65.16/17.52 YES 66.19/17.76 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 66.19/17.76 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 66.19/17.76 66.19/17.76 66.19/17.76 Termination w.r.t. Q of the given QTRS could be proven: 66.19/17.76 66.19/17.76 (0) QTRS 66.19/17.76 (1) QTRS Reverse [EQUIVALENT, 0 ms] 66.19/17.76 (2) QTRS 66.19/17.76 (3) DependencyPairsProof [EQUIVALENT, 279 ms] 66.19/17.76 (4) QDP 66.19/17.76 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 66.19/17.76 (6) QDP 66.19/17.76 (7) QDPOrderProof [EQUIVALENT, 257 ms] 66.19/17.76 (8) QDP 66.19/17.76 (9) QDPOrderProof [EQUIVALENT, 122 ms] 66.19/17.76 (10) QDP 66.19/17.76 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 66.19/17.76 (12) AND 66.19/17.76 (13) QDP 66.19/17.76 (14) QDPOrderProof [EQUIVALENT, 921 ms] 66.19/17.76 (15) QDP 66.19/17.76 (16) DependencyGraphProof [EQUIVALENT, 0 ms] 66.19/17.76 (17) AND 66.19/17.76 (18) QDP 66.19/17.76 (19) UsableRulesProof [EQUIVALENT, 0 ms] 66.19/17.76 (20) QDP 66.19/17.76 (21) QDPSizeChangeProof [EQUIVALENT, 0 ms] 66.19/17.76 (22) YES 66.19/17.76 (23) QDP 66.19/17.76 (24) QDPSizeChangeProof [EQUIVALENT, 0 ms] 66.19/17.76 (25) YES 66.19/17.76 (26) QDP 66.19/17.76 (27) QDPOrderProof [EQUIVALENT, 84 ms] 66.19/17.76 (28) QDP 66.19/17.76 (29) PisEmptyProof [EQUIVALENT, 0 ms] 66.19/17.76 (30) YES 66.19/17.76 (31) QDP 66.19/17.76 (32) QDPOrderProof [EQUIVALENT, 58 ms] 66.19/17.76 (33) QDP 66.19/17.76 (34) PisEmptyProof [EQUIVALENT, 0 ms] 66.19/17.76 (35) YES 66.19/17.76 (36) QDP 66.19/17.76 (37) UsableRulesProof [EQUIVALENT, 0 ms] 66.19/17.76 (38) QDP 66.19/17.76 (39) QDPSizeChangeProof [EQUIVALENT, 0 ms] 66.19/17.76 (40) YES 66.19/17.76 66.19/17.76 66.19/17.76 ---------------------------------------- 66.19/17.76 66.19/17.76 (0) 66.19/17.76 Obligation: 66.19/17.76 Q restricted rewrite system: 66.19/17.76 The TRS R consists of the following rules: 66.19/17.76 66.19/17.76 0(0(0(x1))) -> 0(0(1(0(2(x1))))) 66.19/17.76 0(3(2(x1))) -> 4(3(0(2(x1)))) 66.19/17.76 0(0(4(2(x1)))) -> 0(4(1(0(2(x1))))) 66.19/17.76 0(0(5(2(x1)))) -> 5(0(2(3(0(x1))))) 66.19/17.76 0(1(3(2(x1)))) -> 0(3(1(0(2(x1))))) 66.19/17.76 0(1(3(2(x1)))) -> 3(1(1(0(2(x1))))) 66.19/17.76 0(1(3(2(x1)))) -> 0(1(4(3(1(2(x1)))))) 66.19/17.76 0(4(1(3(x1)))) -> 1(4(3(0(2(2(x1)))))) 66.19/17.76 0(4(2(3(x1)))) -> 5(4(3(0(2(x1))))) 66.19/17.76 0(4(5(2(x1)))) -> 5(0(2(2(4(2(x1)))))) 66.19/17.76 0(5(1(3(x1)))) -> 3(0(1(5(1(2(x1)))))) 66.19/17.76 0(5(3(0(x1)))) -> 5(0(1(4(3(0(x1)))))) 66.19/17.76 0(5(3(2(x1)))) -> 5(1(5(0(2(3(x1)))))) 66.19/17.76 4(0(2(3(x1)))) -> 3(4(3(0(2(x1))))) 66.19/17.76 4(0(2(3(x1)))) -> 4(3(5(0(2(x1))))) 66.19/17.76 4(4(1(3(x1)))) -> 4(3(4(1(2(2(x1)))))) 66.19/17.76 4(5(2(0(x1)))) -> 4(2(1(5(0(2(x1)))))) 66.19/17.76 4(5(2(0(x1)))) -> 5(1(0(2(2(4(x1)))))) 66.19/17.76 5(1(0(0(x1)))) -> 5(1(0(2(0(x1))))) 66.19/17.76 5(1(0(0(x1)))) -> 5(2(1(0(2(0(x1)))))) 66.19/17.76 5(1(3(0(x1)))) -> 5(0(2(1(3(x1))))) 66.19/17.76 5(1(3(2(x1)))) -> 3(0(1(5(1(2(x1)))))) 66.19/17.76 5(1(3(2(x1)))) -> 3(1(1(5(2(2(x1)))))) 66.19/17.76 5(3(0(0(x1)))) -> 5(0(4(3(0(2(x1)))))) 66.19/17.76 0(0(4(1(3(x1))))) -> 4(0(1(0(2(3(x1)))))) 66.19/17.76 0(0(4(5(2(x1))))) -> 5(0(1(0(2(4(x1)))))) 66.19/17.76 0(0(5(3(2(x1))))) -> 0(1(5(0(2(3(x1)))))) 66.19/17.76 0(1(0(5(2(x1))))) -> 1(0(2(5(1(0(x1)))))) 66.19/17.76 0(1(4(5(2(x1))))) -> 2(1(5(0(2(4(x1)))))) 66.19/17.76 0(3(1(4(0(x1))))) -> 4(1(0(1(0(3(x1)))))) 66.19/17.76 0(3(2(0(0(x1))))) -> 0(0(1(0(2(3(x1)))))) 66.19/17.76 0(3(4(0(2(x1))))) -> 4(3(0(2(1(0(x1)))))) 66.19/17.76 0(3(4(0(2(x1))))) -> 4(3(0(2(3(0(x1)))))) 66.19/17.76 0(3(4(4(2(x1))))) -> 4(0(3(4(2(2(x1)))))) 66.19/17.76 0(4(2(5(3(x1))))) -> 0(4(3(5(1(2(x1)))))) 66.19/17.76 0(5(1(2(0(x1))))) -> 3(0(1(5(0(2(x1)))))) 66.19/17.76 4(4(2(2(0(x1))))) -> 4(1(0(2(2(4(x1)))))) 66.19/17.76 4(5(1(2(0(x1))))) -> 5(0(4(1(2(2(x1)))))) 66.19/17.76 4(5(2(3(2(x1))))) -> 5(4(3(5(2(2(x1)))))) 66.19/17.76 5(1(0(3(2(x1))))) -> 5(0(3(1(0(2(x1)))))) 66.19/17.76 5(1(0(5(3(x1))))) -> 5(5(0(1(3(1(x1)))))) 66.19/17.76 5(1(3(0(0(x1))))) -> 3(5(0(1(2(0(x1)))))) 66.19/17.76 5(1(3(0(2(x1))))) -> 3(0(2(1(5(2(x1)))))) 66.19/17.76 5(1(3(0(2(x1))))) -> 5(0(1(0(3(2(x1)))))) 66.19/17.76 5(1(3(0(2(x1))))) -> 5(0(1(1(2(3(x1)))))) 66.19/17.76 5(1(3(2(0(x1))))) -> 5(3(1(5(2(0(x1)))))) 66.19/17.76 5(1(3(2(3(x1))))) -> 3(4(3(5(1(2(x1)))))) 66.19/17.76 5(1(4(5(2(x1))))) -> 5(1(4(1(5(2(x1)))))) 66.19/17.76 5(5(1(3(2(x1))))) -> 3(5(5(4(1(2(x1)))))) 66.19/17.76 66.19/17.76 Q is empty. 66.19/17.76 66.19/17.76 ---------------------------------------- 66.19/17.76 66.19/17.76 (1) QTRS Reverse (EQUIVALENT) 66.19/17.76 We applied the QTRS Reverse Processor [REVERSE]. 66.19/17.76 ---------------------------------------- 66.19/17.76 66.19/17.76 (2) 66.19/17.76 Obligation: 66.19/17.76 Q restricted rewrite system: 66.19/17.76 The TRS R consists of the following rules: 66.19/17.76 66.19/17.76 0(0(0(x1))) -> 2(0(1(0(0(x1))))) 66.19/17.76 2(3(0(x1))) -> 2(0(3(4(x1)))) 66.19/17.76 2(4(0(0(x1)))) -> 2(0(1(4(0(x1))))) 66.19/17.76 2(5(0(0(x1)))) -> 0(3(2(0(5(x1))))) 66.19/17.76 2(3(1(0(x1)))) -> 2(0(1(3(0(x1))))) 66.19/17.76 2(3(1(0(x1)))) -> 2(0(1(1(3(x1))))) 66.19/17.76 2(3(1(0(x1)))) -> 2(1(3(4(1(0(x1)))))) 66.19/17.76 3(1(4(0(x1)))) -> 2(2(0(3(4(1(x1)))))) 66.19/17.76 3(2(4(0(x1)))) -> 2(0(3(4(5(x1))))) 66.19/17.76 2(5(4(0(x1)))) -> 2(4(2(2(0(5(x1)))))) 66.19/17.76 3(1(5(0(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.76 0(3(5(0(x1)))) -> 0(3(4(1(0(5(x1)))))) 66.19/17.76 2(3(5(0(x1)))) -> 3(2(0(5(1(5(x1)))))) 66.19/17.76 3(2(0(4(x1)))) -> 2(0(3(4(3(x1))))) 66.19/17.76 3(2(0(4(x1)))) -> 2(0(5(3(4(x1))))) 66.19/17.76 3(1(4(4(x1)))) -> 2(2(1(4(3(4(x1)))))) 66.19/17.76 0(2(5(4(x1)))) -> 2(0(5(1(2(4(x1)))))) 66.19/17.76 0(2(5(4(x1)))) -> 4(2(2(0(1(5(x1)))))) 66.19/17.76 0(0(1(5(x1)))) -> 0(2(0(1(5(x1))))) 66.19/17.76 0(0(1(5(x1)))) -> 0(2(0(1(2(5(x1)))))) 66.19/17.76 0(3(1(5(x1)))) -> 3(1(2(0(5(x1))))) 66.19/17.76 2(3(1(5(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.76 2(3(1(5(x1)))) -> 2(2(5(1(1(3(x1)))))) 66.19/17.76 0(0(3(5(x1)))) -> 2(0(3(4(0(5(x1)))))) 66.19/17.76 3(1(4(0(0(x1))))) -> 3(2(0(1(0(4(x1)))))) 66.19/17.76 2(5(4(0(0(x1))))) -> 4(2(0(1(0(5(x1)))))) 66.19/17.76 2(3(5(0(0(x1))))) -> 3(2(0(5(1(0(x1)))))) 66.19/17.76 2(5(0(1(0(x1))))) -> 0(1(5(2(0(1(x1)))))) 66.19/17.76 2(5(4(1(0(x1))))) -> 4(2(0(5(1(2(x1)))))) 66.19/17.76 0(4(1(3(0(x1))))) -> 3(0(1(0(1(4(x1)))))) 66.19/17.76 0(0(2(3(0(x1))))) -> 3(2(0(1(0(0(x1)))))) 66.19/17.77 2(0(4(3(0(x1))))) -> 0(1(2(0(3(4(x1)))))) 66.19/17.77 2(0(4(3(0(x1))))) -> 0(3(2(0(3(4(x1)))))) 66.19/17.77 2(4(4(3(0(x1))))) -> 2(2(4(3(0(4(x1)))))) 66.19/17.77 3(5(2(4(0(x1))))) -> 2(1(5(3(4(0(x1)))))) 66.19/17.77 0(2(1(5(0(x1))))) -> 2(0(5(1(0(3(x1)))))) 66.19/17.77 0(2(2(4(4(x1))))) -> 4(2(2(0(1(4(x1)))))) 66.19/17.77 0(2(1(5(4(x1))))) -> 2(2(1(4(0(5(x1)))))) 66.19/17.77 2(3(2(5(4(x1))))) -> 2(2(5(3(4(5(x1)))))) 66.19/17.77 2(3(0(1(5(x1))))) -> 2(0(1(3(0(5(x1)))))) 66.19/17.77 3(5(0(1(5(x1))))) -> 1(3(1(0(5(5(x1)))))) 66.19/17.77 0(0(3(1(5(x1))))) -> 0(2(1(0(5(3(x1)))))) 66.19/17.77 2(0(3(1(5(x1))))) -> 2(5(1(2(0(3(x1)))))) 66.19/17.77 2(0(3(1(5(x1))))) -> 2(3(0(1(0(5(x1)))))) 66.19/17.77 2(0(3(1(5(x1))))) -> 3(2(1(1(0(5(x1)))))) 66.19/17.77 0(2(3(1(5(x1))))) -> 0(2(5(1(3(5(x1)))))) 66.19/17.77 3(2(3(1(5(x1))))) -> 2(1(5(3(4(3(x1)))))) 66.19/17.77 2(5(4(1(5(x1))))) -> 2(5(1(4(1(5(x1)))))) 66.19/17.77 2(3(1(5(5(x1))))) -> 2(1(4(5(5(3(x1)))))) 66.19/17.77 66.19/17.77 Q is empty. 