45.41/12.50 YES 45.58/12.61 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 45.58/12.61 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 45.58/12.61 45.58/12.61 45.58/12.61 Termination w.r.t. Q of the given QTRS could be proven: 45.58/12.61 45.58/12.61 (0) QTRS 45.58/12.61 (1) QTRS Reverse [EQUIVALENT, 0 ms] 45.58/12.61 (2) QTRS 45.58/12.61 (3) DependencyPairsProof [EQUIVALENT, 341 ms] 45.58/12.61 (4) QDP 45.58/12.61 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 45.58/12.61 (6) AND 45.58/12.61 (7) QDP 45.58/12.61 (8) UsableRulesProof [EQUIVALENT, 0 ms] 45.58/12.61 (9) QDP 45.58/12.61 (10) MRRProof [EQUIVALENT, 10 ms] 45.58/12.61 (11) QDP 45.58/12.61 (12) PisEmptyProof [EQUIVALENT, 0 ms] 45.58/12.61 (13) YES 45.58/12.61 (14) QDP 45.58/12.61 (15) QDPOrderProof [EQUIVALENT, 220 ms] 45.58/12.61 (16) QDP 45.58/12.61 (17) QDPOrderProof [EQUIVALENT, 0 ms] 45.58/12.61 (18) QDP 45.58/12.61 (19) PisEmptyProof [EQUIVALENT, 0 ms] 45.58/12.61 (20) YES 45.58/12.61 (21) QDP 45.58/12.61 (22) QDPSizeChangeProof [EQUIVALENT, 0 ms] 45.58/12.61 (23) YES 45.58/12.61 (24) QDP 45.58/12.61 (25) QDPOrderProof [EQUIVALENT, 164 ms] 45.58/12.61 (26) QDP 45.58/12.61 (27) QDPOrderProof [EQUIVALENT, 0 ms] 45.58/12.61 (28) QDP 45.58/12.61 (29) PisEmptyProof [EQUIVALENT, 0 ms] 45.58/12.61 (30) YES 45.58/12.61 45.58/12.61 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (0) 45.58/12.61 Obligation: 45.58/12.61 Q restricted rewrite system: 45.58/12.61 The TRS R consists of the following rules: 45.58/12.61 45.58/12.61 0(0(1(x1))) -> 0(0(2(1(3(x1))))) 45.58/12.61 0(0(1(x1))) -> 0(0(2(2(3(1(x1)))))) 45.58/12.61 0(4(1(x1))) -> 0(4(2(2(1(x1))))) 45.58/12.61 0(4(1(x1))) -> 0(2(3(1(4(2(x1)))))) 45.58/12.61 1(0(1(x1))) -> 0(2(1(1(x1)))) 45.58/12.61 1(0(1(x1))) -> 0(2(1(1(2(x1))))) 45.58/12.61 1(0(1(x1))) -> 1(0(3(2(1(x1))))) 45.58/12.61 1(4(0(x1))) -> 1(0(3(2(4(2(x1)))))) 45.58/12.61 1(4(1(x1))) -> 4(2(1(1(x1)))) 45.58/12.61 1(4(1(x1))) -> 4(2(1(1(2(1(x1)))))) 45.58/12.61 4(0(1(x1))) -> 4(0(2(1(x1)))) 45.58/12.61 5(0(1(x1))) -> 5(0(2(1(x1)))) 45.58/12.61 5(0(1(x1))) -> 0(2(1(5(3(x1))))) 45.58/12.61 5(0(1(x1))) -> 0(3(2(1(5(5(x1)))))) 45.58/12.61 5(0(1(x1))) -> 3(0(2(3(5(1(x1)))))) 45.58/12.61 5(0(5(x1))) -> 0(3(2(5(3(5(x1)))))) 45.58/12.61 5(4(1(x1))) -> 2(4(2(5(1(3(x1)))))) 45.58/12.61 5(4(1(x1))) -> 4(5(2(1(3(3(x1)))))) 45.58/12.61 0(0(4(5(x1)))) -> 0(5(0(2(4(2(x1)))))) 45.58/12.61 0(1(2(0(x1)))) -> 2(0(2(1(0(x1))))) 45.58/12.61 0(2(0(1(x1)))) -> 0(0(2(1(3(x1))))) 45.58/12.61 0(3(4(0(x1)))) -> 3(2(4(0(0(3(x1)))))) 45.58/12.61 0(4(0(4(x1)))) -> 0(0(3(2(4(4(x1)))))) 45.58/12.61 1(0(1(4(x1)))) -> 2(3(1(1(4(0(x1)))))) 45.58/12.61 1(0(3(1(x1)))) -> 4(2(3(1(1(0(x1)))))) 45.58/12.61 1(2(0(4(x1)))) -> 1(4(2(0(3(2(x1)))))) 45.58/12.61 1(3(0(4(x1)))) -> 4(0(3(2(1(3(x1)))))) 45.58/12.61 1(4(1(5(x1)))) -> 2(1(2(5(1(4(x1)))))) 45.58/12.61 4(0(5(1(x1)))) -> 0(2(4(2(1(5(x1)))))) 45.58/12.61 4(1(0(0(x1)))) -> 3(2(1(4(0(0(x1)))))) 45.58/12.61 4(1(0(4(x1)))) -> 4(4(0(2(1(x1))))) 45.58/12.61 5(0(0(1(x1)))) -> 0(3(2(5(1(0(x1)))))) 45.58/12.61 5(0(3(1(x1)))) -> 0(5(3(2(1(x1))))) 45.58/12.61 5(0(3(1(x1)))) -> 0(3(5(2(2(1(x1)))))) 45.58/12.61 5(0(5(1(x1)))) -> 5(2(5(3(1(0(x1)))))) 45.58/12.61 5(2(4(1(x1)))) -> 2(1(4(2(5(x1))))) 45.58/12.61 5(4(1(5(x1)))) -> 4(5(5(2(1(x1))))) 45.58/12.61 5(4(1(5(x1)))) -> 2(1(5(4(2(5(x1)))))) 45.58/12.61 5(4(3(1(x1)))) -> 5(4(2(1(3(x1))))) 45.58/12.61 5(4(3(1(x1)))) -> 3(5(2(4(2(1(x1)))))) 45.58/12.61 5(4(3(1(x1)))) -> 5(2(4(3(2(1(x1)))))) 45.58/12.61 5(5(0(4(x1)))) -> 5(0(2(5(4(2(x1)))))) 45.58/12.61 0(0(3(3(1(x1))))) -> 0(3(0(1(2(3(x1)))))) 45.58/12.61 0(2(4(3(1(x1))))) -> 0(2(3(1(4(2(x1)))))) 45.58/12.61 0(5(4(1(1(x1))))) -> 4(0(2(5(1(1(x1)))))) 45.58/12.61 1(2(4(3(1(x1))))) -> 1(2(3(3(1(4(x1)))))) 45.58/12.61 4(0(0(3(1(x1))))) -> 4(0(3(2(1(0(x1)))))) 45.58/12.61 4(0(0(3(1(x1))))) -> 4(3(2(0(0(1(x1)))))) 45.58/12.61 5(1(0(0(5(x1))))) -> 5(0(0(2(1(5(x1)))))) 45.58/12.61 5(4(2(0(1(x1))))) -> 2(5(4(0(2(1(x1)))))) 45.58/12.61 45.58/12.61 Q is empty. 45.58/12.61 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (1) QTRS Reverse (EQUIVALENT) 45.58/12.61 We applied the QTRS Reverse Processor [REVERSE]. 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (2) 45.58/12.61 Obligation: 45.58/12.61 Q restricted rewrite system: 45.58/12.61 The TRS R consists of the following rules: 45.58/12.61 45.58/12.61 1(0(0(x1))) -> 3(1(2(0(0(x1))))) 45.58/12.61 1(0(0(x1))) -> 1(3(2(2(0(0(x1)))))) 45.58/12.61 1(4(0(x1))) -> 1(2(2(4(0(x1))))) 45.58/12.61 1(4(0(x1))) -> 2(4(1(3(2(0(x1)))))) 45.58/12.61 1(0(1(x1))) -> 1(1(2(0(x1)))) 45.58/12.61 1(0(1(x1))) -> 2(1(1(2(0(x1))))) 45.58/12.61 1(0(1(x1))) -> 1(2(3(0(1(x1))))) 45.58/12.61 0(4(1(x1))) -> 2(4(2(3(0(1(x1)))))) 45.58/12.61 1(4(1(x1))) -> 1(1(2(4(x1)))) 45.58/12.61 1(4(1(x1))) -> 1(2(1(1(2(4(x1)))))) 45.58/12.61 1(0(4(x1))) -> 1(2(0(4(x1)))) 45.58/12.61 1(0(5(x1))) -> 1(2(0(5(x1)))) 45.58/12.61 1(0(5(x1))) -> 3(5(1(2(0(x1))))) 45.58/12.61 1(0(5(x1))) -> 5(5(1(2(3(0(x1)))))) 45.58/12.61 1(0(5(x1))) -> 1(5(3(2(0(3(x1)))))) 45.58/12.61 5(0(5(x1))) -> 5(3(5(2(3(0(x1)))))) 45.58/12.61 1(4(5(x1))) -> 3(1(5(2(4(2(x1)))))) 45.58/12.61 1(4(5(x1))) -> 3(3(1(2(5(4(x1)))))) 45.58/12.61 5(4(0(0(x1)))) -> 2(4(2(0(5(0(x1)))))) 45.58/12.61 0(2(1(0(x1)))) -> 0(1(2(0(2(x1))))) 45.58/12.61 1(0(2(0(x1)))) -> 3(1(2(0(0(x1))))) 45.58/12.61 0(4(3(0(x1)))) -> 3(0(0(4(2(3(x1)))))) 45.58/12.61 4(0(4(0(x1)))) -> 4(4(2(3(0(0(x1)))))) 45.58/12.61 4(1(0(1(x1)))) -> 0(4(1(1(3(2(x1)))))) 45.58/12.61 1(3(0(1(x1)))) -> 0(1(1(3(2(4(x1)))))) 45.58/12.61 4(0(2(1(x1)))) -> 2(3(0(2(4(1(x1)))))) 45.58/12.61 4(0(3(1(x1)))) -> 3(1(2(3(0(4(x1)))))) 45.58/12.61 5(1(4(1(x1)))) -> 4(1(5(2(1(2(x1)))))) 45.58/12.61 1(5(0(4(x1)))) -> 5(1(2(4(2(0(x1)))))) 45.58/12.61 0(0(1(4(x1)))) -> 0(0(4(1(2(3(x1)))))) 45.58/12.61 4(0(1(4(x1)))) -> 1(2(0(4(4(x1))))) 45.58/12.61 1(0(0(5(x1)))) -> 0(1(5(2(3(0(x1)))))) 45.58/12.61 1(3(0(5(x1)))) -> 1(2(3(5(0(x1))))) 45.58/12.61 1(3(0(5(x1)))) -> 1(2(2(5(3(0(x1)))))) 45.58/12.61 1(5(0(5(x1)))) -> 0(1(3(5(2(5(x1)))))) 45.58/12.61 1(4(2(5(x1)))) -> 5(2(4(1(2(x1))))) 45.58/12.61 5(1(4(5(x1)))) -> 1(2(5(5(4(x1))))) 45.58/12.61 5(1(4(5(x1)))) -> 5(2(4(5(1(2(x1)))))) 45.58/12.61 1(3(4(5(x1)))) -> 3(1(2(4(5(x1))))) 45.58/12.61 1(3(4(5(x1)))) -> 1(2(4(2(5(3(x1)))))) 45.58/12.61 1(3(4(5(x1)))) -> 1(2(3(4(2(5(x1)))))) 45.58/12.61 4(0(5(5(x1)))) -> 2(4(5(2(0(5(x1)))))) 45.58/12.61 1(3(3(0(0(x1))))) -> 3(2(1(0(3(0(x1)))))) 45.58/12.61 1(3(4(2(0(x1))))) -> 2(4(1(3(2(0(x1)))))) 45.58/12.61 1(1(4(5(0(x1))))) -> 1(1(5(2(0(4(x1)))))) 45.58/12.61 1(3(4(2(1(x1))))) -> 4(1(3(3(2(1(x1)))))) 45.58/12.61 1(3(0(0(4(x1))))) -> 0(1(2(3(0(4(x1)))))) 45.58/12.61 1(3(0(0(4(x1))))) -> 1(0(0(2(3(4(x1)))))) 45.58/12.61 5(0(0(1(5(x1))))) -> 5(1(2(0(0(5(x1)))))) 45.58/12.61 1(0(2(4(5(x1))))) -> 1(2(0(4(5(2(x1)))))) 45.58/12.61 45.58/12.61 Q is empty. 