43.68/12.05 YES 43.75/12.10 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 43.75/12.10 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 43.75/12.10 43.75/12.10 43.75/12.10 Termination w.r.t. Q of the given QTRS could be proven: 43.75/12.10 43.75/12.10 (0) QTRS 43.75/12.10 (1) QTRS Reverse [EQUIVALENT, 0 ms] 43.75/12.10 (2) QTRS 43.75/12.10 (3) DependencyPairsProof [EQUIVALENT, 332 ms] 43.75/12.10 (4) QDP 43.75/12.10 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 43.75/12.10 (6) AND 43.75/12.10 (7) QDP 43.75/12.10 (8) UsableRulesProof [EQUIVALENT, 0 ms] 43.75/12.10 (9) QDP 43.75/12.10 (10) QDPOrderProof [EQUIVALENT, 0 ms] 43.75/12.10 (11) QDP 43.75/12.10 (12) PisEmptyProof [EQUIVALENT, 0 ms] 43.75/12.10 (13) YES 43.75/12.10 (14) QDP 43.75/12.10 (15) UsableRulesProof [EQUIVALENT, 0 ms] 43.75/12.10 (16) QDP 43.75/12.10 (17) QDPSizeChangeProof [EQUIVALENT, 0 ms] 43.75/12.10 (18) YES 43.75/12.10 (19) QDP 43.75/12.10 (20) QDPOrderProof [EQUIVALENT, 342 ms] 43.75/12.10 (21) QDP 43.75/12.10 (22) DependencyGraphProof [EQUIVALENT, 0 ms] 43.75/12.10 (23) QDP 43.75/12.10 (24) QDPSizeChangeProof [EQUIVALENT, 0 ms] 43.75/12.10 (25) YES 43.75/12.10 43.75/12.10 43.75/12.10 ---------------------------------------- 43.75/12.10 43.75/12.10 (0) 43.75/12.10 Obligation: 43.75/12.10 Q restricted rewrite system: 43.75/12.10 The TRS R consists of the following rules: 43.75/12.10 43.75/12.10 0(0(0(1(x1)))) -> 0(1(0(0(2(3(x1)))))) 43.75/12.10 0(0(4(1(x1)))) -> 4(2(0(1(0(x1))))) 43.75/12.10 0(0(4(1(x1)))) -> 0(2(1(4(2(0(x1)))))) 43.75/12.10 0(0(4(1(x1)))) -> 1(0(5(4(2(0(x1)))))) 43.75/12.10 0(0(4(1(x1)))) -> 3(4(2(0(1(0(x1)))))) 43.75/12.10 0(3(4(0(x1)))) -> 0(3(0(2(4(2(x1)))))) 43.75/12.10 0(3(4(0(x1)))) -> 0(3(4(2(0(2(x1)))))) 43.75/12.10 0(3(4(0(x1)))) -> 0(4(5(3(0(5(x1)))))) 43.75/12.10 0(4(4(0(x1)))) -> 4(4(2(0(0(x1))))) 43.75/12.10 0(4(4(0(x1)))) -> 0(4(0(2(4(2(x1)))))) 43.75/12.10 0(4(4(0(x1)))) -> 4(2(0(0(2(4(x1)))))) 43.75/12.10 0(4(4(0(x1)))) -> 4(2(0(0(4(2(x1)))))) 43.75/12.10 0(4(4(0(x1)))) -> 4(2(0(4(2(0(x1)))))) 43.75/12.10 0(4(4(1(x1)))) -> 0(1(4(4(2(x1))))) 43.75/12.10 0(4(4(1(x1)))) -> 1(4(4(2(0(x1))))) 43.75/12.10 0(4(4(1(x1)))) -> 0(1(4(4(2(3(x1)))))) 43.75/12.10 0(4(4(1(x1)))) -> 0(1(5(4(4(2(x1)))))) 43.75/12.10 0(4(4(1(x1)))) -> 1(5(4(4(2(0(x1)))))) 43.75/12.10 0(4(4(1(x1)))) -> 2(4(0(1(5(4(x1)))))) 43.75/12.10 0(4(4(1(x1)))) -> 2(4(4(1(2(0(x1)))))) 43.75/12.10 0(4(4(1(x1)))) -> 2(4(5(4(0(1(x1)))))) 43.75/12.10 0(4(4(1(x1)))) -> 4(2(4(2(1(0(x1)))))) 43.75/12.10 1(4(4(0(x1)))) -> 5(4(4(2(0(1(x1)))))) 43.75/12.10 1(4(4(1(x1)))) -> 2(1(4(1(2(4(x1)))))) 43.75/12.10 1(4(4(1(x1)))) -> 2(3(4(1(4(1(x1)))))) 43.75/12.10 1(4(4(1(x1)))) -> 2(4(1(2(1(4(x1)))))) 43.75/12.10 1(4(4(1(x1)))) -> 2(4(2(1(4(1(x1)))))) 43.75/12.10 4(0(0(0(x1)))) -> 0(4(2(0(0(5(x1)))))) 43.75/12.10 4(0(0(1(x1)))) -> 2(3(0(1(0(4(x1)))))) 43.75/12.10 4(0(4(0(x1)))) -> 2(4(2(0(0(4(x1)))))) 43.75/12.10 4(0(4(0(x1)))) -> 3(4(2(0(4(0(x1)))))) 43.75/12.10 4(0(4(1(x1)))) -> 2(4(2(0(4(1(x1)))))) 43.75/12.10 4(3(4(0(x1)))) -> 4(0(3(4(2(x1))))) 43.75/12.10 4(3(4(0(x1)))) -> 4(4(2(0(3(x1))))) 43.75/12.10 4(3(4(0(x1)))) -> 0(4(2(3(4(2(x1)))))) 43.75/12.10 4(3(4(0(x1)))) -> 2(3(0(4(4(5(x1)))))) 43.75/12.10 4(3(4(0(x1)))) -> 2(3(4(4(0(2(x1)))))) 43.75/12.10 4(3(4(0(x1)))) -> 2(4(0(3(4(2(x1)))))) 43.75/12.10 4(3(4(0(x1)))) -> 4(2(0(4(2(3(x1)))))) 43.75/12.10 4(3(4(0(x1)))) -> 4(3(0(5(4(2(x1)))))) 43.75/12.10 4(3(4(0(x1)))) -> 4(4(2(3(0(2(x1)))))) 43.75/12.10 4(3(4(0(x1)))) -> 4(4(5(3(0(3(x1)))))) 43.75/12.10 4(3(4(1(x1)))) -> 3(4(2(4(1(x1))))) 43.75/12.10 4(3(4(1(x1)))) -> 1(4(2(4(2(3(x1)))))) 43.75/12.10 4(3(4(1(x1)))) -> 2(3(1(4(5(4(x1)))))) 43.75/12.10 4(3(4(1(x1)))) -> 2(3(4(2(1(4(x1)))))) 43.75/12.10 4(3(4(1(x1)))) -> 2(4(2(3(1(4(x1)))))) 43.75/12.10 4(3(4(1(x1)))) -> 3(1(4(2(4(2(x1)))))) 43.75/12.10 4(3(4(1(x1)))) -> 3(4(2(4(5(1(x1)))))) 43.75/12.10 4(3(4(1(x1)))) -> 3(4(5(3(4(1(x1)))))) 43.75/12.10 4(4(4(0(x1)))) -> 2(4(2(4(4(0(x1)))))) 43.75/12.10 4(4(4(1(x1)))) -> 2(4(2(4(1(4(x1)))))) 43.75/12.10 4(4(4(1(x1)))) -> 4(2(4(1(2(4(x1)))))) 43.75/12.10 4(4(4(1(x1)))) -> 4(5(4(1(5(4(x1)))))) 43.75/12.10 0(0(4(0(0(x1))))) -> 0(4(2(0(0(0(x1)))))) 43.75/12.10 0(1(3(5(0(x1))))) -> 0(1(3(0(5(3(x1)))))) 43.75/12.10 0(3(4(0(0(x1))))) -> 3(4(2(0(0(0(x1)))))) 43.75/12.10 0(3(5(0(1(x1))))) -> 0(1(0(5(2(3(x1)))))) 43.75/12.10 0(3(5(4(1(x1))))) -> 5(1(0(4(5(3(x1)))))) 43.75/12.10 0(4(3(4(0(x1))))) -> 0(2(3(4(4(0(x1)))))) 43.75/12.10 1(4(4(0(0(x1))))) -> 4(1(4(2(0(0(x1)))))) 43.75/12.10 1(4(4(0(1(x1))))) -> 1(4(4(1(2(0(x1)))))) 43.75/12.10 1(4(4(4(1(x1))))) -> 2(4(1(4(4(1(x1)))))) 43.75/12.10 4(0(0(4(1(x1))))) -> 4(4(2(0(1(0(x1)))))) 43.75/12.10 4(0(4(4(0(x1))))) -> 4(0(4(2(4(0(x1)))))) 43.75/12.10 4(1(0(0(1(x1))))) -> 4(2(0(1(1(0(x1)))))) 43.75/12.10 4(1(3(4(0(x1))))) -> 3(1(4(2(4(0(x1)))))) 43.75/12.10 4(3(1(0(0(x1))))) -> 1(4(2(3(0(0(x1)))))) 43.75/12.10 4(3(1(0(0(x1))))) -> 3(1(4(2(0(0(x1)))))) 43.75/12.10 4(3(1(5(0(x1))))) -> 1(4(5(3(0(1(x1)))))) 43.75/12.10 4(3(2(4(0(x1))))) -> 4(2(0(3(4(2(x1)))))) 43.75/12.10 4(3(4(4(1(x1))))) -> 4(4(3(1(4(2(x1)))))) 43.75/12.10 4(3(5(0(0(x1))))) -> 4(0(5(2(3(0(x1)))))) 43.75/12.10 4(3(5(0(1(x1))))) -> 2(3(5(4(0(1(x1)))))) 43.75/12.10 4(3(5(4(0(x1))))) -> 0(3(4(5(4(5(x1)))))) 43.75/12.10 4(3(5(4(0(x1))))) -> 0(4(5(3(4(2(x1)))))) 43.75/12.10 4(3(5(4(0(x1))))) -> 4(4(5(3(0(2(x1)))))) 43.75/12.10 4(3(5(4(0(x1))))) -> 5(4(3(4(2(0(x1)))))) 43.75/12.10 4(3(5(4(1(x1))))) -> 3(4(2(4(5(1(x1)))))) 43.75/12.10 4(3(5(5(0(x1))))) -> 0(2(3(5(4(5(x1)))))) 43.75/12.10 4(3(5(5(1(x1))))) -> 1(5(4(2(5(3(x1)))))) 43.75/12.10 4(4(0(0(1(x1))))) -> 4(0(4(2(0(1(x1)))))) 43.75/12.10 4(4(4(4(1(x1))))) -> 4(4(4(1(2(4(x1)))))) 43.75/12.10 4(4(5(0(1(x1))))) -> 4(5(0(1(4(2(x1)))))) 43.75/12.10 43.75/12.10 Q is empty. 43.75/12.10 43.75/12.10 ---------------------------------------- 43.75/12.10 43.75/12.10 (1) QTRS Reverse (EQUIVALENT) 43.75/12.10 We applied the QTRS Reverse Processor [REVERSE]. 43.75/12.10 ---------------------------------------- 43.75/12.10 43.75/12.10 (2) 43.75/12.10 Obligation: 43.75/12.10 Q restricted rewrite system: 43.75/12.10 The TRS R consists of the following rules: 43.75/12.10 43.75/12.10 1(0(0(0(x1)))) -> 3(2(0(0(1(0(x1)))))) 43.75/12.10 1(4(0(0(x1)))) -> 0(1(0(2(4(x1))))) 43.75/12.10 1(4(0(0(x1)))) -> 0(2(4(1(2(0(x1)))))) 43.75/12.10 1(4(0(0(x1)))) -> 0(2(4(5(0(1(x1)))))) 43.75/12.10 1(4(0(0(x1)))) -> 0(1(0(2(4(3(x1)))))) 43.75/12.10 0(4(3(0(x1)))) -> 2(4(2(0(3(0(x1)))))) 43.75/12.10 0(4(3(0(x1)))) -> 2(0(2(4(3(0(x1)))))) 43.75/12.10 0(4(3(0(x1)))) -> 5(0(3(5(4(0(x1)))))) 43.75/12.10 0(4(4(0(x1)))) -> 0(0(2(4(4(x1))))) 43.75/12.10 0(4(4(0(x1)))) -> 2(4(2(0(4(0(x1)))))) 43.75/12.10 0(4(4(0(x1)))) -> 4(2(0(0(2(4(x1)))))) 43.75/12.10 0(4(4(0(x1)))) -> 2(4(0(0(2(4(x1)))))) 43.75/12.10 0(4(4(0(x1)))) -> 0(2(4(0(2(4(x1)))))) 43.75/12.10 1(4(4(0(x1)))) -> 2(4(4(1(0(x1))))) 43.75/12.10 1(4(4(0(x1)))) -> 0(2(4(4(1(x1))))) 43.75/12.10 1(4(4(0(x1)))) -> 3(2(4(4(1(0(x1)))))) 43.75/12.10 1(4(4(0(x1)))) -> 2(4(4(5(1(0(x1)))))) 43.75/12.10 1(4(4(0(x1)))) -> 0(2(4(4(5(1(x1)))))) 43.75/12.10 1(4(4(0(x1)))) -> 4(5(1(0(4(2(x1)))))) 43.75/12.10 1(4(4(0(x1)))) -> 0(2(1(4(4(2(x1)))))) 43.75/12.10 1(4(4(0(x1)))) -> 1(0(4(5(4(2(x1)))))) 43.75/12.10 1(4(4(0(x1)))) -> 0(1(2(4(2(4(x1)))))) 43.75/12.10 0(4(4(1(x1)))) -> 1(0(2(4(4(5(x1)))))) 43.75/12.10 1(4(4(1(x1)))) -> 4(2(1(4(1(2(x1)))))) 43.75/12.10 