66.19/17.77 66.19/17.77 ---------------------------------------- 66.19/17.77 66.19/17.77 (3) DependencyPairsProof (EQUIVALENT) 66.19/17.77 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 66.19/17.77 ---------------------------------------- 66.19/17.77 66.19/17.77 (4) 66.19/17.77 Obligation: 66.19/17.77 Q DP problem: 66.19/17.77 The TRS P consists of the following rules: 66.19/17.77 66.19/17.77 0^1(0(0(x1))) -> 2^1(0(1(0(0(x1))))) 66.19/17.77 0^1(0(0(x1))) -> 0^1(1(0(0(x1)))) 66.19/17.77 2^1(3(0(x1))) -> 2^1(0(3(4(x1)))) 66.19/17.77 2^1(3(0(x1))) -> 0^1(3(4(x1))) 66.19/17.77 2^1(3(0(x1))) -> 3^1(4(x1)) 66.19/17.77 2^1(4(0(0(x1)))) -> 2^1(0(1(4(0(x1))))) 66.19/17.77 2^1(4(0(0(x1)))) -> 0^1(1(4(0(x1)))) 66.19/17.77 2^1(5(0(0(x1)))) -> 0^1(3(2(0(5(x1))))) 66.19/17.77 2^1(5(0(0(x1)))) -> 3^1(2(0(5(x1)))) 66.19/17.77 2^1(5(0(0(x1)))) -> 2^1(0(5(x1))) 66.19/17.77 2^1(5(0(0(x1)))) -> 0^1(5(x1)) 66.19/17.77 2^1(3(1(0(x1)))) -> 2^1(0(1(3(0(x1))))) 66.19/17.77 2^1(3(1(0(x1)))) -> 0^1(1(3(0(x1)))) 66.19/17.77 2^1(3(1(0(x1)))) -> 3^1(0(x1)) 66.19/17.77 2^1(3(1(0(x1)))) -> 2^1(0(1(1(3(x1))))) 66.19/17.77 2^1(3(1(0(x1)))) -> 0^1(1(1(3(x1)))) 66.19/17.77 2^1(3(1(0(x1)))) -> 3^1(x1) 66.19/17.77 2^1(3(1(0(x1)))) -> 2^1(1(3(4(1(0(x1)))))) 66.19/17.77 2^1(3(1(0(x1)))) -> 3^1(4(1(0(x1)))) 66.19/17.77 3^1(1(4(0(x1)))) -> 2^1(2(0(3(4(1(x1)))))) 66.19/17.77 3^1(1(4(0(x1)))) -> 2^1(0(3(4(1(x1))))) 66.19/17.77 3^1(1(4(0(x1)))) -> 0^1(3(4(1(x1)))) 66.19/17.77 3^1(1(4(0(x1)))) -> 3^1(4(1(x1))) 66.19/17.77 3^1(2(4(0(x1)))) -> 2^1(0(3(4(5(x1))))) 66.19/17.77 3^1(2(4(0(x1)))) -> 0^1(3(4(5(x1)))) 66.19/17.77 3^1(2(4(0(x1)))) -> 3^1(4(5(x1))) 66.19/17.77 2^1(5(4(0(x1)))) -> 2^1(4(2(2(0(5(x1)))))) 66.19/17.77 2^1(5(4(0(x1)))) -> 2^1(2(0(5(x1)))) 66.19/17.77 2^1(5(4(0(x1)))) -> 2^1(0(5(x1))) 66.19/17.77 2^1(5(4(0(x1)))) -> 0^1(5(x1)) 66.19/17.77 3^1(1(5(0(x1)))) -> 2^1(1(5(1(0(3(x1)))))) 66.19/17.77 3^1(1(5(0(x1)))) -> 0^1(3(x1)) 66.19/17.77 3^1(1(5(0(x1)))) -> 3^1(x1) 66.19/17.77 0^1(3(5(0(x1)))) -> 0^1(3(4(1(0(5(x1)))))) 66.19/17.77 0^1(3(5(0(x1)))) -> 3^1(4(1(0(5(x1))))) 66.19/17.77 0^1(3(5(0(x1)))) -> 0^1(5(x1)) 66.19/17.77 2^1(3(5(0(x1)))) -> 3^1(2(0(5(1(5(x1)))))) 66.19/17.77 2^1(3(5(0(x1)))) -> 2^1(0(5(1(5(x1))))) 66.19/17.77 2^1(3(5(0(x1)))) -> 0^1(5(1(5(x1)))) 66.19/17.77 3^1(2(0(4(x1)))) -> 2^1(0(3(4(3(x1))))) 66.19/17.77 3^1(2(0(4(x1)))) -> 0^1(3(4(3(x1)))) 66.19/17.77 3^1(2(0(4(x1)))) -> 3^1(4(3(x1))) 66.19/17.77 3^1(2(0(4(x1)))) -> 3^1(x1) 66.19/17.77 3^1(2(0(4(x1)))) -> 2^1(0(5(3(4(x1))))) 66.19/17.77 3^1(2(0(4(x1)))) -> 0^1(5(3(4(x1)))) 66.19/17.77 3^1(2(0(4(x1)))) -> 3^1(4(x1)) 66.19/17.77 3^1(1(4(4(x1)))) -> 2^1(2(1(4(3(4(x1)))))) 66.19/17.77 3^1(1(4(4(x1)))) -> 2^1(1(4(3(4(x1))))) 66.19/17.77 3^1(1(4(4(x1)))) -> 3^1(4(x1)) 66.19/17.77 0^1(2(5(4(x1)))) -> 2^1(0(5(1(2(4(x1)))))) 66.19/17.77 0^1(2(5(4(x1)))) -> 0^1(5(1(2(4(x1))))) 66.19/17.77 0^1(2(5(4(x1)))) -> 2^1(4(x1)) 66.19/17.77 0^1(2(5(4(x1)))) -> 2^1(2(0(1(5(x1))))) 66.19/17.77 0^1(2(5(4(x1)))) -> 2^1(0(1(5(x1)))) 66.19/17.77 0^1(2(5(4(x1)))) -> 0^1(1(5(x1))) 66.19/17.77 0^1(0(1(5(x1)))) -> 0^1(2(0(1(5(x1))))) 66.19/17.77 0^1(0(1(5(x1)))) -> 2^1(0(1(5(x1)))) 66.19/17.77 0^1(0(1(5(x1)))) -> 0^1(2(0(1(2(5(x1)))))) 66.19/17.77 0^1(0(1(5(x1)))) -> 2^1(0(1(2(5(x1))))) 66.19/17.77 0^1(0(1(5(x1)))) -> 0^1(1(2(5(x1)))) 66.19/17.77 0^1(0(1(5(x1)))) -> 2^1(5(x1)) 66.19/17.77 0^1(3(1(5(x1)))) -> 3^1(1(2(0(5(x1))))) 66.19/17.77 0^1(3(1(5(x1)))) -> 2^1(0(5(x1))) 66.19/17.77 0^1(3(1(5(x1)))) -> 0^1(5(x1)) 66.19/17.77 2^1(3(1(5(x1)))) -> 2^1(1(5(1(0(3(x1)))))) 66.19/17.77 2^1(3(1(5(x1)))) -> 0^1(3(x1)) 66.19/17.77 2^1(3(1(5(x1)))) -> 3^1(x1) 66.19/17.77 2^1(3(1(5(x1)))) -> 2^1(2(5(1(1(3(x1)))))) 66.19/17.77 2^1(3(1(5(x1)))) -> 2^1(5(1(1(3(x1))))) 66.19/17.77 0^1(0(3(5(x1)))) -> 2^1(0(3(4(0(5(x1)))))) 66.19/17.77 0^1(0(3(5(x1)))) -> 0^1(3(4(0(5(x1))))) 66.19/17.77 0^1(0(3(5(x1)))) -> 3^1(4(0(5(x1)))) 66.19/17.77 0^1(0(3(5(x1)))) -> 0^1(5(x1)) 66.19/17.77 3^1(1(4(0(0(x1))))) -> 3^1(2(0(1(0(4(x1)))))) 66.19/17.77 3^1(1(4(0(0(x1))))) -> 2^1(0(1(0(4(x1))))) 66.19/17.77 3^1(1(4(0(0(x1))))) -> 0^1(1(0(4(x1)))) 66.19/17.77 3^1(1(4(0(0(x1))))) -> 0^1(4(x1)) 66.19/17.77 2^1(5(4(0(0(x1))))) -> 2^1(0(1(0(5(x1))))) 66.19/17.77 2^1(5(4(0(0(x1))))) -> 0^1(1(0(5(x1)))) 66.19/17.77 2^1(5(4(0(0(x1))))) -> 0^1(5(x1)) 66.19/17.77 2^1(3(5(0(0(x1))))) -> 3^1(2(0(5(1(0(x1)))))) 66.19/17.77 2^1(3(5(0(0(x1))))) -> 2^1(0(5(1(0(x1))))) 66.19/17.77 2^1(3(5(0(0(x1))))) -> 0^1(5(1(0(x1)))) 66.19/17.77 2^1(5(0(1(0(x1))))) -> 0^1(1(5(2(0(1(x1)))))) 66.19/17.77 2^1(5(0(1(0(x1))))) -> 2^1(0(1(x1))) 66.19/17.77 2^1(5(0(1(0(x1))))) -> 0^1(1(x1)) 66.19/17.77 2^1(5(4(1(0(x1))))) -> 2^1(0(5(1(2(x1))))) 66.19/17.77 2^1(5(4(1(0(x1))))) -> 0^1(5(1(2(x1)))) 66.19/17.77 2^1(5(4(1(0(x1))))) -> 2^1(x1) 66.19/17.77 0^1(4(1(3(0(x1))))) -> 3^1(0(1(0(1(4(x1)))))) 66.19/17.77 0^1(4(1(3(0(x1))))) -> 0^1(1(0(1(4(x1))))) 66.19/17.77 0^1(4(1(3(0(x1))))) -> 0^1(1(4(x1))) 66.19/17.77 0^1(0(2(3(0(x1))))) -> 3^1(2(0(1(0(0(x1)))))) 66.19/17.77 0^1(0(2(3(0(x1))))) -> 2^1(0(1(0(0(x1))))) 66.19/17.77 0^1(0(2(3(0(x1))))) -> 0^1(1(0(0(x1)))) 66.19/17.77 0^1(0(2(3(0(x1))))) -> 0^1(0(x1)) 66.19/17.77 2^1(0(4(3(0(x1))))) -> 0^1(1(2(0(3(4(x1)))))) 66.19/17.77 2^1(0(4(3(0(x1))))) -> 2^1(0(3(4(x1)))) 66.19/17.77 2^1(0(4(3(0(x1))))) -> 0^1(3(4(x1))) 66.19/17.77 2^1(0(4(3(0(x1))))) -> 3^1(4(x1)) 66.19/17.77 2^1(0(4(3(0(x1))))) -> 0^1(3(2(0(3(4(x1)))))) 66.19/17.77 2^1(0(4(3(0(x1))))) -> 3^1(2(0(3(4(x1))))) 66.19/17.77 2^1(4(4(3(0(x1))))) -> 2^1(2(4(3(0(4(x1)))))) 66.19/17.77 2^1(4(4(3(0(x1))))) -> 2^1(4(3(0(4(x1))))) 66.19/17.77 2^1(4(4(3(0(x1))))) -> 3^1(0(4(x1))) 66.19/17.77 2^1(4(4(3(0(x1))))) -> 0^1(4(x1)) 66.19/17.77 3^1(5(2(4(0(x1))))) -> 2^1(1(5(3(4(0(x1)))))) 66.19/17.77 3^1(5(2(4(0(x1))))) -> 3^1(4(0(x1))) 66.19/17.77 0^1(2(1(5(0(x1))))) -> 2^1(0(5(1(0(3(x1)))))) 66.19/17.77 0^1(2(1(5(0(x1))))) -> 0^1(5(1(0(3(x1))))) 66.19/17.77 0^1(2(1(5(0(x1))))) -> 0^1(3(x1)) 66.19/17.77 0^1(2(1(5(0(x1))))) -> 3^1(x1) 66.19/17.77 0^1(2(2(4(4(x1))))) -> 2^1(2(0(1(4(x1))))) 66.19/17.77 0^1(2(2(4(4(x1))))) -> 2^1(0(1(4(x1)))) 66.19/17.77 0^1(2(2(4(4(x1))))) -> 0^1(1(4(x1))) 66.19/17.77 0^1(2(1(5(4(x1))))) -> 2^1(2(1(4(0(5(x1)))))) 66.19/17.77 0^1(2(1(5(4(x1))))) -> 2^1(1(4(0(5(x1))))) 66.19/17.77 0^1(2(1(5(4(x1))))) -> 0^1(5(x1)) 66.19/17.77 2^1(3(2(5(4(x1))))) -> 2^1(2(5(3(4(5(x1)))))) 66.19/17.77 2^1(3(2(5(4(x1))))) -> 2^1(5(3(4(5(x1))))) 66.19/17.77 2^1(3(2(5(4(x1))))) -> 3^1(4(5(x1))) 66.19/17.77 2^1(3(0(1(5(x1))))) -> 2^1(0(1(3(0(5(x1)))))) 66.19/17.77 2^1(3(0(1(5(x1))))) -> 0^1(1(3(0(5(x1))))) 66.19/17.77 2^1(3(0(1(5(x1))))) -> 3^1(0(5(x1))) 66.19/17.77 2^1(3(0(1(5(x1))))) -> 0^1(5(x1)) 66.19/17.77 3^1(5(0(1(5(x1))))) -> 3^1(1(0(5(5(x1))))) 66.19/17.77 3^1(5(0(1(5(x1))))) -> 0^1(5(5(x1))) 66.19/17.77 0^1(0(3(1(5(x1))))) -> 0^1(2(1(0(5(3(x1)))))) 66.19/17.77 0^1(0(3(1(5(x1))))) -> 2^1(1(0(5(3(x1))))) 66.19/17.77 0^1(0(3(1(5(x1))))) -> 0^1(5(3(x1))) 66.19/17.77 0^1(0(3(1(5(x1))))) -> 3^1(x1) 66.19/17.77 2^1(0(3(1(5(x1))))) -> 2^1(5(1(2(0(3(x1)))))) 66.19/17.77 2^1(0(3(1(5(x1))))) -> 2^1(0(3(x1))) 66.19/17.77 2^1(0(3(1(5(x1))))) -> 0^1(3(x1)) 66.19/17.77 2^1(0(3(1(5(x1))))) -> 3^1(x1) 66.19/17.77 2^1(0(3(1(5(x1))))) -> 2^1(3(0(1(0(5(x1)))))) 66.19/17.77 2^1(0(3(1(5(x1))))) -> 3^1(0(1(0(5(x1))))) 66.19/17.77 2^1(0(3(1(5(x1))))) -> 0^1(1(0(5(x1)))) 66.19/17.77 2^1(0(3(1(5(x1))))) -> 0^1(5(x1)) 66.19/17.77 2^1(0(3(1(5(x1))))) -> 3^1(2(1(1(0(5(x1)))))) 66.19/17.77 2^1(0(3(1(5(x1))))) -> 2^1(1(1(0(5(x1))))) 66.19/17.77 0^1(2(3(1(5(x1))))) -> 0^1(2(5(1(3(5(x1)))))) 66.19/17.77 0^1(2(3(1(5(x1))))) -> 2^1(5(1(3(5(x1))))) 66.19/17.77 0^1(2(3(1(5(x1))))) -> 3^1(5(x1)) 66.19/17.77 3^1(2(3(1(5(x1))))) -> 2^1(1(5(3(4(3(x1)))))) 66.19/17.77 3^1(2(3(1(5(x1))))) -> 3^1(4(3(x1))) 66.19/17.77 3^1(2(3(1(5(x1))))) -> 3^1(x1) 66.19/17.77 2^1(5(4(1(5(x1))))) -> 2^1(5(1(4(1(5(x1)))))) 66.19/17.77 2^1(3(1(5(5(x1))))) -> 2^1(1(4(5(5(3(x1)))))) 66.19/17.77 2^1(3(1(5(5(x1))))) -> 3^1(x1) 66.19/17.77 66.19/17.77 The TRS R consists of the following rules: 66.19/17.77 66.19/17.77 0(0(0(x1))) -> 2(0(1(0(0(x1))))) 66.19/17.77 2(3(0(x1))) -> 2(0(3(4(x1)))) 66.19/17.77 2(4(0(0(x1)))) -> 2(0(1(4(0(x1))))) 66.19/17.77 2(5(0(0(x1)))) -> 0(3(2(0(5(x1))))) 66.19/17.77 2(3(1(0(x1)))) -> 2(0(1(3(0(x1))))) 66.19/17.77 2(3(1(0(x1)))) -> 2(0(1(1(3(x1))))) 66.19/17.77 2(3(1(0(x1)))) -> 