45.58/12.61 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (3) DependencyPairsProof (EQUIVALENT) 45.58/12.61 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (4) 45.58/12.61 Obligation: 45.58/12.61 Q DP problem: 45.58/12.61 The TRS P consists of the following rules: 45.58/12.61 45.58/12.61 1^1(0(0(x1))) -> 1^1(2(0(0(x1)))) 45.58/12.61 1^1(0(0(x1))) -> 1^1(3(2(2(0(0(x1)))))) 45.58/12.61 1^1(4(0(x1))) -> 1^1(2(2(4(0(x1))))) 45.58/12.61 1^1(4(0(x1))) -> 4^1(1(3(2(0(x1))))) 45.58/12.61 1^1(4(0(x1))) -> 1^1(3(2(0(x1)))) 45.58/12.61 1^1(0(1(x1))) -> 1^1(1(2(0(x1)))) 45.58/12.61 1^1(0(1(x1))) -> 1^1(2(0(x1))) 45.58/12.61 1^1(0(1(x1))) -> 0^1(x1) 45.58/12.61 1^1(0(1(x1))) -> 1^1(2(3(0(1(x1))))) 45.58/12.61 0^1(4(1(x1))) -> 4^1(2(3(0(1(x1))))) 45.58/12.61 0^1(4(1(x1))) -> 0^1(1(x1)) 45.58/12.61 1^1(4(1(x1))) -> 1^1(1(2(4(x1)))) 45.58/12.61 1^1(4(1(x1))) -> 1^1(2(4(x1))) 45.58/12.61 1^1(4(1(x1))) -> 4^1(x1) 45.58/12.61 1^1(4(1(x1))) -> 1^1(2(1(1(2(4(x1)))))) 45.58/12.61 1^1(0(4(x1))) -> 1^1(2(0(4(x1)))) 45.58/12.61 1^1(0(5(x1))) -> 1^1(2(0(5(x1)))) 45.58/12.61 1^1(0(5(x1))) -> 5^1(1(2(0(x1)))) 45.58/12.61 1^1(0(5(x1))) -> 1^1(2(0(x1))) 45.58/12.61 1^1(0(5(x1))) -> 0^1(x1) 45.58/12.61 1^1(0(5(x1))) -> 5^1(5(1(2(3(0(x1)))))) 45.58/12.61 1^1(0(5(x1))) -> 5^1(1(2(3(0(x1))))) 45.58/12.61 1^1(0(5(x1))) -> 1^1(2(3(0(x1)))) 45.58/12.61 1^1(0(5(x1))) -> 1^1(5(3(2(0(3(x1)))))) 45.58/12.61 1^1(0(5(x1))) -> 5^1(3(2(0(3(x1))))) 45.58/12.61 1^1(0(5(x1))) -> 0^1(3(x1)) 45.58/12.61 5^1(0(5(x1))) -> 5^1(3(5(2(3(0(x1)))))) 45.58/12.61 5^1(0(5(x1))) -> 5^1(2(3(0(x1)))) 45.58/12.61 5^1(0(5(x1))) -> 0^1(x1) 45.58/12.61 1^1(4(5(x1))) -> 1^1(5(2(4(2(x1))))) 45.58/12.61 1^1(4(5(x1))) -> 5^1(2(4(2(x1)))) 45.58/12.61 1^1(4(5(x1))) -> 4^1(2(x1)) 45.58/12.61 1^1(4(5(x1))) -> 1^1(2(5(4(x1)))) 45.58/12.61 1^1(4(5(x1))) -> 5^1(4(x1)) 45.58/12.61 1^1(4(5(x1))) -> 4^1(x1) 45.58/12.61 5^1(4(0(0(x1)))) -> 4^1(2(0(5(0(x1))))) 45.58/12.61 5^1(4(0(0(x1)))) -> 0^1(5(0(x1))) 45.58/12.61 5^1(4(0(0(x1)))) -> 5^1(0(x1)) 45.58/12.61 0^1(2(1(0(x1)))) -> 0^1(1(2(0(2(x1))))) 45.58/12.61 0^1(2(1(0(x1)))) -> 1^1(2(0(2(x1)))) 45.58/12.61 0^1(2(1(0(x1)))) -> 0^1(2(x1)) 45.58/12.61 1^1(0(2(0(x1)))) -> 1^1(2(0(0(x1)))) 45.58/12.61 1^1(0(2(0(x1)))) -> 0^1(0(x1)) 45.58/12.61 0^1(4(3(0(x1)))) -> 0^1(0(4(2(3(x1))))) 45.58/12.61 0^1(4(3(0(x1)))) -> 0^1(4(2(3(x1)))) 45.58/12.61 0^1(4(3(0(x1)))) -> 4^1(2(3(x1))) 45.58/12.61 4^1(0(4(0(x1)))) -> 4^1(4(2(3(0(0(x1)))))) 45.58/12.61 4^1(0(4(0(x1)))) -> 4^1(2(3(0(0(x1))))) 45.58/12.61 4^1(0(4(0(x1)))) -> 0^1(0(x1)) 45.58/12.61 4^1(1(0(1(x1)))) -> 0^1(4(1(1(3(2(x1)))))) 45.58/12.61 4^1(1(0(1(x1)))) -> 4^1(1(1(3(2(x1))))) 45.58/12.61 4^1(1(0(1(x1)))) -> 1^1(1(3(2(x1)))) 45.58/12.61 4^1(1(0(1(x1)))) -> 1^1(3(2(x1))) 45.58/12.61 1^1(3(0(1(x1)))) -> 0^1(1(1(3(2(4(x1)))))) 45.58/12.61 1^1(3(0(1(x1)))) -> 1^1(1(3(2(4(x1))))) 45.58/12.61 1^1(3(0(1(x1)))) -> 1^1(3(2(4(x1)))) 45.58/12.61 1^1(3(0(1(x1)))) -> 4^1(x1) 45.58/12.61 4^1(0(2(1(x1)))) -> 0^1(2(4(1(x1)))) 45.58/12.61 4^1(0(2(1(x1)))) -> 4^1(1(x1)) 45.58/12.61 4^1(0(3(1(x1)))) -> 1^1(2(3(0(4(x1))))) 45.58/12.61 4^1(0(3(1(x1)))) -> 0^1(4(x1)) 45.58/12.61 4^1(0(3(1(x1)))) -> 4^1(x1) 45.58/12.61 5^1(1(4(1(x1)))) -> 4^1(1(5(2(1(2(x1)))))) 45.58/12.61 5^1(1(4(1(x1)))) -> 1^1(5(2(1(2(x1))))) 45.58/12.61 5^1(1(4(1(x1)))) -> 5^1(2(1(2(x1)))) 45.58/12.61 5^1(1(4(1(x1)))) -> 1^1(2(x1)) 45.58/12.61 1^1(5(0(4(x1)))) -> 5^1(1(2(4(2(0(x1)))))) 45.58/12.61 1^1(5(0(4(x1)))) -> 1^1(2(4(2(0(x1))))) 45.58/12.61 1^1(5(0(4(x1)))) -> 4^1(2(0(x1))) 45.58/12.61 1^1(5(0(4(x1)))) -> 0^1(x1) 45.58/12.61 0^1(0(1(4(x1)))) -> 0^1(0(4(1(2(3(x1)))))) 45.58/12.61 0^1(0(1(4(x1)))) -> 0^1(4(1(2(3(x1))))) 45.58/12.61 0^1(0(1(4(x1)))) -> 4^1(1(2(3(x1)))) 45.58/12.61 0^1(0(1(4(x1)))) -> 1^1(2(3(x1))) 45.58/12.61 4^1(0(1(4(x1)))) -> 1^1(2(0(4(4(x1))))) 45.58/12.61 4^1(0(1(4(x1)))) -> 0^1(4(4(x1))) 45.58/12.61 4^1(0(1(4(x1)))) -> 4^1(4(x1)) 45.58/12.61 1^1(0(0(5(x1)))) -> 0^1(1(5(2(3(0(x1)))))) 45.58/12.61 1^1(0(0(5(x1)))) -> 1^1(5(2(3(0(x1))))) 45.58/12.61 1^1(0(0(5(x1)))) -> 5^1(2(3(0(x1)))) 45.58/12.61 1^1(0(0(5(x1)))) -> 0^1(x1) 45.58/12.61 1^1(3(0(5(x1)))) -> 1^1(2(3(5(0(x1))))) 45.58/12.61 1^1(3(0(5(x1)))) -> 5^1(0(x1)) 45.58/12.61 1^1(3(0(5(x1)))) -> 0^1(x1) 45.58/12.61 1^1(3(0(5(x1)))) -> 1^1(2(2(5(3(0(x1)))))) 45.58/12.61 1^1(3(0(5(x1)))) -> 5^1(3(0(x1))) 45.58/12.61 1^1(5(0(5(x1)))) -> 0^1(1(3(5(2(5(x1)))))) 45.58/12.61 1^1(5(0(5(x1)))) -> 1^1(3(5(2(5(x1))))) 45.58/12.61 1^1(5(0(5(x1)))) -> 5^1(2(5(x1))) 45.58/12.61 1^1(4(2(5(x1)))) -> 5^1(2(4(1(2(x1))))) 45.58/12.61 1^1(4(2(5(x1)))) -> 4^1(1(2(x1))) 45.58/12.61 1^1(4(2(5(x1)))) -> 1^1(2(x1)) 45.58/12.61 5^1(1(4(5(x1)))) -> 1^1(2(5(5(4(x1))))) 45.58/12.61 5^1(1(4(5(x1)))) -> 5^1(5(4(x1))) 45.58/12.61 5^1(1(4(5(x1)))) -> 5^1(4(x1)) 45.58/12.61 5^1(1(4(5(x1)))) -> 4^1(x1) 45.58/12.61 5^1(1(4(5(x1)))) -> 5^1(2(4(5(1(2(x1)))))) 45.58/12.61 5^1(1(4(5(x1)))) -> 4^1(5(1(2(x1)))) 45.58/12.61 5^1(1(4(5(x1)))) -> 5^1(1(2(x1))) 45.58/12.61 5^1(1(4(5(x1)))) -> 1^1(2(x1)) 45.58/12.61 1^1(3(4(5(x1)))) -> 1^1(2(4(5(x1)))) 45.58/12.61 1^1(3(4(5(x1)))) -> 1^1(2(4(2(5(3(x1)))))) 45.58/12.61 1^1(3(4(5(x1)))) -> 4^1(2(5(3(x1)))) 45.58/12.61 1^1(3(4(5(x1)))) -> 5^1(3(x1)) 45.58/12.61 1^1(3(4(5(x1)))) -> 1^1(2(3(4(2(5(x1)))))) 45.58/12.61 