45.21/12.35 YES 49.88/13.58 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 49.88/13.58 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 49.88/13.58 49.88/13.58 49.88/13.58 Termination w.r.t. Q of the given QTRS could be proven: 49.88/13.58 49.88/13.58 (0) QTRS 49.88/13.58 (1) QTRS Reverse [EQUIVALENT, 0 ms] 49.88/13.58 (2) QTRS 49.88/13.58 (3) DependencyPairsProof [EQUIVALENT, 358 ms] 49.88/13.58 (4) QDP 49.88/13.58 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 49.88/13.58 (6) AND 49.88/13.58 (7) QDP 49.88/13.58 (8) UsableRulesProof [EQUIVALENT, 0 ms] 49.88/13.58 (9) QDP 49.88/13.58 (10) MRRProof [EQUIVALENT, 37 ms] 49.88/13.58 (11) QDP 49.88/13.58 (12) DependencyGraphProof [EQUIVALENT, 0 ms] 49.88/13.58 (13) TRUE 49.88/13.58 (14) QDP 49.88/13.58 (15) QDPOrderProof [EQUIVALENT, 218 ms] 49.88/13.58 (16) QDP 49.88/13.58 (17) DependencyGraphProof [EQUIVALENT, 0 ms] 49.88/13.58 (18) AND 49.88/13.58 (19) QDP 49.88/13.58 (20) QDPOrderProof [EQUIVALENT, 77 ms] 49.88/13.58 (21) QDP 49.88/13.58 (22) PisEmptyProof [EQUIVALENT, 0 ms] 49.88/13.58 (23) YES 49.88/13.58 (24) QDP 49.88/13.58 (25) QDPOrderProof [EQUIVALENT, 79 ms] 49.88/13.58 (26) QDP 49.88/13.58 (27) PisEmptyProof [EQUIVALENT, 0 ms] 49.88/13.58 (28) YES 49.88/13.58 49.88/13.58 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (0) 49.88/13.58 Obligation: 49.88/13.58 Q restricted rewrite system: 49.88/13.58 The TRS R consists of the following rules: 49.88/13.58 49.88/13.58 0(0(x1)) -> 1(0(2(0(2(x1))))) 49.88/13.58 0(0(x1)) -> 0(2(3(4(0(2(x1)))))) 49.88/13.58 0(3(x1)) -> 0(2(3(2(x1)))) 49.88/13.58 0(3(x1)) -> 2(0(2(1(3(x1))))) 49.88/13.58 0(3(x1)) -> 2(3(0(2(2(x1))))) 49.88/13.58 0(3(x1)) -> 0(2(2(2(3(2(x1)))))) 49.88/13.58 0(0(0(x1))) -> 0(2(0(0(x1)))) 49.88/13.58 0(0(4(x1))) -> 2(4(0(0(3(2(x1)))))) 49.88/13.58 0(0(5(x1))) -> 0(5(2(0(x1)))) 49.88/13.58 0(1(3(x1))) -> 3(2(2(1(0(x1))))) 49.88/13.58 0(3(0(x1))) -> 2(0(2(3(4(0(x1)))))) 49.88/13.58 0(4(3(x1))) -> 3(0(2(4(x1)))) 49.88/13.58 0(4(3(x1))) -> 1(0(2(1(3(4(x1)))))) 49.88/13.58 0(4(3(x1))) -> 2(2(4(3(4(0(x1)))))) 49.88/13.58 0(4(5(x1))) -> 2(4(0(5(2(5(x1)))))) 49.88/13.58 0(4(5(x1))) -> 5(3(2(4(4(0(x1)))))) 49.88/13.58 0(5(0(x1))) -> 0(5(2(4(0(x1))))) 49.88/13.58 0(5(0(x1))) -> 5(0(2(3(2(0(x1)))))) 49.88/13.58 0(5(3(x1))) -> 3(2(5(3(2(0(x1)))))) 49.88/13.58 1(0(0(x1))) -> 1(0(2(5(0(2(x1)))))) 49.88/13.58 1(0(3(x1))) -> 2(1(3(0(5(2(x1)))))) 49.88/13.58 1(0(4(x1))) -> 3(0(2(1(4(x1))))) 49.88/13.58 1(0(4(x1))) -> 2(1(2(1(4(0(x1)))))) 49.88/13.58 3(0(0(x1))) -> 3(0(2(0(x1)))) 49.88/13.58 3(0(3(x1))) -> 3(0(2(3(x1)))) 49.88/13.58 3(0(3(x1))) -> 3(3(2(2(0(x1))))) 49.88/13.58 3(0(4(x1))) -> 2(2(3(4(0(x1))))) 49.88/13.58 3(0(4(x1))) -> 2(3(4(2(0(3(x1)))))) 49.88/13.58 3(0(4(x1))) -> 5(5(2(4(0(3(x1)))))) 49.88/13.58 3(3(4(x1))) -> 3(2(1(3(2(4(x1)))))) 49.88/13.58 3(5(0(x1))) -> 3(5(1(0(2(1(x1)))))) 49.88/13.58 4(0(4(x1))) -> 0(2(4(4(0(x1))))) 49.88/13.58 4(0(4(x1))) -> 1(4(0(2(4(0(x1)))))) 49.88/13.58 0(0(4(3(x1)))) -> 3(0(0(2(1(4(x1)))))) 49.88/13.58 0(0(5(3(x1)))) -> 0(5(2(5(0(3(x1)))))) 49.88/13.58 0(1(0(3(x1)))) -> 2(1(0(3(0(x1))))) 49.88/13.58 0(1(1(0(x1)))) -> 2(1(0(0(1(2(x1)))))) 49.88/13.58 0(4(0(0(x1)))) -> 0(4(0(2(1(0(x1)))))) 49.88/13.58 0(4(1(3(x1)))) -> 2(1(4(2(0(3(x1)))))) 49.88/13.58 0(4(5(3(x1)))) -> 4(3(2(5(0(x1))))) 49.88/13.58 0(4(5(3(x1)))) -> 4(3(5(0(2(4(x1)))))) 49.88/13.58 0(5(0(4(x1)))) -> 5(0(0(2(4(x1))))) 49.88/13.58 0(5(5(3(x1)))) -> 2(3(5(5(0(x1))))) 49.88/13.58 1(3(0(4(x1)))) -> 4(0(1(2(3(x1))))) 49.88/13.58 4(0(3(5(x1)))) -> 5(2(3(4(2(0(x1)))))) 49.88/13.58 0(0(1(0(4(x1))))) -> 0(0(0(1(4(2(x1)))))) 49.88/13.58 0(0(5(5(3(x1))))) -> 0(5(0(3(2(5(x1)))))) 49.88/13.58 3(0(4(3(4(x1))))) -> 3(0(3(4(2(4(x1)))))) 49.88/13.58 3(3(0(4(5(x1))))) -> 1(4(5(3(0(3(x1)))))) 49.88/13.58 3(4(2(0(3(x1))))) -> 2(3(0(2(3(4(x1)))))) 49.88/13.58 49.88/13.58 Q is empty. 49.88/13.58 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (1) QTRS Reverse (EQUIVALENT) 49.88/13.58 We applied the QTRS Reverse Processor [REVERSE]. 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (2) 49.88/13.58 Obligation: 49.88/13.58 Q restricted rewrite system: 49.88/13.58 The TRS R consists of the following rules: 49.88/13.58 49.88/13.58 0(0(x1)) -> 2(0(2(0(1(x1))))) 49.88/13.58 0(0(x1)) -> 2(0(4(3(2(0(x1)))))) 49.88/13.58 3(0(x1)) -> 2(3(2(0(x1)))) 49.88/13.58 3(0(x1)) -> 3(1(2(0(2(x1))))) 49.88/13.58 3(0(x1)) -> 2(2(0(3(2(x1))))) 49.88/13.58 3(0(x1)) -> 2(3(2(2(2(0(x1)))))) 49.88/13.58 0(0(0(x1))) -> 0(0(2(0(x1)))) 49.88/13.58 4(0(0(x1))) -> 2(3(0(0(4(2(x1)))))) 49.88/13.58 5(0(0(x1))) -> 0(2(5(0(x1)))) 49.88/13.58 3(1(0(x1))) -> 0(1(2(2(3(x1))))) 49.88/13.58 0(3(0(x1))) -> 0(4(3(2(0(2(x1)))))) 49.88/13.58 3(4(0(x1))) -> 4(2(0(3(x1)))) 49.88/13.58 3(4(0(x1))) -> 4(3(1(2(0(1(x1)))))) 49.88/13.58 3(4(0(x1))) -> 0(4(3(4(2(2(x1)))))) 49.88/13.58 5(4(0(x1))) -> 5(2(5(0(4(2(x1)))))) 49.88/13.58 5(4(0(x1))) -> 0(4(4(2(3(5(x1)))))) 49.88/13.58 0(5(0(x1))) -> 0(4(2(5(0(x1))))) 49.88/13.58 0(5(0(x1))) -> 0(2(3(2(0(5(x1)))))) 49.88/13.58 3(5(0(x1))) -> 0(2(3(5(2(3(x1)))))) 49.88/13.58 0(0(1(x1))) -> 2(0(5(2(0(1(x1)))))) 49.88/13.58 3(0(1(x1))) -> 2(5(0(3(1(2(x1)))))) 49.88/13.58 4(0(1(x1))) -> 4(1(2(0(3(x1))))) 49.88/13.58 4(0(1(x1))) -> 0(4(1(2(1(2(x1)))))) 49.88/13.58 0(0(3(x1))) -> 0(2(0(3(x1)))) 49.88/13.58 3(0(3(x1))) -> 3(2(0(3(x1)))) 49.88/13.58 3(0(3(x1))) -> 0(2(2(3(3(x1))))) 49.88/13.58 4(0(3(x1))) -> 0(4(3(2(2(x1))))) 49.88/13.58 4(0(3(x1))) -> 3(0(2(4(3(2(x1)))))) 49.88/13.58 4(0(3(x1))) -> 3(0(4(2(5(5(x1)))))) 49.88/13.58 4(3(3(x1))) -> 4(2(3(1(2(3(x1)))))) 49.88/13.58 0(5(3(x1))) -> 1(2(0(1(5(3(x1)))))) 49.88/13.58 4(0(4(x1))) -> 0(4(4(2(0(x1))))) 49.88/13.58 4(0(4(x1))) -> 0(4(2(0(4(1(x1)))))) 49.88/13.58 3(4(0(0(x1)))) -> 4(1(2(0(0(3(x1)))))) 49.88/13.58 3(5(0(0(x1)))) -> 3(0(5(2(5(0(x1)))))) 49.88/13.58 3(0(1(0(x1)))) -> 0(3(0(1(2(x1))))) 49.88/13.58 0(1(1(0(x1)))) -> 2(1(0(0(1(2(x1)))))) 49.88/13.58 0(0(4(0(x1)))) -> 0(1(2(0(4(0(x1)))))) 49.88/13.58 3(1(4(0(x1)))) -> 3(0(2(4(1(2(x1)))))) 49.88/13.58 3(5(4(0(x1)))) -> 0(5(2(3(4(x1))))) 49.88/13.58 3(5(4(0(x1)))) -> 4(2(0(5(3(4(x1)))))) 49.88/13.58 4(0(5(0(x1)))) -> 4(2(0(0(5(x1))))) 49.88/13.58 3(5(5(0(x1)))) -> 0(5(5(3(2(x1))))) 49.88/13.58 4(0(3(1(x1)))) -> 3(2(1(0(4(x1))))) 49.88/13.58 5(3(0(4(x1)))) -> 0(2(4(3(2(5(x1)))))) 49.88/13.58 4(0(1(0(0(x1))))) -> 2(4(1(0(0(0(x1)))))) 49.88/13.58 3(5(5(0(0(x1))))) -> 5(2(3(0(5(0(x1)))))) 49.88/13.58 4(3(4(0(3(x1))))) -> 4(2(4(3(0(3(x1)))))) 49.88/13.58 5(4(0(3(3(x1))))) -> 3(0(3(5(4(1(x1)))))) 49.88/13.58 3(0(2(4(3(x1))))) -> 4(3(2(0(3(2(x1)))))) 49.88/13.58 49.88/13.58 Q is empty. 