2(1(3(4(1(0(x1)))))) 66.19/17.77 3(1(4(0(x1)))) -> 2(2(0(3(4(1(x1)))))) 66.19/17.77 3(2(4(0(x1)))) -> 2(0(3(4(5(x1))))) 66.19/17.77 2(5(4(0(x1)))) -> 2(4(2(2(0(5(x1)))))) 66.19/17.77 3(1(5(0(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.77 0(3(5(0(x1)))) -> 0(3(4(1(0(5(x1)))))) 66.19/17.77 2(3(5(0(x1)))) -> 3(2(0(5(1(5(x1)))))) 66.19/17.77 3(2(0(4(x1)))) -> 2(0(3(4(3(x1))))) 66.19/17.77 3(2(0(4(x1)))) -> 2(0(5(3(4(x1))))) 66.19/17.77 3(1(4(4(x1)))) -> 2(2(1(4(3(4(x1)))))) 66.19/17.77 0(2(5(4(x1)))) -> 2(0(5(1(2(4(x1)))))) 66.19/17.77 0(2(5(4(x1)))) -> 4(2(2(0(1(5(x1)))))) 66.19/17.77 0(0(1(5(x1)))) -> 0(2(0(1(5(x1))))) 66.19/17.77 0(0(1(5(x1)))) -> 0(2(0(1(2(5(x1)))))) 66.19/17.77 0(3(1(5(x1)))) -> 3(1(2(0(5(x1))))) 66.19/17.77 2(3(1(5(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.77 2(3(1(5(x1)))) -> 2(2(5(1(1(3(x1)))))) 66.19/17.77 0(0(3(5(x1)))) -> 2(0(3(4(0(5(x1)))))) 66.19/17.77 3(1(4(0(0(x1))))) -> 3(2(0(1(0(4(x1)))))) 66.19/17.77 2(5(4(0(0(x1))))) -> 4(2(0(1(0(5(x1)))))) 66.19/17.77 2(3(5(0(0(x1))))) -> 3(2(0(5(1(0(x1)))))) 66.19/17.77 2(5(0(1(0(x1))))) -> 0(1(5(2(0(1(x1)))))) 66.19/17.77 2(5(4(1(0(x1))))) -> 4(2(0(5(1(2(x1)))))) 66.19/17.77 0(4(1(3(0(x1))))) -> 3(0(1(0(1(4(x1)))))) 66.19/17.77 0(0(2(3(0(x1))))) -> 3(2(0(1(0(0(x1)))))) 66.19/17.77 2(0(4(3(0(x1))))) -> 0(1(2(0(3(4(x1)))))) 66.19/17.77 2(0(4(3(0(x1))))) -> 0(3(2(0(3(4(x1)))))) 66.19/17.77 2(4(4(3(0(x1))))) -> 2(2(4(3(0(4(x1)))))) 66.19/17.77 3(5(2(4(0(x1))))) -> 2(1(5(3(4(0(x1)))))) 66.19/17.77 0(2(1(5(0(x1))))) -> 2(0(5(1(0(3(x1)))))) 66.19/17.77 0(2(2(4(4(x1))))) -> 4(2(2(0(1(4(x1)))))) 66.19/17.77 0(2(1(5(4(x1))))) -> 2(2(1(4(0(5(x1)))))) 66.19/17.77 2(3(2(5(4(x1))))) -> 2(2(5(3(4(5(x1)))))) 66.19/17.77 2(3(0(1(5(x1))))) -> 2(0(1(3(0(5(x1)))))) 66.19/17.77 3(5(0(1(5(x1))))) -> 1(3(1(0(5(5(x1)))))) 66.19/17.77 0(0(3(1(5(x1))))) -> 0(2(1(0(5(3(x1)))))) 66.19/17.77 2(0(3(1(5(x1))))) -> 2(5(1(2(0(3(x1)))))) 66.19/17.77 2(0(3(1(5(x1))))) -> 2(3(0(1(0(5(x1)))))) 66.19/17.77 2(0(3(1(5(x1))))) -> 3(2(1(1(0(5(x1)))))) 66.19/17.77 0(2(3(1(5(x1))))) -> 0(2(5(1(3(5(x1)))))) 66.19/17.77 3(2(3(1(5(x1))))) -> 2(1(5(3(4(3(x1)))))) 66.19/17.77 2(5(4(1(5(x1))))) -> 2(5(1(4(1(5(x1)))))) 66.19/17.77 2(3(1(5(5(x1))))) -> 2(1(4(5(5(3(x1)))))) 66.19/17.77 66.19/17.77 Q is empty. 66.19/17.77 We have to consider all minimal (P,Q,R)-chains. 66.19/17.77 ---------------------------------------- 66.19/17.77 66.19/17.77 (5) DependencyGraphProof (EQUIVALENT) 66.19/17.77 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 128 less nodes. 66.19/17.77 ---------------------------------------- 66.19/17.77 66.19/17.77 (6) 66.19/17.77 Obligation: 66.19/17.77 Q DP problem: 66.19/17.77 The TRS P consists of the following rules: 66.19/17.77 66.19/17.77 0^1(2(5(4(x1)))) -> 2^1(4(x1)) 66.19/17.77 2^1(4(4(3(0(x1))))) -> 2^1(2(4(3(0(4(x1)))))) 66.19/17.77 2^1(3(1(0(x1)))) -> 3^1(0(x1)) 66.19/17.77 3^1(2(0(4(x1)))) -> 3^1(x1) 66.19/17.77 3^1(1(5(0(x1)))) -> 0^1(3(x1)) 66.19/17.77 0^1(0(1(5(x1)))) -> 2^1(5(x1)) 66.19/17.77 2^1(5(4(1(0(x1))))) -> 2^1(x1) 66.19/17.77 2^1(3(1(0(x1)))) -> 3^1(x1) 66.19/17.77 3^1(1(5(0(x1)))) -> 3^1(x1) 66.19/17.77 3^1(2(3(1(5(x1))))) -> 3^1(x1) 66.19/17.77 2^1(3(1(5(x1)))) -> 0^1(3(x1)) 66.19/17.77 0^1(0(2(3(0(x1))))) -> 0^1(0(x1)) 66.19/17.77 0^1(2(1(5(0(x1))))) -> 0^1(3(x1)) 66.19/17.77 0^1(2(1(5(0(x1))))) -> 3^1(x1) 66.19/17.77 0^1(0(3(1(5(x1))))) -> 3^1(x1) 66.19/17.77 2^1(3(1(5(x1)))) -> 3^1(x1) 66.19/17.77 2^1(4(4(3(0(x1))))) -> 2^1(4(3(0(4(x1))))) 66.19/17.77 2^1(4(4(3(0(x1))))) -> 3^1(0(4(x1))) 66.19/17.77 2^1(0(3(1(5(x1))))) -> 2^1(0(3(x1))) 66.19/17.77 2^1(0(3(1(5(x1))))) -> 0^1(3(x1)) 66.19/17.77 2^1(0(3(1(5(x1))))) -> 3^1(x1) 66.19/17.77 2^1(3(1(5(5(x1))))) -> 3^1(x1) 66.19/17.77 66.19/17.77 The TRS R consists of the following rules: 66.19/17.77 66.19/17.77 0(0(0(x1))) -> 2(0(1(0(0(x1))))) 66.19/17.77 2(3(0(x1))) -> 2(0(3(4(x1)))) 66.19/17.77 2(4(0(0(x1)))) -> 2(0(1(4(0(x1))))) 66.19/17.77 2(5(0(0(x1)))) -> 0(3(2(0(5(x1))))) 66.19/17.77 2(3(1(0(x1)))) -> 2(0(1(3(0(x1))))) 66.19/17.77 2(3(1(0(x1)))) -> 2(0(1(1(3(x1))))) 66.19/17.77 2(3(1(0(x1)))) -> 2(1(3(4(1(0(x1)))))) 66.19/17.77 3(1(4(0(x1)))) -> 2(2(0(3(4(1(x1)))))) 66.19/17.77 3(2(4(0(x1)))) -> 2(0(3(4(5(x1))))) 66.19/17.77 2(5(4(0(x1)))) -> 2(4(2(2(0(5(x1)))))) 66.19/17.77 3(1(5(0(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.77 0(3(5(0(x1)))) -> 0(3(4(1(0(5(x1)))))) 66.19/17.77 2(3(5(0(x1)))) -> 3(2(0(5(1(5(x1)))))) 66.19/17.77 3(2(0(4(x1)))) -> 2(0(3(4(3(x1))))) 66.19/17.77 3(2(0(4(x1)))) -> 2(0(5(3(4(x1))))) 66.19/17.80 3(1(4(4(x1)))) -> 2(2(1(4(3(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 2(0(5(1(2(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 4(2(2(0(1(5(x1)))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(5(x1))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(2(5(x1)))))) 66.19/17.80 0(3(1(5(x1)))) -> 3(1(2(0(5(x1))))) 66.19/17.80 2(3(1(5(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 2(3(1(5(x1)))) -> 2(2(5(1(1(3(x1)))))) 66.19/17.80 0(0(3(5(x1)))) -> 2(0(3(4(0(5(x1)))))) 66.19/17.80 3(1(4(0(0(x1))))) -> 3(2(0(1(0(4(x1)))))) 66.19/17.80 2(5(4(0(0(x1))))) -> 4(2(0(1(0(5(x1)))))) 66.19/17.80 2(3(5(0(0(x1))))) -> 3(2(0(5(1(0(x1)))))) 66.19/17.80 2(5(0(1(0(x1))))) -> 0(1(5(2(0(1(x1)))))) 66.19/17.80 2(5(4(1(0(x1))))) -> 4(2(0(5(1(2(x1)))))) 66.19/17.80 0(4(1(3(0(x1))))) -> 3(0(1(0(1(4(x1)))))) 66.19/17.80 0(0(2(3(0(x1))))) -> 3(2(0(1(0(0(x1)))))) 66.19/17.80 2(0(4(3(0(x1))))) -> 0(1(2(0(3(4(x1)))))) 66.19/17.80 2(0(4(3(0(x1))))) -> 0(3(2(0(3(4(x1)))))) 66.19/17.80 2(4(4(3(0(x1))))) -> 2(2(4(3(0(4(x1)))))) 66.19/17.80 3(5(2(4(0(x1))))) -> 2(1(5(3(4(0(x1)))))) 66.19/17.80 0(2(1(5(0(x1))))) -> 2(0(5(1(0(3(x1)))))) 66.19/17.80 0(2(2(4(4(x1))))) -> 4(2(2(0(1(4(x1)))))) 66.19/17.80 0(2(1(5(4(x1))))) -> 2(2(1(4(0(5(x1)))))) 66.19/17.80 2(3(2(5(4(x1))))) -> 2(2(5(3(4(5(x1)))))) 66.19/17.80 2(3(0(1(5(x1))))) -> 2(0(1(3(0(5(x1)))))) 66.19/17.80 3(5(0(1(5(x1))))) -> 1(3(1(0(5(5(x1)))))) 66.19/17.80 0(0(3(1(5(x1))))) -> 0(2(1(0(5(3(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 2(5(1(2(0(3(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 2(3(0(1(0(5(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 3(2(1(1(0(5(x1)))))) 66.19/17.80 0(2(3(1(5(x1))))) -> 0(2(5(1(3(5(x1)))))) 66.19/17.80 3(2(3(1(5(x1))))) -> 2(1(5(3(4(3(x1)))))) 66.19/17.80 2(5(4(1(5(x1))))) -> 2(5(1(4(1(5(x1)))))) 66.19/17.80 2(3(1(5(5(x1))))) -> 2(1(4(5(5(3(x1)))))) 66.19/17.80 66.19/17.80 Q is empty. 66.19/17.80 We have to consider all minimal (P,Q,R)-chains. 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (7) QDPOrderProof (EQUIVALENT) 66.19/17.80 We use the reduction pair processor [LPAR04,JAR06]. 66.19/17.80 66.19/17.80 66.19/17.80 The following pairs can be oriented strictly and are deleted. 66.19/17.80 66.19/17.80 0^1(2(5(4(x1)))) -> 2^1(4(x1)) 66.19/17.80 The remaining pairs can at least be oriented weakly. 