1^1(3(4(5(x1)))) -> 4^1(2(5(x1))) 45.58/12.61 4^1(0(5(5(x1)))) -> 4^1(5(2(0(5(x1))))) 45.58/12.61 4^1(0(5(5(x1)))) -> 5^1(2(0(5(x1)))) 45.58/12.61 4^1(0(5(5(x1)))) -> 0^1(5(x1)) 45.58/12.61 1^1(3(3(0(0(x1))))) -> 1^1(0(3(0(x1)))) 45.58/12.61 1^1(3(3(0(0(x1))))) -> 0^1(3(0(x1))) 45.58/12.61 1^1(3(4(2(0(x1))))) -> 4^1(1(3(2(0(x1))))) 45.58/12.61 1^1(3(4(2(0(x1))))) -> 1^1(3(2(0(x1)))) 45.58/12.61 1^1(1(4(5(0(x1))))) -> 1^1(1(5(2(0(4(x1)))))) 45.58/12.61 1^1(1(4(5(0(x1))))) -> 1^1(5(2(0(4(x1))))) 45.58/12.61 1^1(1(4(5(0(x1))))) -> 5^1(2(0(4(x1)))) 45.58/12.61 1^1(1(4(5(0(x1))))) -> 0^1(4(x1)) 45.58/12.61 1^1(1(4(5(0(x1))))) -> 4^1(x1) 45.58/12.61 1^1(3(4(2(1(x1))))) -> 4^1(1(3(3(2(1(x1)))))) 45.58/12.61 1^1(3(4(2(1(x1))))) -> 1^1(3(3(2(1(x1))))) 45.58/12.61 1^1(3(0(0(4(x1))))) -> 0^1(1(2(3(0(4(x1)))))) 45.58/12.61 1^1(3(0(0(4(x1))))) -> 1^1(2(3(0(4(x1))))) 45.58/12.61 1^1(3(0(0(4(x1))))) -> 1^1(0(0(2(3(4(x1)))))) 45.58/12.61 1^1(3(0(0(4(x1))))) -> 0^1(0(2(3(4(x1))))) 45.58/12.61 1^1(3(0(0(4(x1))))) -> 0^1(2(3(4(x1)))) 45.58/12.61 5^1(0(0(1(5(x1))))) -> 5^1(1(2(0(0(5(x1)))))) 45.58/12.61 5^1(0(0(1(5(x1))))) -> 1^1(2(0(0(5(x1))))) 45.58/12.61 5^1(0(0(1(5(x1))))) -> 0^1(0(5(x1))) 45.58/12.61 5^1(0(0(1(5(x1))))) -> 0^1(5(x1)) 45.58/12.61 1^1(0(2(4(5(x1))))) -> 1^1(2(0(4(5(2(x1)))))) 45.58/12.61 1^1(0(2(4(5(x1))))) -> 0^1(4(5(2(x1)))) 45.58/12.61 1^1(0(2(4(5(x1))))) -> 4^1(5(2(x1))) 45.58/12.61 1^1(0(2(4(5(x1))))) -> 5^1(2(x1)) 45.58/12.61 45.58/12.61 The TRS R consists of the following rules: 45.58/12.61 45.58/12.61 1(0(0(x1))) -> 3(1(2(0(0(x1))))) 45.58/12.61 1(0(0(x1))) -> 1(3(2(2(0(0(x1)))))) 45.58/12.61 1(4(0(x1))) -> 1(2(2(4(0(x1))))) 45.58/12.61 1(4(0(x1))) -> 2(4(1(3(2(0(x1)))))) 45.58/12.61 1(0(1(x1))) -> 1(1(2(0(x1)))) 45.58/12.61 1(0(1(x1))) -> 2(1(1(2(0(x1))))) 45.58/12.61 1(0(1(x1))) -> 1(2(3(0(1(x1))))) 45.58/12.61 0(4(1(x1))) -> 2(4(2(3(0(1(x1)))))) 45.58/12.61 1(4(1(x1))) -> 1(1(2(4(x1)))) 45.58/12.61 1(4(1(x1))) -> 1(2(1(1(2(4(x1)))))) 45.58/12.61 1(0(4(x1))) -> 1(2(0(4(x1)))) 45.58/12.61 1(0(5(x1))) -> 1(2(0(5(x1)))) 45.58/12.61 1(0(5(x1))) -> 3(5(1(2(0(x1))))) 45.58/12.61 1(0(5(x1))) -> 5(5(1(2(3(0(x1)))))) 45.58/12.61 1(0(5(x1))) -> 1(5(3(2(0(3(x1)))))) 45.58/12.61 5(0(5(x1))) -> 5(3(5(2(3(0(x1)))))) 45.58/12.61 1(4(5(x1))) -> 3(1(5(2(4(2(x1)))))) 45.58/12.61 1(4(5(x1))) -> 3(3(1(2(5(4(x1)))))) 45.58/12.61 5(4(0(0(x1)))) -> 2(4(2(0(5(0(x1)))))) 45.58/12.61 0(2(1(0(x1)))) -> 0(1(2(0(2(x1))))) 45.58/12.61 1(0(2(0(x1)))) -> 3(1(2(0(0(x1))))) 45.58/12.61 0(4(3(0(x1)))) -> 3(0(0(4(2(3(x1)))))) 45.58/12.61 4(0(4(0(x1)))) -> 4(4(2(3(0(0(x1)))))) 45.58/12.61 4(1(0(1(x1)))) -> 0(4(1(1(3(2(x1)))))) 45.58/12.61 1(3(0(1(x1)))) -> 0(1(1(3(2(4(x1)))))) 45.58/12.61 4(0(2(1(x1)))) -> 2(3(0(2(4(1(x1)))))) 45.58/12.61 4(0(3(1(x1)))) -> 3(1(2(3(0(4(x1)))))) 45.58/12.61 5(1(4(1(x1)))) -> 4(1(5(2(1(2(x1)))))) 45.58/12.61 1(5(0(4(x1)))) -> 5(1(2(4(2(0(x1)))))) 45.58/12.61 0(0(1(4(x1)))) -> 0(0(4(1(2(3(x1)))))) 45.58/12.61 4(0(1(4(x1)))) -> 1(2(0(4(4(x1))))) 45.58/12.61 1(0(0(5(x1)))) -> 0(1(5(2(3(0(x1)))))) 45.58/12.61 1(3(0(5(x1)))) -> 1(2(3(5(0(x1))))) 45.58/12.61 1(3(0(5(x1)))) -> 1(2(2(5(3(0(x1)))))) 45.58/12.61 1(5(0(5(x1)))) -> 0(1(3(5(2(5(x1)))))) 45.58/12.61 1(4(2(5(x1)))) -> 5(2(4(1(2(x1))))) 45.58/12.61 5(1(4(5(x1)))) -> 1(2(5(5(4(x1))))) 45.58/12.61 5(1(4(5(x1)))) -> 5(2(4(5(1(2(x1)))))) 45.58/12.61 1(3(4(5(x1)))) -> 3(1(2(4(5(x1))))) 45.58/12.61 1(3(4(5(x1)))) -> 1(2(4(2(5(3(x1)))))) 45.58/12.61 1(3(4(5(x1)))) -> 1(2(3(4(2(5(x1)))))) 45.58/12.61 4(0(5(5(x1)))) -> 2(4(5(2(0(5(x1)))))) 45.58/12.61 1(3(3(0(0(x1))))) -> 3(2(1(0(3(0(x1)))))) 45.58/12.61 1(3(4(2(0(x1))))) -> 2(4(1(3(2(0(x1)))))) 45.58/12.61 1(1(4(5(0(x1))))) -> 1(1(5(2(0(4(x1)))))) 45.58/12.61 1(3(4(2(1(x1))))) -> 4(1(3(3(2(1(x1)))))) 45.58/12.61 1(3(0(0(4(x1))))) -> 0(1(2(3(0(4(x1)))))) 45.58/12.61 1(3(0(0(4(x1))))) -> 1(0(0(2(3(4(x1)))))) 45.58/12.61 5(0(0(1(5(x1))))) -> 5(1(2(0(0(5(x1)))))) 45.58/12.61 1(0(2(4(5(x1))))) -> 1(2(0(4(5(2(x1)))))) 45.58/12.61 45.58/12.61 Q is empty. 45.58/12.61 We have to consider all minimal (P,Q,R)-chains. 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (5) DependencyGraphProof (EQUIVALENT) 45.58/12.61 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 123 less nodes. 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (6) 45.58/12.61 Complex Obligation (AND) 45.58/12.61 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (7) 45.58/12.61 Obligation: 45.58/12.61 Q DP problem: 45.58/12.61 The TRS P consists of the following rules: 45.58/12.61 45.58/12.61 0^1(2(1(0(x1)))) -> 0^1(2(x1)) 45.58/12.61 45.58/12.61 The TRS R consists of the following rules: 45.58/12.61 45.58/12.61 1(0(0(x1))) -> 3(1(2(0(0(x1))))) 45.58/12.61 1(0(0(x1))) -> 1(3(2(2(0(0(x1)))))) 45.58/12.61 1(4(0(x1))) -> 1(2(2(4(0(x1))))) 45.58/12.61 1(4(0(x1))) -> 2(4(1(3(2(0(x1)))))) 45.58/12.61 1(0(1(x1))) -> 1(1(2(0(x1)))) 45.58/12.61 1(0(1(x1))) -> 2(1(1(2(0(x1))))) 45.58/12.61 1(0(1(x1))) -> 1(2(3(0(1(x1))))) 45.58/12.61 0(4(1(x1))) -> 2(4(2(3(0(1(x1)))))) 45.58/12.61 1(4(1(x1))) -> 1(1(2(4(x1)))) 45.58/12.61 1(4(1(x1))) -> 1(2(1(1(2(4(x1)))))) 45.58/12.61 1(0(4(x1))) -> 1(2(0(4(x1)))) 45.58/12.61 1(0(5(x1))) -> 1(2(0(5(x1)))) 45.58/12.61 1(0(5(x1))) -> 3(5(1(2(0(x1))))) 45.58/12.61 1(0(5(x1))) -> 5(5(1(2(3(0(x1)))))) 45.58/12.61 1(0(5(x1))) -> 1(5(3(2(0(3(x1)))))) 45.58/12.61 5(0(5(x1))) -> 5(3(5(2(3(0(x1)))))) 45.58/12.61 1(4(5(x1))) -> 3(1(5(2(4(2(x1)))))) 45.58/12.61 1(4(5(x1))) -> 3(3(1(2(5(4(x1)))))) 45.58/12.61 5(4(0(0(x1)))) -> 2(4(2(0(5(0(x1)))))) 45.58/12.61 0(2(1(0(x1)))) -> 0(1(2(0(2(x1))))) 45.58/12.61 1(0(2(0(x1)))) -> 3(1(2(0(0(x1))))) 45.58/12.61 0(4(3(0(x1)))) -> 3(0(0(4(2(3(x1)))))) 45.58/12.61 4(0(4(0(x1)))) -> 4(4(2(3(0(0(x1)))))) 45.58/12.61 4(1(0(1(x1)))) -> 0(4(1(1(3(2(x1)))))) 45.58/12.61 1(3(0(1(x1)))) -> 0(1(1(3(2(4(x1)))))) 45.58/12.61 4(0(2(1(x1)))) -> 2(3(0(2(4(1(x1)))))) 45.58/12.61 4(0(3(1(x1)))) -> 3(1(2(3(0(4(x1)))))) 45.58/12.61 5(1(4(1(x1)))) -> 4(1(5(2(1(2(x1)))))) 45.58/12.61 1(5(0(4(x1)))) -> 5(1(2(4(2(0(x1)))))) 45.58/12.61 0(0(1(4(x1)))) -> 