49.88/13.58 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (3) DependencyPairsProof (EQUIVALENT) 49.88/13.58 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (4) 49.88/13.58 Obligation: 49.88/13.58 Q DP problem: 49.88/13.58 The TRS P consists of the following rules: 49.88/13.58 49.88/13.58 0^1(0(x1)) -> 0^1(2(0(1(x1)))) 49.88/13.58 0^1(0(x1)) -> 0^1(1(x1)) 49.88/13.58 0^1(0(x1)) -> 0^1(4(3(2(0(x1))))) 49.88/13.58 0^1(0(x1)) -> 4^1(3(2(0(x1)))) 49.88/13.58 0^1(0(x1)) -> 3^1(2(0(x1))) 49.88/13.58 3^1(0(x1)) -> 3^1(2(0(x1))) 49.88/13.58 3^1(0(x1)) -> 3^1(1(2(0(2(x1))))) 49.88/13.58 3^1(0(x1)) -> 0^1(2(x1)) 49.88/13.58 3^1(0(x1)) -> 0^1(3(2(x1))) 49.88/13.58 3^1(0(x1)) -> 3^1(2(x1)) 49.88/13.58 3^1(0(x1)) -> 3^1(2(2(2(0(x1))))) 49.88/13.58 0^1(0(0(x1))) -> 0^1(0(2(0(x1)))) 49.88/13.58 0^1(0(0(x1))) -> 0^1(2(0(x1))) 49.88/13.58 4^1(0(0(x1))) -> 3^1(0(0(4(2(x1))))) 49.88/13.58 4^1(0(0(x1))) -> 0^1(0(4(2(x1)))) 49.88/13.58 4^1(0(0(x1))) -> 0^1(4(2(x1))) 49.88/13.58 4^1(0(0(x1))) -> 4^1(2(x1)) 49.88/13.58 5^1(0(0(x1))) -> 0^1(2(5(0(x1)))) 49.88/13.58 5^1(0(0(x1))) -> 5^1(0(x1)) 49.88/13.58 3^1(1(0(x1))) -> 0^1(1(2(2(3(x1))))) 49.88/13.58 3^1(1(0(x1))) -> 3^1(x1) 49.88/13.58 0^1(3(0(x1))) -> 0^1(4(3(2(0(2(x1)))))) 49.88/13.58 0^1(3(0(x1))) -> 4^1(3(2(0(2(x1))))) 49.88/13.58 0^1(3(0(x1))) -> 3^1(2(0(2(x1)))) 49.88/13.58 0^1(3(0(x1))) -> 0^1(2(x1)) 49.88/13.58 3^1(4(0(x1))) -> 4^1(2(0(3(x1)))) 49.88/13.58 3^1(4(0(x1))) -> 0^1(3(x1)) 49.88/13.58 3^1(4(0(x1))) -> 3^1(x1) 49.88/13.58 3^1(4(0(x1))) -> 4^1(3(1(2(0(1(x1)))))) 49.88/13.58 3^1(4(0(x1))) -> 3^1(1(2(0(1(x1))))) 49.88/13.58 3^1(4(0(x1))) -> 0^1(1(x1)) 49.88/13.58 3^1(4(0(x1))) -> 0^1(4(3(4(2(2(x1)))))) 49.88/13.58 3^1(4(0(x1))) -> 4^1(3(4(2(2(x1))))) 49.88/13.58 3^1(4(0(x1))) -> 3^1(4(2(2(x1)))) 49.88/13.58 3^1(4(0(x1))) -> 4^1(2(2(x1))) 49.88/13.58 5^1(4(0(x1))) -> 5^1(2(5(0(4(2(x1)))))) 49.88/13.58 5^1(4(0(x1))) -> 5^1(0(4(2(x1)))) 49.88/13.58 5^1(4(0(x1))) -> 0^1(4(2(x1))) 49.88/13.58 5^1(4(0(x1))) -> 4^1(2(x1)) 49.88/13.58 5^1(4(0(x1))) -> 0^1(4(4(2(3(5(x1)))))) 49.88/13.58 5^1(4(0(x1))) -> 4^1(4(2(3(5(x1))))) 49.88/13.58 5^1(4(0(x1))) -> 4^1(2(3(5(x1)))) 49.88/13.58 5^1(4(0(x1))) -> 3^1(5(x1)) 49.88/13.58 5^1(4(0(x1))) -> 5^1(x1) 49.88/13.58 0^1(5(0(x1))) -> 0^1(4(2(5(0(x1))))) 49.88/13.58 0^1(5(0(x1))) -> 4^1(2(5(0(x1)))) 49.88/13.58 0^1(5(0(x1))) -> 0^1(2(3(2(0(5(x1)))))) 49.88/13.58 0^1(5(0(x1))) -> 3^1(2(0(5(x1)))) 49.88/13.58 0^1(5(0(x1))) -> 0^1(5(x1)) 49.88/13.58 0^1(5(0(x1))) -> 5^1(x1) 49.88/13.58 3^1(5(0(x1))) -> 0^1(2(3(5(2(3(x1)))))) 49.88/13.58 3^1(5(0(x1))) -> 3^1(5(2(3(x1)))) 49.88/13.58 3^1(5(0(x1))) -> 5^1(2(3(x1))) 49.88/13.58 3^1(5(0(x1))) -> 3^1(x1) 49.88/13.58 0^1(0(1(x1))) -> 0^1(5(2(0(1(x1))))) 49.88/13.58 0^1(0(1(x1))) -> 5^1(2(0(1(x1)))) 49.88/13.58 3^1(0(1(x1))) -> 5^1(0(3(1(2(x1))))) 49.88/13.58 3^1(0(1(x1))) -> 0^1(3(1(2(x1)))) 49.88/13.58 3^1(0(1(x1))) -> 3^1(1(2(x1))) 49.88/13.58 4^1(0(1(x1))) -> 4^1(1(2(0(3(x1))))) 49.88/13.58 4^1(0(1(x1))) -> 0^1(3(x1)) 49.88/13.58 4^1(0(1(x1))) -> 3^1(x1) 49.88/13.58 4^1(0(1(x1))) -> 0^1(4(1(2(1(2(x1)))))) 49.88/13.58 4^1(0(1(x1))) -> 4^1(1(2(1(2(x1))))) 49.88/13.58 0^1(0(3(x1))) -> 0^1(2(0(3(x1)))) 49.88/13.58 3^1(0(3(x1))) -> 3^1(2(0(3(x1)))) 49.88/13.58 3^1(0(3(x1))) -> 0^1(2(2(3(3(x1))))) 49.88/13.58 3^1(0(3(x1))) -> 3^1(3(x1)) 49.88/13.58 4^1(0(3(x1))) -> 0^1(4(3(2(2(x1))))) 49.88/13.58 4^1(0(3(x1))) -> 4^1(3(2(2(x1)))) 49.88/13.58 4^1(0(3(x1))) -> 3^1(2(2(x1))) 49.88/13.58 4^1(0(3(x1))) -> 3^1(0(2(4(3(2(x1)))))) 49.88/13.58 4^1(0(3(x1))) -> 0^1(2(4(3(2(x1))))) 49.88/13.58 4^1(0(3(x1))) -> 4^1(3(2(x1))) 49.88/13.58 4^1(0(3(x1))) -> 3^1(2(x1)) 49.88/13.58 4^1(0(3(x1))) -> 3^1(0(4(2(5(5(x1)))))) 49.88/13.58 4^1(0(3(x1))) -> 0^1(4(2(5(5(x1))))) 49.88/13.58 4^1(0(3(x1))) -> 4^1(2(5(5(x1)))) 49.88/13.58 4^1(0(3(x1))) -> 5^1(5(x1)) 49.88/13.58 4^1(0(3(x1))) -> 5^1(x1) 49.88/13.58 4^1(3(3(x1))) -> 4^1(2(3(1(2(3(x1)))))) 49.88/13.58 4^1(3(3(x1))) -> 3^1(1(2(3(x1)))) 49.88/13.58 0^1(5(3(x1))) -> 0^1(1(5(3(x1)))) 49.88/13.58 4^1(0(4(x1))) -> 0^1(4(4(2(0(x1))))) 49.88/13.58 4^1(0(4(x1))) -> 4^1(4(2(0(x1)))) 49.88/13.58 4^1(0(4(x1))) -> 4^1(2(0(x1))) 49.88/13.58 4^1(0(4(x1))) -> 0^1(x1) 49.88/13.58 4^1(0(4(x1))) -> 0^1(4(2(0(4(1(x1)))))) 49.88/13.58 4^1(0(4(x1))) -> 4^1(2(0(4(1(x1))))) 49.88/13.58 4^1(0(4(x1))) -> 0^1(4(1(x1))) 49.88/13.58 4^1(0(4(x1))) -> 4^1(1(x1)) 49.88/13.58 3^1(4(0(0(x1)))) -> 4^1(1(2(0(0(3(x1)))))) 49.88/13.58 3^1(4(0(0(x1)))) -> 0^1(0(3(x1))) 49.88/13.58 3^1(4(0(0(x1)))) -> 0^1(3(x1)) 49.88/13.58 3^1(4(0(0(x1)))) -> 3^1(x1) 49.88/13.58 3^1(5(0(0(x1)))) -> 3^1(0(5(2(5(0(x1)))))) 49.88/13.58 3^1(5(0(0(x1)))) -> 0^1(5(2(5(0(x1))))) 49.88/13.58 3^1(5(0(0(x1)))) -> 5^1(2(5(0(x1)))) 49.88/13.58 3^1(5(0(0(x1)))) -> 5^1(0(x1)) 49.88/13.58 3^1(0(1(0(x1)))) -> 0^1(3(0(1(2(x1))))) 49.88/13.58 3^1(0(1(0(x1)))) -> 3^1(0(1(2(x1)))) 49.88/13.58 3^1(0(1(0(x1)))) -> 0^1(1(2(x1))) 49.88/13.58 0^1(1(1(0(x1)))) -> 0^1(0(1(2(x1)))) 49.88/13.58 0^1(1(1(0(x1)))) -> 0^1(1(2(x1))) 49.88/13.58 0^1(0(4(0(x1)))) -> 0^1(1(2(0(4(0(x1)))))) 49.88/13.58 3^1(1(4(0(x1)))) -> 3^1(0(2(4(1(2(x1)))))) 49.88/13.58 3^1(1(4(0(x1)))) -> 0^1(2(4(1(2(x1))))) 49.88/13.58 3^1(1(4(0(x1)))) -> 4^1(1(2(x1))) 49.88/13.58 3^1(5(4(0(x1)))) -> 0^1(5(2(3(4(x1))))) 49.88/13.58 3^1(5(4(0(x1)))) -> 5^1(2(3(4(x1)))) 49.88/13.58 3^1(5(4(0(x1)))) -> 3^1(4(x1)) 49.88/13.58 3^1(5(4(0(x1)))) -> 4^1(x1) 49.88/13.58 3^1(5(4(0(x1)))) -> 