66.19/17.80 Used ordering: Polynomial interpretation [POLO]: 66.19/17.80 66.19/17.80 POL(0(x_1)) = 0 66.19/17.80 POL(0^1(x_1)) = x_1 66.19/17.80 POL(1(x_1)) = 0 66.19/17.80 POL(2(x_1)) = x_1 66.19/17.80 POL(2^1(x_1)) = 0 66.19/17.80 POL(3(x_1)) = 0 66.19/17.80 POL(3^1(x_1)) = 0 66.19/17.80 POL(4(x_1)) = 0 66.19/17.80 POL(5(x_1)) = 1 66.19/17.80 66.19/17.80 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 66.19/17.80 66.19/17.80 0(4(1(3(0(x1))))) -> 3(0(1(0(1(4(x1)))))) 66.19/17.80 3(1(4(0(x1)))) -> 2(2(0(3(4(1(x1)))))) 66.19/17.80 3(2(4(0(x1)))) -> 2(0(3(4(5(x1))))) 66.19/17.80 3(1(5(0(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(3(4(3(x1))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(5(3(4(x1))))) 66.19/17.80 3(1(4(4(x1)))) -> 2(2(1(4(3(4(x1)))))) 66.19/17.80 3(1(4(0(0(x1))))) -> 3(2(0(1(0(4(x1)))))) 66.19/17.80 3(5(2(4(0(x1))))) -> 2(1(5(3(4(0(x1)))))) 66.19/17.80 3(5(0(1(5(x1))))) -> 1(3(1(0(5(5(x1)))))) 66.19/17.80 3(2(3(1(5(x1))))) -> 2(1(5(3(4(3(x1)))))) 66.19/17.80 0(0(0(x1))) -> 2(0(1(0(0(x1))))) 66.19/17.80 0(3(5(0(x1)))) -> 0(3(4(1(0(5(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 2(0(5(1(2(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 4(2(2(0(1(5(x1)))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(5(x1))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(2(5(x1)))))) 66.19/17.80 0(3(1(5(x1)))) -> 3(1(2(0(5(x1))))) 66.19/17.80 0(0(3(5(x1)))) -> 2(0(3(4(0(5(x1)))))) 66.19/17.80 0(0(2(3(0(x1))))) -> 3(2(0(1(0(0(x1)))))) 66.19/17.80 0(2(1(5(0(x1))))) -> 2(0(5(1(0(3(x1)))))) 66.19/17.80 0(2(2(4(4(x1))))) -> 4(2(2(0(1(4(x1)))))) 66.19/17.80 0(2(1(5(4(x1))))) -> 2(2(1(4(0(5(x1)))))) 66.19/17.80 0(0(3(1(5(x1))))) -> 0(2(1(0(5(3(x1)))))) 66.19/17.80 0(2(3(1(5(x1))))) -> 0(2(5(1(3(5(x1)))))) 66.19/17.80 66.19/17.80 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (8) 66.19/17.80 Obligation: 66.19/17.80 Q DP problem: 66.19/17.80 The TRS P consists of the following rules: 66.19/17.80 66.19/17.80 2^1(4(4(3(0(x1))))) -> 2^1(2(4(3(0(4(x1)))))) 66.19/17.80 2^1(3(1(0(x1)))) -> 3^1(0(x1)) 66.19/17.80 3^1(2(0(4(x1)))) -> 3^1(x1) 66.19/17.80 3^1(1(5(0(x1)))) -> 0^1(3(x1)) 66.19/17.80 0^1(0(1(5(x1)))) -> 2^1(5(x1)) 66.19/17.80 2^1(5(4(1(0(x1))))) -> 2^1(x1) 66.19/17.80 2^1(3(1(0(x1)))) -> 3^1(x1) 66.19/17.80 3^1(1(5(0(x1)))) -> 3^1(x1) 66.19/17.80 3^1(2(3(1(5(x1))))) -> 3^1(x1) 66.19/17.80 2^1(3(1(5(x1)))) -> 0^1(3(x1)) 66.19/17.80 0^1(0(2(3(0(x1))))) -> 0^1(0(x1)) 66.19/17.80 0^1(2(1(5(0(x1))))) -> 0^1(3(x1)) 66.19/17.80 0^1(2(1(5(0(x1))))) -> 3^1(x1) 66.19/17.80 0^1(0(3(1(5(x1))))) -> 3^1(x1) 66.19/17.80 2^1(3(1(5(x1)))) -> 3^1(x1) 66.19/17.80 2^1(4(4(3(0(x1))))) -> 2^1(4(3(0(4(x1))))) 66.19/17.80 2^1(4(4(3(0(x1))))) -> 3^1(0(4(x1))) 66.19/17.80 2^1(0(3(1(5(x1))))) -> 2^1(0(3(x1))) 66.19/17.80 2^1(0(3(1(5(x1))))) -> 0^1(3(x1)) 66.19/17.80 2^1(0(3(1(5(x1))))) -> 3^1(x1) 66.19/17.80 2^1(3(1(5(5(x1))))) -> 3^1(x1) 66.19/17.80 66.19/17.80 The TRS R consists of the following rules: 66.19/17.80 66.19/17.80 0(0(0(x1))) -> 2(0(1(0(0(x1))))) 66.19/17.80 2(3(0(x1))) -> 2(0(3(4(x1)))) 66.19/17.80 2(4(0(0(x1)))) -> 2(0(1(4(0(x1))))) 66.19/17.80 2(5(0(0(x1)))) -> 0(3(2(0(5(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(0(1(3(0(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(0(1(1(3(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(1(3(4(1(0(x1)))))) 66.19/17.80 3(1(4(0(x1)))) -> 2(2(0(3(4(1(x1)))))) 66.19/17.80 3(2(4(0(x1)))) -> 2(0(3(4(5(x1))))) 66.19/17.80 2(5(4(0(x1)))) -> 2(4(2(2(0(5(x1)))))) 66.19/17.80 3(1(5(0(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 0(3(5(0(x1)))) -> 0(3(4(1(0(5(x1)))))) 66.19/17.80 2(3(5(0(x1)))) -> 3(2(0(5(1(5(x1)))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(3(4(3(x1))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(5(3(4(x1))))) 66.19/17.80 3(1(4(4(x1)))) -> 2(2(1(4(3(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 2(0(5(1(2(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 4(2(2(0(1(5(x1)))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(5(x1))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(2(5(x1)))))) 66.19/17.80 0(3(1(5(x1)))) -> 3(1(2(0(5(x1))))) 66.19/17.80 2(3(1(5(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 2(3(1(5(x1)))) -> 2(2(5(1(1(3(x1)))))) 66.19/17.80 0(0(3(5(x1)))) -> 2(0(3(4(0(5(x1)))))) 66.19/17.80 3(1(4(0(0(x1))))) -> 3(2(0(1(0(4(x1)))))) 66.19/17.80 2(5(4(0(0(x1))))) -> 4(2(0(1(0(5(x1)))))) 66.19/17.80 2(3(5(0(0(x1))))) -> 3(2(0(5(1(0(x1)))))) 66.19/17.80 2(5(0(1(0(x1))))) -> 0(1(5(2(0(1(x1)))))) 66.19/17.80 2(5(4(1(0(x1))))) -> 4(2(0(5(1(2(x1)))))) 66.19/17.80 0(4(1(3(0(x1))))) -> 3(0(1(0(1(4(x1)))))) 66.19/17.80 0(0(2(3(0(x1))))) -> 3(2(0(1(0(0(x1)))))) 66.19/17.80 2(0(4(3(0(x1))))) -> 0(1(2(0(3(4(x1)))))) 66.19/17.80 2(0(4(3(0(x1))))) -> 0(3(2(0(3(4(x1)))))) 66.19/17.80 2(4(4(3(0(x1))))) -> 2(2(4(3(0(4(x1)))))) 66.19/17.80 3(5(2(4(0(x1))))) -> 2(1(5(3(4(0(x1)))))) 66.19/17.80 0(2(1(5(0(x1))))) -> 2(0(5(1(0(3(x1)))))) 66.19/17.80 0(2(2(4(4(x1))))) -> 4(2(2(0(1(4(x1)))))) 66.19/17.80 0(2(1(5(4(x1))))) -> 2(2(1(4(0(5(x1)))))) 66.19/17.80 2(3(2(5(4(x1))))) -> 2(2(5(3(4(5(x1)))))) 66.19/17.80 2(3(0(1(5(x1))))) -> 2(0(1(3(0(5(x1)))))) 66.19/17.80 3(5(0(1(5(x1))))) -> 1(3(1(0(5(5(x1)))))) 66.19/17.80 0(0(3(1(5(x1))))) -> 0(2(1(0(5(3(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 2(5(1(2(0(3(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 2(3(0(1(0(5(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 3(2(1(1(0(5(x1)))))) 66.19/17.80 0(2(3(1(5(x1))))) -> 0(2(5(1(3(5(x1)))))) 66.19/17.80 3(2(3(1(5(x1))))) -> 2(1(5(3(4(3(x1)))))) 66.19/17.80 2(5(4(1(5(x1))))) -> 2(5(1(4(1(5(x1)))))) 66.19/17.80 2(3(1(5(5(x1))))) -> 2(1(4(5(5(3(x1)))))) 66.19/17.80 66.19/17.80 Q is empty. 66.19/17.80 We have to consider all minimal (P,Q,R)-chains. 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (9) QDPOrderProof (EQUIVALENT) 66.19/17.80 We use the reduction pair processor [LPAR04,JAR06]. 66.19/17.80 66.19/17.80 66.19/17.80 The following pairs can be oriented strictly and are deleted. 66.19/17.80 66.19/17.80 0^1(0(1(5(x1)))) -> 2^1(5(x1)) 66.19/17.80 0^1(0(3(1(5(x1))))) -> 3^1(x1) 66.19/17.80 The remaining pairs can at least be oriented weakly. 66.19/17.80 Used ordering: Polynomial interpretation [POLO]: 66.19/17.80 66.19/17.80 POL(0(x_1)) = 1 66.19/17.80 POL(0^1(x_1)) = x_1 66.19/17.80 POL(1(x_1)) = 0 66.19/17.80 POL(2(x_1)) = 0 66.19/17.80 POL(2^1(x_1)) = 0 66.19/17.80 POL(3(x_1)) = 0 66.19/17.80 POL(3^1(x_1)) = 0 66.19/17.80 POL(4(x_1)) = 0 66.19/17.80 POL(5(x_1)) = 0 66.19/17.80 66.19/17.80 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 66.19/17.80 66.19/17.80 0(4(1(3(0(x1))))) -> 3(0(1(0(1(4(x1)))))) 66.19/17.80 3(1(4(0(x1)))) -> 2(2(0(3(4(1(x1)))))) 66.19/17.80 3(2(4(0(x1)))) -> 2(0(3(4(5(x1))))) 66.19/17.80 3(1(5(0(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(3(4(3(x1))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(5(3(4(x1))))) 66.19/17.80 3(1(4(4(x1)))) -> 2(2(1(4(3(4(x1)))))) 66.19/17.80 3(1(4(0(0(x1))))) -> 3(2(0(1(0(4(x1)))))) 66.19/17.80 3(5(2(4(0(x1))))) -> 2(1(5(3(4(0(x1)))))) 66.19/17.80 3(5(0(1(5(x1))))) -> 1(3(1(0(5(5(x1)))))) 66.19/17.80 3(2(3(1(5(x1))))) -> 2(1(5(3(4(3(x1)))))) 66.19/17.80 0(0(0(x1))) -> 2(0(1(0(0(x1))))) 66.19/17.80 0(3(5(0(x1)))) -> 0(3(4(1(0(5(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 2(0(5(1(2(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 4(2(2(0(1(5(x1)))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(5(x1))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(2(5(x1)))))) 66.19/17.80 0(3(1(5(x1)))) -> 3(1(2(0(5(x1))))) 66.19/17.80 0(0(3(5(x1)))) -> 2(0(3(4(0(5(x1)))))) 66.19/17.80 0(0(2(3(0(x1))))) -> 3(2(0(1(0(0(x1)))))) 66.19/17.80 0(2(1(5(0(x1))))) -> 2(0(5(1(0(3(x1)))))) 66.19/17.80 0(2(2(4(4(x1))))) -> 4(2(2(0(1(4(x1)))))) 66.19/17.80 0(2(1(5(4(x1))))) -> 2(2(1(4(0(5(x1)))))) 66.19/17.80 0(0(3(1(5(x1))))) -> 0(2(1(0(5(3(x1)))))) 66.19/17.80 0(2(3(1(5(x1))))) -> 0(2(5(1(3(5(x1)))))) 66.19/17.80 66.19/17.80 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (10) 66.19/17.80 Obligation: 66.19/17.80 Q DP problem: 66.19/17.80 The TRS P consists of the following rules: 66.19/17.80 66.19/17.80 2^1(4(4(3(0(x1))))) -> 2^1(2(4(3(0(4(x1)))))) 66.19/17.80 2^1(3(1(0(x1)))) -> 3^1(0(x1)) 66.19/17.80 3^1(2(0(4(x1)))) -> 3^1(x1) 66.19/17.80 3^1(1(5(0(x1)))) -> 0^1(3(x1)) 66.19/17.80 2^1(5(4(1(0(x1))))) -> 2^1(x1) 66.19/17.80 2^1(3(1(0(x1)))) -> 3^1(x1) 66.19/17.80 3^1(1(5(0(x1)))) -> 3^1(x1) 66.19/17.80 3^1(2(3(1(5(x1))))) -> 3^1(x1) 66.19/17.80 2^1(3(1(5(x1)))) -> 0^1(3(x1)) 66.19/17.80 0^1(0(2(3(0(x1))))) -> 0^1(0(x1)) 66.19/17.80 0^1(2(1(5(0(x1))))) -> 0^1(3(x1)) 66.19/17.80 0^1(2(1(5(0(x1))))) -> 3^1(x1) 66.19/17.80 2^1(3(1(5(x1)))) -> 3^1(x1) 66.19/17.80 2^1(4(4(3(0(x1))))) -> 2^1(4(3(0(4(x1))))) 66.19/17.80 2^1(4(4(3(0(x1))))) -> 3^1(0(4(x1))) 66.19/17.80 2^1(0(3(1(5(x1))))) -> 2^1(0(3(x1))) 66.19/17.80 2^1(0(3(1(5(x1))))) -> 