0(0(4(1(2(3(x1)))))) 45.58/12.61 4(0(1(4(x1)))) -> 1(2(0(4(4(x1))))) 45.58/12.61 1(0(0(5(x1)))) -> 0(1(5(2(3(0(x1)))))) 45.58/12.61 1(3(0(5(x1)))) -> 1(2(3(5(0(x1))))) 45.58/12.61 1(3(0(5(x1)))) -> 1(2(2(5(3(0(x1)))))) 45.58/12.61 1(5(0(5(x1)))) -> 0(1(3(5(2(5(x1)))))) 45.58/12.61 1(4(2(5(x1)))) -> 5(2(4(1(2(x1))))) 45.58/12.61 5(1(4(5(x1)))) -> 1(2(5(5(4(x1))))) 45.58/12.61 5(1(4(5(x1)))) -> 5(2(4(5(1(2(x1)))))) 45.58/12.61 1(3(4(5(x1)))) -> 3(1(2(4(5(x1))))) 45.58/12.61 1(3(4(5(x1)))) -> 1(2(4(2(5(3(x1)))))) 45.58/12.61 1(3(4(5(x1)))) -> 1(2(3(4(2(5(x1)))))) 45.58/12.61 4(0(5(5(x1)))) -> 2(4(5(2(0(5(x1)))))) 45.58/12.61 1(3(3(0(0(x1))))) -> 3(2(1(0(3(0(x1)))))) 45.58/12.61 1(3(4(2(0(x1))))) -> 2(4(1(3(2(0(x1)))))) 45.58/12.61 1(1(4(5(0(x1))))) -> 1(1(5(2(0(4(x1)))))) 45.58/12.61 1(3(4(2(1(x1))))) -> 4(1(3(3(2(1(x1)))))) 45.58/12.61 1(3(0(0(4(x1))))) -> 0(1(2(3(0(4(x1)))))) 45.58/12.61 1(3(0(0(4(x1))))) -> 1(0(0(2(3(4(x1)))))) 45.58/12.61 5(0(0(1(5(x1))))) -> 5(1(2(0(0(5(x1)))))) 45.58/12.61 1(0(2(4(5(x1))))) -> 1(2(0(4(5(2(x1)))))) 45.58/12.61 45.58/12.61 Q is empty. 45.58/12.61 We have to consider all minimal (P,Q,R)-chains. 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (8) UsableRulesProof (EQUIVALENT) 45.58/12.61 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. 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (9) 45.58/12.61 Obligation: 45.58/12.61 Q DP problem: 45.58/12.61 The TRS P consists of the following rules: 45.58/12.61 45.58/12.61 0^1(2(1(0(x1)))) -> 0^1(2(x1)) 45.58/12.61 45.58/12.61 R is empty. 45.58/12.61 Q is empty. 45.58/12.61 We have to consider all minimal (P,Q,R)-chains. 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (10) MRRProof (EQUIVALENT) 45.58/12.61 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. 45.58/12.61 45.58/12.61 Strictly oriented dependency pairs: 45.58/12.61 45.58/12.61 0^1(2(1(0(x1)))) -> 0^1(2(x1)) 45.58/12.61 45.58/12.61 45.58/12.61 Used ordering: Polynomial interpretation [POLO]: 45.58/12.61 45.58/12.61 POL(0(x_1)) = 2*x_1 45.58/12.61 POL(0^1(x_1)) = x_1 45.58/12.61 POL(1(x_1)) = 1 + 3*x_1 45.58/12.61 POL(2(x_1)) = 2*x_1 45.58/12.61 45.58/12.61 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (11) 45.58/12.61 Obligation: 45.58/12.61 Q DP problem: 45.58/12.61 P is empty. 45.58/12.61 R is empty. 45.58/12.61 Q is empty. 45.58/12.61 We have to consider all minimal (P,Q,R)-chains. 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (12) PisEmptyProof (EQUIVALENT) 45.58/12.61 The TRS P is empty. Hence, there is no (P,Q,R) chain. 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (13) 45.58/12.61 YES 45.58/12.61 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (14) 45.58/12.61 Obligation: 45.58/12.61 Q DP problem: 45.58/12.61 The TRS P consists of the following rules: 45.58/12.61 45.58/12.61 0^1(0(1(4(x1)))) -> 0^1(0(4(1(2(3(x1)))))) 45.58/12.61 0^1(4(1(x1))) -> 0^1(1(x1)) 45.58/12.61 0^1(0(1(4(x1)))) -> 0^1(4(1(2(3(x1))))) 45.58/12.61 45.58/12.61 The TRS R consists of the following rules: 45.58/12.61 45.58/12.61 1(0(0(x1))) -> 3(1(2(0(0(x1))))) 45.58/12.61 1(0(0(x1))) -> 1(3(2(2(0(0(x1)))))) 45.58/12.61 1(4(0(x1))) -> 1(2(2(4(0(x1))))) 45.58/12.61 1(4(0(x1))) -> 2(4(1(3(2(0(x1)))))) 45.58/12.61 1(0(1(x1))) -> 1(1(2(0(x1)))) 45.58/12.61 1(0(1(x1))) -> 2(1(1(2(0(x1))))) 45.58/12.61 1(0(1(x1))) -> 1(2(3(0(1(x1))))) 45.58/12.61 0(4(1(x1))) -> 2(4(2(3(0(1(x1)))))) 45.58/12.61 1(4(1(x1))) -> 1(1(2(4(x1)))) 45.58/12.61 1(4(1(x1))) -> 1(2(1(1(2(4(x1)))))) 45.58/12.61 1(0(4(x1))) -> 1(2(0(4(x1)))) 45.58/12.61 1(0(5(x1))) -> 1(2(0(5(x1)))) 45.58/12.61 1(0(5(x1))) -> 3(5(1(2(0(x1))))) 45.58/12.61 1(0(5(x1))) -> 5(5(1(2(3(0(x1)))))) 45.58/12.61 1(0(5(x1))) -> 1(5(3(2(0(3(x1)))))) 45.58/12.61 5(0(5(x1))) -> 5(3(5(2(3(0(x1)))))) 45.58/12.61 1(4(5(x1))) -> 3(1(5(2(4(2(x1)))))) 45.58/12.61 1(4(5(x1))) -> 3(3(1(2(5(4(x1)))))) 45.58/12.61 5(4(0(0(x1)))) -> 2(4(2(0(5(0(x1)))))) 45.58/12.61 0(2(1(0(x1)))) -> 0(1(2(0(2(x1))))) 45.58/12.61 1(0(2(0(x1)))) -> 3(1(2(0(0(x1))))) 45.58/12.61 0(4(3(0(x1)))) -> 3(0(0(4(2(3(x1)))))) 45.58/12.61 4(0(4(0(x1)))) -> 4(4(2(3(0(0(x1)))))) 45.58/12.61 4(1(0(1(x1)))) -> 0(4(1(1(3(2(x1)))))) 45.58/12.61 1(3(0(1(x1)))) -> 0(1(1(3(2(4(x1)))))) 45.58/12.61 4(0(2(1(x1)))) -> 2(3(0(2(4(1(x1)))))) 45.58/12.61 4(0(3(1(x1)))) -> 3(1(2(3(0(4(x1)))))) 45.58/12.61 5(1(4(1(x1)))) -> 4(1(5(2(1(2(x1)))))) 45.58/12.61 1(5(0(4(x1)))) -> 5(1(2(4(2(0(x1)))))) 45.58/12.61 0(0(1(4(x1)))) -> 0(0(4(1(2(3(x1)))))) 45.58/12.61 4(0(1(4(x1)))) -> 1(2(0(4(4(x1))))) 45.58/12.61 1(0(0(5(x1)))) -> 0(1(5(2(3(0(x1)))))) 45.58/12.61 1(3(0(5(x1)))) -> 1(2(3(5(0(x1))))) 45.58/12.61 1(3(0(5(x1)))) -> 1(2(2(5(3(0(x1)))))) 45.58/12.61 1(5(0(5(x1)))) -> 0(1(3(5(2(5(x1)))))) 45.58/12.61 1(4(2(5(x1)))) -> 5(2(4(1(2(x1))))) 45.58/12.61 5(1(4(5(x1)))) -> 1(2(5(5(4(x1))))) 45.58/12.61 5(1(4(5(x1)))) -> 5(2(4(5(1(2(x1)))))) 45.58/12.61 1(3(4(5(x1)))) -> 3(1(2(4(5(x1))))) 45.58/12.61 1(3(4(5(x1)))) -> 1(2(4(2(5(3(x1)))))) 45.58/12.61 1(3(4(5(x1)))) -> 1(2(3(4(2(5(x1)))))) 45.58/12.61 4(0(5(5(x1)))) -> 2(4(5(2(0(5(x1)))))) 45.58/12.61 1(3(3(0(0(x1))))) -> 3(2(1(0(3(0(x1)))))) 45.58/12.61 1(3(4(2(0(x1))))) -> 2(4(1(3(2(0(x1)))))) 45.58/12.61 1(1(4(5(0(x1))))) -> 1(1(5(2(0(4(x1)))))) 45.58/12.61 1(3(4(2(1(x1))))) -> 4(1(3(3(2(1(x1)))))) 45.58/12.61 1(3(0(0(4(x1))))) -> 0(1(2(3(0(4(x1)))))) 45.58/12.61 1(3(0(0(4(x1))))) -> 1(0(0(2(3(4(x1)))))) 45.58/12.61 5(0(0(1(5(x1))))) -> 5(1(2(0(0(5(x1)))))) 45.58/12.61 1(0(2(4(5(x1))))) -> 1(2(0(4(5(2(x1)))))) 45.58/12.61 45.58/12.61 Q is empty. 45.58/12.61 We have to consider all minimal (P,Q,R)-chains. 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (15) QDPOrderProof (EQUIVALENT) 45.58/12.61 We use the reduction pair processor [LPAR04,JAR06]. 45.58/12.61 45.58/12.61 45.58/12.61 The following pairs can be oriented strictly and are deleted. 45.58/12.61 45.58/12.61 0^1(4(1(x1))) -> 0^1(1(x1)) 45.58/12.61 0^1(0(1(4(x1)))) -> 0^1(4(1(2(3(x1))))) 45.58/12.61 The remaining pairs can at least be oriented weakly. 