4^1(2(0(5(3(4(x1)))))) 49.88/13.58 3^1(5(4(0(x1)))) -> 0^1(5(3(4(x1)))) 49.88/13.58 3^1(5(4(0(x1)))) -> 5^1(3(4(x1))) 49.88/13.58 4^1(0(5(0(x1)))) -> 4^1(2(0(0(5(x1))))) 49.88/13.58 4^1(0(5(0(x1)))) -> 0^1(0(5(x1))) 49.88/13.58 4^1(0(5(0(x1)))) -> 0^1(5(x1)) 49.88/13.58 4^1(0(5(0(x1)))) -> 5^1(x1) 49.88/13.58 3^1(5(5(0(x1)))) -> 0^1(5(5(3(2(x1))))) 49.88/13.58 3^1(5(5(0(x1)))) -> 5^1(5(3(2(x1)))) 49.88/13.58 3^1(5(5(0(x1)))) -> 5^1(3(2(x1))) 49.88/13.58 3^1(5(5(0(x1)))) -> 3^1(2(x1)) 49.88/13.58 4^1(0(3(1(x1)))) -> 3^1(2(1(0(4(x1))))) 49.88/13.58 4^1(0(3(1(x1)))) -> 0^1(4(x1)) 49.88/13.58 4^1(0(3(1(x1)))) -> 4^1(x1) 49.88/13.58 5^1(3(0(4(x1)))) -> 0^1(2(4(3(2(5(x1)))))) 49.88/13.58 5^1(3(0(4(x1)))) -> 4^1(3(2(5(x1)))) 49.88/13.58 5^1(3(0(4(x1)))) -> 3^1(2(5(x1))) 49.88/13.58 5^1(3(0(4(x1)))) -> 5^1(x1) 49.88/13.58 4^1(0(1(0(0(x1))))) -> 4^1(1(0(0(0(x1))))) 49.88/13.58 4^1(0(1(0(0(x1))))) -> 0^1(0(0(x1))) 49.88/13.58 3^1(5(5(0(0(x1))))) -> 5^1(2(3(0(5(0(x1)))))) 49.88/13.58 3^1(5(5(0(0(x1))))) -> 3^1(0(5(0(x1)))) 49.88/13.58 3^1(5(5(0(0(x1))))) -> 0^1(5(0(x1))) 49.88/13.58 3^1(5(5(0(0(x1))))) -> 5^1(0(x1)) 49.88/13.58 4^1(3(4(0(3(x1))))) -> 4^1(2(4(3(0(3(x1)))))) 49.88/13.58 4^1(3(4(0(3(x1))))) -> 4^1(3(0(3(x1)))) 49.88/13.58 4^1(3(4(0(3(x1))))) -> 3^1(0(3(x1))) 49.88/13.58 5^1(4(0(3(3(x1))))) -> 3^1(0(3(5(4(1(x1)))))) 49.88/13.58 5^1(4(0(3(3(x1))))) -> 0^1(3(5(4(1(x1))))) 49.88/13.58 5^1(4(0(3(3(x1))))) -> 3^1(5(4(1(x1)))) 49.88/13.58 5^1(4(0(3(3(x1))))) -> 5^1(4(1(x1))) 49.88/13.58 5^1(4(0(3(3(x1))))) -> 4^1(1(x1)) 49.88/13.58 3^1(0(2(4(3(x1))))) -> 4^1(3(2(0(3(2(x1)))))) 49.88/13.58 3^1(0(2(4(3(x1))))) -> 3^1(2(0(3(2(x1))))) 49.88/13.58 3^1(0(2(4(3(x1))))) -> 0^1(3(2(x1))) 49.88/13.58 3^1(0(2(4(3(x1))))) -> 3^1(2(x1)) 49.88/13.58 49.88/13.58 The TRS R consists of the following rules: 49.88/13.58 49.88/13.58 0(0(x1)) -> 2(0(2(0(1(x1))))) 49.88/13.58 0(0(x1)) -> 2(0(4(3(2(0(x1)))))) 49.88/13.58 3(0(x1)) -> 2(3(2(0(x1)))) 49.88/13.58 3(0(x1)) -> 3(1(2(0(2(x1))))) 49.88/13.58 3(0(x1)) -> 2(2(0(3(2(x1))))) 49.88/13.58 3(0(x1)) -> 2(3(2(2(2(0(x1)))))) 49.88/13.58 0(0(0(x1))) -> 0(0(2(0(x1)))) 49.88/13.58 4(0(0(x1))) -> 2(3(0(0(4(2(x1)))))) 49.88/13.58 5(0(0(x1))) -> 0(2(5(0(x1)))) 49.88/13.58 3(1(0(x1))) -> 0(1(2(2(3(x1))))) 49.88/13.58 0(3(0(x1))) -> 0(4(3(2(0(2(x1)))))) 49.88/13.58 3(4(0(x1))) -> 4(2(0(3(x1)))) 49.88/13.58 3(4(0(x1))) -> 4(3(1(2(0(1(x1)))))) 49.88/13.58 3(4(0(x1))) -> 0(4(3(4(2(2(x1)))))) 49.88/13.58 5(4(0(x1))) -> 5(2(5(0(4(2(x1)))))) 49.88/13.58 5(4(0(x1))) -> 0(4(4(2(3(5(x1)))))) 49.88/13.58 0(5(0(x1))) -> 0(4(2(5(0(x1))))) 49.88/13.58 0(5(0(x1))) -> 0(2(3(2(0(5(x1)))))) 49.88/13.58 3(5(0(x1))) -> 0(2(3(5(2(3(x1)))))) 49.88/13.58 0(0(1(x1))) -> 2(0(5(2(0(1(x1)))))) 49.88/13.58 3(0(1(x1))) -> 2(5(0(3(1(2(x1)))))) 49.88/13.58 4(0(1(x1))) -> 4(1(2(0(3(x1))))) 49.88/13.58 4(0(1(x1))) -> 0(4(1(2(1(2(x1)))))) 49.88/13.58 0(0(3(x1))) -> 0(2(0(3(x1)))) 49.88/13.58 3(0(3(x1))) -> 3(2(0(3(x1)))) 49.88/13.58 3(0(3(x1))) -> 0(2(2(3(3(x1))))) 49.88/13.58 4(0(3(x1))) -> 0(4(3(2(2(x1))))) 49.88/13.58 4(0(3(x1))) -> 3(0(2(4(3(2(x1)))))) 49.88/13.58 4(0(3(x1))) -> 3(0(4(2(5(5(x1)))))) 49.88/13.58 4(3(3(x1))) -> 4(2(3(1(2(3(x1)))))) 49.88/13.58 0(5(3(x1))) -> 1(2(0(1(5(3(x1)))))) 49.88/13.58 4(0(4(x1))) -> 0(4(4(2(0(x1))))) 49.88/13.58 4(0(4(x1))) -> 0(4(2(0(4(1(x1)))))) 49.88/13.58 3(4(0(0(x1)))) -> 4(1(2(0(0(3(x1)))))) 49.88/13.58 3(5(0(0(x1)))) -> 3(0(5(2(5(0(x1)))))) 49.88/13.58 3(0(1(0(x1)))) -> 0(3(0(1(2(x1))))) 49.88/13.58 0(1(1(0(x1)))) -> 2(1(0(0(1(2(x1)))))) 49.88/13.58 0(0(4(0(x1)))) -> 0(1(2(0(4(0(x1)))))) 49.88/13.58 3(1(4(0(x1)))) -> 3(0(2(4(1(2(x1)))))) 49.88/13.58 3(5(4(0(x1)))) -> 0(5(2(3(4(x1))))) 49.88/13.58 3(5(4(0(x1)))) -> 4(2(0(5(3(4(x1)))))) 49.88/13.58 4(0(5(0(x1)))) -> 4(2(0(0(5(x1))))) 49.88/13.58 3(5(5(0(x1)))) -> 0(5(5(3(2(x1))))) 49.88/13.58 4(0(3(1(x1)))) -> 3(2(1(0(4(x1))))) 49.88/13.58 5(3(0(4(x1)))) -> 0(2(4(3(2(5(x1)))))) 49.88/13.58 4(0(1(0(0(x1))))) -> 2(4(1(0(0(0(x1)))))) 49.88/13.58 3(5(5(0(0(x1))))) -> 5(2(3(0(5(0(x1)))))) 49.88/13.58 4(3(4(0(3(x1))))) -> 4(2(4(3(0(3(x1)))))) 49.88/13.58 5(4(0(3(3(x1))))) -> 3(0(3(5(4(1(x1)))))) 49.88/13.58 3(0(2(4(3(x1))))) -> 4(3(2(0(3(2(x1)))))) 49.88/13.58 49.88/13.58 Q is empty. 49.88/13.58 We have to consider all minimal (P,Q,R)-chains. 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (5) DependencyGraphProof (EQUIVALENT) 49.88/13.58 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 108 less nodes. 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (6) 49.88/13.58 Complex Obligation (AND) 49.88/13.58 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (7) 49.88/13.58 Obligation: 49.88/13.58 Q DP problem: 49.88/13.58 The TRS P consists of the following rules: 49.88/13.58 49.88/13.58 0^1(0(x1)) -> 0^1(1(x1)) 49.88/13.58 0^1(1(1(0(x1)))) -> 0^1(0(1(2(x1)))) 49.88/13.58 49.88/13.58 The TRS R consists of the following rules: 49.88/13.58 49.88/13.58 0(0(x1)) -> 2(0(2(0(1(x1))))) 49.88/13.58 0(0(x1)) -> 2(0(4(3(2(0(x1)))))) 49.88/13.58 3(0(x1)) -> 2(3(2(0(x1)))) 49.88/13.58 3(0(x1)) -> 3(1(2(0(2(x1))))) 49.88/13.58 3(0(x1)) -> 2(2(0(3(2(x1))))) 49.88/13.58 3(0(x1)) -> 2(3(2(2(2(0(x1)))))) 49.88/13.58 0(0(0(x1))) -> 0(0(2(0(x1)))) 49.88/13.58 4(0(0(x1))) -> 2(3(0(0(4(2(x1)))))) 49.88/13.58 5(0(0(x1))) -> 0(2(5(0(x1)))) 49.88/13.58 3(1(0(x1))) -> 0(1(2(2(3(x1))))) 49.88/13.58 0(3(0(x1))) -> 0(4(3(2(0(2(x1)))))) 49.88/13.58 3(4(0(x1))) -> 4(2(0(3(x1)))) 49.88/13.58 3(4(0(x1))) -> 4(3(1(2(0(1(x1)))))) 49.88/13.58 3(4(0(x1))) -> 0(4(3(4(2(2(x1)))))) 49.88/13.58 5(4(0(x1))) -> 5(2(5(0(4(2(x1)))))) 49.88/13.58 5(4(0(x1))) -> 0(4(4(2(3(5(x1)))))) 49.88/13.58 0(5(0(x1))) -> 0(4(2(5(0(x1))))) 49.88/13.58 0(5(0(x1))) -> 0(2(3(2(0(5(x1)))))) 49.88/13.58 3(5(0(x1))) -> 0(2(3(5(2(3(x1)))))) 49.88/13.58 0(0(1(x1))) -> 2(0(5(2(0(1(x1)))))) 49.88/13.58 3(0(1(x1))) -> 2(5(0(3(1(2(x1)))))) 49.88/13.58 4(0(1(x1))) -> 4(1(2(0(3(x1))))) 49.88/13.58 4(0(1(x1))) -> 0(4(1(2(1(2(x1)))))) 49.88/13.58 0(0(3(x1))) -> 0(2(0(3(x1)))) 49.88/13.58 