1(4(4(1(x1)))) -> 1(4(1(4(3(2(x1)))))) 43.75/12.10 1(4(4(1(x1)))) -> 4(1(2(1(4(2(x1)))))) 43.75/12.10 1(4(4(1(x1)))) -> 1(4(1(2(4(2(x1)))))) 43.75/12.10 0(0(0(4(x1)))) -> 5(0(0(2(4(0(x1)))))) 43.75/12.10 1(0(0(4(x1)))) -> 4(0(1(0(3(2(x1)))))) 43.75/12.10 0(4(0(4(x1)))) -> 4(0(0(2(4(2(x1)))))) 43.75/12.10 0(4(0(4(x1)))) -> 0(4(0(2(4(3(x1)))))) 43.75/12.10 1(4(0(4(x1)))) -> 1(4(0(2(4(2(x1)))))) 43.75/12.10 0(4(3(4(x1)))) -> 2(4(3(0(4(x1))))) 43.75/12.10 0(4(3(4(x1)))) -> 3(0(2(4(4(x1))))) 43.75/12.10 0(4(3(4(x1)))) -> 2(4(3(2(4(0(x1)))))) 43.75/12.10 0(4(3(4(x1)))) -> 5(4(4(0(3(2(x1)))))) 43.75/12.10 0(4(3(4(x1)))) -> 2(0(4(4(3(2(x1)))))) 43.75/12.10 0(4(3(4(x1)))) -> 2(4(3(0(4(2(x1)))))) 43.75/12.10 0(4(3(4(x1)))) -> 3(2(4(0(2(4(x1)))))) 43.75/12.10 0(4(3(4(x1)))) -> 2(4(5(0(3(4(x1)))))) 43.75/12.10 0(4(3(4(x1)))) -> 2(0(3(2(4(4(x1)))))) 43.75/12.10 0(4(3(4(x1)))) -> 3(0(3(5(4(4(x1)))))) 43.75/12.10 1(4(3(4(x1)))) -> 1(4(2(4(3(x1))))) 43.75/12.10 1(4(3(4(x1)))) -> 3(2(4(2(4(1(x1)))))) 43.75/12.10 1(4(3(4(x1)))) -> 4(5(4(1(3(2(x1)))))) 43.75/12.10 1(4(3(4(x1)))) -> 4(1(2(4(3(2(x1)))))) 43.75/12.10 1(4(3(4(x1)))) -> 4(1(3(2(4(2(x1)))))) 43.75/12.10 1(4(3(4(x1)))) -> 2(4(2(4(1(3(x1)))))) 43.75/12.10 1(4(3(4(x1)))) -> 1(5(4(2(4(3(x1)))))) 43.75/12.10 1(4(3(4(x1)))) -> 1(4(3(5(4(3(x1)))))) 43.75/12.10 0(4(4(4(x1)))) -> 0(4(4(2(4(2(x1)))))) 43.75/12.10 1(4(4(4(x1)))) -> 4(1(4(2(4(2(x1)))))) 43.75/12.10 1(4(4(4(x1)))) -> 4(2(1(4(2(4(x1)))))) 43.75/12.10 1(4(4(4(x1)))) -> 4(5(1(4(5(4(x1)))))) 43.75/12.10 0(0(4(0(0(x1))))) -> 0(0(0(2(4(0(x1)))))) 43.75/12.10 0(5(3(1(0(x1))))) -> 3(5(0(3(1(0(x1)))))) 43.75/12.10 0(0(4(3(0(x1))))) -> 0(0(0(2(4(3(x1)))))) 43.75/12.10 1(0(5(3(0(x1))))) -> 3(2(5(0(1(0(x1)))))) 43.75/12.10 1(4(5(3(0(x1))))) -> 3(5(4(0(1(5(x1)))))) 43.75/12.10 0(4(3(4(0(x1))))) -> 0(4(4(3(2(0(x1)))))) 43.75/12.10 0(0(4(4(1(x1))))) -> 0(0(2(4(1(4(x1)))))) 43.75/12.10 1(0(4(4(1(x1))))) -> 0(2(1(4(4(1(x1)))))) 43.75/12.10 1(4(4(4(1(x1))))) -> 1(4(4(1(4(2(x1)))))) 43.75/12.10 1(4(0(0(4(x1))))) -> 0(1(0(2(4(4(x1)))))) 43.75/12.10 0(4(4(0(4(x1))))) -> 0(4(2(4(0(4(x1)))))) 43.75/12.10 1(0(0(1(4(x1))))) -> 0(1(1(0(2(4(x1)))))) 43.75/12.10 0(4(3(1(4(x1))))) -> 0(4(2(4(1(3(x1)))))) 43.75/12.10 0(0(1(3(4(x1))))) -> 0(0(3(2(4(1(x1)))))) 43.75/12.10 0(0(1(3(4(x1))))) -> 0(0(2(4(1(3(x1)))))) 43.75/12.10 0(5(1(3(4(x1))))) -> 1(0(3(5(4(1(x1)))))) 43.75/12.10 0(4(2(3(4(x1))))) -> 2(4(3(0(2(4(x1)))))) 43.75/12.10 1(4(4(3(4(x1))))) -> 2(4(1(3(4(4(x1)))))) 43.75/12.10 0(0(5(3(4(x1))))) -> 0(3(2(5(0(4(x1)))))) 43.75/12.10 1(0(5(3(4(x1))))) -> 1(0(4(5(3(2(x1)))))) 43.75/12.10 0(4(5(3(4(x1))))) -> 5(4(5(4(3(0(x1)))))) 43.75/12.10 0(4(5(3(4(x1))))) -> 2(4(3(5(4(0(x1)))))) 43.75/12.10 0(4(5(3(4(x1))))) -> 2(0(3(5(4(4(x1)))))) 43.75/12.10 0(4(5(3(4(x1))))) -> 0(2(4(3(4(5(x1)))))) 43.75/12.10 1(4(5(3(4(x1))))) -> 1(5(4(2(4(3(x1)))))) 43.75/12.10 0(5(5(3(4(x1))))) -> 5(4(5(3(2(0(x1)))))) 43.75/12.10 1(5(5(3(4(x1))))) -> 3(5(2(4(5(1(x1)))))) 43.75/12.10 1(0(0(4(4(x1))))) -> 1(0(2(4(0(4(x1)))))) 43.75/12.10 1(4(4(4(4(x1))))) -> 4(2(1(4(4(4(x1)))))) 43.75/12.10 1(0(5(4(4(x1))))) -> 2(4(1(0(5(4(x1)))))) 43.75/12.10 43.75/12.10 Q is empty. 43.75/12.10 43.75/12.10 ---------------------------------------- 43.75/12.10 43.75/12.10 (3) DependencyPairsProof (EQUIVALENT) 43.75/12.10 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 43.75/12.10 ---------------------------------------- 43.75/12.10 43.75/12.10 (4) 43.75/12.10 Obligation: 43.75/12.10 Q DP problem: 43.75/12.10 The TRS P consists of the following rules: 43.75/12.10 43.75/12.10 1^1(0(0(0(x1)))) -> 0^1(0(1(0(x1)))) 43.75/12.10 1^1(0(0(0(x1)))) -> 0^1(1(0(x1))) 43.75/12.10 1^1(0(0(0(x1)))) -> 1^1(0(x1)) 43.75/12.10 1^1(4(0(0(x1)))) -> 0^1(1(0(2(4(x1))))) 43.75/12.10 1^1(4(0(0(x1)))) -> 1^1(0(2(4(x1)))) 43.75/12.10 1^1(4(0(0(x1)))) -> 0^1(2(4(x1))) 43.75/12.10 1^1(4(0(0(x1)))) -> 0^1(2(4(1(2(0(x1)))))) 43.75/12.10 1^1(4(0(0(x1)))) -> 1^1(2(0(x1))) 43.75/12.10 1^1(4(0(0(x1)))) -> 0^1(2(4(5(0(1(x1)))))) 43.75/12.10 1^1(4(0(0(x1)))) -> 0^1(1(x1)) 43.75/12.10 1^1(4(0(0(x1)))) -> 1^1(x1) 43.75/12.10 1^1(4(0(0(x1)))) -> 0^1(1(0(2(4(3(x1)))))) 43.75/12.10 1^1(4(0(0(x1)))) -> 1^1(0(2(4(3(x1))))) 43.75/12.10 1^1(4(0(0(x1)))) -> 0^1(2(4(3(x1)))) 43.75/12.10 0^1(4(3(0(x1)))) -> 0^1(3(0(x1))) 43.75/12.10 0^1(4(3(0(x1)))) -> 0^1(2(4(3(0(x1))))) 43.75/12.10 0^1(4(3(0(x1)))) -> 0^1(3(5(4(0(x1))))) 43.75/12.10 0^1(4(4(0(x1)))) -> 0^1(0(2(4(4(x1))))) 43.75/12.10 0^1(4(4(0(x1)))) -> 0^1(2(4(4(x1)))) 43.75/12.10 0^1(4(4(0(x1)))) -> 0^1(4(0(x1))) 43.75/12.10 0^1(4(4(0(x1)))) -> 0^1(0(2(4(x1)))) 43.75/12.10 0^1(4(4(0(x1)))) -> 0^1(2(4(x1))) 43.75/12.10 0^1(4(4(0(x1)))) -> 0^1(2(4(0(2(4(x1)))))) 43.75/12.10 1^1(4(4(0(x1)))) -> 1^1(0(x1)) 43.75/12.10 1^1(4(4(0(x1)))) -> 0^1(2(4(4(1(x1))))) 43.75/12.10 1^1(4(4(0(x1)))) -> 1^1(x1) 43.75/12.10 1^1(4(4(0(x1)))) -> 0^1(2(4(4(5(1(x1)))))) 43.75/12.10 1^1(4(4(0(x1)))) -> 1^1(0(4(2(x1)))) 43.75/12.10 1^1(4(4(0(x1)))) -> 0^1(4(2(x1))) 43.75/12.10 1^1(4(4(0(x1)))) -> 0^1(2(1(4(4(2(x1)))))) 43.75/12.10 1^1(4(4(0(x1)))) -> 1^1(4(4(2(x1)))) 43.75/12.10 1^1(4(4(0(x1)))) -> 1^1(0(4(5(4(2(x1)))))) 43.75/12.10 1^1(4(4(0(x1)))) -> 0^1(4(5(4(2(x1))))) 43.75/12.10 1^1(4(4(0(x1)))) -> 0^1(1(2(4(2(4(x1)))))) 43.75/12.10 1^1(4(4(0(x1)))) -> 1^1(2(4(2(4(x1))))) 43.75/12.10 0^1(4(4(1(x1)))) -> 1^1(0(2(4(4(5(x1)))))) 43.75/12.10 0^1(4(4(1(x1)))) -> 0^1(2(4(4(5(x1))))) 43.75/12.10 1^1(4(4(1(x1)))) -> 1^1(4(1(2(x1)))) 43.75/12.10 1^1(4(4(1(x1)))) -> 1^1(2(x1)) 43.75/12.10 1^1(4(4(1(x1)))) -> 1^1(4(1(4(3(2(x1)))))) 43.75/12.10 1^1(4(4(1(x1)))) -> 1^1(4(3(2(x1)))) 43.75/12.10 1^1(4(4(1(x1)))) -> 1^1(2(1(4(2(x1))))) 43.75/12.10 1^1(4(4(1(x1)))) -> 1^1(4(2(x1))) 43.75/12.10 1^1(4(4(1(x1)))) -> 1^1(4(1(2(4(2(x1)))))) 43.75/12.10 1^1(4(4(1(x1)))) -> 1^1(2(4(2(x1)))) 43.75/12.10 0^1(0(0(4(x1)))) -> 0^1(0(2(4(0(x1))))) 43.75/12.10 0^1(0(0(4(x1)))) -> 0^1(2(4(0(x1)))) 43.75/12.10 0^1(0(0(4(x1)))) -> 0^1(x1) 43.75/12.10 1^1(0(0(4(x1)))) -> 0^1(1(0(3(2(x1))))) 43.75/12.10 1^1(0(0(4(x1)))) -> 1^1(0(3(2(x1)))) 43.75/12.10 1^1(0(0(4(x1)))) -> 0^1(3(2(x1))) 43.75/12.10 0^1(4(0(4(x1)))) -> 0^1(0(2(4(2(x1))))) 43.75/12.10 0^1(4(0(4(x1)))) -> 0^1(2(4(2(x1)))) 43.75/12.10 0^1(4(0(4(x1)))) -> 0^1(4(0(2(4(3(x1)))))) 43.75/12.10 0^1(4(0(4(x1)))) -> 0^1(2(4(3(x1)))) 43.75/12.10 1^1(4(0(4(x1)))) -> 1^1(4(0(2(4(2(x1)))))) 43.75/12.10 1^1(4(0(4(x1)))) -> 0^1(2(4(2(x1)))) 43.75/12.10 0^1(4(3(4(x1)))) -> 0^1(4(x1)) 43.75/12.10 0^1(4(3(4(x1)))) -> 0^1(2(4(4(x1)))) 43.75/12.10 0^1(4(3(4(x1)))) -> 0^1(x1) 43.75/12.10 0^1(4(3(4(x1)))) -> 0^1(3(2(x1))) 43.75/12.10 0^1(4(3(4(x1)))) -> 0^1(4(4(3(2(x1))))) 43.75/12.10 0^1(4(3(4(x1)))) -> 0^1(4(2(x1))) 43.75/12.10 0^1(4(3(4(x1)))) -> 0^1(2(4(x1))) 43.75/12.10 0^1(4(3(4(x1)))) -> 0^1(3(4(x1))) 43.75/12.10 0^1(4(3(4(x1)))) -> 0^1(3(2(4(4(x1))))) 43.75/12.10 0^1(4(3(4(x1)))) -> 0^1(3(5(4(4(x1))))) 43.75/12.10 1^1(4(3(4(x1)))) -> 1^1(4(2(4(3(x1))))) 43.75/12.10 1^1(4(3(4(x1)))) -> 1^1(x1) 43.75/12.10 1^1(4(3(4(x1)))) -> 1^1(3(2(x1))) 43.75/12.10 1^1(4(3(4(x1)))) -> 1^1(2(4(3(2(x1))))) 43.75/12.10 1^1(4(3(4(x1)))) -> 1^1(3(2(4(2(x1))))) 43.75/12.10 1^1(4(3(4(x1)))) -> 1^1(3(x1)) 43.75/12.10 1^1(4(3(4(x1)))) -> 1^1(5(4(2(4(3(x1)))))) 43.75/12.10 1^1(4(3(4(x1)))) -> 1^1(4(3(5(4(3(x1)))))) 43.75/12.10 0^1(4(4(4(x1)))) -> 0^1(4(4(2(4(2(x1)))))) 43.75/12.10 1^1(4(4(4(x1)))) -> 1^1(4(2(4(2(x1))))) 43.75/12.10 1^1(4(4(4(x1)))) -> 1^1(4(2(4(x1)))) 