0^1(3(x1)) 66.19/17.80 2^1(0(3(1(5(x1))))) -> 3^1(x1) 66.19/17.80 2^1(3(1(5(5(x1))))) -> 3^1(x1) 66.19/17.80 66.19/17.80 The TRS R consists of the following rules: 66.19/17.80 66.19/17.80 0(0(0(x1))) -> 2(0(1(0(0(x1))))) 66.19/17.80 2(3(0(x1))) -> 2(0(3(4(x1)))) 66.19/17.80 2(4(0(0(x1)))) -> 2(0(1(4(0(x1))))) 66.19/17.80 2(5(0(0(x1)))) -> 0(3(2(0(5(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(0(1(3(0(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(0(1(1(3(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(1(3(4(1(0(x1)))))) 66.19/17.80 3(1(4(0(x1)))) -> 2(2(0(3(4(1(x1)))))) 66.19/17.80 3(2(4(0(x1)))) -> 2(0(3(4(5(x1))))) 66.19/17.80 2(5(4(0(x1)))) -> 2(4(2(2(0(5(x1)))))) 66.19/17.80 3(1(5(0(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 0(3(5(0(x1)))) -> 0(3(4(1(0(5(x1)))))) 66.19/17.80 2(3(5(0(x1)))) -> 3(2(0(5(1(5(x1)))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(3(4(3(x1))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(5(3(4(x1))))) 66.19/17.80 3(1(4(4(x1)))) -> 2(2(1(4(3(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 2(0(5(1(2(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 4(2(2(0(1(5(x1)))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(5(x1))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(2(5(x1)))))) 66.19/17.80 0(3(1(5(x1)))) -> 3(1(2(0(5(x1))))) 66.19/17.80 2(3(1(5(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 2(3(1(5(x1)))) -> 2(2(5(1(1(3(x1)))))) 66.19/17.80 0(0(3(5(x1)))) -> 2(0(3(4(0(5(x1)))))) 66.19/17.80 3(1(4(0(0(x1))))) -> 3(2(0(1(0(4(x1)))))) 66.19/17.80 2(5(4(0(0(x1))))) -> 4(2(0(1(0(5(x1)))))) 66.19/17.80 2(3(5(0(0(x1))))) -> 3(2(0(5(1(0(x1)))))) 66.19/17.80 2(5(0(1(0(x1))))) -> 0(1(5(2(0(1(x1)))))) 66.19/17.80 2(5(4(1(0(x1))))) -> 4(2(0(5(1(2(x1)))))) 66.19/17.80 0(4(1(3(0(x1))))) -> 3(0(1(0(1(4(x1)))))) 66.19/17.80 0(0(2(3(0(x1))))) -> 3(2(0(1(0(0(x1)))))) 66.19/17.80 2(0(4(3(0(x1))))) -> 0(1(2(0(3(4(x1)))))) 66.19/17.80 2(0(4(3(0(x1))))) -> 0(3(2(0(3(4(x1)))))) 66.19/17.80 2(4(4(3(0(x1))))) -> 2(2(4(3(0(4(x1)))))) 66.19/17.80 3(5(2(4(0(x1))))) -> 2(1(5(3(4(0(x1)))))) 66.19/17.80 0(2(1(5(0(x1))))) -> 2(0(5(1(0(3(x1)))))) 66.19/17.80 0(2(2(4(4(x1))))) -> 4(2(2(0(1(4(x1)))))) 66.19/17.80 0(2(1(5(4(x1))))) -> 2(2(1(4(0(5(x1)))))) 66.19/17.80 2(3(2(5(4(x1))))) -> 2(2(5(3(4(5(x1)))))) 66.19/17.80 2(3(0(1(5(x1))))) -> 2(0(1(3(0(5(x1)))))) 66.19/17.80 3(5(0(1(5(x1))))) -> 1(3(1(0(5(5(x1)))))) 66.19/17.80 0(0(3(1(5(x1))))) -> 0(2(1(0(5(3(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 2(5(1(2(0(3(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 2(3(0(1(0(5(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 3(2(1(1(0(5(x1)))))) 66.19/17.80 0(2(3(1(5(x1))))) -> 0(2(5(1(3(5(x1)))))) 66.19/17.80 3(2(3(1(5(x1))))) -> 2(1(5(3(4(3(x1)))))) 66.19/17.80 2(5(4(1(5(x1))))) -> 2(5(1(4(1(5(x1)))))) 66.19/17.80 2(3(1(5(5(x1))))) -> 2(1(4(5(5(3(x1)))))) 66.19/17.80 66.19/17.80 Q is empty. 66.19/17.80 We have to consider all minimal (P,Q,R)-chains. 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (11) DependencyGraphProof (EQUIVALENT) 66.19/17.80 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 8 less nodes. 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (12) 66.19/17.80 Complex Obligation (AND) 66.19/17.80 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (13) 66.19/17.80 Obligation: 66.19/17.80 Q DP problem: 66.19/17.80 The TRS P consists of the following rules: 66.19/17.80 66.19/17.80 3^1(1(5(0(x1)))) -> 0^1(3(x1)) 66.19/17.80 0^1(0(2(3(0(x1))))) -> 0^1(0(x1)) 66.19/17.80 0^1(2(1(5(0(x1))))) -> 0^1(3(x1)) 66.19/17.80 0^1(2(1(5(0(x1))))) -> 3^1(x1) 66.19/17.80 3^1(1(5(0(x1)))) -> 3^1(x1) 66.19/17.80 3^1(2(0(4(x1)))) -> 3^1(x1) 66.19/17.80 3^1(2(3(1(5(x1))))) -> 3^1(x1) 66.19/17.80 66.19/17.80 The TRS R consists of the following rules: 66.19/17.80 66.19/17.80 0(0(0(x1))) -> 2(0(1(0(0(x1))))) 66.19/17.80 2(3(0(x1))) -> 2(0(3(4(x1)))) 66.19/17.80 2(4(0(0(x1)))) -> 2(0(1(4(0(x1))))) 66.19/17.80 2(5(0(0(x1)))) -> 0(3(2(0(5(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(0(1(3(0(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(0(1(1(3(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(1(3(4(1(0(x1)))))) 66.19/17.80 3(1(4(0(x1)))) -> 2(2(0(3(4(1(x1)))))) 66.19/17.80 3(2(4(0(x1)))) -> 2(0(3(4(5(x1))))) 66.19/17.80 2(5(4(0(x1)))) -> 2(4(2(2(0(5(x1)))))) 66.19/17.80 3(1(5(0(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 0(3(5(0(x1)))) -> 0(3(4(1(0(5(x1)))))) 66.19/17.80 2(3(5(0(x1)))) -> 3(2(0(5(1(5(x1)))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(3(4(3(x1))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(5(3(4(x1))))) 66.19/17.80 3(1(4(4(x1)))) -> 2(2(1(4(3(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 2(0(5(1(2(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 4(2(2(0(1(5(x1)))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(5(x1))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(2(5(x1)))))) 66.19/17.80 0(3(1(5(x1)))) -> 3(1(2(0(5(x1))))) 66.19/17.80 2(3(1(5(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 2(3(1(5(x1)))) -> 2(2(5(1(1(3(x1)))))) 66.19/17.80 0(0(3(5(x1)))) -> 2(0(3(4(0(5(x1)))))) 66.19/17.80 3(1(4(0(0(x1))))) -> 3(2(0(1(0(4(x1)))))) 66.19/17.80 2(5(4(0(0(x1))))) -> 4(2(0(1(0(5(x1)))))) 66.19/17.80 2(3(5(0(0(x1))))) -> 3(2(0(5(1(0(x1)))))) 66.19/17.80 2(5(0(1(0(x1))))) -> 0(1(5(2(0(1(x1)))))) 66.19/17.80 2(5(4(1(0(x1))))) -> 4(2(0(5(1(2(x1)))))) 66.19/17.80 0(4(1(3(0(x1))))) -> 3(0(1(0(1(4(x1)))))) 66.19/17.80 0(0(2(3(0(x1))))) -> 3(2(0(1(0(0(x1)))))) 66.19/17.80 2(0(4(3(0(x1))))) -> 0(1(2(0(3(4(x1)))))) 66.19/17.80 2(0(4(3(0(x1))))) -> 0(3(2(0(3(4(x1)))))) 66.19/17.80 2(4(4(3(0(x1))))) -> 2(2(4(3(0(4(x1)))))) 66.19/17.80 3(5(2(4(0(x1))))) -> 2(1(5(3(4(0(x1)))))) 66.19/17.80 0(2(1(5(0(x1))))) -> 2(0(5(1(0(3(x1)))))) 66.19/17.80 0(2(2(4(4(x1))))) -> 4(2(2(0(1(4(x1)))))) 66.19/17.80 0(2(1(5(4(x1))))) -> 2(2(1(4(0(5(x1)))))) 66.19/17.80 2(3(2(5(4(x1))))) -> 2(2(5(3(4(5(x1)))))) 66.19/17.80 2(3(0(1(5(x1))))) -> 2(0(1(3(0(5(x1)))))) 66.19/17.80 3(5(0(1(5(x1))))) -> 1(3(1(0(5(5(x1)))))) 66.19/17.80 0(0(3(1(5(x1))))) -> 0(2(1(0(5(3(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 2(5(1(2(0(3(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 2(3(0(1(0(5(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 3(2(1(1(0(5(x1)))))) 66.19/17.80 0(2(3(1(5(x1))))) -> 0(2(5(1(3(5(x1)))))) 66.19/17.80 3(2(3(1(5(x1))))) -> 2(1(5(3(4(3(x1)))))) 66.19/17.80 2(5(4(1(5(x1))))) -> 2(5(1(4(1(5(x1)))))) 66.19/17.80 2(3(1(5(5(x1))))) -> 2(1(4(5(5(3(x1)))))) 66.19/17.80 66.19/17.80 Q is empty. 66.19/17.80 We have to consider all minimal (P,Q,R)-chains. 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (14) QDPOrderProof (EQUIVALENT) 66.19/17.80 We use the reduction pair processor [LPAR04,JAR06]. 66.19/17.80 66.19/17.80 66.19/17.80 The following pairs can be oriented strictly and are deleted. 66.19/17.80 66.19/17.80 3^1(1(5(0(x1)))) -> 0^1(3(x1)) 66.19/17.80 0^1(2(1(5(0(x1))))) -> 0^1(3(x1)) 66.19/17.80 The remaining pairs can at least be oriented weakly. 