3(0(3(x1))) -> 3(2(0(3(x1)))) 49.88/13.58 3(0(3(x1))) -> 0(2(2(3(3(x1))))) 49.88/13.58 4(0(3(x1))) -> 0(4(3(2(2(x1))))) 49.88/13.58 4(0(3(x1))) -> 3(0(2(4(3(2(x1)))))) 49.88/13.58 4(0(3(x1))) -> 3(0(4(2(5(5(x1)))))) 49.88/13.58 4(3(3(x1))) -> 4(2(3(1(2(3(x1)))))) 49.88/13.58 0(5(3(x1))) -> 1(2(0(1(5(3(x1)))))) 49.88/13.58 4(0(4(x1))) -> 0(4(4(2(0(x1))))) 49.88/13.58 4(0(4(x1))) -> 0(4(2(0(4(1(x1)))))) 49.88/13.58 3(4(0(0(x1)))) -> 4(1(2(0(0(3(x1)))))) 49.88/13.58 3(5(0(0(x1)))) -> 3(0(5(2(5(0(x1)))))) 49.88/13.58 3(0(1(0(x1)))) -> 0(3(0(1(2(x1))))) 49.88/13.58 0(1(1(0(x1)))) -> 2(1(0(0(1(2(x1)))))) 49.88/13.58 0(0(4(0(x1)))) -> 0(1(2(0(4(0(x1)))))) 49.88/13.58 3(1(4(0(x1)))) -> 3(0(2(4(1(2(x1)))))) 49.88/13.58 3(5(4(0(x1)))) -> 0(5(2(3(4(x1))))) 49.88/13.58 3(5(4(0(x1)))) -> 4(2(0(5(3(4(x1)))))) 49.88/13.58 4(0(5(0(x1)))) -> 4(2(0(0(5(x1))))) 49.88/13.58 3(5(5(0(x1)))) -> 0(5(5(3(2(x1))))) 49.88/13.58 4(0(3(1(x1)))) -> 3(2(1(0(4(x1))))) 49.88/13.58 5(3(0(4(x1)))) -> 0(2(4(3(2(5(x1)))))) 49.88/13.58 4(0(1(0(0(x1))))) -> 2(4(1(0(0(0(x1)))))) 49.88/13.58 3(5(5(0(0(x1))))) -> 5(2(3(0(5(0(x1)))))) 49.88/13.58 4(3(4(0(3(x1))))) -> 4(2(4(3(0(3(x1)))))) 49.88/13.58 5(4(0(3(3(x1))))) -> 3(0(3(5(4(1(x1)))))) 49.88/13.58 3(0(2(4(3(x1))))) -> 4(3(2(0(3(2(x1)))))) 49.88/13.58 49.88/13.58 Q is empty. 49.88/13.58 We have to consider all minimal (P,Q,R)-chains. 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (8) UsableRulesProof (EQUIVALENT) 49.88/13.58 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. 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (9) 49.88/13.58 Obligation: 49.88/13.58 Q DP problem: 49.88/13.58 The TRS P consists of the following rules: 49.88/13.58 49.88/13.58 0^1(0(x1)) -> 0^1(1(x1)) 49.88/13.58 0^1(1(1(0(x1)))) -> 0^1(0(1(2(x1)))) 49.88/13.58 49.88/13.58 R is empty. 49.88/13.58 Q is empty. 49.88/13.58 We have to consider all minimal (P,Q,R)-chains. 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (10) MRRProof (EQUIVALENT) 49.88/13.58 By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. 49.88/13.58 49.88/13.58 Strictly oriented dependency pairs: 49.88/13.58 49.88/13.58 0^1(1(1(0(x1)))) -> 0^1(0(1(2(x1)))) 49.88/13.58 49.88/13.58 49.88/13.58 Used ordering: Polynomial interpretation [POLO]: 49.88/13.58 49.88/13.58 POL(0(x_1)) = 3 + 3*x_1 49.88/13.58 POL(0^1(x_1)) = 2*x_1 49.88/13.58 POL(1(x_1)) = 3 + 3*x_1 49.88/13.58 POL(2(x_1)) = x_1 49.88/13.58 49.88/13.58 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (11) 49.88/13.58 Obligation: 49.88/13.58 Q DP problem: 49.88/13.58 The TRS P consists of the following rules: 49.88/13.58 49.88/13.58 0^1(0(x1)) -> 0^1(1(x1)) 49.88/13.58 49.88/13.58 R is empty. 49.88/13.58 Q is empty. 49.88/13.58 We have to consider all minimal (P,Q,R)-chains. 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (12) DependencyGraphProof (EQUIVALENT) 49.88/13.58 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (13) 49.88/13.58 TRUE 49.88/13.58 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (14) 49.88/13.58 Obligation: 49.88/13.58 Q DP problem: 49.88/13.58 The TRS P consists of the following rules: 49.88/13.58 49.88/13.58 0^1(5(0(x1))) -> 0^1(5(x1)) 49.88/13.58 0^1(5(0(x1))) -> 5^1(x1) 49.88/13.58 5^1(0(0(x1))) -> 5^1(0(x1)) 49.88/13.58 5^1(4(0(x1))) -> 3^1(5(x1)) 49.88/13.58 3^1(1(0(x1))) -> 3^1(x1) 49.88/13.58 3^1(4(0(x1))) -> 0^1(3(x1)) 49.88/13.58 3^1(4(0(x1))) -> 3^1(x1) 49.88/13.58 3^1(5(0(x1))) -> 3^1(x1) 49.88/13.58 3^1(0(3(x1))) -> 3^1(3(x1)) 49.88/13.58 3^1(4(0(0(x1)))) -> 0^1(0(3(x1))) 49.88/13.58 3^1(4(0(0(x1)))) -> 0^1(3(x1)) 49.88/13.58 3^1(4(0(0(x1)))) -> 3^1(x1) 49.88/13.58 3^1(5(0(0(x1)))) -> 5^1(0(x1)) 49.88/13.58 5^1(4(0(x1))) -> 5^1(x1) 49.88/13.58 5^1(3(0(4(x1)))) -> 5^1(x1) 49.88/13.58 5^1(4(0(3(3(x1))))) -> 3^1(0(3(5(4(1(x1)))))) 49.88/13.58 3^1(0(1(0(x1)))) -> 0^1(3(0(1(2(x1))))) 49.88/13.58 3^1(5(4(0(x1)))) -> 3^1(4(x1)) 49.88/13.58 3^1(5(4(0(x1)))) -> 4^1(x1) 49.88/13.58 4^1(0(0(x1))) -> 3^1(0(0(4(2(x1))))) 49.88/13.58 3^1(5(4(0(x1)))) -> 0^1(5(3(4(x1)))) 49.88/13.58 3^1(5(4(0(x1)))) -> 5^1(3(4(x1))) 49.88/13.58 3^1(5(5(0(0(x1))))) -> 3^1(0(5(0(x1)))) 49.88/13.58 3^1(5(5(0(0(x1))))) -> 0^1(5(0(x1))) 49.88/13.58 3^1(5(5(0(0(x1))))) -> 5^1(0(x1)) 49.88/13.58 4^1(0(1(x1))) -> 0^1(3(x1)) 49.88/13.58 4^1(0(1(x1))) -> 3^1(x1) 49.88/13.58 4^1(0(3(x1))) -> 5^1(5(x1)) 49.88/13.58 4^1(0(3(x1))) -> 5^1(x1) 49.88/13.58 4^1(0(4(x1))) -> 0^1(x1) 49.88/13.58 4^1(0(5(0(x1)))) -> 0^1(0(5(x1))) 49.88/13.58 4^1(0(5(0(x1)))) -> 0^1(5(x1)) 49.88/13.58 4^1(0(5(0(x1)))) -> 5^1(x1) 49.88/13.58 4^1(0(3(1(x1)))) -> 0^1(4(x1)) 49.88/13.58 4^1(0(3(1(x1)))) -> 4^1(x1) 49.88/13.58 4^1(0(1(0(0(x1))))) -> 0^1(0(0(x1))) 49.88/13.58 4^1(3(4(0(3(x1))))) -> 4^1(3(0(3(x1)))) 49.88/13.58 4^1(3(4(0(3(x1))))) -> 3^1(0(3(x1))) 49.88/13.58 49.88/13.58 The TRS R consists of the following rules: 49.88/13.58 49.88/13.58 0(0(x1)) -> 2(0(2(0(1(x1))))) 49.88/13.58 0(0(x1)) -> 2(0(4(3(2(0(x1)))))) 49.88/13.58 3(0(x1)) -> 2(3(2(0(x1)))) 49.88/13.58 3(0(x1)) -> 3(1(2(0(2(x1))))) 49.88/13.58 3(0(x1)) -> 2(2(0(3(2(x1))))) 49.88/13.58 3(0(x1)) -> 2(3(2(2(2(0(x1)))))) 49.88/13.58 0(0(0(x1))) -> 0(0(2(0(x1)))) 49.88/13.58 4(0(0(x1))) -> 2(3(0(0(4(2(x1)))))) 49.88/13.58 5(0(0(x1))) -> 0(2(5(0(x1)))) 49.88/13.58 3(1(0(x1))) -> 0(1(2(2(3(x1))))) 49.88/13.58 0(3(0(x1))) -> 0(4(3(2(0(2(x1)))))) 49.88/13.58 3(4(0(x1))) -> 4(2(0(3(x1)))) 49.88/13.58 3(4(0(x1))) -> 4(3(1(2(0(1(x1)))))) 49.88/13.58 3(4(0(x1))) -> 0(4(3(4(2(2(x1)))))) 49.88/13.58 5(4(0(x1))) -> 5(2(5(0(4(2(x1)))))) 49.88/13.58 5(4(0(x1))) -> 0(4(4(2(3(5(x1)))))) 49.88/13.58 0(5(0(x1))) -> 0(4(2(5(0(x1))))) 49.88/13.58 0(5(0(x1))) -> 0(2(3(2(0(5(x1)))))) 49.88/13.58 3(5(0(x1))) -> 0(2(3(5(2(3(x1)))))) 49.88/13.58 0(0(1(x1))) -> 2(0(5(2(0(1(x1)))))) 49.88/13.58 3(0(1(x1))) -> 2(5(0(3(1(2(x1)))))) 49.88/13.58 4(0(1(x1))) -> 4(1(2(0(3(x1))))) 49.88/13.58 4(0(1(x1))) -> 0(4(1(2(1(2(x1)))))) 49.88/13.58 0(0(3(x1))) -> 0(2(0(3(x1)))) 49.88/13.58 3(0(3(x1))) -> 3(2(0(3(x1)))) 49.88/13.58 3(0(3(x1))) -> 0(2(2(3(3(x1))))) 