43.75/12.10 1^1(4(4(4(x1)))) -> 1^1(4(5(4(x1)))) 43.75/12.10 0^1(0(4(0(0(x1))))) -> 0^1(0(0(2(4(0(x1)))))) 43.75/12.10 0^1(0(4(0(0(x1))))) -> 0^1(0(2(4(0(x1))))) 43.75/12.10 0^1(0(4(0(0(x1))))) -> 0^1(2(4(0(x1)))) 43.75/12.10 0^1(5(3(1(0(x1))))) -> 0^1(3(1(0(x1)))) 43.75/12.10 0^1(0(4(3(0(x1))))) -> 0^1(0(0(2(4(3(x1)))))) 43.75/12.10 0^1(0(4(3(0(x1))))) -> 0^1(0(2(4(3(x1))))) 43.75/12.10 0^1(0(4(3(0(x1))))) -> 0^1(2(4(3(x1)))) 43.75/12.10 1^1(0(5(3(0(x1))))) -> 0^1(1(0(x1))) 43.75/12.10 1^1(0(5(3(0(x1))))) -> 1^1(0(x1)) 43.75/12.10 1^1(4(5(3(0(x1))))) -> 0^1(1(5(x1))) 43.75/12.10 1^1(4(5(3(0(x1))))) -> 1^1(5(x1)) 43.75/12.10 0^1(4(3(4(0(x1))))) -> 0^1(4(4(3(2(0(x1)))))) 43.75/12.10 0^1(0(4(4(1(x1))))) -> 0^1(0(2(4(1(4(x1)))))) 43.75/12.10 0^1(0(4(4(1(x1))))) -> 0^1(2(4(1(4(x1))))) 43.75/12.10 0^1(0(4(4(1(x1))))) -> 1^1(4(x1)) 43.75/12.10 1^1(0(4(4(1(x1))))) -> 0^1(2(1(4(4(1(x1)))))) 43.75/12.10 1^1(0(4(4(1(x1))))) -> 1^1(4(4(1(x1)))) 43.75/12.10 1^1(4(4(4(1(x1))))) -> 1^1(4(4(1(4(2(x1)))))) 43.75/12.10 1^1(4(4(4(1(x1))))) -> 1^1(4(2(x1))) 43.75/12.10 1^1(4(0(0(4(x1))))) -> 0^1(1(0(2(4(4(x1)))))) 43.75/12.10 1^1(4(0(0(4(x1))))) -> 1^1(0(2(4(4(x1))))) 43.75/12.10 1^1(4(0(0(4(x1))))) -> 0^1(2(4(4(x1)))) 43.75/12.10 0^1(4(4(0(4(x1))))) -> 0^1(4(2(4(0(4(x1)))))) 43.75/12.10 1^1(0(0(1(4(x1))))) -> 0^1(1(1(0(2(4(x1)))))) 43.75/12.10 1^1(0(0(1(4(x1))))) -> 1^1(1(0(2(4(x1))))) 43.75/12.10 1^1(0(0(1(4(x1))))) -> 1^1(0(2(4(x1)))) 43.75/12.10 1^1(0(0(1(4(x1))))) -> 0^1(2(4(x1))) 43.75/12.10 0^1(4(3(1(4(x1))))) -> 0^1(4(2(4(1(3(x1)))))) 43.75/12.10 0^1(4(3(1(4(x1))))) -> 1^1(3(x1)) 43.75/12.10 0^1(0(1(3(4(x1))))) -> 0^1(0(3(2(4(1(x1)))))) 43.75/12.10 0^1(0(1(3(4(x1))))) -> 0^1(3(2(4(1(x1))))) 43.75/12.10 0^1(0(1(3(4(x1))))) -> 1^1(x1) 43.75/12.10 0^1(0(1(3(4(x1))))) -> 0^1(0(2(4(1(3(x1)))))) 43.75/12.10 0^1(0(1(3(4(x1))))) -> 0^1(2(4(1(3(x1))))) 43.75/12.10 0^1(0(1(3(4(x1))))) -> 1^1(3(x1)) 43.75/12.10 0^1(5(1(3(4(x1))))) -> 1^1(0(3(5(4(1(x1)))))) 43.75/12.10 0^1(5(1(3(4(x1))))) -> 0^1(3(5(4(1(x1))))) 43.75/12.10 0^1(5(1(3(4(x1))))) -> 1^1(x1) 43.75/12.10 0^1(4(2(3(4(x1))))) -> 0^1(2(4(x1))) 43.75/12.10 1^1(4(4(3(4(x1))))) -> 1^1(3(4(4(x1)))) 43.75/12.10 0^1(0(5(3(4(x1))))) -> 0^1(3(2(5(0(4(x1)))))) 43.75/12.10 0^1(0(5(3(4(x1))))) -> 0^1(4(x1)) 43.75/12.10 1^1(0(5(3(4(x1))))) -> 1^1(0(4(5(3(2(x1)))))) 43.75/12.10 1^1(0(5(3(4(x1))))) -> 0^1(4(5(3(2(x1))))) 43.75/12.10 0^1(4(5(3(4(x1))))) -> 0^1(x1) 43.75/12.10 0^1(4(5(3(4(x1))))) -> 0^1(3(5(4(4(x1))))) 43.75/12.10 0^1(4(5(3(4(x1))))) -> 0^1(2(4(3(4(5(x1)))))) 43.75/12.10 1^1(4(5(3(4(x1))))) -> 1^1(5(4(2(4(3(x1)))))) 43.75/12.10 0^1(5(5(3(4(x1))))) -> 0^1(x1) 43.75/12.10 1^1(5(5(3(4(x1))))) -> 1^1(x1) 43.75/12.10 1^1(0(0(4(4(x1))))) -> 1^1(0(2(4(0(4(x1)))))) 43.75/12.10 1^1(0(0(4(4(x1))))) -> 0^1(2(4(0(4(x1))))) 43.75/12.10 1^1(0(0(4(4(x1))))) -> 0^1(4(x1)) 43.75/12.10 1^1(4(4(4(4(x1))))) -> 1^1(4(4(4(x1)))) 43.75/12.10 1^1(0(5(4(4(x1))))) -> 1^1(0(5(4(x1)))) 43.75/12.10 1^1(0(5(4(4(x1))))) -> 0^1(5(4(x1))) 43.75/12.10 43.75/12.10 The TRS R consists of the following rules: 43.75/12.10 43.75/12.10 1(0(0(0(x1)))) -> 3(2(0(0(1(0(x1)))))) 43.75/12.10 1(4(0(0(x1)))) -> 0(1(0(2(4(x1))))) 43.75/12.10 1(4(0(0(x1)))) -> 0(2(4(1(2(0(x1)))))) 43.75/12.10 1(4(0(0(x1)))) -> 0(2(4(5(0(1(x1)))))) 43.75/12.10 1(4(0(0(x1)))) -> 0(1(0(2(4(3(x1)))))) 43.75/12.10 0(4(3(0(x1)))) -> 2(4(2(0(3(0(x1)))))) 43.75/12.10 0(4(3(0(x1)))) -> 2(0(2(4(3(0(x1)))))) 43.75/12.10 0(4(3(0(x1)))) -> 5(0(3(5(4(0(x1)))))) 43.75/12.10 0(4(4(0(x1)))) -> 0(0(2(4(4(x1))))) 43.75/12.10 0(4(4(0(x1)))) -> 2(4(2(0(4(0(x1)))))) 43.75/12.10 0(4(4(0(x1)))) -> 4(2(0(0(2(4(x1)))))) 43.75/12.10 0(4(4(0(x1)))) -> 2(4(0(0(2(4(x1)))))) 43.75/12.10 0(4(4(0(x1)))) -> 0(2(4(0(2(4(x1)))))) 43.75/12.10 1(4(4(0(x1)))) -> 2(4(4(1(0(x1))))) 43.75/12.10 1(4(4(0(x1)))) -> 0(2(4(4(1(x1))))) 43.75/12.10 1(4(4(0(x1)))) -> 3(2(4(4(1(0(x1)))))) 43.75/12.10 1(4(4(0(x1)))) -> 2(4(4(5(1(0(x1)))))) 43.75/12.10 1(4(4(0(x1)))) -> 0(2(4(4(5(1(x1)))))) 43.75/12.10 1(4(4(0(x1)))) -> 4(5(1(0(4(2(x1)))))) 43.75/12.10 1(4(4(0(x1)))) -> 0(2(1(4(4(2(x1)))))) 43.75/12.10 1(4(4(0(x1)))) -> 1(0(4(5(4(2(x1)))))) 43.75/12.10 1(4(4(0(x1)))) -> 0(1(2(4(2(4(x1)))))) 43.75/12.10 0(4(4(1(x1)))) -> 1(0(2(4(4(5(x1)))))) 43.75/12.10 1(4(4(1(x1)))) -> 4(2(1(4(1(2(x1)))))) 43.75/12.10 1(4(4(1(x1)))) -> 1(4(1(4(3(2(x1)))))) 43.75/12.10 1(4(4(1(x1)))) -> 4(1(2(1(4(2(x1)))))) 43.75/12.10 1(4(4(1(x1)))) -> 1(4(1(2(4(2(x1)))))) 43.75/12.10 0(0(0(4(x1)))) -> 5(0(0(2(4(0(x1)))))) 43.75/12.10 1(0(0(4(x1)))) -> 4(0(1(0(3(2(x1)))))) 43.75/12.10 0(4(0(4(x1)))) -> 4(0(0(2(4(2(x1)))))) 43.75/12.10 0(4(0(4(x1)))) -> 0(4(0(2(4(3(x1)))))) 43.75/12.10 1(4(0(4(x1)))) -> 1(4(0(2(4(2(x1)))))) 43.75/12.10 0(4(3(4(x1)))) -> 2(4(3(0(4(x1))))) 43.75/12.10 0(4(3(4(x1)))) -> 3(0(2(4(4(x1))))) 43.75/12.10 0(4(3(4(x1)))) -> 2(4(3(2(4(0(x1)))))) 43.75/12.10 0(4(3(4(x1)))) -> 5(4(4(0(3(2(x1)))))) 43.75/12.10 0(4(3(4(x1)))) -> 2(0(4(4(3(2(x1)))))) 43.75/12.10 0(4(3(4(x1)))) -> 2(4(3(0(4(2(x1)))))) 43.75/12.10 0(4(3(4(x1)))) -> 3(2(4(0(2(4(x1)))))) 43.75/12.10 0(4(3(4(x1)))) -> 2(4(5(0(3(4(x1)))))) 43.75/12.10 0(4(3(4(x1)))) -> 2(0(3(2(4(4(x1)))))) 43.75/12.10 0(4(3(4(x1)))) -> 3(0(3(5(4(4(x1)))))) 43.75/12.10 1(4(3(4(x1)))) -> 1(4(2(4(3(x1))))) 43.75/12.10 1(4(3(4(x1)))) -> 3(2(4(2(4(1(x1)))))) 43.75/12.10 1(4(3(4(x1)))) -> 4(5(4(1(3(2(x1)))))) 43.75/12.10 1(4(3(4(x1)))) -> 4(1(2(4(3(2(x1)))))) 43.75/12.10 1(4(3(4(x1)))) -> 4(1(3(2(4(2(x1)))))) 43.75/12.10 1(4(3(4(x1)))) -> 2(4(2(4(1(3(x1)))))) 43.75/12.10 1(4(3(4(x1)))) -> 1(5(4(2(4(3(x1)))))) 43.75/12.10 1(4(3(4(x1)))) -> 1(4(3(5(4(3(x1)))))) 43.75/12.10 0(4(4(4(x1)))) -> 0(4(4(2(4(2(x1)))))) 43.75/12.10 1(4(4(4(x1)))) -> 4(1(4(2(4(2(x1)))))) 43.75/12.10 1(4(4(4(x1)))) -> 4(2(1(4(2(4(x1)))))) 43.75/12.10 1(4(4(4(x1)))) -> 4(5(1(4(5(4(x1)))))) 43.75/12.10 0(0(4(0(0(x1))))) -> 0(0(0(2(4(0(x1)))))) 43.75/12.10 0(5(3(1(0(x1))))) -> 3(5(0(3(1(0(x1)))))) 43.75/12.10 0(0(4(3(0(x1))))) -> 0(0(0(2(4(3(x1)))))) 43.75/12.10 1(0(5(3(0(x1))))) -> 3(2(5(0(1(0(x1)))))) 43.75/12.10 1(4(5(3(0(x1))))) -> 3(5(4(0(1(5(x1)))))) 43.75/12.10 0(4(3(4(0(x1))))) -> 0(4(4(3(2(0(x1)))))) 43.75/12.10 0(0(4(4(1(x1))))) -> 0(0(2(4(1(4(x1)))))) 43.75/12.10 1(0(4(4(1(x1))))) -> 0(2(1(4(4(1(x1)))))) 43.75/12.10 1(4(4(4(1(x1))))) -> 1(4(4(1(4(2(x1)))))) 43.75/12.10 1(4(0(0(4(x1))))) -> 0(1(0(2(4(4(x1)))))) 43.75/12.10 0(4(4(0(4(x1))))) -> 0(4(2(4(0(4(x1)))))) 43.75/12.10 1(0(0(1(4(x1))))) -> 0(1(1(0(2(4(x1)))))) 43.75/12.10 0(4(3(1(4(x1))))) -> 0(4(2(4(1(3(x1)))))) 43.75/12.10 0(0(1(3(4(x1))))) -> 0(0(3(2(4(1(x1)))))) 43.75/12.10 0(0(1(3(4(x1))))) -> 0(0(2(4(1(3(x1)))))) 43.75/12.10 0(5(1(3(4(x1))))) -> 1(0(3(5(4(1(x1)))))) 43.75/12.10 0(4(2(3(4(x1))))) -> 2(4(3(0(2(4(x1)))))) 43.75/12.10 1(4(4(3(4(x1))))) -> 2(4(1(3(4(4(x1)))))) 43.75/12.10 0(0(5(3(4(x1))))) -> 0(3(2(5(0(4(x1)))))) 43.75/12.10 1(0(5(3(4(x1))))) -> 1(0(4(5(3(2(x1)))))) 43.75/12.10 0(4(5(3(4(x1))))) -> 5(4(5(4(3(0(x1)))))) 43.75/12.10 0(4(5(3(4(x1))))) -> 2(4(3(5(4(0(x1)))))) 43.75/12.10 0(4(5(3(4(x1))))) -> 2(0(3(5(4(4(x1)))))) 43.75/12.10 0(4(5(3(4(x1))))) -> 0(2(4(3(4(5(x1)))))) 43.75/12.10 1(4(5(3(4(x1))))) -> 1(5(4(2(4(3(x1)))))) 43.75/12.10 0(5(5(3(4(x1))))) -> 5(4(5(3(2(0(x1)))))) 43.75/12.10 1(5(5(3(4(x1))))) -> 3(5(2(4(5(1(x1)))))) 43.75/12.10 1(0(0(4(4(x1))))) -> 1(0(2(4(0(4(x1)))))) 43.75/12.10 1(4(4(4(4(x1))))) -> 4(2(1(4(4(4(x1)))))) 43.75/12.10 1(0(5(4(4(x1))))) -> 2(4(1(0(5(4(x1)))))) 43.75/12.10 43.75/12.10 Q is empty. 