45.58/12.61 Used ordering: Polynomial interpretation [POLO]: 45.58/12.61 45.58/12.61 POL(0(x_1)) = 1 + x_1 45.58/12.61 POL(0^1(x_1)) = x_1 45.58/12.61 POL(1(x_1)) = x_1 45.58/12.61 POL(2(x_1)) = 0 45.58/12.61 POL(3(x_1)) = x_1 45.58/12.61 POL(4(x_1)) = 1 + x_1 45.58/12.61 POL(5(x_1)) = x_1 45.58/12.61 45.58/12.61 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 45.58/12.61 45.58/12.61 0(4(1(x1))) -> 2(4(2(3(0(1(x1)))))) 45.58/12.61 1(0(0(x1))) -> 3(1(2(0(0(x1))))) 45.58/12.61 1(0(0(x1))) -> 1(3(2(2(0(0(x1)))))) 45.58/12.61 1(4(0(x1))) -> 1(2(2(4(0(x1))))) 45.58/12.61 1(4(0(x1))) -> 2(4(1(3(2(0(x1)))))) 45.58/12.61 1(0(1(x1))) -> 1(1(2(0(x1)))) 45.58/12.61 1(0(1(x1))) -> 2(1(1(2(0(x1))))) 45.58/12.61 1(0(1(x1))) -> 1(2(3(0(1(x1))))) 45.58/12.61 1(4(1(x1))) -> 1(1(2(4(x1)))) 45.58/12.61 1(4(1(x1))) -> 1(2(1(1(2(4(x1)))))) 45.58/12.61 1(0(4(x1))) -> 1(2(0(4(x1)))) 45.58/12.61 1(0(5(x1))) -> 1(2(0(5(x1)))) 45.58/12.61 1(0(5(x1))) -> 3(5(1(2(0(x1))))) 45.58/12.61 1(0(5(x1))) -> 5(5(1(2(3(0(x1)))))) 45.58/12.61 1(0(5(x1))) -> 1(5(3(2(0(3(x1)))))) 45.58/12.61 1(4(5(x1))) -> 3(1(5(2(4(2(x1)))))) 45.58/12.61 1(4(5(x1))) -> 3(3(1(2(5(4(x1)))))) 45.58/12.61 1(0(2(0(x1)))) -> 3(1(2(0(0(x1))))) 45.58/12.61 1(3(0(1(x1)))) -> 0(1(1(3(2(4(x1)))))) 45.58/12.61 1(5(0(4(x1)))) -> 5(1(2(4(2(0(x1)))))) 45.58/12.61 1(0(0(5(x1)))) -> 0(1(5(2(3(0(x1)))))) 45.58/12.61 1(3(0(5(x1)))) -> 1(2(3(5(0(x1))))) 45.58/12.61 1(3(0(5(x1)))) -> 1(2(2(5(3(0(x1)))))) 45.58/12.61 1(5(0(5(x1)))) -> 0(1(3(5(2(5(x1)))))) 45.58/12.61 1(4(2(5(x1)))) -> 5(2(4(1(2(x1))))) 45.58/12.61 1(3(4(5(x1)))) -> 3(1(2(4(5(x1))))) 45.58/12.61 1(3(4(5(x1)))) -> 1(2(4(2(5(3(x1)))))) 45.58/12.61 1(3(4(5(x1)))) -> 1(2(3(4(2(5(x1)))))) 45.58/12.61 1(3(3(0(0(x1))))) -> 3(2(1(0(3(0(x1)))))) 45.58/12.61 1(3(4(2(0(x1))))) -> 2(4(1(3(2(0(x1)))))) 45.58/12.61 1(1(4(5(0(x1))))) -> 1(1(5(2(0(4(x1)))))) 45.58/12.61 1(3(4(2(1(x1))))) -> 4(1(3(3(2(1(x1)))))) 45.58/12.61 1(3(0(0(4(x1))))) -> 0(1(2(3(0(4(x1)))))) 45.58/12.61 1(3(0(0(4(x1))))) -> 1(0(0(2(3(4(x1)))))) 45.58/12.61 1(0(2(4(5(x1))))) -> 1(2(0(4(5(2(x1)))))) 45.58/12.61 45.58/12.61 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (16) 45.58/12.61 Obligation: 45.58/12.61 Q DP problem: 45.58/12.61 The TRS P consists of the following rules: 45.58/12.61 45.58/12.61 0^1(0(1(4(x1)))) -> 0^1(0(4(1(2(3(x1)))))) 45.58/12.61 45.58/12.61 The TRS R consists of the following rules: 45.58/12.61 45.58/12.61 1(0(0(x1))) -> 3(1(2(0(0(x1))))) 45.58/12.61 1(0(0(x1))) -> 1(3(2(2(0(0(x1)))))) 45.58/12.61 1(4(0(x1))) -> 1(2(2(4(0(x1))))) 45.58/12.61 1(4(0(x1))) -> 2(4(1(3(2(0(x1)))))) 45.58/12.61 1(0(1(x1))) -> 1(1(2(0(x1)))) 45.58/12.61 1(0(1(x1))) -> 2(1(1(2(0(x1))))) 45.58/12.61 1(0(1(x1))) -> 1(2(3(0(1(x1))))) 45.58/12.61 0(4(1(x1))) -> 2(4(2(3(0(1(x1)))))) 45.58/12.61 1(4(1(x1))) -> 1(1(2(4(x1)))) 45.58/12.61 1(4(1(x1))) -> 1(2(1(1(2(4(x1)))))) 45.58/12.61 1(0(4(x1))) -> 1(2(0(4(x1)))) 45.58/12.61 1(0(5(x1))) -> 1(2(0(5(x1)))) 45.58/12.61 1(0(5(x1))) -> 3(5(1(2(0(x1))))) 45.58/12.61 1(0(5(x1))) -> 5(5(1(2(3(0(x1)))))) 45.58/12.61 1(0(5(x1))) -> 1(5(3(2(0(3(x1)))))) 45.58/12.61 5(0(5(x1))) -> 5(3(5(2(3(0(x1)))))) 45.58/12.61 1(4(5(x1))) -> 3(1(5(2(4(2(x1)))))) 45.58/12.61 1(4(5(x1))) -> 3(3(1(2(5(4(x1)))))) 45.58/12.61 5(4(0(0(x1)))) -> 2(4(2(0(5(0(x1)))))) 45.58/12.61 0(2(1(0(x1)))) -> 0(1(2(0(2(x1))))) 45.58/12.61 1(0(2(0(x1)))) -> 3(1(2(0(0(x1))))) 45.58/12.61 0(4(3(0(x1)))) -> 3(0(0(4(2(3(x1)))))) 45.58/12.61 4(0(4(0(x1)))) -> 4(4(2(3(0(0(x1)))))) 45.58/12.61 4(1(0(1(x1)))) -> 0(4(1(1(3(2(x1)))))) 45.58/12.61 1(3(0(1(x1)))) -> 0(1(1(3(2(4(x1)))))) 45.58/12.61 4(0(2(1(x1)))) -> 2(3(0(2(4(1(x1)))))) 45.58/12.61 4(0(3(1(x1)))) -> 3(1(2(3(0(4(x1)))))) 45.58/12.61 5(1(4(1(x1)))) -> 4(1(5(2(1(2(x1)))))) 45.58/12.61 1(5(0(4(x1)))) -> 5(1(2(4(2(0(x1)))))) 45.58/12.61 0(0(1(4(x1)))) -> 0(0(4(1(2(3(x1)))))) 45.58/12.61 4(0(1(4(x1)))) -> 1(2(0(4(4(x1))))) 45.58/12.61 1(0(0(5(x1)))) -> 0(1(5(2(3(0(x1)))))) 45.58/12.61 1(3(0(5(x1)))) -> 1(2(3(5(0(x1))))) 45.58/12.61 1(3(0(5(x1)))) -> 1(2(2(5(3(0(x1)))))) 45.58/12.61 1(5(0(5(x1)))) -> 0(1(3(5(2(5(x1)))))) 45.58/12.61 1(4(2(5(x1)))) -> 5(2(4(1(2(x1))))) 45.58/12.61 5(1(4(5(x1)))) -> 1(2(5(5(4(x1))))) 45.58/12.61 5(1(4(5(x1)))) -> 5(2(4(5(1(2(x1)))))) 45.58/12.61 1(3(4(5(x1)))) -> 3(1(2(4(5(x1))))) 45.58/12.61 1(3(4(5(x1)))) -> 1(2(4(2(5(3(x1)))))) 45.58/12.61 1(3(4(5(x1)))) -> 1(2(3(4(2(5(x1)))))) 45.58/12.61 4(0(5(5(x1)))) -> 2(4(5(2(0(5(x1)))))) 45.58/12.61 1(3(3(0(0(x1))))) -> 3(2(1(0(3(0(x1)))))) 45.58/12.61 1(3(4(2(0(x1))))) -> 2(4(1(3(2(0(x1)))))) 45.58/12.61 1(1(4(5(0(x1))))) -> 1(1(5(2(0(4(x1)))))) 45.58/12.61 1(3(4(2(1(x1))))) -> 4(1(3(3(2(1(x1)))))) 45.58/12.61 1(3(0(0(4(x1))))) -> 0(1(2(3(0(4(x1)))))) 45.58/12.61 1(3(0(0(4(x1))))) -> 1(0(0(2(3(4(x1)))))) 45.58/12.61 5(0(0(1(5(x1))))) -> 5(1(2(0(0(5(x1)))))) 45.58/12.61 1(0(2(4(5(x1))))) -> 1(2(0(4(5(2(x1)))))) 45.58/12.61 45.58/12.61 Q is empty. 45.58/12.61 We have to consider all minimal (P,Q,R)-chains. 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (17) QDPOrderProof (EQUIVALENT) 45.58/12.61 We use the reduction pair processor [LPAR04,JAR06]. 45.58/12.61 45.58/12.61 45.58/12.61 The following pairs can be oriented strictly and are deleted. 45.58/12.61 45.58/12.61 0^1(0(1(4(x1)))) -> 0^1(0(4(1(2(3(x1)))))) 45.58/12.61 The remaining pairs can at least be oriented weakly. 45.58/12.61 Used ordering: Polynomial interpretation [POLO]: 45.58/12.61 45.58/12.61 POL(0(x_1)) = 1 + x_1 45.58/12.61 POL(0^1(x_1)) = x_1 45.58/12.61 POL(1(x_1)) = 1 45.58/12.61 POL(2(x_1)) = 1 45.58/12.61 POL(3(x_1)) = 0 45.58/12.61 POL(4(x_1)) = 0 45.58/12.61 POL(5(x_1)) = 0 45.58/12.61 45.58/12.61 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 45.58/12.61 45.58/12.61 0(4(1(x1))) -> 2(4(2(3(0(1(x1)))))) 45.58/12.61 45.58/12.61 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (18) 45.58/12.61 Obligation: 45.58/12.61 Q DP problem: 45.58/12.61 P is empty. 