49.88/13.58 4(0(3(x1))) -> 0(4(3(2(2(x1))))) 49.88/13.58 4(0(3(x1))) -> 3(0(2(4(3(2(x1)))))) 49.88/13.58 4(0(3(x1))) -> 3(0(4(2(5(5(x1)))))) 49.88/13.58 4(3(3(x1))) -> 4(2(3(1(2(3(x1)))))) 49.88/13.58 0(5(3(x1))) -> 1(2(0(1(5(3(x1)))))) 49.88/13.58 4(0(4(x1))) -> 0(4(4(2(0(x1))))) 49.88/13.58 4(0(4(x1))) -> 0(4(2(0(4(1(x1)))))) 49.88/13.58 3(4(0(0(x1)))) -> 4(1(2(0(0(3(x1)))))) 49.88/13.58 3(5(0(0(x1)))) -> 3(0(5(2(5(0(x1)))))) 49.88/13.58 3(0(1(0(x1)))) -> 0(3(0(1(2(x1))))) 49.88/13.58 0(1(1(0(x1)))) -> 2(1(0(0(1(2(x1)))))) 49.88/13.58 0(0(4(0(x1)))) -> 0(1(2(0(4(0(x1)))))) 49.88/13.58 3(1(4(0(x1)))) -> 3(0(2(4(1(2(x1)))))) 49.88/13.58 3(5(4(0(x1)))) -> 0(5(2(3(4(x1))))) 49.88/13.58 3(5(4(0(x1)))) -> 4(2(0(5(3(4(x1)))))) 49.88/13.58 4(0(5(0(x1)))) -> 4(2(0(0(5(x1))))) 49.88/13.58 3(5(5(0(x1)))) -> 0(5(5(3(2(x1))))) 49.88/13.58 4(0(3(1(x1)))) -> 3(2(1(0(4(x1))))) 49.88/13.58 5(3(0(4(x1)))) -> 0(2(4(3(2(5(x1)))))) 49.88/13.58 4(0(1(0(0(x1))))) -> 2(4(1(0(0(0(x1)))))) 49.88/13.58 3(5(5(0(0(x1))))) -> 5(2(3(0(5(0(x1)))))) 49.88/13.58 4(3(4(0(3(x1))))) -> 4(2(4(3(0(3(x1)))))) 49.88/13.58 5(4(0(3(3(x1))))) -> 3(0(3(5(4(1(x1)))))) 49.88/13.58 3(0(2(4(3(x1))))) -> 4(3(2(0(3(2(x1)))))) 49.88/13.58 49.88/13.58 Q is empty. 49.88/13.58 We have to consider all minimal (P,Q,R)-chains. 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (15) QDPOrderProof (EQUIVALENT) 49.88/13.58 We use the reduction pair processor [LPAR04,JAR06]. 49.88/13.58 49.88/13.58 49.88/13.58 The following pairs can be oriented strictly and are deleted. 49.88/13.58 49.88/13.58 0^1(5(0(x1))) -> 0^1(5(x1)) 49.88/13.58 0^1(5(0(x1))) -> 5^1(x1) 49.88/13.58 5^1(0(0(x1))) -> 5^1(0(x1)) 49.88/13.58 5^1(4(0(x1))) -> 3^1(5(x1)) 49.88/13.58 3^1(1(0(x1))) -> 3^1(x1) 49.88/13.58 3^1(4(0(x1))) -> 0^1(3(x1)) 49.88/13.58 3^1(4(0(x1))) -> 3^1(x1) 49.88/13.58 3^1(5(0(x1))) -> 3^1(x1) 49.88/13.58 3^1(0(3(x1))) -> 3^1(3(x1)) 49.88/13.58 3^1(4(0(0(x1)))) -> 0^1(0(3(x1))) 49.88/13.58 3^1(4(0(0(x1)))) -> 0^1(3(x1)) 49.88/13.58 3^1(4(0(0(x1)))) -> 3^1(x1) 49.88/13.58 3^1(5(0(0(x1)))) -> 5^1(0(x1)) 49.88/13.58 5^1(4(0(x1))) -> 5^1(x1) 49.88/13.58 5^1(3(0(4(x1)))) -> 5^1(x1) 49.88/13.58 3^1(0(1(0(x1)))) -> 0^1(3(0(1(2(x1))))) 49.88/13.58 3^1(5(4(0(x1)))) -> 3^1(4(x1)) 49.88/13.58 4^1(0(0(x1))) -> 3^1(0(0(4(2(x1))))) 49.88/13.58 3^1(5(4(0(x1)))) -> 0^1(5(3(4(x1)))) 49.88/13.58 3^1(5(4(0(x1)))) -> 5^1(3(4(x1))) 49.88/13.58 3^1(5(5(0(0(x1))))) -> 0^1(5(0(x1))) 49.88/13.58 3^1(5(5(0(0(x1))))) -> 5^1(0(x1)) 49.88/13.58 4^1(0(1(x1))) -> 0^1(3(x1)) 49.88/13.58 4^1(0(1(x1))) -> 3^1(x1) 49.88/13.58 4^1(0(3(x1))) -> 5^1(5(x1)) 49.88/13.58 4^1(0(3(x1))) -> 5^1(x1) 49.88/13.58 4^1(0(4(x1))) -> 0^1(x1) 49.88/13.58 4^1(0(5(0(x1)))) -> 0^1(0(5(x1))) 49.88/13.58 4^1(0(5(0(x1)))) -> 0^1(5(x1)) 49.88/13.58 4^1(0(5(0(x1)))) -> 5^1(x1) 49.88/13.58 4^1(0(3(1(x1)))) -> 0^1(4(x1)) 49.88/13.58 4^1(0(3(1(x1)))) -> 4^1(x1) 49.88/13.58 4^1(0(1(0(0(x1))))) -> 0^1(0(0(x1))) 49.88/13.58 4^1(3(4(0(3(x1))))) -> 3^1(0(3(x1))) 49.88/13.58 The remaining pairs can at least be oriented weakly. 49.88/13.58 Used ordering: Polynomial interpretation [POLO]: 49.88/13.58 49.88/13.58 POL(0(x_1)) = 1 + x_1 49.88/13.58 POL(0^1(x_1)) = x_1 49.88/13.58 POL(1(x_1)) = x_1 49.88/13.58 POL(2(x_1)) = 0 49.88/13.58 POL(3(x_1)) = x_1 49.88/13.58 POL(3^1(x_1)) = x_1 49.88/13.58 POL(4(x_1)) = x_1 49.88/13.58 POL(4^1(x_1)) = 1 + x_1 49.88/13.58 POL(5(x_1)) = x_1 49.88/13.58 POL(5^1(x_1)) = x_1 49.88/13.58 49.88/13.58 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 49.88/13.58 49.88/13.58 5(0(0(x1))) -> 0(2(5(0(x1)))) 49.88/13.58 5(4(0(x1))) -> 5(2(5(0(4(2(x1)))))) 49.88/13.58 5(4(0(x1))) -> 0(4(4(2(3(5(x1)))))) 49.88/13.58 5(3(0(4(x1)))) -> 0(2(4(3(2(5(x1)))))) 49.88/13.58 5(4(0(3(3(x1))))) -> 3(0(3(5(4(1(x1)))))) 49.88/13.58 0(0(x1)) -> 2(0(2(0(1(x1))))) 49.88/13.58 0(0(x1)) -> 2(0(4(3(2(0(x1)))))) 49.88/13.58 0(0(0(x1))) -> 0(0(2(0(x1)))) 49.88/13.58 0(3(0(x1))) -> 0(4(3(2(0(2(x1)))))) 49.88/13.58 0(5(0(x1))) -> 0(4(2(5(0(x1))))) 49.88/13.58 0(5(0(x1))) -> 0(2(3(2(0(5(x1)))))) 49.88/13.58 0(0(1(x1))) -> 2(0(5(2(0(1(x1)))))) 49.88/13.58 0(0(3(x1))) -> 0(2(0(3(x1)))) 49.88/13.58 0(5(3(x1))) -> 1(2(0(1(5(3(x1)))))) 49.88/13.58 0(1(1(0(x1)))) -> 2(1(0(0(1(2(x1)))))) 49.88/13.58 0(0(4(0(x1)))) -> 0(1(2(0(4(0(x1)))))) 49.88/13.58 3(0(x1)) -> 2(3(2(0(x1)))) 49.88/13.58 3(0(x1)) -> 3(1(2(0(2(x1))))) 49.88/13.58 3(0(x1)) -> 2(2(0(3(2(x1))))) 49.88/13.58 3(0(x1)) -> 2(3(2(2(2(0(x1)))))) 49.88/13.58 3(1(0(x1))) -> 0(1(2(2(3(x1))))) 49.88/13.58 3(4(0(x1))) -> 4(2(0(3(x1)))) 49.88/13.58 3(4(0(x1))) -> 4(3(1(2(0(1(x1)))))) 49.88/13.58 3(4(0(x1))) -> 0(4(3(4(2(2(x1)))))) 49.88/13.58 3(5(0(x1))) -> 0(2(3(5(2(3(x1)))))) 49.88/13.58 3(0(1(x1))) -> 2(5(0(3(1(2(x1)))))) 49.88/13.58 3(0(3(x1))) -> 3(2(0(3(x1)))) 49.88/13.58 3(0(3(x1))) -> 0(2(2(3(3(x1))))) 49.88/13.58 3(4(0(0(x1)))) -> 4(1(2(0(0(3(x1)))))) 49.88/13.58 3(5(0(0(x1)))) -> 3(0(5(2(5(0(x1)))))) 49.88/13.58 3(0(1(0(x1)))) -> 0(3(0(1(2(x1))))) 49.88/13.58 3(1(4(0(x1)))) -> 3(0(2(4(1(2(x1)))))) 49.88/13.58 3(5(4(0(x1)))) -> 0(5(2(3(4(x1))))) 49.88/13.58 3(5(4(0(x1)))) -> 4(2(0(5(3(4(x1)))))) 49.88/13.58 3(5(5(0(x1)))) -> 0(5(5(3(2(x1))))) 49.88/13.58 3(5(5(0(0(x1))))) -> 5(2(3(0(5(0(x1)))))) 49.88/13.58 3(0(2(4(3(x1))))) -> 4(3(2(0(3(2(x1)))))) 49.88/13.58 4(0(0(x1))) -> 2(3(0(0(4(2(x1)))))) 49.88/13.58 4(0(1(x1))) -> 4(1(2(0(3(x1))))) 49.88/13.58 4(0(1(x1))) -> 0(4(1(2(1(2(x1)))))) 49.88/13.58 4(0(3(x1))) -> 0(4(3(2(2(x1))))) 49.88/13.58 4(0(3(x1))) -> 3(0(2(4(3(2(x1)))))) 49.88/13.58 4(0(3(x1))) -> 3(0(4(2(5(5(x1)))))) 49.88/13.58 4(3(3(x1))) -> 4(2(3(1(2(3(x1)))))) 49.88/13.58 4(0(4(x1))) -> 0(4(4(2(0(x1))))) 49.88/13.58 4(0(4(x1))) -> 0(4(2(0(4(1(x1)))))) 49.88/13.58 4(0(5(0(x1)))) -> 4(2(0(0(5(x1))))) 49.88/13.58 4(0(3(1(x1)))) -> 3(2(1(0(4(x1))))) 49.88/13.58 4(0(1(0(0(x1))))) -> 2(4(1(0(0(0(x1)))))) 49.88/13.58 4(3(4(0(3(x1))))) -> 4(2(4(3(0(3(x1)))))) 49.88/13.58 49.88/13.58 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (16) 49.88/13.58 Obligation: 