EQUIVALENT) 64.53/17.31 We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. 64.53/17.31 ---------------------------------------- 64.53/17.31 64.53/17.31 (29) 64.53/17.31 Obligation: 64.53/17.31 Q DP problem: 64.53/17.31 The TRS P consists of the following rules: 64.53/17.31 64.53/17.31 0^1(1(3(5(x1)))) -> 0^1(x1) 64.53/17.31 64.53/17.31 R is empty. 64.53/17.31 Q is empty. 64.53/17.31 We have to consider all minimal (P,Q,R)-chains. 64.53/17.31 ---------------------------------------- 64.53/17.31 64.53/17.31 (30) QDPSizeChangeProof (EQUIVALENT) 64.53/17.31 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 64.53/17.31 64.53/17.31 From the DPs we obtained the following set of size-change graphs: 64.53/17.31 *0^1(1(3(5(x1)))) -> 0^1(x1) 64.53/17.31 The graph contains the following edges 1 > 1 64.53/17.31 64.53/17.31 64.53/17.31 ---------------------------------------- 64.53/17.31 64.53/17.31 (31) 64.53/17.31 YES 64.72/17.39 EOF