45.58/12.61 The TRS R consists of the following rules: 45.58/12.61 45.58/12.61 1(0(0(x1))) -> 3(1(2(0(0(x1))))) 45.58/12.61 1(0(0(x1))) -> 1(3(2(2(0(0(x1)))))) 45.58/12.61 1(4(0(x1))) -> 1(2(2(4(0(x1))))) 45.58/12.61 1(4(0(x1))) -> 2(4(1(3(2(0(x1)))))) 45.58/12.61 1(0(1(x1))) -> 1(1(2(0(x1)))) 45.58/12.61 1(0(1(x1))) -> 2(1(1(2(0(x1))))) 45.58/12.61 1(0(1(x1))) -> 1(2(3(0(1(x1))))) 45.58/12.61 0(4(1(x1))) -> 2(4(2(3(0(1(x1)))))) 45.58/12.61 1(4(1(x1))) -> 1(1(2(4(x1)))) 45.58/12.61 1(4(1(x1))) -> 1(2(1(1(2(4(x1)))))) 45.58/12.61 1(0(4(x1))) -> 1(2(0(4(x1)))) 45.58/12.61 1(0(5(x1))) -> 1(2(0(5(x1)))) 45.58/12.61 1(0(5(x1))) -> 3(5(1(2(0(x1))))) 45.58/12.61 1(0(5(x1))) -> 5(5(1(2(3(0(x1)))))) 45.58/12.61 1(0(5(x1))) -> 1(5(3(2(0(3(x1)))))) 45.58/12.61 5(0(5(x1))) -> 5(3(5(2(3(0(x1)))))) 45.58/12.61 1(4(5(x1))) -> 3(1(5(2(4(2(x1)))))) 45.58/12.61 1(4(5(x1))) -> 3(3(1(2(5(4(x1)))))) 45.58/12.61 5(4(0(0(x1)))) -> 2(4(2(0(5(0(x1)))))) 45.58/12.61 0(2(1(0(x1)))) -> 0(1(2(0(2(x1))))) 45.58/12.61 1(0(2(0(x1)))) -> 3(1(2(0(0(x1))))) 45.58/12.61 0(4(3(0(x1)))) -> 3(0(0(4(2(3(x1)))))) 45.58/12.61 4(0(4(0(x1)))) -> 4(4(2(3(0(0(x1)))))) 45.58/12.61 4(1(0(1(x1)))) -> 0(4(1(1(3(2(x1)))))) 45.58/12.61 1(3(0(1(x1)))) -> 0(1(1(3(2(4(x1)))))) 45.58/12.61 4(0(2(1(x1)))) -> 2(3(0(2(4(1(x1)))))) 45.58/12.61 4(0(3(1(x1)))) -> 3(1(2(3(0(4(x1)))))) 45.58/12.61 5(1(4(1(x1)))) -> 4(1(5(2(1(2(x1)))))) 45.58/12.61 1(5(0(4(x1)))) -> 5(1(2(4(2(0(x1)))))) 45.58/12.61 0(0(1(4(x1)))) -> 0(0(4(1(2(3(x1)))))) 45.58/12.61 4(0(1(4(x1)))) -> 1(2(0(4(4(x1))))) 45.58/12.61 1(0(0(5(x1)))) -> 0(1(5(2(3(0(x1)))))) 45.58/12.61 1(3(0(5(x1)))) -> 1(2(3(5(0(x1))))) 45.58/12.61 1(3(0(5(x1)))) -> 1(2(2(5(3(0(x1)))))) 45.58/12.61 1(5(0(5(x1)))) -> 0(1(3(5(2(5(x1)))))) 45.58/12.61 1(4(2(5(x1)))) -> 5(2(4(1(2(x1))))) 45.58/12.61 5(1(4(5(x1)))) -> 1(2(5(5(4(x1))))) 45.58/12.61 5(1(4(5(x1)))) -> 5(2(4(5(1(2(x1)))))) 45.58/12.61 1(3(4(5(x1)))) -> 3(1(2(4(5(x1))))) 45.58/12.61 1(3(4(5(x1)))) -> 1(2(4(2(5(3(x1)))))) 45.58/12.61 1(3(4(5(x1)))) -> 1(2(3(4(2(5(x1)))))) 45.58/12.61 4(0(5(5(x1)))) -> 2(4(5(2(0(5(x1)))))) 45.58/12.61 1(3(3(0(0(x1))))) -> 3(2(1(0(3(0(x1)))))) 45.58/12.61 1(3(4(2(0(x1))))) -> 2(4(1(3(2(0(x1)))))) 45.58/12.61 1(1(4(5(0(x1))))) -> 1(1(5(2(0(4(x1)))))) 45.58/12.61 1(3(4(2(1(x1))))) -> 4(1(3(3(2(1(x1)))))) 45.58/12.61 1(3(0(0(4(x1))))) -> 0(1(2(3(0(4(x1)))))) 45.58/12.61 1(3(0(0(4(x1))))) -> 1(0(0(2(3(4(x1)))))) 45.58/12.61 5(0(0(1(5(x1))))) -> 5(1(2(0(0(5(x1)))))) 45.58/12.61 1(0(2(4(5(x1))))) -> 1(2(0(4(5(2(x1)))))) 45.58/12.61 45.58/12.61 Q is empty. 45.58/12.61 We have to consider all minimal (P,Q,R)-chains. 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (19) PisEmptyProof (EQUIVALENT) 45.58/12.61 The TRS P is empty. Hence, there is no (P,Q,R) chain. 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (20) 45.58/12.61 YES 45.58/12.61 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (21) 45.58/12.61 Obligation: 45.58/12.61 Q DP problem: 45.58/12.61 The TRS P consists of the following rules: 45.58/12.61 45.58/12.61 4^1(0(3(1(x1)))) -> 4^1(x1) 45.58/12.61 4^1(0(2(1(x1)))) -> 4^1(1(x1)) 45.58/12.61 4^1(0(1(4(x1)))) -> 4^1(4(x1)) 45.58/12.61 45.58/12.61 The TRS R consists of the following rules: 45.58/12.61 45.58/12.61 1(0(0(x1))) -> 3(1(2(0(0(x1))))) 45.58/12.61 1(0(0(x1))) -> 1(3(2(2(0(0(x1)))))) 45.58/12.61 1(4(0(x1))) -> 1(2(2(4(0(x1))))) 45.58/12.61 1(4(0(x1))) -> 2(4(1(3(2(0(x1)))))) 45.58/12.61 1(0(1(x1))) -> 1(1(2(0(x1)))) 45.58/12.61 1(0(1(x1))) -> 2(1(1(2(0(x1))))) 45.58/12.61 1(0(1(x1))) -> 1(2(3(0(1(x1))))) 45.58/12.61 0(4(1(x1))) -> 2(4(2(3(0(1(x1)))))) 45.58/12.61 1(4(1(x1))) -> 1(1(2(4(x1)))) 45.58/12.61 1(4(1(x1))) -> 1(2(1(1(2(4(x1)))))) 45.58/12.61 1(0(4(x1))) -> 1(2(0(4(x1)))) 45.58/12.61 1(0(5(x1))) -> 1(2(0(5(x1)))) 45.58/12.61 1(0(5(x1))) -> 3(5(1(2(0(x1))))) 45.58/12.61 1(0(5(x1))) -> 5(5(1(2(3(0(x1)))))) 45.58/12.61 1(0(5(x1))) -> 1(5(3(2(0(3(x1)))))) 45.58/12.61 5(0(5(x1))) -> 5(3(5(2(3(0(x1)))))) 45.58/12.61 1(4(5(x1))) -> 3(1(5(2(4(2(x1)))))) 45.58/12.61 1(4(5(x1))) -> 3(3(1(2(5(4(x1)))))) 45.58/12.61 5(4(0(0(x1)))) -> 2(4(2(0(5(0(x1)))))) 45.58/12.61 0(2(1(0(x1)))) -> 0(1(2(0(2(x1))))) 45.58/12.61 1(0(2(0(x1)))) -> 3(1(2(0(0(x1))))) 45.58/12.61 0(4(3(0(x1)))) -> 3(0(0(4(2(3(x1)))))) 45.58/12.61 4(0(4(0(x1)))) -> 4(4(2(3(0(0(x1)))))) 45.58/12.61 4(1(0(1(x1)))) -> 0(4(1(1(3(2(x1)))))) 45.58/12.61 1(3(0(1(x1)))) -> 0(1(1(3(2(4(x1)))))) 45.58/12.61 4(0(2(1(x1)))) -> 2(3(0(2(4(1(x1)))))) 45.58/12.61 4(0(3(1(x1)))) -> 3(1(2(3(0(4(x1)))))) 45.58/12.61 5(1(4(1(x1)))) -> 4(1(5(2(1(2(x1)))))) 45.58/12.61 1(5(0(4(x1)))) -> 5(1(2(4(2(0(x1)))))) 45.58/12.61 0(0(1(4(x1)))) -> 0(0(4(1(2(3(x1)))))) 45.58/12.61 4(0(1(4(x1)))) -> 1(2(0(4(4(x1))))) 45.58/12.61 1(0(0(5(x1)))) -> 0(1(5(2(3(0(x1)))))) 45.58/12.61 1(3(0(5(x1)))) -> 1(2(3(5(0(x1))))) 45.58/12.61 1(3(0(5(x1)))) -> 1(2(2(5(3(0(x1)))))) 45.58/12.61 1(5(0(5(x1)))) -> 0(1(3(5(2(5(x1)))))) 45.58/12.61 1(4(2(5(x1)))) -> 5(2(4(1(2(x1))))) 45.58/12.61 5(1(4(5(x1)))) -> 1(2(5(5(4(x1))))) 45.58/12.61 5(1(4(5(x1)))) -> 5(2(4(5(1(2(x1)))))) 45.58/12.61 1(3(4(5(x1)))) -> 3(1(2(4(5(x1))))) 45.58/12.61 1(3(4(5(x1)))) -> 1(2(4(2(5(3(x1)))))) 45.58/12.61 1(3(4(5(x1)))) -> 1(2(3(4(2(5(x1)))))) 45.58/12.61 4(0(5(5(x1)))) -> 2(4(5(2(0(5(x1)))))) 45.58/12.61 1(3(3(0(0(x1))))) -> 3(2(1(0(3(0(x1)))))) 45.58/12.61 1(3(4(2(0(x1))))) -> 2(4(1(3(2(0(x1)))))) 45.58/12.61 1(1(4(5(0(x1))))) -> 1(1(5(2(0(4(x1)))))) 45.58/12.61 1(3(4(2(1(x1))))) -> 4(1(3(3(2(1(x1)))))) 45.58/12.61 1(3(0(0(4(x1))))) -> 0(1(2(3(0(4(x1)))))) 45.58/12.61 1(3(0(0(4(x1))))) -> 1(0(0(2(3(4(x1)))))) 45.58/12.61 5(0(0(1(5(x1))))) -> 5(1(2(0(0(5(x1)))))) 45.58/12.61 1(0(2(4(5(x1))))) -> 1(2(0(4(5(2(x1)))))) 45.58/12.61 45.58/12.61 Q is empty. 45.58/12.61 We have to consider all minimal (P,Q,R)-chains. 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (22) QDPSizeChangeProof (EQUIVALENT) 45.58/12.61 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. 