49.88/13.58 Q DP problem: 49.88/13.58 The TRS P consists of the following rules: 49.88/13.58 49.88/13.58 5^1(4(0(3(3(x1))))) -> 3^1(0(3(5(4(1(x1)))))) 49.88/13.58 3^1(5(4(0(x1)))) -> 4^1(x1) 49.88/13.58 3^1(5(5(0(0(x1))))) -> 3^1(0(5(0(x1)))) 49.88/13.58 4^1(3(4(0(3(x1))))) -> 4^1(3(0(3(x1)))) 49.88/13.58 49.88/13.58 The TRS R consists of the following rules: 49.88/13.58 49.88/13.58 0(0(x1)) -> 2(0(2(0(1(x1))))) 49.88/13.58 0(0(x1)) -> 2(0(4(3(2(0(x1)))))) 49.88/13.58 3(0(x1)) -> 2(3(2(0(x1)))) 49.88/13.58 3(0(x1)) -> 3(1(2(0(2(x1))))) 49.88/13.58 3(0(x1)) -> 2(2(0(3(2(x1))))) 49.88/13.58 3(0(x1)) -> 2(3(2(2(2(0(x1)))))) 49.88/13.58 0(0(0(x1))) -> 0(0(2(0(x1)))) 49.88/13.58 4(0(0(x1))) -> 2(3(0(0(4(2(x1)))))) 49.88/13.58 5(0(0(x1))) -> 0(2(5(0(x1)))) 49.88/13.58 3(1(0(x1))) -> 0(1(2(2(3(x1))))) 49.88/13.58 0(3(0(x1))) -> 0(4(3(2(0(2(x1)))))) 49.88/13.58 3(4(0(x1))) -> 4(2(0(3(x1)))) 49.88/13.58 3(4(0(x1))) -> 4(3(1(2(0(1(x1)))))) 49.88/13.58 3(4(0(x1))) -> 0(4(3(4(2(2(x1)))))) 49.88/13.58 5(4(0(x1))) -> 5(2(5(0(4(2(x1)))))) 49.88/13.58 5(4(0(x1))) -> 0(4(4(2(3(5(x1)))))) 49.88/13.58 0(5(0(x1))) -> 0(4(2(5(0(x1))))) 49.88/13.58 0(5(0(x1))) -> 0(2(3(2(0(5(x1)))))) 49.88/13.58 3(5(0(x1))) -> 0(2(3(5(2(3(x1)))))) 49.88/13.58 0(0(1(x1))) -> 2(0(5(2(0(1(x1)))))) 49.88/13.58 3(0(1(x1))) -> 2(5(0(3(1(2(x1)))))) 49.88/13.58 4(0(1(x1))) -> 4(1(2(0(3(x1))))) 49.88/13.58 4(0(1(x1))) -> 0(4(1(2(1(2(x1)))))) 49.88/13.58 0(0(3(x1))) -> 0(2(0(3(x1)))) 49.88/13.58 3(0(3(x1))) -> 3(2(0(3(x1)))) 49.88/13.58 3(0(3(x1))) -> 0(2(2(3(3(x1))))) 49.88/13.58 4(0(3(x1))) -> 0(4(3(2(2(x1))))) 49.88/13.58 4(0(3(x1))) -> 3(0(2(4(3(2(x1)))))) 49.88/13.58 4(0(3(x1))) -> 3(0(4(2(5(5(x1)))))) 49.88/13.58 4(3(3(x1))) -> 4(2(3(1(2(3(x1)))))) 49.88/13.58 0(5(3(x1))) -> 1(2(0(1(5(3(x1)))))) 49.88/13.58 4(0(4(x1))) -> 0(4(4(2(0(x1))))) 49.88/13.58 4(0(4(x1))) -> 0(4(2(0(4(1(x1)))))) 49.88/13.58 3(4(0(0(x1)))) -> 4(1(2(0(0(3(x1)))))) 49.88/13.58 3(5(0(0(x1)))) -> 3(0(5(2(5(0(x1)))))) 49.88/13.58 3(0(1(0(x1)))) -> 0(3(0(1(2(x1))))) 49.88/13.58 0(1(1(0(x1)))) -> 2(1(0(0(1(2(x1)))))) 49.88/13.58 0(0(4(0(x1)))) -> 0(1(2(0(4(0(x1)))))) 49.88/13.58 3(1(4(0(x1)))) -> 3(0(2(4(1(2(x1)))))) 49.88/13.58 3(5(4(0(x1)))) -> 0(5(2(3(4(x1))))) 49.88/13.58 3(5(4(0(x1)))) -> 4(2(0(5(3(4(x1)))))) 49.88/13.58 4(0(5(0(x1)))) -> 4(2(0(0(5(x1))))) 49.88/13.58 3(5(5(0(x1)))) -> 0(5(5(3(2(x1))))) 49.88/13.58 4(0(3(1(x1)))) -> 3(2(1(0(4(x1))))) 49.88/13.58 5(3(0(4(x1)))) -> 0(2(4(3(2(5(x1)))))) 49.88/13.58 4(0(1(0(0(x1))))) -> 2(4(1(0(0(0(x1)))))) 49.88/13.58 3(5(5(0(0(x1))))) -> 5(2(3(0(5(0(x1)))))) 49.88/13.58 4(3(4(0(3(x1))))) -> 4(2(4(3(0(3(x1)))))) 49.88/13.58 5(4(0(3(3(x1))))) -> 3(0(3(5(4(1(x1)))))) 49.88/13.58 3(0(2(4(3(x1))))) -> 4(3(2(0(3(2(x1)))))) 49.88/13.58 49.88/13.58 Q is empty. 49.88/13.58 We have to consider all minimal (P,Q,R)-chains. 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (17) DependencyGraphProof (EQUIVALENT) 49.88/13.58 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes. 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (18) 49.88/13.58 Complex Obligation (AND) 49.88/13.58 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (19) 49.88/13.58 Obligation: 49.88/13.58 Q DP problem: 49.88/13.58 The TRS P consists of the following rules: 49.88/13.58 49.88/13.58 4^1(3(4(0(3(x1))))) -> 4^1(3(0(3(x1)))) 49.88/13.58 49.88/13.58 The TRS R consists of the following rules: 49.88/13.58 49.88/13.58 0(0(x1)) -> 2(0(2(0(1(x1))))) 49.88/13.58 0(0(x1)) -> 2(0(4(3(2(0(x1)))))) 49.88/13.58 3(0(x1)) -> 2(3(2(0(x1)))) 49.88/13.58 3(0(x1)) -> 3(1(2(0(2(x1))))) 49.88/13.58 3(0(x1)) -> 2(2(0(3(2(x1))))) 49.88/13.58 3(0(x1)) -> 2(3(2(2(2(0(x1)))))) 49.88/13.58 0(0(0(x1))) -> 0(0(2(0(x1)))) 49.88/13.58 4(0(0(x1))) -> 2(3(0(0(4(2(x1)))))) 49.88/13.58 5(0(0(x1))) -> 0(2(5(0(x1)))) 49.88/13.58 3(1(0(x1))) -> 0(1(2(2(3(x1))))) 49.88/13.58 0(3(0(x1))) -> 0(4(3(2(0(2(x1)))))) 49.88/13.58 3(4(0(x1))) -> 4(2(0(3(x1)))) 49.88/13.58 3(4(0(x1))) -> 4(3(1(2(0(1(x1)))))) 49.88/13.58 3(4(0(x1))) -> 0(4(3(4(2(2(x1)))))) 49.88/13.58 5(4(0(x1))) -> 5(2(5(0(4(2(x1)))))) 49.88/13.58 5(4(0(x1))) -> 0(4(4(2(3(5(x1)))))) 49.88/13.58 0(5(0(x1))) -> 0(4(2(5(0(x1))))) 49.88/13.58 0(5(0(x1))) -> 0(2(3(2(0(5(x1)))))) 49.88/13.58 3(5(0(x1))) -> 0(2(3(5(2(3(x1)))))) 49.88/13.58 0(0(1(x1))) -> 2(0(5(2(0(1(x1)))))) 49.88/13.58 3(0(1(x1))) -> 2(5(0(3(1(2(x1)))))) 49.88/13.58 4(0(1(x1))) -> 4(1(2(0(3(x1))))) 49.88/13.58 4(0(1(x1))) -> 0(4(1(2(1(2(x1)))))) 49.88/13.58 0(0(3(x1))) -> 0(2(0(3(x1)))) 49.88/13.58 3(0(3(x1))) -> 3(2(0(3(x1)))) 49.88/13.58 3(0(3(x1))) -> 0(2(2(3(3(x1))))) 49.88/13.58 4(0(3(x1))) -> 0(4(3(2(2(x1))))) 49.88/13.58 4(0(3(x1))) -> 3(0(2(4(3(2(x1)))))) 49.88/13.58 4(0(3(x1))) -> 3(0(4(2(5(5(x1)))))) 49.88/13.58 4(3(3(x1))) -> 4(2(3(1(2(3(x1)))))) 49.88/13.58 0(5(3(x1))) -> 1(2(0(1(5(3(x1)))))) 49.88/13.58 4(0(4(x1))) -> 0(4(4(2(0(x1))))) 49.88/13.58 4(0(4(x1))) -> 0(4(2(0(4(1(x1)))))) 49.88/13.58 3(4(0(0(x1)))) -> 4(1(2(0(0(3(x1)))))) 49.88/13.58 3(5(0(0(x1)))) -> 3(0(5(2(5(0(x1)))))) 49.88/13.58 3(0(1(0(x1)))) -> 0(3(0(1(2(x1))))) 49.88/13.58 0(1(1(0(x1)))) -> 2(1(0(0(1(2(x1)))))) 49.88/13.58 0(0(4(0(x1)))) -> 0(1(2(0(4(0(x1)))))) 49.88/13.58 3(1(4(0(x1)))) -> 3(0(2(4(1(2(x1)))))) 49.88/13.58 3(5(4(0(x1)))) -> 0(5(2(3(4(x1))))) 49.88/13.58 3(5(4(0(x1)))) -> 4(2(0(5(3(4(x1)))))) 49.88/13.58 4(0(5(0(x1)))) -> 4(2(0(0(5(x1))))) 49.88/13.58 3(5(5(0(x1)))) -> 0(5(5(3(2(x1))))) 49.88/13.58 4(0(3(1(x1)))) -> 3(2(1(0(4(x1))))) 49.88/13.58 5(3(0(4(x1)))) -> 0(2(4(3(2(5(x1)))))) 49.88/13.58 4(0(1(0(0(x1))))) -> 2(4(1(0(0(0(x1)))))) 49.88/13.58 3(5(5(0(0(x1))))) -> 5(2(3(0(5(0(x1)))))) 49.88/13.58 4(3(4(0(3(x1))))) -> 4(2(4(3(0(3(x1)))))) 49.88/13.58 5(4(0(3(3(x1))))) -> 3(0(3(5(4(1(x1)))))) 49.88/13.58 3(0(2(4(3(x1))))) -> 4(3(2(0(3(2(x1)))))) 49.88/13.58 49.88/13.58 Q is empty. 49.88/13.58 We have to consider all minimal (P,Q,R)-chains. 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (20) QDPOrderProof (EQUIVALENT) 49.88/13.58 We use the reduction pair processor [LPAR04,JAR06]. 