66.19/17.80 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 66.19/17.80 66.19/17.80 POL( 0^1_1(x_1) ) = max{0, 2x_1 - 1} 66.19/17.80 POL( 3_1(x_1) ) = 0 66.19/17.80 POL( 1_1(x_1) ) = max{0, 2x_1 - 2} 66.19/17.80 POL( 4_1(x_1) ) = max{0, -2} 66.19/17.80 POL( 0_1(x_1) ) = 1 66.19/17.80 POL( 2_1(x_1) ) = max{0, x_1 - 1} 66.19/17.80 POL( 5_1(x_1) ) = 2x_1 66.19/17.80 POL( 3^1_1(x_1) ) = 1 66.19/17.80 66.19/17.80 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 66.19/17.80 66.19/17.80 3(1(4(0(x1)))) -> 2(2(0(3(4(1(x1)))))) 66.19/17.80 3(2(4(0(x1)))) -> 2(0(3(4(5(x1))))) 66.19/17.80 3(1(5(0(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(3(4(3(x1))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(5(3(4(x1))))) 66.19/17.80 3(1(4(4(x1)))) -> 2(2(1(4(3(4(x1)))))) 66.19/17.80 3(1(4(0(0(x1))))) -> 3(2(0(1(0(4(x1)))))) 66.19/17.80 3(5(2(4(0(x1))))) -> 2(1(5(3(4(0(x1)))))) 66.19/17.80 3(5(0(1(5(x1))))) -> 1(3(1(0(5(5(x1)))))) 66.19/17.80 3(2(3(1(5(x1))))) -> 2(1(5(3(4(3(x1)))))) 66.19/17.80 0(0(0(x1))) -> 2(0(1(0(0(x1))))) 66.19/17.80 0(3(5(0(x1)))) -> 0(3(4(1(0(5(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 2(0(5(1(2(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 4(2(2(0(1(5(x1)))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(5(x1))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(2(5(x1)))))) 66.19/17.80 0(3(1(5(x1)))) -> 3(1(2(0(5(x1))))) 66.19/17.80 0(0(3(5(x1)))) -> 2(0(3(4(0(5(x1)))))) 66.19/17.80 0(4(1(3(0(x1))))) -> 3(0(1(0(1(4(x1)))))) 66.19/17.80 0(0(2(3(0(x1))))) -> 3(2(0(1(0(0(x1)))))) 66.19/17.80 0(2(1(5(0(x1))))) -> 2(0(5(1(0(3(x1)))))) 66.19/17.80 0(2(2(4(4(x1))))) -> 4(2(2(0(1(4(x1)))))) 66.19/17.80 0(2(1(5(4(x1))))) -> 2(2(1(4(0(5(x1)))))) 66.19/17.80 0(0(3(1(5(x1))))) -> 0(2(1(0(5(3(x1)))))) 66.19/17.80 0(2(3(1(5(x1))))) -> 0(2(5(1(3(5(x1)))))) 66.19/17.80 66.19/17.80 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (15) 66.19/17.80 Obligation: 66.19/17.80 Q DP problem: 66.19/17.80 The TRS P consists of the following rules: 66.19/17.80 66.19/17.80 0^1(0(2(3(0(x1))))) -> 0^1(0(x1)) 66.19/17.80 0^1(2(1(5(0(x1))))) -> 3^1(x1) 66.19/17.80 3^1(1(5(0(x1)))) -> 3^1(x1) 66.19/17.80 3^1(2(0(4(x1)))) -> 3^1(x1) 66.19/17.80 3^1(2(3(1(5(x1))))) -> 3^1(x1) 66.19/17.80 66.19/17.80 The TRS R consists of the following rules: 66.19/17.80 66.19/17.80 0(0(0(x1))) -> 2(0(1(0(0(x1))))) 66.19/17.80 2(3(0(x1))) -> 2(0(3(4(x1)))) 66.19/17.80 2(4(0(0(x1)))) -> 2(0(1(4(0(x1))))) 66.19/17.80 2(5(0(0(x1)))) -> 0(3(2(0(5(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(0(1(3(0(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(0(1(1(3(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(1(3(4(1(0(x1)))))) 66.19/17.80 3(1(4(0(x1)))) -> 2(2(0(3(4(1(x1)))))) 66.19/17.80 3(2(4(0(x1)))) -> 2(0(3(4(5(x1))))) 66.19/17.80 2(5(4(0(x1)))) -> 2(4(2(2(0(5(x1)))))) 66.19/17.80 3(1(5(0(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 0(3(5(0(x1)))) -> 0(3(4(1(0(5(x1)))))) 66.19/17.80 2(3(5(0(x1)))) -> 3(2(0(5(1(5(x1)))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(3(4(3(x1))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(5(3(4(x1))))) 66.19/17.80 3(1(4(4(x1)))) -> 2(2(1(4(3(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 2(0(5(1(2(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 4(2(2(0(1(5(x1)))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(5(x1))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(2(5(x1)))))) 66.19/17.80 0(3(1(5(x1)))) -> 3(1(2(0(5(x1))))) 66.19/17.80 2(3(1(5(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 2(3(1(5(x1)))) -> 2(2(5(1(1(3(x1)))))) 66.19/17.80 0(0(3(5(x1)))) -> 2(0(3(4(0(5(x1)))))) 66.19/17.80 3(1(4(0(0(x1))))) -> 3(2(0(1(0(4(x1)))))) 66.19/17.80 2(5(4(0(0(x1))))) -> 4(2(0(1(0(5(x1)))))) 66.19/17.80 2(3(5(0(0(x1))))) -> 3(2(0(5(1(0(x1)))))) 66.19/17.80 2(5(0(1(0(x1))))) -> 0(1(5(2(0(1(x1)))))) 66.19/17.80 2(5(4(1(0(x1))))) -> 4(2(0(5(1(2(x1)))))) 66.19/17.80 0(4(1(3(0(x1))))) -> 3(0(1(0(1(4(x1)))))) 66.19/17.80 0(0(2(3(0(x1))))) -> 3(2(0(1(0(0(x1)))))) 66.19/17.80 2(0(4(3(0(x1))))) -> 0(1(2(0(3(4(x1)))))) 66.19/17.80 2(0(4(3(0(x1))))) -> 0(3(2(0(3(4(x1)))))) 66.19/17.80 2(4(4(3(0(x1))))) -> 2(2(4(3(0(4(x1)))))) 66.19/17.80 3(5(2(4(0(x1))))) -> 2(1(5(3(4(0(x1)))))) 66.19/17.80 0(2(1(5(0(x1))))) -> 2(0(5(1(0(3(x1)))))) 66.19/17.80 0(2(2(4(4(x1))))) -> 4(2(2(0(1(4(x1)))))) 66.19/17.80 0(2(1(5(4(x1))))) -> 2(2(1(4(0(5(x1)))))) 66.19/17.80 2(3(2(5(4(x1))))) -> 2(2(5(3(4(5(x1)))))) 66.19/17.80 2(3(0(1(5(x1))))) -> 2(0(1(3(0(5(x1)))))) 66.19/17.80 3(5(0(1(5(x1))))) -> 1(3(1(0(5(5(x1)))))) 66.19/17.80 0(0(3(1(5(x1))))) -> 0(2(1(0(5(3(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 2(5(1(2(0(3(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 2(3(0(1(0(5(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 3(2(1(1(0(5(x1)))))) 66.19/17.80 0(2(3(1(5(x1))))) -> 0(2(5(1(3(5(x1)))))) 66.19/17.80 3(2(3(1(5(x1))))) -> 2(1(5(3(4(3(x1)))))) 66.19/17.80 2(5(4(1(5(x1))))) -> 2(5(1(4(1(5(x1)))))) 66.19/17.80 2(3(1(5(5(x1))))) -> 2(1(4(5(5(3(x1)))))) 66.19/17.80 66.19/17.80 Q is empty. 66.19/17.80 We have to consider all minimal (P,Q,R)-chains. 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (16) DependencyGraphProof (EQUIVALENT) 66.19/17.80 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (17) 66.19/17.80 Complex Obligation (AND) 66.19/17.80 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (18) 66.19/17.80 Obligation: 66.19/17.80 Q DP problem: 66.19/17.80 The TRS P consists of the following rules: 66.19/17.80 66.19/17.80 3^1(2(0(4(x1)))) -> 3^1(x1) 66.19/17.80 3^1(1(5(0(x1)))) -> 3^1(x1) 66.19/17.80 3^1(2(3(1(5(x1))))) -> 3^1(x1) 66.19/17.80 66.19/17.80 The TRS R consists of the following rules: 66.19/17.80 66.19/17.80 0(0(0(x1))) -> 2(0(1(0(0(x1))))) 66.19/17.80 2(3(0(x1))) -> 2(0(3(4(x1)))) 66.19/17.80 2(4(0(0(x1)))) -> 2(0(1(4(0(x1))))) 66.19/17.80 2(5(0(0(x1)))) -> 0(3(2(0(5(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(0(1(3(0(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(0(1(1(3(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(1(3(4(1(0(x1)))))) 66.19/17.80 3(1(4(0(x1)))) -> 2(2(0(3(4(1(x1)))))) 66.19/17.80 3(2(4(0(x1)))) -> 2(0(3(4(5(x1))))) 66.19/17.80 2(5(4(0(x1)))) -> 2(4(2(2(0(5(x1)))))) 66.19/17.80 3(1(5(0(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 0(3(5(0(x1)))) -> 0(3(4(1(0(5(x1)))))) 66.19/17.80 2(3(5(0(x1)))) -> 3(2(0(5(1(5(x1)))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(3(4(3(x1))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(5(3(4(x1))))) 66.19/17.80 3(1(4(4(x1)))) -> 2(2(1(4(3(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 2(0(5(1(2(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 4(2(2(0(1(5(x1)))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(5(x1))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(2(5(x1)))))) 66.19/17.80 0(3(1(5(x1)))) -> 3(1(2(0(5(x1))))) 66.19/17.80 2(3(1(5(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 2(3(1(5(x1)))) -> 2(2(5(1(1(3(x1)))))) 66.19/17.80 0(0(3(5(x1)))) -> 2(0(3(4(0(5(x1)))))) 66.19/17.80 3(1(4(0(0(x1))))) -> 3(2(0(1(0(4(x1)))))) 66.19/17.80 2(5(4(0(0(x1))))) -> 4(2(0(1(0(5(x1)))))) 66.19/17.80 2(3(5(0(0(x1))))) -> 3(2(0(5(1(0(x1)))))) 66.19/17.80 2(5(0(1(0(x1))))) -> 0(1(5(2(0(1(x1)))))) 66.19/17.80 2(5(4(1(0(x1))))) -> 4(2(0(5(1(2(x1)))))) 66.19/17.80 0(4(1(3(0(x1))))) -> 3(0(1(0(1(4(x1)))))) 66.19/17.80 0(0(2(3(0(x1))))) -> 3(2(0(1(0(0(x1)))))) 66.19/17.80 2(0(4(3(0(x1))))) -> 0(1(2(0(3(4(x1)))))) 66.19/17.80 2(0(4(3(0(x1))))) -> 0(3(2(0(3(4(x1)))))) 66.19/17.80 2(4(4(3(0(x1))))) -> 2(2(4(3(0(4(x1)))))) 66.19/17.80 3(5(2(4(0(x1))))) -> 2(1(5(3(4(0(x1)))))) 66.19/17.80 0(2(1(5(0(x1))))) -> 2(0(5(1(0(3(x1)))))) 66.19/17.80 0(2(2(4(4(x1))))) -> 4(2(2(0(1(4(x1)))))) 66.19/17.80 0(2(1(5(4(x1))))) -> 2(2(1(4(0(5(x1)))))) 66.19/17.80 2(3(2(5(4(x1))))) -> 2(2(5(3(4(5(x1)))))) 66.19/17.80 2(3(0(1(5(x1))))) -> 2(0(1(3(0(5(x1)))))) 66.19/17.80 3(5(0(1(5(x1))))) -> 1(3(1(0(5(5(x1)))))) 66.19/17.80 0(0(3(1(5(x1))))) -> 0(2(1(0(5(3(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 2(5(1(2(0(3(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 2(3(0(1(0(5(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 3(2(1(1(0(5(x1)))))) 66.19/17.80 0(2(3(1(5(x1))))) -> 0(2(5(1(3(5(x1)))))) 66.19/17.80 3(2(3(1(5(x1))))) -> 2(1(5(3(4(3(x1)))))) 66.19/17.80 2(5(4(1(5(x1))))) -> 2(5(1(4(1(5(x1)))))) 66.19/17.80 2(3(1(5(5(x1))))) -> 2(1(4(5(5(3(x1)))))) 66.19/17.80 66.19/17.80 Q is empty. 66.19/17.80 We have to consider all minimal (P,Q,R)-chains. 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (19) UsableRulesProof (EQUIVALENT) 66.19/17.80 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. 