45.58/12.61 45.58/12.61 From the DPs we obtained the following set of size-change graphs: 45.58/12.61 *4^1(0(3(1(x1)))) -> 4^1(x1) 45.58/12.61 The graph contains the following edges 1 > 1 45.58/12.61 45.58/12.61 45.58/12.61 *4^1(0(2(1(x1)))) -> 4^1(1(x1)) 45.58/12.61 The graph contains the following edges 1 > 1 45.58/12.61 45.58/12.61 45.58/12.61 *4^1(0(1(4(x1)))) -> 4^1(4(x1)) 45.58/12.61 The graph contains the following edges 1 > 1 45.58/12.61 45.58/12.61 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (23) 45.58/12.61 YES 45.58/12.61 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (24) 45.58/12.61 Obligation: 45.58/12.61 Q DP problem: 45.58/12.61 The TRS P consists of the following rules: 45.58/12.61 45.58/12.61 5^1(1(4(5(x1)))) -> 5^1(5(4(x1))) 45.58/12.61 5^1(4(0(0(x1)))) -> 5^1(0(x1)) 45.58/12.61 5^1(1(4(5(x1)))) -> 5^1(4(x1)) 45.58/12.61 45.58/12.61 The TRS R consists of the following rules: 45.58/12.61 45.58/12.61 1(0(0(x1))) -> 3(1(2(0(0(x1))))) 45.58/12.61 1(0(0(x1))) -> 1(3(2(2(0(0(x1)))))) 45.58/12.61 1(4(0(x1))) -> 1(2(2(4(0(x1))))) 45.58/12.61 1(4(0(x1))) -> 2(4(1(3(2(0(x1)))))) 45.58/12.61 1(0(1(x1))) -> 1(1(2(0(x1)))) 45.58/12.61 1(0(1(x1))) -> 2(1(1(2(0(x1))))) 45.58/12.61 1(0(1(x1))) -> 1(2(3(0(1(x1))))) 45.58/12.61 0(4(1(x1))) -> 2(4(2(3(0(1(x1)))))) 45.58/12.61 1(4(1(x1))) -> 1(1(2(4(x1)))) 45.58/12.61 1(4(1(x1))) -> 1(2(1(1(2(4(x1)))))) 45.58/12.61 1(0(4(x1))) -> 1(2(0(4(x1)))) 45.58/12.61 1(0(5(x1))) -> 1(2(0(5(x1)))) 45.58/12.61 1(0(5(x1))) -> 3(5(1(2(0(x1))))) 45.58/12.61 1(0(5(x1))) -> 5(5(1(2(3(0(x1)))))) 45.58/12.61 1(0(5(x1))) -> 1(5(3(2(0(3(x1)))))) 45.58/12.61 5(0(5(x1))) -> 5(3(5(2(3(0(x1)))))) 45.58/12.61 1(4(5(x1))) -> 3(1(5(2(4(2(x1)))))) 45.58/12.61 1(4(5(x1))) -> 3(3(1(2(5(4(x1)))))) 45.58/12.61 5(4(0(0(x1)))) -> 2(4(2(0(5(0(x1)))))) 45.58/12.61 0(2(1(0(x1)))) -> 0(1(2(0(2(x1))))) 45.58/12.61 1(0(2(0(x1)))) -> 3(1(2(0(0(x1))))) 45.58/12.61 0(4(3(0(x1)))) -> 3(0(0(4(2(3(x1)))))) 45.58/12.61 4(0(4(0(x1)))) -> 4(4(2(3(0(0(x1)))))) 45.58/12.61 4(1(0(1(x1)))) -> 0(4(1(1(3(2(x1)))))) 45.58/12.61 1(3(0(1(x1)))) -> 0(1(1(3(2(4(x1)))))) 45.58/12.61 4(0(2(1(x1)))) -> 2(3(0(2(4(1(x1)))))) 45.58/12.61 4(0(3(1(x1)))) -> 3(1(2(3(0(4(x1)))))) 45.58/12.61 5(1(4(1(x1)))) -> 4(1(5(2(1(2(x1)))))) 45.58/12.61 1(5(0(4(x1)))) -> 5(1(2(4(2(0(x1)))))) 45.58/12.61 0(0(1(4(x1)))) -> 0(0(4(1(2(3(x1)))))) 45.58/12.61 4(0(1(4(x1)))) -> 1(2(0(4(4(x1))))) 45.58/12.61 1(0(0(5(x1)))) -> 0(1(5(2(3(0(x1)))))) 45.58/12.61 1(3(0(5(x1)))) -> 1(2(3(5(0(x1))))) 45.58/12.61 1(3(0(5(x1)))) -> 1(2(2(5(3(0(x1)))))) 45.58/12.61 1(5(0(5(x1)))) -> 0(1(3(5(2(5(x1)))))) 45.58/12.61 1(4(2(5(x1)))) -> 5(2(4(1(2(x1))))) 45.58/12.61 5(1(4(5(x1)))) -> 1(2(5(5(4(x1))))) 45.58/12.61 5(1(4(5(x1)))) -> 5(2(4(5(1(2(x1)))))) 45.58/12.61 1(3(4(5(x1)))) -> 3(1(2(4(5(x1))))) 45.58/12.61 1(3(4(5(x1)))) -> 1(2(4(2(5(3(x1)))))) 45.58/12.61 1(3(4(5(x1)))) -> 1(2(3(4(2(5(x1)))))) 45.58/12.61 4(0(5(5(x1)))) -> 2(4(5(2(0(5(x1)))))) 45.58/12.61 1(3(3(0(0(x1))))) -> 3(2(1(0(3(0(x1)))))) 45.58/12.61 1(3(4(2(0(x1))))) -> 2(4(1(3(2(0(x1)))))) 45.58/12.61 1(1(4(5(0(x1))))) -> 1(1(5(2(0(4(x1)))))) 45.58/12.61 1(3(4(2(1(x1))))) -> 4(1(3(3(2(1(x1)))))) 45.58/12.61 1(3(0(0(4(x1))))) -> 0(1(2(3(0(4(x1)))))) 45.58/12.61 1(3(0(0(4(x1))))) -> 1(0(0(2(3(4(x1)))))) 45.58/12.61 5(0(0(1(5(x1))))) -> 5(1(2(0(0(5(x1)))))) 45.58/12.61 1(0(2(4(5(x1))))) -> 1(2(0(4(5(2(x1)))))) 45.58/12.61 45.58/12.61 Q is empty. 45.58/12.61 We have to consider all minimal (P,Q,R)-chains. 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (25) QDPOrderProof (EQUIVALENT) 45.58/12.61 We use the reduction pair processor [LPAR04,JAR06]. 45.58/12.61 45.58/12.61 45.58/12.61 The following pairs can be oriented strictly and are deleted. 45.58/12.61 45.58/12.61 5^1(4(0(0(x1)))) -> 5^1(0(x1)) 45.58/12.61 The remaining pairs can at least be oriented weakly. 45.58/12.61 Used ordering: Polynomial interpretation [POLO]: 45.58/12.61 45.58/12.61 POL(0(x_1)) = 1 + x_1 45.58/12.61 POL(1(x_1)) = x_1 45.58/12.61 POL(2(x_1)) = 0 45.58/12.61 POL(3(x_1)) = 0 45.58/12.61 POL(4(x_1)) = x_1 45.58/12.61 POL(5(x_1)) = x_1 45.58/12.61 POL(5^1(x_1)) = x_1 45.58/12.61 45.58/12.61 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 45.58/12.61 45.58/12.61 4(0(4(0(x1)))) -> 4(4(2(3(0(0(x1)))))) 45.58/12.61 4(1(0(1(x1)))) -> 0(4(1(1(3(2(x1)))))) 45.58/12.61 4(0(2(1(x1)))) -> 2(3(0(2(4(1(x1)))))) 45.58/12.61 4(0(3(1(x1)))) -> 3(1(2(3(0(4(x1)))))) 45.58/12.61 4(0(1(4(x1)))) -> 1(2(0(4(4(x1))))) 45.58/12.61 4(0(5(5(x1)))) -> 2(4(5(2(0(5(x1)))))) 45.58/12.61 5(0(5(x1))) -> 5(3(5(2(3(0(x1)))))) 45.58/12.61 5(4(0(0(x1)))) -> 2(4(2(0(5(0(x1)))))) 45.58/12.61 5(1(4(1(x1)))) -> 4(1(5(2(1(2(x1)))))) 45.58/12.61 5(1(4(5(x1)))) -> 1(2(5(5(4(x1))))) 45.58/12.61 5(1(4(5(x1)))) -> 5(2(4(5(1(2(x1)))))) 45.58/12.61 5(0(0(1(5(x1))))) -> 5(1(2(0(0(5(x1)))))) 45.58/12.61 0(4(1(x1))) -> 2(4(2(3(0(1(x1)))))) 45.58/12.61 0(2(1(0(x1)))) -> 0(1(2(0(2(x1))))) 45.58/12.61 0(4(3(0(x1)))) -> 3(0(0(4(2(3(x1)))))) 45.58/12.61 0(0(1(4(x1)))) -> 0(0(4(1(2(3(x1)))))) 45.58/12.61 45.58/12.61 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (26) 45.58/12.61 Obligation: 45.58/12.61 Q DP problem: 45.58/12.61 The TRS P consists of the following rules: 45.58/12.61 45.58/12.61 5^1(1(4(5(x1)))) -> 5^1(5(4(x1))) 45.58/12.61 5^1(1(4(5(x1)))) -> 5^1(4(x1)) 45.58/12.61 45.58/12.61 The TRS R consists of the following rules: 45.58/12.61 45.58/12.61 1(0(0(x1))) -> 3(1(2(0(0(x1))))) 45.58/12.61 1(0(0(x1))) -> 1(3(2(2(0(0(x1)))))) 45.58/12.61 1(4(0(x1))) -> 1(2(2(4(0(x1))))) 45.58/12.61 1(4(0(x1))) -> 2(4(1(3(2(0(x1)))))) 45.58/12.61 1(0(1(x1))) -> 1(1(2(0(x1)))) 45.58/12.61 1(0(1(x1))) -> 2(1(1(2(0(x1))))) 45.58/12.61 1(0(1(x1))) -> 1(2(3(0(1(x1))))) 45.58/12.61 0(4(1(x1))) -> 2(4(2(3(0(1(x1)))))) 45.58/12.61 1(4(1(x1))) -> 1(1(2(4(x1)))) 45.58/12.61 1(4(1(x1))) -> 1(2(1(1(2(4(x1)))))) 45.58/12.61 1(0(4(x1))) -> 1(2(0(4(x1)))) 45.58/12.61 1(0(5(x1))) -> 1(2(0(5(x1)))) 45.58/12.61 1(0(5(x1))) -> 3(5(1(2(0(x1))))) 45.58/12.61 1(0(5(x1))) -> 5(5(1(2(3(0(x1)))))) 45.58/12.61 1(0(5(x1))) -> 1(5(3(2(0(3(x1)))))) 45.58/12.61 5(0(5(x1))) -> 5(3(5(2(3(0(x1)))))) 45.58/12.61 1(4(5(x1))) -> 3(1(5(2(4(2(x1)))))) 45.58/12.61 1(4(5(x1))) -> 3(3(1(2(5(4(x1)))))) 45.58/12.61 5(4(0(0(x1)))) -> 2(4(2(0(5(0(x1)))))) 45.58/12.61 0(2(1(0(x1)))) -> 0(1(2(0(2(x1))))) 45.58/12.61 1(0(2(0(x1)))) -> 3(1(2(0(0(x1))))) 45.58/12.61 