49.88/13.58 49.88/13.58 49.88/13.58 The following pairs can be oriented strictly and are deleted. 49.88/13.58 49.88/13.58 4^1(3(4(0(3(x1))))) -> 4^1(3(0(3(x1)))) 49.88/13.58 The remaining pairs can at least be oriented weakly. 49.88/13.58 Used ordering: Polynomial interpretation [POLO]: 49.88/13.58 49.88/13.58 POL(0(x_1)) = 0 49.88/13.58 POL(1(x_1)) = 0 49.88/13.58 POL(2(x_1)) = x_1 49.88/13.58 POL(3(x_1)) = 1 + x_1 49.88/13.58 POL(4(x_1)) = 1 49.88/13.58 POL(4^1(x_1)) = x_1 49.88/13.58 POL(5(x_1)) = 1 + x_1 49.88/13.58 49.88/13.58 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 49.88/13.58 49.88/13.58 3(0(x1)) -> 2(3(2(0(x1)))) 49.88/13.58 3(0(x1)) -> 3(1(2(0(2(x1))))) 49.88/13.58 3(0(x1)) -> 2(2(0(3(2(x1))))) 49.88/13.58 3(0(x1)) -> 2(3(2(2(2(0(x1)))))) 49.88/13.58 3(1(0(x1))) -> 0(1(2(2(3(x1))))) 49.88/13.58 3(4(0(x1))) -> 4(2(0(3(x1)))) 49.88/13.58 3(4(0(x1))) -> 4(3(1(2(0(1(x1)))))) 49.88/13.58 3(4(0(x1))) -> 0(4(3(4(2(2(x1)))))) 49.88/13.58 3(5(0(x1))) -> 0(2(3(5(2(3(x1)))))) 49.88/13.58 3(0(1(x1))) -> 2(5(0(3(1(2(x1)))))) 49.88/13.58 3(0(3(x1))) -> 3(2(0(3(x1)))) 49.88/13.58 3(0(3(x1))) -> 0(2(2(3(3(x1))))) 49.88/13.58 3(4(0(0(x1)))) -> 4(1(2(0(0(3(x1)))))) 49.88/13.58 3(5(0(0(x1)))) -> 3(0(5(2(5(0(x1)))))) 49.88/13.58 3(0(1(0(x1)))) -> 0(3(0(1(2(x1))))) 49.88/13.58 3(1(4(0(x1)))) -> 3(0(2(4(1(2(x1)))))) 49.88/13.58 3(5(4(0(x1)))) -> 0(5(2(3(4(x1))))) 49.88/13.58 3(5(4(0(x1)))) -> 4(2(0(5(3(4(x1)))))) 49.88/13.58 3(5(5(0(x1)))) -> 0(5(5(3(2(x1))))) 49.88/13.58 3(5(5(0(0(x1))))) -> 5(2(3(0(5(0(x1)))))) 49.88/13.58 3(0(2(4(3(x1))))) -> 4(3(2(0(3(2(x1)))))) 49.88/13.58 0(0(x1)) -> 2(0(2(0(1(x1))))) 49.88/13.58 0(0(x1)) -> 2(0(4(3(2(0(x1)))))) 49.88/13.58 0(0(0(x1))) -> 0(0(2(0(x1)))) 49.88/13.58 0(3(0(x1))) -> 0(4(3(2(0(2(x1)))))) 49.88/13.58 0(5(0(x1))) -> 0(4(2(5(0(x1))))) 49.88/13.58 0(5(0(x1))) -> 0(2(3(2(0(5(x1)))))) 49.88/13.58 0(0(1(x1))) -> 2(0(5(2(0(1(x1)))))) 49.88/13.58 0(0(3(x1))) -> 0(2(0(3(x1)))) 49.88/13.58 0(5(3(x1))) -> 1(2(0(1(5(3(x1)))))) 49.88/13.58 0(1(1(0(x1)))) -> 2(1(0(0(1(2(x1)))))) 49.88/13.58 0(0(4(0(x1)))) -> 0(1(2(0(4(0(x1)))))) 49.88/13.58 49.88/13.58 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (21) 49.88/13.58 Obligation: 49.88/13.58 Q DP problem: 49.88/13.58 P is empty. 49.88/13.58 The TRS R consists of the following rules: 49.88/13.58 49.88/13.58 0(0(x1)) -> 2(0(2(0(1(x1))))) 49.88/13.58 0(0(x1)) -> 2(0(4(3(2(0(x1)))))) 49.88/13.58 3(0(x1)) -> 2(3(2(0(x1)))) 49.88/13.58 3(0(x1)) -> 3(1(2(0(2(x1))))) 49.88/13.58 3(0(x1)) -> 2(2(0(3(2(x1))))) 49.88/13.58 3(0(x1)) -> 2(3(2(2(2(0(x1)))))) 49.88/13.58 0(0(0(x1))) -> 0(0(2(0(x1)))) 49.88/13.58 4(0(0(x1))) -> 2(3(0(0(4(2(x1)))))) 49.88/13.58 5(0(0(x1))) -> 0(2(5(0(x1)))) 49.88/13.58 3(1(0(x1))) -> 0(1(2(2(3(x1))))) 49.88/13.58 0(3(0(x1))) -> 0(4(3(2(0(2(x1)))))) 49.88/13.58 3(4(0(x1))) -> 4(2(0(3(x1)))) 49.88/13.58 3(4(0(x1))) -> 4(3(1(2(0(1(x1)))))) 49.88/13.58 3(4(0(x1))) -> 0(4(3(4(2(2(x1)))))) 49.88/13.58 5(4(0(x1))) -> 5(2(5(0(4(2(x1)))))) 49.88/13.58 5(4(0(x1))) -> 0(4(4(2(3(5(x1)))))) 49.88/13.58 0(5(0(x1))) -> 0(4(2(5(0(x1))))) 49.88/13.58 0(5(0(x1))) -> 0(2(3(2(0(5(x1)))))) 49.88/13.58 3(5(0(x1))) -> 0(2(3(5(2(3(x1)))))) 49.88/13.58 0(0(1(x1))) -> 2(0(5(2(0(1(x1)))))) 49.88/13.58 3(0(1(x1))) -> 2(5(0(3(1(2(x1)))))) 49.88/13.58 4(0(1(x1))) -> 4(1(2(0(3(x1))))) 49.88/13.58 4(0(1(x1))) -> 0(4(1(2(1(2(x1)))))) 49.88/13.58 0(0(3(x1))) -> 0(2(0(3(x1)))) 49.88/13.58 3(0(3(x1))) -> 3(2(0(3(x1)))) 49.88/13.58 3(0(3(x1))) -> 0(2(2(3(3(x1))))) 49.88/13.58 4(0(3(x1))) -> 0(4(3(2(2(x1))))) 49.88/13.58 4(0(3(x1))) -> 3(0(2(4(3(2(x1)))))) 49.88/13.58 4(0(3(x1))) -> 3(0(4(2(5(5(x1)))))) 49.88/13.58 4(3(3(x1))) -> 4(2(3(1(2(3(x1)))))) 49.88/13.58 0(5(3(x1))) -> 1(2(0(1(5(3(x1)))))) 49.88/13.58 4(0(4(x1))) -> 0(4(4(2(0(x1))))) 49.88/13.58 4(0(4(x1))) -> 0(4(2(0(4(1(x1)))))) 49.88/13.58 3(4(0(0(x1)))) -> 4(1(2(0(0(3(x1)))))) 49.88/13.58 3(5(0(0(x1)))) -> 3(0(5(2(5(0(x1)))))) 49.88/13.58 3(0(1(0(x1)))) -> 0(3(0(1(2(x1))))) 49.88/13.58 0(1(1(0(x1)))) -> 2(1(0(0(1(2(x1)))))) 49.88/13.58 0(0(4(0(x1)))) -> 0(1(2(0(4(0(x1)))))) 49.88/13.58 3(1(4(0(x1)))) -> 3(0(2(4(1(2(x1)))))) 49.88/13.58 3(5(4(0(x1)))) -> 0(5(2(3(4(x1))))) 49.88/13.58 3(5(4(0(x1)))) -> 4(2(0(5(3(4(x1)))))) 49.88/13.58 4(0(5(0(x1)))) -> 4(2(0(0(5(x1))))) 49.88/13.58 3(5(5(0(x1)))) -> 0(5(5(3(2(x1))))) 49.88/13.58 4(0(3(1(x1)))) -> 3(2(1(0(4(x1))))) 49.88/13.58 5(3(0(4(x1)))) -> 0(2(4(3(2(5(x1)))))) 49.88/13.58 4(0(1(0(0(x1))))) -> 2(4(1(0(0(0(x1)))))) 49.88/13.58 3(5(5(0(0(x1))))) -> 5(2(3(0(5(0(x1)))))) 49.88/13.58 4(3(4(0(3(x1))))) -> 4(2(4(3(0(3(x1)))))) 49.88/13.58 5(4(0(3(3(x1))))) -> 3(0(3(5(4(1(x1)))))) 49.88/13.58 3(0(2(4(3(x1))))) -> 4(3(2(0(3(2(x1)))))) 49.88/13.58 49.88/13.58 Q is empty. 49.88/13.58 We have to consider all minimal (P,Q,R)-chains. 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (22) PisEmptyProof (EQUIVALENT) 49.88/13.58 The TRS P is empty. Hence, there is no (P,Q,R) chain. 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (23) 49.88/13.58 YES 49.88/13.58 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (24) 49.88/13.58 Obligation: 49.88/13.58 Q DP problem: 49.88/13.58 The TRS P consists of the following rules: 49.88/13.58 49.88/13.58 3^1(5(5(0(0(x1))))) -> 3^1(0(5(0(x1)))) 49.88/13.58 49.88/13.58 The TRS R consists of the following rules: 49.88/13.58 49.88/13.58 0(0(x1)) -> 2(0(2(0(1(x1))))) 49.88/13.58 0(0(x1)) -> 2(0(4(3(2(0(x1)))))) 49.88/13.58 3(0(x1)) -> 2(3(2(0(x1)))) 49.88/13.58 3(0(x1)) -> 3(1(2(0(2(x1))))) 49.88/13.58 3(0(x1)) -> 2(2(0(3(2(x1))))) 49.88/13.58 3(0(x1)) -> 2(3(2(2(2(0(x1)))))) 49.88/13.58 0(0(0(x1))) -> 0(0(2(0(x1)))) 49.88/13.58 4(0(0(x1))) -> 2(3(0(0(4(2(x1)))))) 49.88/13.58 5(0(0(x1))) -> 0(2(5(0(x1)))) 49.88/13.58 3(1(0(x1))) -> 0(1(2(2(3(x1))))) 49.88/13.58 0(3(0(x1))) -> 0(4(3(2(0(2(x1)))))) 49.88/13.58 3(4(0(x1))) -> 4(2(0(3(x1)))) 49.88/13.58 3(4(0(x1))) -> 4(3(1(2(0(1(x1)))))) 49.88/13.58 3(4(0(x1))) -> 0(4(3(4(2(2(x1)))))) 49.88/13.58 5(4(0(x1))) -> 5(2(5(0(4(2(x1)))))) 