43.75/12.10 We have to consider all minimal (P,Q,R)-chains. 43.75/12.10 ---------------------------------------- 43.75/12.10 43.75/12.10 (5) DependencyGraphProof (EQUIVALENT) 43.75/12.10 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 107 less nodes. 43.75/12.10 ---------------------------------------- 43.75/12.10 43.75/12.10 (6) 43.75/12.10 Complex Obligation (AND) 43.75/12.10 43.75/12.10 ---------------------------------------- 43.75/12.10 43.75/12.10 (7) 43.75/12.10 Obligation: 43.75/12.10 Q DP problem: 43.75/12.10 The TRS P consists of the following rules: 43.75/12.10 43.75/12.10 1^1(0(5(4(4(x1))))) -> 1^1(0(5(4(x1)))) 43.75/12.10 43.75/12.10 The TRS R consists of the following rules: 43.75/12.10 43.75/12.10 1(0(0(0(x1)))) -> 3(2(0(0(1(0(x1)))))) 43.75/12.10 1(4(0(0(x1)))) -> 0(1(0(2(4(x1))))) 43.75/12.10 1(4(0(0(x1)))) -> 0(2(4(1(2(0(x1)))))) 43.75/12.10 1(4(0(0(x1)))) -> 0(2(4(5(0(1(x1)))))) 43.75/12.10 1(4(0(0(x1)))) -> 0(1(0(2(4(3(x1)))))) 43.75/12.10 0(4(3(0(x1)))) -> 2(4(2(0(3(0(x1)))))) 43.75/12.10 0(4(3(0(x1)))) -> 2(0(2(4(3(0(x1)))))) 43.75/12.10 0(4(3(0(x1)))) -> 5(0(3(5(4(0(x1)))))) 43.75/12.10 0(4(4(0(x1)))) -> 0(0(2(4(4(x1))))) 43.75/12.10 0(4(4(0(x1)))) -> 2(4(2(0(4(0(x1)))))) 43.75/12.10 0(4(4(0(x1)))) -> 4(2(0(0(2(4(x1)))))) 43.75/12.10 0(4(4(0(x1)))) -> 2(4(0(0(2(4(x1)))))) 43.75/12.10 0(4(4(0(x1)))) -> 0(2(4(0(2(4(x1)))))) 43.75/12.10 1(4(4(0(x1)))) -> 2(4(4(1(0(x1))))) 43.75/12.10 1(4(4(0(x1)))) -> 0(2(4(4(1(x1))))) 43.75/12.10 1(4(4(0(x1)))) -> 3(2(4(4(1(0(x1)))))) 43.75/12.10 1(4(4(0(x1)))) -> 2(4(4(5(1(0(x1)))))) 43.75/12.10 1(4(4(0(x1)))) -> 0(2(4(4(5(1(x1)))))) 43.75/12.10 1(4(4(0(x1)))) -> 4(5(1(0(4(2(x1)))))) 43.75/12.10 1(4(4(0(x1)))) -> 0(2(1(4(4(2(x1)))))) 43.75/12.10 1(4(4(0(x1)))) -> 1(0(4(5(4(2(x1)))))) 43.75/12.10 1(4(4(0(x1)))) -> 0(1(2(4(2(4(x1)))))) 43.75/12.10 0(4(4(1(x1)))) -> 1(0(2(4(4(5(x1)))))) 43.75/12.10 1(4(4(1(x1)))) -> 4(2(1(4(1(2(x1)))))) 43.75/12.10 1(4(4(1(x1)))) -> 1(4(1(4(3(2(x1)))))) 43.75/12.10 1(4(4(1(x1)))) -> 4(1(2(1(4(2(x1)))))) 43.75/12.10 1(4(4(1(x1)))) -> 1(4(1(2(4(2(x1)))))) 43.75/12.10 0(0(0(4(x1)))) -> 5(0(0(2(4(0(x1)))))) 43.75/12.10 1(0(0(4(x1)))) -> 4(0(1(0(3(2(x1)))))) 43.75/12.10 0(4(0(4(x1)))) -> 4(0(0(2(4(2(x1)))))) 43.75/12.10 0(4(0(4(x1)))) -> 0(4(0(2(4(3(x1)))))) 43.75/12.10 1(4(0(4(x1)))) -> 1(4(0(2(4(2(x1)))))) 43.75/12.10 0(4(3(4(x1)))) -> 2(4(3(0(4(x1))))) 43.75/12.10 0(4(3(4(x1)))) -> 3(0(2(4(4(x1))))) 43.75/12.10 0(4(3(4(x1)))) -> 2(4(3(2(4(0(x1)))))) 43.75/12.10 0(4(3(4(x1)))) -> 5(4(4(0(3(2(x1)))))) 43.75/12.10 0(4(3(4(x1)))) -> 2(0(4(4(3(2(x1)))))) 43.75/12.10 0(4(3(4(x1)))) -> 2(4(3(0(4(2(x1)))))) 43.75/12.10 0(4(3(4(x1)))) -> 3(2(4(0(2(4(x1)))))) 43.75/12.10 0(4(3(4(x1)))) -> 2(4(5(0(3(4(x1)))))) 43.75/12.10 0(4(3(4(x1)))) -> 2(0(3(2(4(4(x1)))))) 43.75/12.10 0(4(3(4(x1)))) -> 3(0(3(5(4(4(x1)))))) 43.75/12.10 1(4(3(4(x1)))) -> 1(4(2(4(3(x1))))) 43.75/12.10 1(4(3(4(x1)))) -> 3(2(4(2(4(1(x1)))))) 43.75/12.10 1(4(3(4(x1)))) -> 4(5(4(1(3(2(x1)))))) 43.75/12.10 1(4(3(4(x1)))) -> 4(1(2(4(3(2(x1)))))) 43.75/12.10 1(4(3(4(x1)))) -> 4(1(3(2(4(2(x1)))))) 43.75/12.10 1(4(3(4(x1)))) -> 2(4(2(4(1(3(x1)))))) 43.75/12.10 1(4(3(4(x1)))) -> 1(5(4(2(4(3(x1)))))) 43.75/12.10 1(4(3(4(x1)))) -> 1(4(3(5(4(3(x1)))))) 43.75/12.10 0(4(4(4(x1)))) -> 0(4(4(2(4(2(x1)))))) 43.75/12.10 1(4(4(4(x1)))) -> 4(1(4(2(4(2(x1)))))) 43.75/12.10 1(4(4(4(x1)))) -> 4(2(1(4(2(4(x1)))))) 43.75/12.10 1(4(4(4(x1)))) -> 4(5(1(4(5(4(x1)))))) 43.75/12.10 0(0(4(0(0(x1))))) -> 0(0(0(2(4(0(x1)))))) 43.75/12.10 0(5(3(1(0(x1))))) -> 3(5(0(3(1(0(x1)))))) 43.75/12.10 0(0(4(3(0(x1))))) -> 0(0(0(2(4(3(x1)))))) 43.75/12.10 1(0(5(3(0(x1))))) -> 3(2(5(0(1(0(x1)))))) 43.75/12.10 1(4(5(3(0(x1))))) -> 3(5(4(0(1(5(x1)))))) 43.75/12.10 0(4(3(4(0(x1))))) -> 0(4(4(3(2(0(x1)))))) 43.75/12.10 0(0(4(4(1(x1))))) -> 0(0(2(4(1(4(x1)))))) 43.75/12.10 1(0(4(4(1(x1))))) -> 0(2(1(4(4(1(x1)))))) 43.75/12.10 1(4(4(4(1(x1))))) -> 1(4(4(1(4(2(x1)))))) 43.75/12.10 1(4(0(0(4(x1))))) -> 0(1(0(2(4(4(x1)))))) 43.75/12.10 0(4(4(0(4(x1))))) -> 0(4(2(4(0(4(x1)))))) 43.75/12.10 1(0(0(1(4(x1))))) -> 0(1(1(0(2(4(x1)))))) 43.75/12.10 0(4(3(1(4(x1))))) -> 0(4(2(4(1(3(x1)))))) 43.75/12.10 0(0(1(3(4(x1))))) -> 0(0(3(2(4(1(x1)))))) 43.75/12.10 0(0(1(3(4(x1))))) -> 0(0(2(4(1(3(x1)))))) 43.75/12.10 0(5(1(3(4(x1))))) -> 1(0(3(5(4(1(x1)))))) 43.75/12.10 0(4(2(3(4(x1))))) -> 2(4(3(0(2(4(x1)))))) 43.75/12.10 1(4(4(3(4(x1))))) -> 2(4(1(3(4(4(x1)))))) 43.75/12.10 0(0(5(3(4(x1))))) -> 0(3(2(5(0(4(x1)))))) 43.75/12.10 1(0(5(3(4(x1))))) -> 1(0(4(5(3(2(x1)))))) 43.75/12.10 0(4(5(3(4(x1))))) -> 5(4(5(4(3(0(x1)))))) 43.75/12.10 0(4(5(3(4(x1))))) -> 2(4(3(5(4(0(x1)))))) 43.75/12.10 0(4(5(3(4(x1))))) -> 2(0(3(5(4(4(x1)))))) 43.75/12.10 0(4(5(3(4(x1))))) -> 0(2(4(3(4(5(x1)))))) 43.75/12.10 1(4(5(3(4(x1))))) -> 1(5(4(2(4(3(x1)))))) 43.75/12.10 0(5(5(3(4(x1))))) -> 5(4(5(3(2(0(x1)))))) 43.75/12.10 1(5(5(3(4(x1))))) -> 3(5(2(4(5(1(x1)))))) 43.75/12.10 1(0(0(4(4(x1))))) -> 1(0(2(4(0(4(x1)))))) 43.75/12.10 1(4(4(4(4(x1))))) -> 4(2(1(4(4(4(x1)))))) 43.75/12.10 1(0(5(4(4(x1))))) -> 2(4(1(0(5(4(x1)))))) 43.75/12.10 43.75/12.10 Q is empty. 43.75/12.10 We have to consider all minimal (P,Q,R)-chains. 43.75/12.10 ---------------------------------------- 43.75/12.10 43.75/12.10 (8) UsableRulesProof (EQUIVALENT) 43.75/12.10 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. 43.75/12.10 ---------------------------------------- 43.75/12.10 43.75/12.10 (9) 43.75/12.10 Obligation: 43.75/12.10 Q DP problem: 43.75/12.10 The TRS P consists of the following rules: 43.75/12.10 43.75/12.10 1^1(0(5(4(4(x1))))) -> 1^1(0(5(4(x1)))) 43.75/12.10 43.75/12.10 R is empty. 43.75/12.10 Q is empty. 43.75/12.10 We have to consider all minimal (P,Q,R)-chains. 43.75/12.10 ---------------------------------------- 43.75/12.10 43.75/12.10 (10) QDPOrderProof (EQUIVALENT) 43.75/12.10 We use the reduction pair processor [LPAR04,JAR06]. 43.75/12.10 43.75/12.10 43.75/12.10 The following pairs can be oriented strictly and are deleted. 43.75/12.10 43.75/12.10 1^1(0(5(4(4(x1))))) -> 1^1(0(5(4(x1)))) 43.75/12.10 The remaining pairs can at least be oriented weakly. 43.75/12.10 Used ordering: Polynomial interpretation [POLO]: 43.75/12.10 43.75/12.10 POL(0(x_1)) = x_1 43.75/12.10 POL(1^1(x_1)) = x_1 43.75/12.10 POL(4(x_1)) = 1 + x_1 43.75/12.10 POL(5(x_1)) = x_1 43.75/12.10 43.75/12.10 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 43.75/12.10 none 43.75/12.10 43.75/12.10 43.75/12.10 ---------------------------------------- 43.75/12.10 43.75/12.10 (11) 43.75/12.10 Obligation: 43.75/12.10 Q DP problem: 43.75/12.10 P is empty. 43.75/12.10 R is empty. 43.75/12.10 Q is empty. 43.75/12.10 We have to consider all minimal (P,Q,R)-chains. 43.75/12.10 ---------------------------------------- 43.75/12.10 43.75/12.10 (12) PisEmptyProof (EQUIVALENT) 43.75/12.10 The TRS P is empty. Hence, there is no (P,Q,R) chain. 