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (20) 66.19/17.80 Obligation: 66.19/17.80 Q DP problem: 66.19/17.80 The TRS P consists of the following rules: 66.19/17.80 66.19/17.80 3^1(2(0(4(x1)))) -> 3^1(x1) 66.19/17.80 3^1(1(5(0(x1)))) -> 3^1(x1) 66.19/17.80 3^1(2(3(1(5(x1))))) -> 3^1(x1) 66.19/17.80 66.19/17.80 R is empty. 66.19/17.80 Q is empty. 66.19/17.80 We have to consider all minimal (P,Q,R)-chains. 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (21) QDPSizeChangeProof (EQUIVALENT) 66.19/17.80 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. 66.19/17.80 66.19/17.80 From the DPs we obtained the following set of size-change graphs: 66.19/17.80 *3^1(2(0(4(x1)))) -> 3^1(x1) 66.19/17.80 The graph contains the following edges 1 > 1 66.19/17.80 66.19/17.80 66.19/17.80 *3^1(1(5(0(x1)))) -> 3^1(x1) 66.19/17.80 The graph contains the following edges 1 > 1 66.19/17.80 66.19/17.80 66.19/17.80 *3^1(2(3(1(5(x1))))) -> 3^1(x1) 66.19/17.80 The graph contains the following edges 1 > 1 66.19/17.80 66.19/17.80 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (22) 66.19/17.80 YES 66.19/17.80 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (23) 66.19/17.80 Obligation: 66.19/17.80 Q DP problem: 66.19/17.80 The TRS P consists of the following rules: 66.19/17.80 66.19/17.80 0^1(0(2(3(0(x1))))) -> 0^1(0(x1)) 66.19/17.80 66.19/17.80 The TRS R consists of the following rules: 66.19/17.80 66.19/17.80 0(0(0(x1))) -> 2(0(1(0(0(x1))))) 66.19/17.80 2(3(0(x1))) -> 2(0(3(4(x1)))) 66.19/17.80 2(4(0(0(x1)))) -> 2(0(1(4(0(x1))))) 66.19/17.80 2(5(0(0(x1)))) -> 0(3(2(0(5(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(0(1(3(0(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(0(1(1(3(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(1(3(4(1(0(x1)))))) 66.19/17.80 3(1(4(0(x1)))) -> 2(2(0(3(4(1(x1)))))) 66.19/17.80 3(2(4(0(x1)))) -> 2(0(3(4(5(x1))))) 66.19/17.80 2(5(4(0(x1)))) -> 2(4(2(2(0(5(x1)))))) 66.19/17.80 3(1(5(0(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 0(3(5(0(x1)))) -> 0(3(4(1(0(5(x1)))))) 66.19/17.80 2(3(5(0(x1)))) -> 3(2(0(5(1(5(x1)))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(3(4(3(x1))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(5(3(4(x1))))) 66.19/17.80 3(1(4(4(x1)))) -> 2(2(1(4(3(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 2(0(5(1(2(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 4(2(2(0(1(5(x1)))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(5(x1))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(2(5(x1)))))) 66.19/17.80 0(3(1(5(x1)))) -> 3(1(2(0(5(x1))))) 66.19/17.80 2(3(1(5(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 2(3(1(5(x1)))) -> 2(2(5(1(1(3(x1)))))) 66.19/17.80 0(0(3(5(x1)))) -> 2(0(3(4(0(5(x1)))))) 66.19/17.80 3(1(4(0(0(x1))))) -> 3(2(0(1(0(4(x1)))))) 66.19/17.80 2(5(4(0(0(x1))))) -> 4(2(0(1(0(5(x1)))))) 66.19/17.80 2(3(5(0(0(x1))))) -> 3(2(0(5(1(0(x1)))))) 66.19/17.80 2(5(0(1(0(x1))))) -> 0(1(5(2(0(1(x1)))))) 66.19/17.80 2(5(4(1(0(x1))))) -> 4(2(0(5(1(2(x1)))))) 66.19/17.80 0(4(1(3(0(x1))))) -> 3(0(1(0(1(4(x1)))))) 66.19/17.80 0(0(2(3(0(x1))))) -> 3(2(0(1(0(0(x1)))))) 66.19/17.80 2(0(4(3(0(x1))))) -> 0(1(2(0(3(4(x1)))))) 66.19/17.80 2(0(4(3(0(x1))))) -> 0(3(2(0(3(4(x1)))))) 66.19/17.80 2(4(4(3(0(x1))))) -> 2(2(4(3(0(4(x1)))))) 66.19/17.80 3(5(2(4(0(x1))))) -> 2(1(5(3(4(0(x1)))))) 66.19/17.80 0(2(1(5(0(x1))))) -> 2(0(5(1(0(3(x1)))))) 66.19/17.80 0(2(2(4(4(x1))))) -> 4(2(2(0(1(4(x1)))))) 66.19/17.80 0(2(1(5(4(x1))))) -> 2(2(1(4(0(5(x1)))))) 66.19/17.80 2(3(2(5(4(x1))))) -> 2(2(5(3(4(5(x1)))))) 66.19/17.80 2(3(0(1(5(x1))))) -> 2(0(1(3(0(5(x1)))))) 66.19/17.80 3(5(0(1(5(x1))))) -> 1(3(1(0(5(5(x1)))))) 66.19/17.80 0(0(3(1(5(x1))))) -> 0(2(1(0(5(3(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 2(5(1(2(0(3(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 2(3(0(1(0(5(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 3(2(1(1(0(5(x1)))))) 66.19/17.80 0(2(3(1(5(x1))))) -> 0(2(5(1(3(5(x1)))))) 66.19/17.80 3(2(3(1(5(x1))))) -> 2(1(5(3(4(3(x1)))))) 66.19/17.80 2(5(4(1(5(x1))))) -> 2(5(1(4(1(5(x1)))))) 66.19/17.80 2(3(1(5(5(x1))))) -> 2(1(4(5(5(3(x1)))))) 66.19/17.80 66.19/17.80 Q is empty. 66.19/17.80 We have to consider all minimal (P,Q,R)-chains. 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (24) QDPSizeChangeProof (EQUIVALENT) 66.19/17.80 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. 66.19/17.80 66.19/17.80 From the DPs we obtained the following set of size-change graphs: 66.19/17.80 *0^1(0(2(3(0(x1))))) -> 0^1(0(x1)) 66.19/17.80 The graph contains the following edges 1 > 1 66.19/17.80 66.19/17.80 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (25) 66.19/17.80 YES 66.19/17.80 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (26) 66.19/17.80 Obligation: 66.19/17.80 Q DP problem: 66.19/17.80 The TRS P consists of the following rules: 66.19/17.80 66.19/17.80 2^1(4(4(3(0(x1))))) -> 2^1(4(3(0(4(x1))))) 66.19/17.80 2^1(4(4(3(0(x1))))) -> 2^1(2(4(3(0(4(x1)))))) 66.19/17.80 66.19/17.80 The TRS R consists of the following rules: 66.19/17.80 66.19/17.80 0(0(0(x1))) -> 2(0(1(0(0(x1))))) 66.19/17.80 2(3(0(x1))) -> 2(0(3(4(x1)))) 66.19/17.80 2(4(0(0(x1)))) -> 2(0(1(4(0(x1))))) 66.19/17.80 2(5(0(0(x1)))) -> 0(3(2(0(5(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(0(1(3(0(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(0(1(1(3(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(1(3(4(1(0(x1)))))) 66.19/17.80 3(1(4(0(x1)))) -> 2(2(0(3(4(1(x1)))))) 66.19/17.80 3(2(4(0(x1)))) -> 2(0(3(4(5(x1))))) 66.19/17.80 2(5(4(0(x1)))) -> 2(4(2(2(0(5(x1)))))) 66.19/17.80 3(1(5(0(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 0(3(5(0(x1)))) -> 0(3(4(1(0(5(x1)))))) 66.19/17.80 2(3(5(0(x1)))) -> 3(2(0(5(1(5(x1)))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(3(4(3(x1))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(5(3(4(x1))))) 66.19/17.80 3(1(4(4(x1)))) -> 2(2(1(4(3(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 2(0(5(1(2(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 4(2(2(0(1(5(x1)))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(5(x1))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(2(5(x1)))))) 66.19/17.80 0(3(1(5(x1)))) -> 3(1(2(0(5(x1))))) 66.19/17.80 2(3(1(5(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 2(3(1(5(x1)))) -> 2(2(5(1(1(3(x1)))))) 66.19/17.80 0(0(3(5(x1)))) -> 2(0(3(4(0(5(x1)))))) 66.19/17.80 3(1(4(0(0(x1))))) -> 3(2(0(1(0(4(x1)))))) 66.19/17.80 2(5(4(0(0(x1))))) -> 4(2(0(1(0(5(x1)))))) 66.19/17.80 2(3(5(0(0(x1))))) -> 3(2(0(5(1(0(x1)))))) 66.19/17.80 2(5(0(1(0(x1))))) -> 0(1(5(2(0(1(x1)))))) 66.19/17.80 2(5(4(1(0(x1))))) -> 4(2(0(5(1(2(x1)))))) 66.19/17.80 0(4(1(3(0(x1))))) -> 3(0(1(0(1(4(x1)))))) 66.19/17.80 0(0(2(3(0(x1))))) -> 3(2(0(1(0(0(x1)))))) 66.19/17.80 2(0(4(3(0(x1))))) -> 0(1(2(0(3(4(x1)))))) 66.19/17.80 2(0(4(3(0(x1))))) -> 0(3(2(0(3(4(x1)))))) 66.19/17.80 2(4(4(3(0(x1))))) -> 2(2(4(3(0(4(x1)))))) 66.19/17.80 3(5(2(4(0(x1))))) -> 2(1(5(3(4(0(x1)))))) 66.19/17.80 0(2(1(5(0(x1))))) -> 2(0(5(1(0(3(x1)))))) 66.19/17.80 0(2(2(4(4(x1))))) -> 4(2(2(0(1(4(x1)))))) 66.19/17.80 0(2(1(5(4(x1))))) -> 2(2(1(4(0(5(x1)))))) 66.19/17.80 2(3(2(5(4(x1))))) -> 2(2(5(3(4(5(x1)))))) 66.19/17.80 2(3(0(1(5(x1))))) -> 2(0(1(3(0(5(x1)))))) 66.19/17.80 3(5(0(1(5(x1))))) -> 1(3(1(0(5(5(x1)))))) 66.19/17.80 0(0(3(1(5(x1))))) -> 0(2(1(0(5(3(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 2(5(1(2(0(3(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 2(3(0(1(0(5(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 3(2(1(1(0(5(x1)))))) 66.19/17.80 0(2(3(1(5(x1))))) -> 0(2(5(1(3(5(x1)))))) 66.19/17.80 3(2(3(1(5(x1))))) -> 2(1(5(3(4(3(x1)))))) 66.19/17.80 2(5(4(1(5(x1))))) -> 2(5(1(4(1(5(x1)))))) 66.19/17.80 2(3(1(5(5(x1))))) -> 2(1(4(5(5(3(x1)))))) 66.19/17.80 66.19/17.80 Q is empty. 66.19/17.80 We have to consider all minimal (P,Q,R)-chains. 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (27) QDPOrderProof (EQUIVALENT) 66.19/17.80 We use the reduction pair processor [LPAR04,JAR06]. 