49.88/13.58 5(4(0(x1))) -> 0(4(4(2(3(5(x1)))))) 49.88/13.58 0(5(0(x1))) -> 0(4(2(5(0(x1))))) 49.88/13.58 0(5(0(x1))) -> 0(2(3(2(0(5(x1)))))) 49.88/13.58 3(5(0(x1))) -> 0(2(3(5(2(3(x1)))))) 49.88/13.58 0(0(1(x1))) -> 2(0(5(2(0(1(x1)))))) 49.88/13.58 3(0(1(x1))) -> 2(5(0(3(1(2(x1)))))) 49.88/13.58 4(0(1(x1))) -> 4(1(2(0(3(x1))))) 49.88/13.58 4(0(1(x1))) -> 0(4(1(2(1(2(x1)))))) 49.88/13.58 0(0(3(x1))) -> 0(2(0(3(x1)))) 49.88/13.58 3(0(3(x1))) -> 3(2(0(3(x1)))) 49.88/13.58 3(0(3(x1))) -> 0(2(2(3(3(x1))))) 49.88/13.58 4(0(3(x1))) -> 0(4(3(2(2(x1))))) 49.88/13.58 4(0(3(x1))) -> 3(0(2(4(3(2(x1)))))) 49.88/13.58 4(0(3(x1))) -> 3(0(4(2(5(5(x1)))))) 49.88/13.58 4(3(3(x1))) -> 4(2(3(1(2(3(x1)))))) 49.88/13.58 0(5(3(x1))) -> 1(2(0(1(5(3(x1)))))) 49.88/13.58 4(0(4(x1))) -> 0(4(4(2(0(x1))))) 49.88/13.58 4(0(4(x1))) -> 0(4(2(0(4(1(x1)))))) 49.88/13.58 3(4(0(0(x1)))) -> 4(1(2(0(0(3(x1)))))) 49.88/13.58 3(5(0(0(x1)))) -> 3(0(5(2(5(0(x1)))))) 49.88/13.58 3(0(1(0(x1)))) -> 0(3(0(1(2(x1))))) 49.88/13.58 0(1(1(0(x1)))) -> 2(1(0(0(1(2(x1)))))) 49.88/13.58 0(0(4(0(x1)))) -> 0(1(2(0(4(0(x1)))))) 49.88/13.58 3(1(4(0(x1)))) -> 3(0(2(4(1(2(x1)))))) 49.88/13.58 3(5(4(0(x1)))) -> 0(5(2(3(4(x1))))) 49.88/13.58 3(5(4(0(x1)))) -> 4(2(0(5(3(4(x1)))))) 49.88/13.58 4(0(5(0(x1)))) -> 4(2(0(0(5(x1))))) 49.88/13.58 3(5(5(0(x1)))) -> 0(5(5(3(2(x1))))) 49.88/13.58 4(0(3(1(x1)))) -> 3(2(1(0(4(x1))))) 49.88/13.58 5(3(0(4(x1)))) -> 0(2(4(3(2(5(x1)))))) 49.88/13.58 4(0(1(0(0(x1))))) -> 2(4(1(0(0(0(x1)))))) 49.88/13.58 3(5(5(0(0(x1))))) -> 5(2(3(0(5(0(x1)))))) 49.88/13.58 4(3(4(0(3(x1))))) -> 4(2(4(3(0(3(x1)))))) 49.88/13.58 5(4(0(3(3(x1))))) -> 3(0(3(5(4(1(x1)))))) 49.88/13.58 3(0(2(4(3(x1))))) -> 4(3(2(0(3(2(x1)))))) 49.88/13.58 49.88/13.58 Q is empty. 49.88/13.58 We have to consider all minimal (P,Q,R)-chains. 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (25) QDPOrderProof (EQUIVALENT) 49.88/13.58 We use the reduction pair processor [LPAR04,JAR06]. 49.88/13.58 49.88/13.58 49.88/13.58 The following pairs can be oriented strictly and are deleted. 49.88/13.58 49.88/13.58 3^1(5(5(0(0(x1))))) -> 3^1(0(5(0(x1)))) 49.88/13.58 The remaining pairs can at least be oriented weakly. 49.88/13.58 Used ordering: Polynomial interpretation [POLO]: 49.88/13.58 49.88/13.58 POL(0(x_1)) = 0 49.88/13.58 POL(1(x_1)) = 0 49.88/13.58 POL(2(x_1)) = 0 49.88/13.58 POL(3(x_1)) = 1 + x_1 49.88/13.58 POL(3^1(x_1)) = x_1 49.88/13.58 POL(4(x_1)) = 0 49.88/13.58 POL(5(x_1)) = 1 49.88/13.58 49.88/13.58 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 49.88/13.58 49.88/13.58 0(0(x1)) -> 2(0(2(0(1(x1))))) 49.88/13.58 0(0(x1)) -> 2(0(4(3(2(0(x1)))))) 49.88/13.58 0(0(0(x1))) -> 0(0(2(0(x1)))) 49.88/13.58 0(3(0(x1))) -> 0(4(3(2(0(2(x1)))))) 49.88/13.58 0(5(0(x1))) -> 0(4(2(5(0(x1))))) 49.88/13.58 0(5(0(x1))) -> 0(2(3(2(0(5(x1)))))) 49.88/13.58 0(0(1(x1))) -> 2(0(5(2(0(1(x1)))))) 49.88/13.58 0(0(3(x1))) -> 0(2(0(3(x1)))) 49.88/13.58 0(5(3(x1))) -> 1(2(0(1(5(3(x1)))))) 49.88/13.58 0(1(1(0(x1)))) -> 2(1(0(0(1(2(x1)))))) 49.88/13.58 0(0(4(0(x1)))) -> 0(1(2(0(4(0(x1)))))) 49.88/13.58 49.88/13.58 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (26) 49.88/13.58 Obligation: 49.88/13.58 Q DP problem: 49.88/13.58 P is empty. 49.88/13.58 The TRS R consists of the following rules: 49.88/13.58 49.88/13.58 0(0(x1)) -> 2(0(2(0(1(x1))))) 49.88/13.58 0(0(x1)) -> 2(0(4(3(2(0(x1)))))) 49.88/13.58 3(0(x1)) -> 2(3(2(0(x1)))) 49.88/13.58 3(0(x1)) -> 3(1(2(0(2(x1))))) 49.88/13.58 3(0(x1)) -> 2(2(0(3(2(x1))))) 49.88/13.58 3(0(x1)) -> 2(3(2(2(2(0(x1)))))) 49.88/13.58 0(0(0(x1))) -> 0(0(2(0(x1)))) 49.88/13.58 4(0(0(x1))) -> 2(3(0(0(4(2(x1)))))) 49.88/13.58 5(0(0(x1))) -> 0(2(5(0(x1)))) 49.88/13.58 3(1(0(x1))) -> 0(1(2(2(3(x1))))) 49.88/13.58 0(3(0(x1))) -> 0(4(3(2(0(2(x1)))))) 49.88/13.58 3(4(0(x1))) -> 4(2(0(3(x1)))) 49.88/13.58 3(4(0(x1))) -> 4(3(1(2(0(1(x1)))))) 49.88/13.58 3(4(0(x1))) -> 0(4(3(4(2(2(x1)))))) 49.88/13.58 5(4(0(x1))) -> 5(2(5(0(4(2(x1)))))) 49.88/13.58 5(4(0(x1))) -> 0(4(4(2(3(5(x1)))))) 49.88/13.58 0(5(0(x1))) -> 0(4(2(5(0(x1))))) 49.88/13.58 0(5(0(x1))) -> 0(2(3(2(0(5(x1)))))) 49.88/13.58 3(5(0(x1))) -> 0(2(3(5(2(3(x1)))))) 49.88/13.58 0(0(1(x1))) -> 2(0(5(2(0(1(x1)))))) 49.88/13.58 3(0(1(x1))) -> 2(5(0(3(1(2(x1)))))) 49.88/13.58 4(0(1(x1))) -> 4(1(2(0(3(x1))))) 49.88/13.58 4(0(1(x1))) -> 0(4(1(2(1(2(x1)))))) 49.88/13.58 0(0(3(x1))) -> 0(2(0(3(x1)))) 49.88/13.58 3(0(3(x1))) -> 3(2(0(3(x1)))) 49.88/13.58 3(0(3(x1))) -> 0(2(2(3(3(x1))))) 49.88/13.58 4(0(3(x1))) -> 0(4(3(2(2(x1))))) 49.88/13.58 4(0(3(x1))) -> 3(0(2(4(3(2(x1)))))) 49.88/13.58 4(0(3(x1))) -> 3(0(4(2(5(5(x1)))))) 49.88/13.58 4(3(3(x1))) -> 4(2(3(1(2(3(x1)))))) 49.88/13.58 0(5(3(x1))) -> 1(2(0(1(5(3(x1)))))) 49.88/13.58 4(0(4(x1))) -> 0(4(4(2(0(x1))))) 49.88/13.58 4(0(4(x1))) -> 0(4(2(0(4(1(x1)))))) 49.88/13.58 3(4(0(0(x1)))) -> 4(1(2(0(0(3(x1)))))) 49.88/13.58 3(5(0(0(x1)))) -> 3(0(5(2(5(0(x1)))))) 49.88/13.58 3(0(1(0(x1)))) -> 0(3(0(1(2(x1))))) 49.88/13.58 0(1(1(0(x1)))) -> 2(1(0(0(1(2(x1)))))) 49.88/13.58 0(0(4(0(x1)))) -> 0(1(2(0(4(0(x1)))))) 49.88/13.58 3(1(4(0(x1)))) -> 3(0(2(4(1(2(x1)))))) 49.88/13.58 3(5(4(0(x1)))) -> 0(5(2(3(4(x1))))) 49.88/13.58 3(5(4(0(x1)))) -> 4(2(0(5(3(4(x1)))))) 49.88/13.58 4(0(5(0(x1)))) -> 4(2(0(0(5(x1))))) 49.88/13.58 3(5(5(0(x1)))) -> 0(5(5(3(2(x1))))) 49.88/13.58 4(0(3(1(x1)))) -> 3(2(1(0(4(x1))))) 49.88/13.58 5(3(0(4(x1)))) -> 0(2(4(3(2(5(x1)))))) 49.88/13.58 4(0(1(0(0(x1))))) -> 2(4(1(0(0(0(x1)))))) 49.88/13.58 3(5(5(0(0(x1))))) -> 5(2(3(0(5(0(x1)))))) 49.88/13.58 4(3(4(0(3(x1))))) -> 4(2(4(3(0(3(x1)))))) 49.88/13.58 5(4(0(3(3(x1))))) -> 3(0(3(5(4(1(x1)))))) 49.88/13.58 3(0(2(4(3(x1))))) -> 4(3(2(0(3(2(x1)))))) 49.88/13.58 49.88/13.58 Q is empty. 49.88/13.58 We have to consider all minimal (P,Q,R)-chains. 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (27) PisEmptyProof (EQUIVALENT) 49.88/13.58 The TRS P is empty. Hence, there is no (P,Q,R) chain. 49.88/13.58 ---------------------------------------- 49.88/13.58 49.88/13.58 (28) 49.88/13.58 YES 50.11/13.65 EOF