43.75/12.10 ---------------------------------------- 43.75/12.10 43.75/12.10 (13) 43.75/12.10 YES 43.75/12.10 43.75/12.10 ---------------------------------------- 43.75/12.10 43.75/12.10 (14) 43.75/12.10 Obligation: 43.75/12.10 Q DP problem: 43.75/12.10 The TRS P consists of the following rules: 43.75/12.10 43.75/12.10 1^1(4(4(4(4(x1))))) -> 1^1(4(4(4(x1)))) 43.75/12.10 43.75/12.10 The TRS R consists of the following rules: 43.75/12.10 43.75/12.10 1(0(0(0(x1)))) -> 3(2(0(0(1(0(x1)))))) 43.75/12.10 1(4(0(0(x1)))) -> 0(1(0(2(4(x1))))) 43.75/12.10 1(4(0(0(x1)))) -> 0(2(4(1(2(0(x1)))))) 43.75/12.10 1(4(0(0(x1)))) -> 0(2(4(5(0(1(x1)))))) 43.75/12.10 1(4(0(0(x1)))) -> 0(1(0(2(4(3(x1)))))) 43.75/12.10 0(4(3(0(x1)))) -> 2(4(2(0(3(0(x1)))))) 43.75/12.10 0(4(3(0(x1)))) -> 2(0(2(4(3(0(x1)))))) 43.75/12.10 0(4(3(0(x1)))) -> 5(0(3(5(4(0(x1)))))) 43.75/12.10 0(4(4(0(x1)))) -> 0(0(2(4(4(x1))))) 43.75/12.10 0(4(4(0(x1)))) -> 2(4(2(0(4(0(x1)))))) 43.75/12.10 0(4(4(0(x1)))) -> 4(2(0(0(2(4(x1)))))) 43.75/12.10 0(4(4(0(x1)))) -> 2(4(0(0(2(4(x1)))))) 43.75/12.10 0(4(4(0(x1)))) -> 0(2(4(0(2(4(x1)))))) 43.75/12.10 1(4(4(0(x1)))) -> 2(4(4(1(0(x1))))) 43.75/12.10 1(4(4(0(x1)))) -> 0(2(4(4(1(x1))))) 43.75/12.10 1(4(4(0(x1)))) -> 3(2(4(4(1(0(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 2(4(4(5(1(0(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 0(2(4(4(5(1(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 4(5(1(0(4(2(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 0(2(1(4(4(2(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 1(0(4(5(4(2(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 0(1(2(4(2(4(x1)))))) 43.75/12.11 0(4(4(1(x1)))) -> 1(0(2(4(4(5(x1)))))) 43.75/12.11 1(4(4(1(x1)))) -> 4(2(1(4(1(2(x1)))))) 43.75/12.11 1(4(4(1(x1)))) -> 1(4(1(4(3(2(x1)))))) 43.75/12.11 1(4(4(1(x1)))) -> 4(1(2(1(4(2(x1)))))) 43.75/12.11 1(4(4(1(x1)))) -> 1(4(1(2(4(2(x1)))))) 43.75/12.11 0(0(0(4(x1)))) -> 5(0(0(2(4(0(x1)))))) 43.75/12.11 1(0(0(4(x1)))) -> 4(0(1(0(3(2(x1)))))) 43.75/12.11 0(4(0(4(x1)))) -> 4(0(0(2(4(2(x1)))))) 43.75/12.11 0(4(0(4(x1)))) -> 0(4(0(2(4(3(x1)))))) 43.75/12.11 1(4(0(4(x1)))) -> 1(4(0(2(4(2(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(4(3(0(4(x1))))) 43.75/12.11 0(4(3(4(x1)))) -> 3(0(2(4(4(x1))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(4(3(2(4(0(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 5(4(4(0(3(2(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(0(4(4(3(2(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(4(3(0(4(2(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 3(2(4(0(2(4(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(4(5(0(3(4(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(0(3(2(4(4(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 3(0(3(5(4(4(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 1(4(2(4(3(x1))))) 43.75/12.11 1(4(3(4(x1)))) -> 3(2(4(2(4(1(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 4(5(4(1(3(2(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 4(1(2(4(3(2(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 4(1(3(2(4(2(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 2(4(2(4(1(3(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 1(5(4(2(4(3(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 1(4(3(5(4(3(x1)))))) 43.75/12.11 0(4(4(4(x1)))) -> 0(4(4(2(4(2(x1)))))) 43.75/12.11 1(4(4(4(x1)))) -> 4(1(4(2(4(2(x1)))))) 43.75/12.11 1(4(4(4(x1)))) -> 4(2(1(4(2(4(x1)))))) 43.75/12.11 1(4(4(4(x1)))) -> 4(5(1(4(5(4(x1)))))) 43.75/12.11 0(0(4(0(0(x1))))) -> 0(0(0(2(4(0(x1)))))) 43.75/12.11 0(5(3(1(0(x1))))) -> 3(5(0(3(1(0(x1)))))) 43.75/12.11 0(0(4(3(0(x1))))) -> 0(0(0(2(4(3(x1)))))) 43.75/12.11 1(0(5(3(0(x1))))) -> 3(2(5(0(1(0(x1)))))) 43.75/12.11 1(4(5(3(0(x1))))) -> 3(5(4(0(1(5(x1)))))) 43.75/12.11 0(4(3(4(0(x1))))) -> 0(4(4(3(2(0(x1)))))) 43.75/12.11 0(0(4(4(1(x1))))) -> 0(0(2(4(1(4(x1)))))) 43.75/12.11 1(0(4(4(1(x1))))) -> 0(2(1(4(4(1(x1)))))) 43.75/12.11 1(4(4(4(1(x1))))) -> 1(4(4(1(4(2(x1)))))) 43.75/12.11 1(4(0(0(4(x1))))) -> 0(1(0(2(4(4(x1)))))) 43.75/12.11 0(4(4(0(4(x1))))) -> 0(4(2(4(0(4(x1)))))) 43.75/12.11 1(0(0(1(4(x1))))) -> 0(1(1(0(2(4(x1)))))) 43.75/12.11 0(4(3(1(4(x1))))) -> 0(4(2(4(1(3(x1)))))) 43.75/12.11 0(0(1(3(4(x1))))) -> 0(0(3(2(4(1(x1)))))) 43.75/12.11 0(0(1(3(4(x1))))) -> 0(0(2(4(1(3(x1)))))) 43.75/12.11 0(5(1(3(4(x1))))) -> 1(0(3(5(4(1(x1)))))) 43.75/12.11 0(4(2(3(4(x1))))) -> 2(4(3(0(2(4(x1)))))) 43.75/12.11 1(4(4(3(4(x1))))) -> 2(4(1(3(4(4(x1)))))) 43.75/12.11 0(0(5(3(4(x1))))) -> 0(3(2(5(0(4(x1)))))) 43.75/12.11 1(0(5(3(4(x1))))) -> 1(0(4(5(3(2(x1)))))) 43.75/12.11 0(4(5(3(4(x1))))) -> 5(4(5(4(3(0(x1)))))) 43.75/12.11 0(4(5(3(4(x1))))) -> 2(4(3(5(4(0(x1)))))) 43.75/12.11 0(4(5(3(4(x1))))) -> 2(0(3(5(4(4(x1)))))) 43.75/12.11 0(4(5(3(4(x1))))) -> 0(2(4(3(4(5(x1)))))) 43.75/12.11 1(4(5(3(4(x1))))) -> 1(5(4(2(4(3(x1)))))) 43.75/12.11 0(5(5(3(4(x1))))) -> 5(4(5(3(2(0(x1)))))) 43.75/12.11 1(5(5(3(4(x1))))) -> 3(5(2(4(5(1(x1)))))) 43.75/12.11 1(0(0(4(4(x1))))) -> 1(0(2(4(0(4(x1)))))) 43.75/12.11 1(4(4(4(4(x1))))) -> 4(2(1(4(4(4(x1)))))) 43.75/12.11 1(0(5(4(4(x1))))) -> 2(4(1(0(5(4(x1)))))) 43.75/12.11 43.75/12.11 Q is empty. 43.75/12.11 We have to consider all minimal (P,Q,R)-chains. 43.75/12.11 ---------------------------------------- 43.75/12.11 43.75/12.11 (15) UsableRulesProof (EQUIVALENT) 43.75/12.11 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. 43.75/12.11 ---------------------------------------- 43.75/12.11 43.75/12.11 (16) 43.75/12.11 Obligation: 43.75/12.11 Q DP problem: 43.75/12.11 The TRS P consists of the following rules: 43.75/12.11 43.75/12.11 1^1(4(4(4(4(x1))))) -> 1^1(4(4(4(x1)))) 43.75/12.11 43.75/12.11 R is empty. 43.75/12.11 Q is empty. 43.75/12.11 We have to consider all minimal (P,Q,R)-chains. 43.75/12.11 ---------------------------------------- 43.75/12.11 43.75/12.11 (17) QDPSizeChangeProof (EQUIVALENT) 43.75/12.11 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. 43.75/12.11 43.75/12.11 From the DPs we obtained the following set of size-change graphs: 43.75/12.11 *1^1(4(4(4(4(x1))))) -> 1^1(4(4(4(x1)))) 43.75/12.11 The graph contains the following edges 1 > 1 43.75/12.11 43.75/12.11 43.75/12.11 ---------------------------------------- 43.75/12.11 43.75/12.11 (18) 43.75/12.11 YES 43.75/12.11 43.75/12.11 ---------------------------------------- 43.75/12.11 43.75/12.11 (19) 43.75/12.11 Obligation: 43.75/12.11 Q DP problem: 43.75/12.11 The TRS P consists of the following rules: 43.75/12.11 43.75/12.11 0^1(4(4(0(x1)))) -> 0^1(4(0(x1))) 43.75/12.11 0^1(4(3(4(x1)))) -> 0^1(4(x1)) 43.75/12.11 0^1(4(3(4(x1)))) -> 0^1(x1) 43.75/12.11 0^1(0(0(4(x1)))) -> 0^1(x1) 43.75/12.11 0^1(0(4(4(1(x1))))) -> 1^1(4(x1)) 43.75/12.11 1^1(4(0(0(x1)))) -> 0^1(1(x1)) 43.75/12.11 0^1(0(1(3(4(x1))))) -> 1^1(x1) 43.75/12.11 1^1(0(0(0(x1)))) -> 0^1(0(1(0(x1)))) 43.75/12.11 0^1(5(1(3(4(x1))))) -> 1^1(x1) 43.75/12.11 1^1(0(0(0(x1)))) -> 0^1(1(0(x1))) 43.75/12.11 0^1(0(5(3(4(x1))))) -> 0^1(4(x1)) 43.75/12.11 0^1(4(5(3(4(x1))))) -> 0^1(x1) 43.75/12.11 0^1(5(5(3(4(x1))))) -> 0^1(x1) 43.75/12.11 1^1(0(0(0(x1)))) -> 1^1(0(x1)) 43.75/12.11 1^1(4(0(0(x1)))) -> 1^1(x1) 43.75/12.11 1^1(4(4(0(x1)))) -> 1^1(0(x1)) 43.75/12.11 1^1(4(4(0(x1)))) -> 1^1(x1) 43.75/12.11 1^1(4(4(0(x1)))) -> 1^1(0(4(2(x1)))) 43.75/12.11 1^1(4(3(4(x1)))) -> 1^1(x1) 43.75/12.11 1^1(0(5(3(0(x1))))) -> 0^1(1(0(x1))) 43.75/12.11 1^1(0(5(3(0(x1))))) -> 1^1(0(x1)) 43.75/12.11 1^1(4(5(3(0(x1))))) -> 0^1(1(5(x1))) 43.75/12.11 1^1(4(5(3(0(x1))))) -> 1^1(5(x1)) 43.75/12.11 1^1(5(5(3(4(x1))))) -> 1^1(x1) 43.75/12.11 1^1(0(4(4(1(x1))))) -> 1^1(4(4(1(x1)))) 43.75/12.11 1^1(0(0(4(4(x1))))) -> 0^1(4(x1)) 43.75/12.11 43.75/12.11 The TRS R consists of the following rules: 43.75/12.11 43.75/12.11 1(0(0(0(x1)))) -> 3(2(0(0(1(0(x1)))))) 43.75/12.11 1(4(0(0(x1)))) -> 0(1(0(2(4(x1))))) 43.75/12.11 1(4(0(0(x1)))) -> 0(2(4(1(2(0(x1)))))) 43.75/12.11 1(4(0(0(x1)))) -> 