0(4(3(0(x1)))) -> 3(0(0(4(2(3(x1)))))) 45.58/12.61 4(0(4(0(x1)))) -> 4(4(2(3(0(0(x1)))))) 45.58/12.61 4(1(0(1(x1)))) -> 0(4(1(1(3(2(x1)))))) 45.58/12.61 1(3(0(1(x1)))) -> 0(1(1(3(2(4(x1)))))) 45.58/12.61 4(0(2(1(x1)))) -> 2(3(0(2(4(1(x1)))))) 45.58/12.61 4(0(3(1(x1)))) -> 3(1(2(3(0(4(x1)))))) 45.58/12.61 5(1(4(1(x1)))) -> 4(1(5(2(1(2(x1)))))) 45.58/12.61 1(5(0(4(x1)))) -> 5(1(2(4(2(0(x1)))))) 45.58/12.61 0(0(1(4(x1)))) -> 0(0(4(1(2(3(x1)))))) 45.58/12.61 4(0(1(4(x1)))) -> 1(2(0(4(4(x1))))) 45.58/12.61 1(0(0(5(x1)))) -> 0(1(5(2(3(0(x1)))))) 45.58/12.61 1(3(0(5(x1)))) -> 1(2(3(5(0(x1))))) 45.58/12.61 1(3(0(5(x1)))) -> 1(2(2(5(3(0(x1)))))) 45.58/12.61 1(5(0(5(x1)))) -> 0(1(3(5(2(5(x1)))))) 45.58/12.61 1(4(2(5(x1)))) -> 5(2(4(1(2(x1))))) 45.58/12.61 5(1(4(5(x1)))) -> 1(2(5(5(4(x1))))) 45.58/12.61 5(1(4(5(x1)))) -> 5(2(4(5(1(2(x1)))))) 45.58/12.61 1(3(4(5(x1)))) -> 3(1(2(4(5(x1))))) 45.58/12.61 1(3(4(5(x1)))) -> 1(2(4(2(5(3(x1)))))) 45.58/12.61 1(3(4(5(x1)))) -> 1(2(3(4(2(5(x1)))))) 45.58/12.61 4(0(5(5(x1)))) -> 2(4(5(2(0(5(x1)))))) 45.58/12.61 1(3(3(0(0(x1))))) -> 3(2(1(0(3(0(x1)))))) 45.58/12.61 1(3(4(2(0(x1))))) -> 2(4(1(3(2(0(x1)))))) 45.58/12.61 1(1(4(5(0(x1))))) -> 1(1(5(2(0(4(x1)))))) 45.58/12.61 1(3(4(2(1(x1))))) -> 4(1(3(3(2(1(x1)))))) 45.58/12.61 1(3(0(0(4(x1))))) -> 0(1(2(3(0(4(x1)))))) 45.58/12.61 1(3(0(0(4(x1))))) -> 1(0(0(2(3(4(x1)))))) 45.58/12.61 5(0(0(1(5(x1))))) -> 5(1(2(0(0(5(x1)))))) 45.58/12.61 1(0(2(4(5(x1))))) -> 1(2(0(4(5(2(x1)))))) 45.58/12.61 45.58/12.61 Q is empty. 45.58/12.61 We have to consider all minimal (P,Q,R)-chains. 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (27) QDPOrderProof (EQUIVALENT) 45.58/12.61 We use the reduction pair processor [LPAR04,JAR06]. 45.58/12.61 45.58/12.61 45.58/12.61 The following pairs can be oriented strictly and are deleted. 45.58/12.61 45.58/12.61 5^1(1(4(5(x1)))) -> 5^1(5(4(x1))) 45.58/12.61 5^1(1(4(5(x1)))) -> 5^1(4(x1)) 45.58/12.61 The remaining pairs can at least be oriented weakly. 45.58/12.61 Used ordering: Polynomial interpretation [POLO]: 45.58/12.61 45.58/12.61 POL(0(x_1)) = 1 + x_1 45.58/12.61 POL(1(x_1)) = 1 + x_1 45.58/12.61 POL(2(x_1)) = 0 45.58/12.61 POL(3(x_1)) = 0 45.58/12.61 POL(4(x_1)) = 1 + x_1 45.58/12.61 POL(5(x_1)) = x_1 45.58/12.61 POL(5^1(x_1)) = x_1 45.58/12.61 45.58/12.61 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 45.58/12.61 45.58/12.61 4(0(4(0(x1)))) -> 4(4(2(3(0(0(x1)))))) 45.58/12.61 4(1(0(1(x1)))) -> 0(4(1(1(3(2(x1)))))) 45.58/12.61 4(0(2(1(x1)))) -> 2(3(0(2(4(1(x1)))))) 45.58/12.61 4(0(3(1(x1)))) -> 3(1(2(3(0(4(x1)))))) 45.58/12.61 4(0(1(4(x1)))) -> 1(2(0(4(4(x1))))) 45.58/12.61 4(0(5(5(x1)))) -> 2(4(5(2(0(5(x1)))))) 45.58/12.61 5(0(5(x1))) -> 5(3(5(2(3(0(x1)))))) 45.58/12.61 5(4(0(0(x1)))) -> 2(4(2(0(5(0(x1)))))) 45.58/12.61 5(1(4(1(x1)))) -> 4(1(5(2(1(2(x1)))))) 45.58/12.61 5(1(4(5(x1)))) -> 1(2(5(5(4(x1))))) 45.58/12.61 5(1(4(5(x1)))) -> 5(2(4(5(1(2(x1)))))) 45.58/12.61 5(0(0(1(5(x1))))) -> 5(1(2(0(0(5(x1)))))) 45.58/12.61 0(4(1(x1))) -> 2(4(2(3(0(1(x1)))))) 45.58/12.61 45.58/12.61 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (28) 45.58/12.61 Obligation: 45.58/12.61 Q DP problem: 45.58/12.61 P is empty. 45.58/12.61 The TRS R consists of the following rules: 45.58/12.61 45.58/12.61 1(0(0(x1))) -> 3(1(2(0(0(x1))))) 45.58/12.61 1(0(0(x1))) -> 1(3(2(2(0(0(x1)))))) 45.58/12.61 1(4(0(x1))) -> 1(2(2(4(0(x1))))) 45.58/12.61 1(4(0(x1))) -> 2(4(1(3(2(0(x1)))))) 45.58/12.61 1(0(1(x1))) -> 1(1(2(0(x1)))) 45.58/12.61 1(0(1(x1))) -> 2(1(1(2(0(x1))))) 45.58/12.61 1(0(1(x1))) -> 1(2(3(0(1(x1))))) 45.58/12.61 0(4(1(x1))) -> 2(4(2(3(0(1(x1)))))) 45.58/12.61 1(4(1(x1))) -> 1(1(2(4(x1)))) 45.58/12.61 1(4(1(x1))) -> 1(2(1(1(2(4(x1)))))) 45.58/12.61 1(0(4(x1))) -> 1(2(0(4(x1)))) 45.58/12.61 1(0(5(x1))) -> 1(2(0(5(x1)))) 45.58/12.61 1(0(5(x1))) -> 3(5(1(2(0(x1))))) 45.58/12.61 1(0(5(x1))) -> 5(5(1(2(3(0(x1)))))) 45.58/12.61 1(0(5(x1))) -> 1(5(3(2(0(3(x1)))))) 45.58/12.61 5(0(5(x1))) -> 5(3(5(2(3(0(x1)))))) 45.58/12.61 1(4(5(x1))) -> 3(1(5(2(4(2(x1)))))) 45.58/12.61 1(4(5(x1))) -> 3(3(1(2(5(4(x1)))))) 45.58/12.61 5(4(0(0(x1)))) -> 2(4(2(0(5(0(x1)))))) 45.58/12.61 0(2(1(0(x1)))) -> 0(1(2(0(2(x1))))) 45.58/12.61 1(0(2(0(x1)))) -> 3(1(2(0(0(x1))))) 45.58/12.61 0(4(3(0(x1)))) -> 3(0(0(4(2(3(x1)))))) 45.58/12.61 4(0(4(0(x1)))) -> 4(4(2(3(0(0(x1)))))) 45.58/12.61 4(1(0(1(x1)))) -> 0(4(1(1(3(2(x1)))))) 45.58/12.61 1(3(0(1(x1)))) -> 0(1(1(3(2(4(x1)))))) 45.58/12.61 4(0(2(1(x1)))) -> 2(3(0(2(4(1(x1)))))) 45.58/12.61 4(0(3(1(x1)))) -> 3(1(2(3(0(4(x1)))))) 45.58/12.61 5(1(4(1(x1)))) -> 4(1(5(2(1(2(x1)))))) 45.58/12.61 1(5(0(4(x1)))) -> 5(1(2(4(2(0(x1)))))) 45.58/12.61 0(0(1(4(x1)))) -> 0(0(4(1(2(3(x1)))))) 45.58/12.61 4(0(1(4(x1)))) -> 1(2(0(4(4(x1))))) 45.58/12.61 1(0(0(5(x1)))) -> 0(1(5(2(3(0(x1)))))) 45.58/12.61 1(3(0(5(x1)))) -> 1(2(3(5(0(x1))))) 45.58/12.61 1(3(0(5(x1)))) -> 1(2(2(5(3(0(x1)))))) 45.58/12.61 1(5(0(5(x1)))) -> 0(1(3(5(2(5(x1)))))) 45.58/12.61 1(4(2(5(x1)))) -> 5(2(4(1(2(x1))))) 45.58/12.61 5(1(4(5(x1)))) -> 1(2(5(5(4(x1))))) 45.58/12.61 5(1(4(5(x1)))) -> 5(2(4(5(1(2(x1)))))) 45.58/12.61 1(3(4(5(x1)))) -> 3(1(2(4(5(x1))))) 45.58/12.61 1(3(4(5(x1)))) -> 1(2(4(2(5(3(x1)))))) 45.58/12.61 1(3(4(5(x1)))) -> 1(2(3(4(2(5(x1)))))) 45.58/12.61 4(0(5(5(x1)))) -> 2(4(5(2(0(5(x1)))))) 45.58/12.61 1(3(3(0(0(x1))))) -> 3(2(1(0(3(0(x1)))))) 45.58/12.61 1(3(4(2(0(x1))))) -> 2(4(1(3(2(0(x1)))))) 45.58/12.61 1(1(4(5(0(x1))))) -> 1(1(5(2(0(4(x1)))))) 45.58/12.61 1(3(4(2(1(x1))))) -> 4(1(3(3(2(1(x1)))))) 45.58/12.61 1(3(0(0(4(x1))))) -> 0(1(2(3(0(4(x1)))))) 45.58/12.61 1(3(0(0(4(x1))))) -> 1(0(0(2(3(4(x1)))))) 45.58/12.61 5(0(0(1(5(x1))))) -> 5(1(2(0(0(5(x1)))))) 45.58/12.61 1(0(2(4(5(x1))))) -> 1(2(0(4(5(2(x1)))))) 45.58/12.61 45.58/12.61 Q is empty. 45.58/12.61 We have to consider all minimal (P,Q,R)-chains. 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (29) PisEmptyProof (EQUIVALENT) 45.58/12.61 The TRS P is empty. Hence, there is no (P,Q,R) chain. 45.58/12.61 ---------------------------------------- 45.58/12.61 45.58/12.61 (30) 45.58/12.61 YES 46.19/12.76 EOF