0(2(4(5(0(1(x1)))))) 43.75/12.11 1(4(0(0(x1)))) -> 0(1(0(2(4(3(x1)))))) 43.75/12.11 0(4(3(0(x1)))) -> 2(4(2(0(3(0(x1)))))) 43.75/12.11 0(4(3(0(x1)))) -> 2(0(2(4(3(0(x1)))))) 43.75/12.11 0(4(3(0(x1)))) -> 5(0(3(5(4(0(x1)))))) 43.75/12.11 0(4(4(0(x1)))) -> 0(0(2(4(4(x1))))) 43.75/12.11 0(4(4(0(x1)))) -> 2(4(2(0(4(0(x1)))))) 43.75/12.11 0(4(4(0(x1)))) -> 4(2(0(0(2(4(x1)))))) 43.75/12.11 0(4(4(0(x1)))) -> 2(4(0(0(2(4(x1)))))) 43.75/12.11 0(4(4(0(x1)))) -> 0(2(4(0(2(4(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 2(4(4(1(0(x1))))) 43.75/12.11 1(4(4(0(x1)))) -> 0(2(4(4(1(x1))))) 43.75/12.11 1(4(4(0(x1)))) -> 3(2(4(4(1(0(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 2(4(4(5(1(0(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 0(2(4(4(5(1(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 4(5(1(0(4(2(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 0(2(1(4(4(2(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 1(0(4(5(4(2(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 0(1(2(4(2(4(x1)))))) 43.75/12.11 0(4(4(1(x1)))) -> 1(0(2(4(4(5(x1)))))) 43.75/12.11 1(4(4(1(x1)))) -> 4(2(1(4(1(2(x1)))))) 43.75/12.11 1(4(4(1(x1)))) -> 1(4(1(4(3(2(x1)))))) 43.75/12.11 1(4(4(1(x1)))) -> 4(1(2(1(4(2(x1)))))) 43.75/12.11 1(4(4(1(x1)))) -> 1(4(1(2(4(2(x1)))))) 43.75/12.11 0(0(0(4(x1)))) -> 5(0(0(2(4(0(x1)))))) 43.75/12.11 1(0(0(4(x1)))) -> 4(0(1(0(3(2(x1)))))) 43.75/12.11 0(4(0(4(x1)))) -> 4(0(0(2(4(2(x1)))))) 43.75/12.11 0(4(0(4(x1)))) -> 0(4(0(2(4(3(x1)))))) 43.75/12.11 1(4(0(4(x1)))) -> 1(4(0(2(4(2(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(4(3(0(4(x1))))) 43.75/12.11 0(4(3(4(x1)))) -> 3(0(2(4(4(x1))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(4(3(2(4(0(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 5(4(4(0(3(2(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(0(4(4(3(2(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(4(3(0(4(2(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 3(2(4(0(2(4(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(4(5(0(3(4(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(0(3(2(4(4(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 3(0(3(5(4(4(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 1(4(2(4(3(x1))))) 43.75/12.11 1(4(3(4(x1)))) -> 3(2(4(2(4(1(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 4(5(4(1(3(2(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 4(1(2(4(3(2(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 4(1(3(2(4(2(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 2(4(2(4(1(3(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 1(5(4(2(4(3(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 1(4(3(5(4(3(x1)))))) 43.75/12.11 0(4(4(4(x1)))) -> 0(4(4(2(4(2(x1)))))) 43.75/12.11 1(4(4(4(x1)))) -> 4(1(4(2(4(2(x1)))))) 43.75/12.11 1(4(4(4(x1)))) -> 4(2(1(4(2(4(x1)))))) 43.75/12.11 1(4(4(4(x1)))) -> 4(5(1(4(5(4(x1)))))) 43.75/12.11 0(0(4(0(0(x1))))) -> 0(0(0(2(4(0(x1)))))) 43.75/12.11 0(5(3(1(0(x1))))) -> 3(5(0(3(1(0(x1)))))) 43.75/12.11 0(0(4(3(0(x1))))) -> 0(0(0(2(4(3(x1)))))) 43.75/12.11 1(0(5(3(0(x1))))) -> 3(2(5(0(1(0(x1)))))) 43.75/12.11 1(4(5(3(0(x1))))) -> 3(5(4(0(1(5(x1)))))) 43.75/12.11 0(4(3(4(0(x1))))) -> 0(4(4(3(2(0(x1)))))) 43.75/12.11 0(0(4(4(1(x1))))) -> 0(0(2(4(1(4(x1)))))) 43.75/12.11 1(0(4(4(1(x1))))) -> 0(2(1(4(4(1(x1)))))) 43.75/12.11 1(4(4(4(1(x1))))) -> 1(4(4(1(4(2(x1)))))) 43.75/12.11 1(4(0(0(4(x1))))) -> 0(1(0(2(4(4(x1)))))) 43.75/12.11 0(4(4(0(4(x1))))) -> 0(4(2(4(0(4(x1)))))) 43.75/12.11 1(0(0(1(4(x1))))) -> 0(1(1(0(2(4(x1)))))) 43.75/12.11 0(4(3(1(4(x1))))) -> 0(4(2(4(1(3(x1)))))) 43.75/12.11 0(0(1(3(4(x1))))) -> 0(0(3(2(4(1(x1)))))) 43.75/12.11 0(0(1(3(4(x1))))) -> 0(0(2(4(1(3(x1)))))) 43.75/12.11 0(5(1(3(4(x1))))) -> 1(0(3(5(4(1(x1)))))) 43.75/12.11 0(4(2(3(4(x1))))) -> 2(4(3(0(2(4(x1)))))) 43.75/12.11 1(4(4(3(4(x1))))) -> 2(4(1(3(4(4(x1)))))) 43.75/12.11 0(0(5(3(4(x1))))) -> 0(3(2(5(0(4(x1)))))) 43.75/12.11 1(0(5(3(4(x1))))) -> 1(0(4(5(3(2(x1)))))) 43.75/12.11 0(4(5(3(4(x1))))) -> 5(4(5(4(3(0(x1)))))) 43.75/12.11 0(4(5(3(4(x1))))) -> 2(4(3(5(4(0(x1)))))) 43.75/12.11 0(4(5(3(4(x1))))) -> 2(0(3(5(4(4(x1)))))) 43.75/12.11 0(4(5(3(4(x1))))) -> 0(2(4(3(4(5(x1)))))) 43.75/12.11 1(4(5(3(4(x1))))) -> 1(5(4(2(4(3(x1)))))) 43.75/12.11 0(5(5(3(4(x1))))) -> 5(4(5(3(2(0(x1)))))) 43.75/12.11 1(5(5(3(4(x1))))) -> 3(5(2(4(5(1(x1)))))) 43.75/12.11 1(0(0(4(4(x1))))) -> 1(0(2(4(0(4(x1)))))) 43.75/12.11 1(4(4(4(4(x1))))) -> 4(2(1(4(4(4(x1)))))) 43.75/12.11 1(0(5(4(4(x1))))) -> 2(4(1(0(5(4(x1)))))) 43.75/12.11 43.75/12.11 Q is empty. 43.75/12.11 We have to consider all minimal (P,Q,R)-chains. 43.75/12.11 ---------------------------------------- 43.75/12.11 43.75/12.11 (20) QDPOrderProof (EQUIVALENT) 43.75/12.11 We use the reduction pair processor [LPAR04,JAR06]. 43.75/12.11 43.75/12.11 43.75/12.11 The following pairs can be oriented strictly and are deleted. 43.75/12.11 43.75/12.11 0^1(4(4(0(x1)))) -> 0^1(4(0(x1))) 43.75/12.11 0^1(4(3(4(x1)))) -> 0^1(4(x1)) 43.75/12.11 0^1(4(3(4(x1)))) -> 0^1(x1) 43.75/12.11 0^1(0(0(4(x1)))) -> 0^1(x1) 43.75/12.11 0^1(0(4(4(1(x1))))) -> 1^1(4(x1)) 43.75/12.11 1^1(4(0(0(x1)))) -> 0^1(1(x1)) 43.75/12.11 0^1(0(1(3(4(x1))))) -> 1^1(x1) 43.75/12.11 0^1(5(1(3(4(x1))))) -> 1^1(x1) 43.75/12.11 0^1(4(5(3(4(x1))))) -> 0^1(x1) 43.75/12.11 0^1(5(5(3(4(x1))))) -> 0^1(x1) 43.75/12.11 1^1(4(0(0(x1)))) -> 1^1(x1) 43.75/12.11 1^1(4(4(0(x1)))) -> 1^1(0(x1)) 43.75/12.11 1^1(4(4(0(x1)))) -> 1^1(x1) 43.75/12.11 1^1(4(4(0(x1)))) -> 1^1(0(4(2(x1)))) 43.75/12.11 1^1(4(3(4(x1)))) -> 1^1(x1) 43.75/12.11 1^1(4(5(3(0(x1))))) -> 0^1(1(5(x1))) 43.75/12.11 1^1(4(5(3(0(x1))))) -> 1^1(5(x1)) 43.75/12.11 1^1(5(5(3(4(x1))))) -> 1^1(x1) 43.75/12.11 1^1(0(0(4(4(x1))))) -> 0^1(4(x1)) 43.75/12.11 The remaining pairs can at least be oriented weakly. 43.75/12.11 Used ordering: Polynomial interpretation [POLO]: 43.75/12.11 43.75/12.11 POL(0(x_1)) = x_1 43.75/12.11 POL(0^1(x_1)) = x_1 43.75/12.11 POL(1(x_1)) = x_1 43.75/12.11 POL(1^1(x_1)) = x_1 43.75/12.11 POL(2(x_1)) = 0 43.75/12.11 POL(3(x_1)) = x_1 43.75/12.11 POL(4(x_1)) = 1 + x_1 43.75/12.11 POL(5(x_1)) = x_1 43.75/12.11 43.75/12.11 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 43.75/12.11 43.75/12.11 0(4(3(0(x1)))) -> 2(4(2(0(3(0(x1)))))) 43.75/12.11 0(4(3(0(x1)))) -> 2(0(2(4(3(0(x1)))))) 43.75/12.11 0(4(3(0(x1)))) -> 5(0(3(5(4(0(x1)))))) 43.75/12.11 0(4(4(0(x1)))) -> 0(0(2(4(4(x1))))) 43.75/12.11 0(4(4(0(x1)))) -> 2(4(2(0(4(0(x1)))))) 43.75/12.11 0(4(4(0(x1)))) -> 4(2(0(0(2(4(x1)))))) 43.75/12.11 0(4(4(0(x1)))) -> 2(4(0(0(2(4(x1)))))) 43.75/12.11 0(4(4(0(x1)))) -> 0(2(4(0(2(4(x1)))))) 43.75/12.11 0(4(4(1(x1)))) -> 1(0(2(4(4(5(x1)))))) 43.75/12.11 0(0(0(4(x1)))) -> 5(0(0(2(4(0(x1)))))) 43.75/12.11 0(4(0(4(x1)))) -> 4(0(0(2(4(2(x1)))))) 43.75/12.11 0(4(0(4(x1)))) -> 0(4(0(2(4(3(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(4(3(0(4(x1))))) 43.75/12.11 0(4(3(4(x1)))) -> 3(0(2(4(4(x1))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(4(3(2(4(0(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 5(4(4(0(3(2(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(0(4(4(3(2(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(4(3(0(4(2(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 3(2(4(0(2(4(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(4(5(0(3(4(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(0(3(2(4(4(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 3(0(3(5(4(4(x1)))))) 43.75/12.11 0(4(4(4(x1)))) -> 0(4(4(2(4(2(x1)))))) 43.75/12.11 0(0(4(0(0(x1))))) -> 0(0(0(2(4(0(x1)))))) 43.75/12.11 0(5(3(1(0(x1))))) -> 3(5(0(3(1(0(x1)))))) 43.75/12.11 0(0(4(3(0(x1))))) -> 0(0(0(2(4(3(x1)))))) 43.75/12.11 0(4(3(4(0(x1))))) -> 0(4(4(3(2(0(x1)))))) 43.75/12.11 0(0(4(4(1(x1))))) -> 0(0(2(4(1(4(x1)))))) 43.75/12.11 0(4(4(0(4(x1))))) -> 0(4(2(4(0(4(x1)))))) 43.75/12.11 0(4(3(1(4(x1))))) -> 0(4(2(4(1(3(x1)))))) 43.75/12.11 0(0(1(3(4(x1))))) -> 0(0(3(2(4(1(x1)))))) 43.75/12.11 0(0(1(3(4(x1))))) -> 0(0(2(4(1(3(x1)))))) 43.75/12.11 0(5(1(3(4(x1))))) -> 1(0(3(5(4(1(x1)))))) 43.75/12.11 0(4(2(3(4(x1))))) -> 2(4(3(0(2(4(x1)))))) 43.75/12.11 0(0(5(3(4(x1))))) -> 0(3(2(5(0(4(x1)))))) 43.75/12.11 0(4(5(3(4(x1))))) -> 5(4(5(4(3(0(x1)))))) 43.75/12.11 0(4(5(3(4(x1))))) -> 2(4(3(5(4(0(x1)))))) 43.75/12.11 0(4(5(3(4(x1))))) -> 2(0(3(5(4(4(x1)))))) 43.75/12.11 0(4(5(3(4(x1))))) -> 0(2(4(3(4(5(x1)))))) 43.75/12.11 0(5(5(3(4(x1))))) -> 5(4(5(3(2(0(x1)))))) 43.75/12.11 1(0(0(0(x1)))) -> 3(2(0(0(1(0(x1)))))) 43.75/12.11 1(4(0(0(x1)))) -> 0(1(0(2(4(x1))))) 43.75/12.11 1(4(0(0(x1)))) -> 0(2(4(1(2(0(x1)))))) 43.75/12.11 1(4(0(0(x1)))) -> 0(2(4(5(0(1(x1)))))) 43.75/12.11 1(4(0(0(x1)))) -> 0(1(0(2(4(3(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 2(4(4(1(0(x1))))) 43.75/12.11 1(4(4(0(x1)))) -> 0(2(4(4(1(x1))))) 43.75/12.11 1(4(4(0(x1)))) -> 3(2(4(4(1(0(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 2(4(4(5(1(0(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 0(2(4(4(5(1(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 4(5(1(0(4(2(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 0(2(1(4(4(2(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 1(0(4(5(4(2(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 0(1(2(4(2(4(x1)))))) 43.75/12.11 1(4(4(1(x1)))) -> 4(2(1(4(1(2(x1)))))) 43.75/12.11 1(4(4(1(x1)))) -> 1(4(1(4(3(2(x1)))))) 43.75/12.11 1(4(4(1(x1)))) -> 4(1(2(1(4(2(x1)))))) 43.75/12.11 1(4(4(1(x1)))) -> 1(4(1(2(4(2(x1)))))) 43.75/12.11 1(0(0(4(x1)))) -> 4(0(1(0(3(2(x1)))))) 43.75/12.11 1(4(0(4(x1)))) -> 1(4(0(2(4(2(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 1(4(2(4(3(x1))))) 43.75/12.11 1(4(3(4(x1)))) -> 3(2(4(2(4(1(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 4(5(4(1(3(2(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 4(1(2(4(3(2(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 4(1(3(2(4(2(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 2(4(2(4(1(3(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 1(5(4(2(4(3(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 1(4(3(5(4(3(x1)))))) 43.75/12.11 1(4(4(4(x1)))) -> 4(1(4(2(4(2(x1)))))) 43.75/12.11 1(4(4(4(x1)))) -> 4(2(1(4(2(4(x1)))))) 43.75/12.11 1(4(4(4(x1)))) -> 4(5(1(4(5(4(x1)))))) 43.75/12.11 1(0(5(3(0(x1))))) -> 3(2(5(0(1(0(x1)))))) 43.75/12.11 1(4(5(3(0(x1))))) -> 3(5(4(0(1(5(x1)))))) 43.75/12.11 1(0(4(4(1(x1))))) -> 0(2(1(4(4(1(x1)))))) 43.75/12.11 1(4(4(4(1(x1))))) -> 1(4(4(1(4(2(x1)))))) 43.75/12.11 1(4(0(0(4(x1))))) -> 0(1(0(2(4(4(x1)))))) 43.75/12.11 1(0(0(1(4(x1))))) -> 0(1(1(0(2(4(x1)))))) 43.75/12.11 1(4(4(3(4(x1))))) -> 2(4(1(3(4(4(x1)))))) 43.75/12.11 1(0(5(3(4(x1))))) -> 1(0(4(5(3(2(x1)))))) 43.75/12.11 1(4(5(3(4(x1))))) -> 1(5(4(2(4(3(x1)))))) 43.75/12.11 1(5(5(3(4(x1))))) -> 3(5(2(4(5(1(x1)))))) 43.75/12.11 1(0(0(4(4(x1))))) -> 1(0(2(4(0(4(x1)))))) 43.75/12.11 1(4(4(4(4(x1))))) -> 4(2(1(4(4(4(x1)))))) 43.75/12.11 1(0(5(4(4(x1))))) -> 2(4(1(0(5(4(x1)))))) 43.75/12.11 43.75/12.11 43.75/12.11 ---------------------------------------- 43.75/12.11 43.75/12.11 (21) 43.75/12.11 Obligation: 43.75/12.11 Q DP problem: 43.75/12.11 The TRS P consists of the following rules: 43.75/12.11 43.75/12.11 1^1(0(0(0(x1)))) -> 0^1(0(1(0(x1)))) 43.75/12.11 1^1(0(0(0(x1)))) -> 0^1(1(0(x1))) 43.75/12.11 0^1(0(5(3(4(x1))))) -> 0^1(4(x1)) 43.75/12.11 1^1(0(0(0(x1)))) -> 1^1(0(x1)) 43.75/12.11 1^1(0(5(3(0(x1))))) -> 0^1(1(0(x1))) 43.75/12.11 1^1(0(5(3(0(x1))))) -> 1^1(0(x1)) 43.75/12.11 1^1(0(4(4(1(x1))))) -> 1^1(4(4(1(x1)))) 43.75/12.11 43.75/12.11 The TRS R consists of the following rules: 43.75/12.11 43.75/12.11 1(0(0(0(x1)))) -> 3(2(0(0(1(0(x1)))))) 43.75/12.11 1(4(0(0(x1)))) -> 0(1(0(2(4(x1))))) 43.75/12.11 1(4(0(0(x1)))) -> 0(2(4(1(2(0(x1)))))) 43.75/12.11 1(4(0(0(x1)))) -> 0(2(4(5(0(1(x1)))))) 43.75/12.11 1(4(0(0(x1)))) -> 0(1(0(2(4(3(x1)))))) 43.75/12.11 0(4(3(0(x1)))) -> 2(4(2(0(3(0(x1)))))) 43.75/12.11 0(4(3(0(x1)))) -> 2(0(2(4(3(0(x1)))))) 43.75/12.11 0(4(3(0(x1)))) -> 5(0(3(5(4(0(x1)))))) 43.75/12.11 0(4(4(0(x1)))) -> 0(0(2(4(4(x1))))) 43.75/12.11 0(4(4(0(x1)))) -> 2(4(2(0(4(0(x1)))))) 43.75/12.11 0(4(4(0(x1)))) -> 4(2(0(0(2(4(x1)))))) 43.75/12.11 0(4(4(0(x1)))) -> 2(4(0(0(2(4(x1)))))) 43.75/12.11 0(4(4(0(x1)))) -> 0(2(4(0(2(4(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 2(4(4(1(0(x1))))) 43.75/12.11 1(4(4(0(x1)))) -> 0(2(4(4(1(x1))))) 43.75/12.11 1(4(4(0(x1)))) -> 3(2(4(4(1(0(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 2(4(4(5(1(0(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 0(2(4(4(5(1(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 4(5(1(0(4(2(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 0(2(1(4(4(2(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 1(0(4(5(4(2(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 0(1(2(4(2(4(x1)))))) 43.75/12.11 0(4(4(1(x1)))) -> 1(0(2(4(4(5(x1)))))) 43.75/12.11 1(4(4(1(x1)))) -> 4(2(1(4(1(2(x1)))))) 43.75/12.11 1(4(4(1(x1)))) -> 1(4(1(4(3(2(x1)))))) 43.75/12.11 1(4(4(1(x1)))) -> 4(1(2(1(4(2(x1)))))) 43.75/12.11 1(4(4(1(x1)))) -> 1(4(1(2(4(2(x1)))))) 43.75/12.11 0(0(0(4(x1)))) -> 5(0(0(2(4(0(x1)))))) 43.75/12.11 1(0(0(4(x1)))) -> 4(0(1(0(3(2(x1)))))) 43.75/12.11 0(4(0(4(x1)))) -> 4(0(0(2(4(2(x1)))))) 43.75/12.11 0(4(0(4(x1)))) -> 0(4(0(2(4(3(x1)))))) 43.75/12.11 1(4(0(4(x1)))) -> 1(4(0(2(4(2(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(4(3(0(4(x1))))) 43.75/12.11 0(4(3(4(x1)))) -> 3(0(2(4(4(x1))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(4(3(2(4(0(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 5(4(4(0(3(2(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(0(4(4(3(2(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(4(3(0(4(2(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 3(2(4(0(2(4(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(4(5(0(3(4(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(0(3(2(4(4(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 3(0(3(5(4(4(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 1(4(2(4(3(x1))))) 43.75/12.11 1(4(3(4(x1)))) -> 3(2(4(2(4(1(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 4(5(4(1(3(2(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 4(1(2(4(3(2(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 4(1(3(2(4(2(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 2(4(2(4(1(3(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 1(5(4(2(4(3(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 1(4(3(5(4(3(x1)))))) 43.75/12.11 0(4(4(4(x1)))) -> 0(4(4(2(4(2(x1)))))) 43.75/12.11 1(4(4(4(x1)))) -> 4(1(4(2(4(2(x1)))))) 43.75/12.11 1(4(4(4(x1)))) -> 4(2(1(4(2(4(x1)))))) 43.75/12.11 1(4(4(4(x1)))) -> 4(5(1(4(5(4(x1)))))) 43.75/12.11 0(0(4(0(0(x1))))) -> 0(0(0(2(4(0(x1)))))) 43.75/12.11 0(5(3(1(0(x1))))) -> 3(5(0(3(1(0(x1)))))) 43.75/12.11 0(0(4(3(0(x1))))) -> 0(0(0(2(4(3(x1)))))) 43.75/12.11 1(0(5(3(0(x1))))) -> 3(2(5(0(1(0(x1)))))) 43.75/12.11 1(4(5(3(0(x1))))) -> 3(5(4(0(1(5(x1)))))) 43.75/12.11 0(4(3(4(0(x1))))) -> 0(4(4(3(2(0(x1)))))) 43.75/12.11 0(0(4(4(1(x1))))) -> 0(0(2(4(1(4(x1)))))) 43.75/12.11 1(0(4(4(1(x1))))) -> 0(2(1(4(4(1(x1)))))) 43.75/12.11 1(4(4(4(1(x1))))) -> 1(4(4(1(4(2(x1)))))) 43.75/12.11 1(4(0(0(4(x1))))) -> 0(1(0(2(4(4(x1)))))) 43.75/12.11 0(4(4(0(4(x1))))) -> 0(4(2(4(0(4(x1)))))) 43.75/12.11 1(0(0(1(4(x1))))) -> 0(1(1(0(2(4(x1)))))) 43.75/12.11 0(4(3(1(4(x1))))) -> 0(4(2(4(1(3(x1)))))) 43.75/12.11 0(0(1(3(4(x1))))) -> 0(0(3(2(4(1(x1)))))) 43.75/12.11 0(0(1(3(4(x1))))) -> 0(0(2(4(1(3(x1)))))) 43.75/12.11 0(5(1(3(4(x1))))) -> 1(0(3(5(4(1(x1)))))) 43.75/12.11 0(4(2(3(4(x1))))) -> 2(4(3(0(2(4(x1)))))) 43.75/12.11 1(4(4(3(4(x1))))) -> 2(4(1(3(4(4(x1)))))) 43.75/12.11 0(0(5(3(4(x1))))) -> 0(3(2(5(0(4(x1)))))) 43.75/12.11 1(0(5(3(4(x1))))) -> 1(0(4(5(3(2(x1)))))) 43.75/12.11 0(4(5(3(4(x1))))) -> 5(4(5(4(3(0(x1)))))) 43.75/12.11 0(4(5(3(4(x1))))) -> 2(4(3(5(4(0(x1)))))) 43.75/12.11 0(4(5(3(4(x1))))) -> 2(0(3(5(4(4(x1)))))) 43.75/12.11 0(4(5(3(4(x1))))) -> 0(2(4(3(4(5(x1)))))) 43.75/12.11 1(4(5(3(4(x1))))) -> 1(5(4(2(4(3(x1)))))) 43.75/12.11 0(5(5(3(4(x1))))) -> 5(4(5(3(2(0(x1)))))) 43.75/12.11 1(5(5(3(4(x1))))) -> 3(5(2(4(5(1(x1)))))) 43.75/12.11 1(0(0(4(4(x1))))) -> 1(0(2(4(0(4(x1)))))) 43.75/12.11 1(4(4(4(4(x1))))) -> 4(2(1(4(4(4(x1)))))) 43.75/12.11 1(0(5(4(4(x1))))) -> 2(4(1(0(5(4(x1)))))) 43.75/12.11 43.75/12.11 Q is empty. 43.75/12.11 We have to consider all minimal (P,Q,R)-chains. 43.75/12.11 ---------------------------------------- 43.75/12.11 43.75/12.11 (22) DependencyGraphProof (EQUIVALENT) 43.75/12.11 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes. 43.75/12.11 ---------------------------------------- 43.75/12.11 43.75/12.11 (23) 43.75/12.11 Obligation: 43.75/12.11 Q DP problem: 43.75/12.11 The TRS P consists of the following rules: 43.75/12.11 43.75/12.11 1^1(0(5(3(0(x1))))) -> 1^1(0(x1)) 43.75/12.11 1^1(0(0(0(x1)))) -> 1^1(0(x1)) 43.75/12.11 43.75/12.11 The TRS R consists of the following rules: 43.75/12.11 43.75/12.11 1(0(0(0(x1)))) -> 3(2(0(0(1(0(x1)))))) 43.75/12.11 1(4(0(0(x1)))) -> 0(1(0(2(4(x1))))) 43.75/12.11 1(4(0(0(x1)))) -> 0(2(4(1(2(0(x1)))))) 43.75/12.11 1(4(0(0(x1)))) -> 0(2(4(5(0(1(x1)))))) 43.75/12.11 1(4(0(0(x1)))) -> 0(1(0(2(4(3(x1)))))) 43.75/12.11 0(4(3(0(x1)))) -> 2(4(2(0(3(0(x1)))))) 43.75/12.11 0(4(3(0(x1)))) -> 2(0(2(4(3(0(x1)))))) 43.75/12.11 0(4(3(0(x1)))) -> 5(0(3(5(4(0(x1)))))) 43.75/12.11 0(4(4(0(x1)))) -> 0(0(2(4(4(x1))))) 43.75/12.11 0(4(4(0(x1)))) -> 2(4(2(0(4(0(x1)))))) 43.75/12.11 0(4(4(0(x1)))) -> 4(2(0(0(2(4(x1)))))) 43.75/12.11 0(4(4(0(x1)))) -> 2(4(0(0(2(4(x1)))))) 43.75/12.11 0(4(4(0(x1)))) -> 0(2(4(0(2(4(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 2(4(4(1(0(x1))))) 43.75/12.11 1(4(4(0(x1)))) -> 0(2(4(4(1(x1))))) 43.75/12.11 1(4(4(0(x1)))) -> 3(2(4(4(1(0(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 2(4(4(5(1(0(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 0(2(4(4(5(1(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 4(5(1(0(4(2(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 0(2(1(4(4(2(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 1(0(4(5(4(2(x1)))))) 43.75/12.11 1(4(4(0(x1)))) -> 0(1(2(4(2(4(x1)))))) 43.75/12.11 0(4(4(1(x1)))) -> 1(0(2(4(4(5(x1)))))) 43.75/12.11 1(4(4(1(x1)))) -> 4(2(1(4(1(2(x1)))))) 43.75/12.11 1(4(4(1(x1)))) -> 1(4(1(4(3(2(x1)))))) 43.75/12.11 1(4(4(1(x1)))) -> 4(1(2(1(4(2(x1)))))) 43.75/12.11 1(4(4(1(x1)))) -> 1(4(1(2(4(2(x1)))))) 43.75/12.11 0(0(0(4(x1)))) -> 5(0(0(2(4(0(x1)))))) 43.75/12.11 1(0(0(4(x1)))) -> 4(0(1(0(3(2(x1)))))) 43.75/12.11 0(4(0(4(x1)))) -> 4(0(0(2(4(2(x1)))))) 43.75/12.11 0(4(0(4(x1)))) -> 0(4(0(2(4(3(x1)))))) 43.75/12.11 1(4(0(4(x1)))) -> 1(4(0(2(4(2(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(4(3(0(4(x1))))) 43.75/12.11 0(4(3(4(x1)))) -> 3(0(2(4(4(x1))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(4(3(2(4(0(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 5(4(4(0(3(2(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(0(4(4(3(2(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(4(3(0(4(2(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 3(2(4(0(2(4(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(4(5(0(3(4(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 2(0(3(2(4(4(x1)))))) 43.75/12.11 0(4(3(4(x1)))) -> 3(0(3(5(4(4(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 1(4(2(4(3(x1))))) 43.75/12.11 1(4(3(4(x1)))) -> 3(2(4(2(4(1(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 4(5(4(1(3(2(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 4(1(2(4(3(2(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 4(1(3(2(4(2(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 2(4(2(4(1(3(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 1(5(4(2(4(3(x1)))))) 43.75/12.11 1(4(3(4(x1)))) -> 1(4(3(5(4(3(x1)))))) 43.75/12.11 0(4(4(4(x1)))) -> 0(4(4(2(4(2(x1)))))) 43.75/12.11 1(4(4(4(x1)))) -> 4(1(4(2(4(2(x1)))))) 43.75/12.11 1(4(4(4(x1)))) -> 4(2(1(4(2(4(x1)))))) 43.75/12.11 1(4(4(4(x1)))) -> 4(5(1(4(5(4(x1)))))) 43.75/12.11 0(0(4(0(0(x1))))) -> 0(0(0(2(4(0(x1)))))) 43.75/12.11 0(5(3(1(0(x1))))) -> 3(5(0(3(1(0(x1)))))) 43.75/12.11 0(0(4(3(0(x1))))) -> 0(0(0(2(4(3(x1)))))) 43.75/12.11 1(0(5(3(0(x1))))) -> 3(2(5(0(1(0(x1)))))) 43.75/12.11 1(4(5(3(0(x1))))) -> 3(5(4(0(1(5(x1)))))) 43.75/12.11 0(4(3(4(0(x1))))) -> 0(4(4(3(2(0(x1)))))) 43.75/12.11 0(0(4(4(1(x1))))) -> 0(0(2(4(1(4(x1)))))) 43.75/12.11 1(0(4(4(1(x1))))) -> 0(2(1(4(4(1(x1)))))) 43.75/12.11 1(4(4(4(1(x1))))) -> 1(4(4(1(4(2(x1)))))) 43.75/12.11 1(4(0(0(4(x1))))) -> 0(1(0(2(4(4(x1)))))) 43.75/12.11 0(4(4(0(4(x1))))) -> 0(4(2(4(0(4(x1)))))) 43.75/12.11 1(0(0(1(4(x1))))) -> 0(1(1(0(2(4(x1)))))) 43.75/12.11 0(4(3(1(4(x1))))) -> 0(4(2(4(1(3(x1)))))) 43.75/12.11 0(0(1(3(4(x1))))) -> 0(0(3(2(4(1(x1)))))) 43.75/12.11 0(0(1(3(4(x1))))) -> 0(0(2(4(1(3(x1)))))) 43.75/12.11 0(5(1(3(4(x1))))) -> 1(0(3(5(4(1(x1)))))) 43.75/12.11 0(4(2(3(4(x1))))) -> 2(4(3(0(2(4(x1)))))) 43.75/12.11 1(4(4(3(4(x1))))) -> 2(4(1(3(4(4(x1)))))) 43.75/12.11 0(0(5(3(4(x1))))) -> 0(3(2(5(0(4(x1)))))) 43.75/12.11 1(0(5(3(4(x1))))) -> 1(0(4(5(3(2(x1)))))) 43.75/12.11 0(4(5(3(4(x1))))) -> 5(4(5(4(3(0(x1)))))) 43.75/12.11 0(4(5(3(4(x1))))) -> 2(4(3(5(4(0(x1)))))) 43.75/12.11 0(4(5(3(4(x1))))) -> 2(0(3(5(4(4(x1)))))) 43.75/12.11 0(4(5(3(4(x1))))) -> 0(2(4(3(4(5(x1)))))) 43.75/12.11 1(4(5(3(4(x1))))) -> 1(5(4(2(4(3(x1)))))) 43.75/12.11 0(5(5(3(4(x1))))) -> 5(4(5(3(2(0(x1)))))) 43.75/12.11 1(5(5(3(4(x1))))) -> 3(5(2(4(5(1(x1)))))) 43.75/12.11 1(0(0(4(4(x1))))) -> 1(0(2(4(0(4(x1)))))) 43.75/12.11 1(4(4(4(4(x1))))) -> 4(2(1(4(4(4(x1)))))) 43.75/12.11 1(0(5(4(4(x1))))) -> 2(4(1(0(5(4(x1)))))) 43.75/12.11 43.75/12.11 Q is empty. 43.75/12.11 We have to consider all minimal (P,Q,R)-chains. 43.75/12.11 ---------------------------------------- 43.75/12.11 43.75/12.11 (24) QDPSizeChangeProof (EQUIVALENT) 43.75/12.11 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. 43.75/12.11 43.75/12.11 From the DPs we obtained the following set of size-change graphs: 43.75/12.11 *1^1(0(5(3(0(x1))))) -> 1^1(0(x1)) 43.75/12.11 The graph contains the following edges 1 > 1 43.75/12.11 43.75/12.11 43.75/12.11 *1^1(0(0(0(x1)))) -> 1^1(0(x1)) 43.75/12.11 The graph contains the following edges 1 > 1 43.75/12.11 43.75/12.11 43.75/12.11 ---------------------------------------- 43.75/12.11 43.75/12.11 (25) 43.75/12.11 YES 43.96/12.15 EOF