66.19/17.80 66.19/17.80 66.19/17.80 The following pairs can be oriented strictly and are deleted. 66.19/17.80 66.19/17.80 2^1(4(4(3(0(x1))))) -> 2^1(4(3(0(4(x1))))) 66.19/17.80 2^1(4(4(3(0(x1))))) -> 2^1(2(4(3(0(4(x1)))))) 66.19/17.80 The remaining pairs can at least be oriented weakly. 66.19/17.80 Used ordering: Polynomial interpretation [POLO]: 66.19/17.80 66.19/17.80 POL(0(x_1)) = 0 66.19/17.80 POL(1(x_1)) = 0 66.19/17.80 POL(2(x_1)) = x_1 66.19/17.80 POL(2^1(x_1)) = x_1 66.19/17.80 POL(3(x_1)) = 0 66.19/17.80 POL(4(x_1)) = 1 + x_1 66.19/17.80 POL(5(x_1)) = x_1 66.19/17.80 66.19/17.80 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 66.19/17.80 66.19/17.80 3(1(4(0(x1)))) -> 2(2(0(3(4(1(x1)))))) 66.19/17.80 3(2(4(0(x1)))) -> 2(0(3(4(5(x1))))) 66.19/17.80 3(1(5(0(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(3(4(3(x1))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(5(3(4(x1))))) 66.19/17.80 3(1(4(4(x1)))) -> 2(2(1(4(3(4(x1)))))) 66.19/17.80 3(1(4(0(0(x1))))) -> 3(2(0(1(0(4(x1)))))) 66.19/17.80 3(5(2(4(0(x1))))) -> 2(1(5(3(4(0(x1)))))) 66.19/17.80 3(5(0(1(5(x1))))) -> 1(3(1(0(5(5(x1)))))) 66.19/17.80 3(2(3(1(5(x1))))) -> 2(1(5(3(4(3(x1)))))) 66.19/17.80 2(4(0(0(x1)))) -> 2(0(1(4(0(x1))))) 66.19/17.80 2(4(4(3(0(x1))))) -> 2(2(4(3(0(4(x1)))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(0(1(3(0(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(0(1(1(3(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(1(3(4(1(0(x1)))))) 66.19/17.80 2(3(1(5(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 2(3(1(5(x1)))) -> 2(2(5(1(1(3(x1)))))) 66.19/17.80 2(3(5(0(0(x1))))) -> 3(2(0(5(1(0(x1)))))) 66.19/17.80 2(5(4(1(0(x1))))) -> 4(2(0(5(1(2(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 2(5(1(2(0(3(x1)))))) 66.19/17.80 2(3(1(5(5(x1))))) -> 2(1(4(5(5(3(x1)))))) 66.19/17.80 2(5(0(0(x1)))) -> 0(3(2(0(5(x1))))) 66.19/17.80 2(5(4(0(x1)))) -> 2(4(2(2(0(5(x1)))))) 66.19/17.80 2(5(4(0(0(x1))))) -> 4(2(0(1(0(5(x1)))))) 66.19/17.80 2(5(0(1(0(x1))))) -> 0(1(5(2(0(1(x1)))))) 66.19/17.80 2(5(4(1(5(x1))))) -> 2(5(1(4(1(5(x1)))))) 66.19/17.80 2(3(0(x1))) -> 2(0(3(4(x1)))) 66.19/17.80 2(3(5(0(x1)))) -> 3(2(0(5(1(5(x1)))))) 66.19/17.80 2(0(4(3(0(x1))))) -> 0(1(2(0(3(4(x1)))))) 66.19/17.80 2(0(4(3(0(x1))))) -> 0(3(2(0(3(4(x1)))))) 66.19/17.80 2(3(2(5(4(x1))))) -> 2(2(5(3(4(5(x1)))))) 66.19/17.80 2(3(0(1(5(x1))))) -> 2(0(1(3(0(5(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 2(3(0(1(0(5(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 3(2(1(1(0(5(x1)))))) 66.19/17.80 66.19/17.80 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (28) 66.19/17.80 Obligation: 66.19/17.80 Q DP problem: 66.19/17.80 P is empty. 66.19/17.80 The TRS R consists of the following rules: 66.19/17.80 66.19/17.80 0(0(0(x1))) -> 2(0(1(0(0(x1))))) 66.19/17.80 2(3(0(x1))) -> 2(0(3(4(x1)))) 66.19/17.80 2(4(0(0(x1)))) -> 2(0(1(4(0(x1))))) 66.19/17.80 2(5(0(0(x1)))) -> 0(3(2(0(5(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(0(1(3(0(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(0(1(1(3(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(1(3(4(1(0(x1)))))) 66.19/17.80 3(1(4(0(x1)))) -> 2(2(0(3(4(1(x1)))))) 66.19/17.80 3(2(4(0(x1)))) -> 2(0(3(4(5(x1))))) 66.19/17.80 2(5(4(0(x1)))) -> 2(4(2(2(0(5(x1)))))) 66.19/17.80 3(1(5(0(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 0(3(5(0(x1)))) -> 0(3(4(1(0(5(x1)))))) 66.19/17.80 2(3(5(0(x1)))) -> 3(2(0(5(1(5(x1)))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(3(4(3(x1))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(5(3(4(x1))))) 66.19/17.80 3(1(4(4(x1)))) -> 2(2(1(4(3(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 2(0(5(1(2(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 4(2(2(0(1(5(x1)))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(5(x1))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(2(5(x1)))))) 66.19/17.80 0(3(1(5(x1)))) -> 3(1(2(0(5(x1))))) 66.19/17.80 2(3(1(5(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 2(3(1(5(x1)))) -> 2(2(5(1(1(3(x1)))))) 66.19/17.80 0(0(3(5(x1)))) -> 2(0(3(4(0(5(x1)))))) 66.19/17.80 3(1(4(0(0(x1))))) -> 3(2(0(1(0(4(x1)))))) 66.19/17.80 2(5(4(0(0(x1))))) -> 4(2(0(1(0(5(x1)))))) 66.19/17.80 2(3(5(0(0(x1))))) -> 3(2(0(5(1(0(x1)))))) 66.19/17.80 2(5(0(1(0(x1))))) -> 0(1(5(2(0(1(x1)))))) 66.19/17.80 2(5(4(1(0(x1))))) -> 4(2(0(5(1(2(x1)))))) 66.19/17.80 0(4(1(3(0(x1))))) -> 3(0(1(0(1(4(x1)))))) 66.19/17.80 0(0(2(3(0(x1))))) -> 3(2(0(1(0(0(x1)))))) 66.19/17.80 2(0(4(3(0(x1))))) -> 0(1(2(0(3(4(x1)))))) 66.19/17.80 2(0(4(3(0(x1))))) -> 0(3(2(0(3(4(x1)))))) 66.19/17.80 2(4(4(3(0(x1))))) -> 2(2(4(3(0(4(x1)))))) 66.19/17.80 3(5(2(4(0(x1))))) -> 2(1(5(3(4(0(x1)))))) 66.19/17.80 0(2(1(5(0(x1))))) -> 2(0(5(1(0(3(x1)))))) 66.19/17.80 0(2(2(4(4(x1))))) -> 4(2(2(0(1(4(x1)))))) 66.19/17.80 0(2(1(5(4(x1))))) -> 2(2(1(4(0(5(x1)))))) 66.19/17.80 2(3(2(5(4(x1))))) -> 2(2(5(3(4(5(x1)))))) 66.19/17.80 2(3(0(1(5(x1))))) -> 2(0(1(3(0(5(x1)))))) 66.19/17.80 3(5(0(1(5(x1))))) -> 1(3(1(0(5(5(x1)))))) 66.19/17.80 0(0(3(1(5(x1))))) -> 0(2(1(0(5(3(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 2(5(1(2(0(3(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 2(3(0(1(0(5(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 3(2(1(1(0(5(x1)))))) 66.19/17.80 0(2(3(1(5(x1))))) -> 0(2(5(1(3(5(x1)))))) 66.19/17.80 3(2(3(1(5(x1))))) -> 2(1(5(3(4(3(x1)))))) 66.19/17.80 2(5(4(1(5(x1))))) -> 2(5(1(4(1(5(x1)))))) 66.19/17.80 2(3(1(5(5(x1))))) -> 2(1(4(5(5(3(x1)))))) 66.19/17.80 66.19/17.80 Q is empty. 66.19/17.80 We have to consider all minimal (P,Q,R)-chains. 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (29) PisEmptyProof (EQUIVALENT) 66.19/17.80 The TRS P is empty. Hence, there is no (P,Q,R) chain. 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (30) 66.19/17.80 YES 66.19/17.80 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (31) 66.19/17.80 Obligation: 66.19/17.80 Q DP problem: 66.19/17.80 The TRS P consists of the following rules: 66.19/17.80 66.19/17.80 2^1(0(3(1(5(x1))))) -> 2^1(0(3(x1))) 66.19/17.80 66.19/17.80 The TRS R consists of the following rules: 66.19/17.80 66.19/17.80 0(0(0(x1))) -> 2(0(1(0(0(x1))))) 66.19/17.80 2(3(0(x1))) -> 2(0(3(4(x1)))) 66.19/17.80 2(4(0(0(x1)))) -> 2(0(1(4(0(x1))))) 66.19/17.80 2(5(0(0(x1)))) -> 0(3(2(0(5(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(0(1(3(0(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(0(1(1(3(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(1(3(4(1(0(x1)))))) 66.19/17.80 3(1(4(0(x1)))) -> 2(2(0(3(4(1(x1)))))) 66.19/17.80 3(2(4(0(x1)))) -> 2(0(3(4(5(x1))))) 66.19/17.80 2(5(4(0(x1)))) -> 2(4(2(2(0(5(x1)))))) 66.19/17.80 3(1(5(0(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 0(3(5(0(x1)))) -> 0(3(4(1(0(5(x1)))))) 66.19/17.80 2(3(5(0(x1)))) -> 3(2(0(5(1(5(x1)))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(3(4(3(x1))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(5(3(4(x1))))) 66.19/17.80 3(1(4(4(x1)))) -> 2(2(1(4(3(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 2(0(5(1(2(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 4(2(2(0(1(5(x1)))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(5(x1))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(2(5(x1)))))) 66.19/17.80 0(3(1(5(x1)))) -> 3(1(2(0(5(x1))))) 66.19/17.80 2(3(1(5(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 2(3(1(5(x1)))) -> 2(2(5(1(1(3(x1)))))) 66.19/17.80 0(0(3(5(x1)))) -> 2(0(3(4(0(5(x1)))))) 66.19/17.80 3(1(4(0(0(x1))))) -> 3(2(0(1(0(4(x1)))))) 66.19/17.80 2(5(4(0(0(x1))))) -> 4(2(0(1(0(5(x1)))))) 66.19/17.80 2(3(5(0(0(x1))))) -> 3(2(0(5(1(0(x1)))))) 66.19/17.80 2(5(0(1(0(x1))))) -> 0(1(5(2(0(1(x1)))))) 66.19/17.80 2(5(4(1(0(x1))))) -> 4(2(0(5(1(2(x1)))))) 66.19/17.80 0(4(1(3(0(x1))))) -> 3(0(1(0(1(4(x1)))))) 66.19/17.80 0(0(2(3(0(x1))))) -> 3(2(0(1(0(0(x1)))))) 66.19/17.80 2(0(4(3(0(x1))))) -> 0(1(2(0(3(4(x1)))))) 66.19/17.80 2(0(4(3(0(x1))))) -> 0(3(2(0(3(4(x1)))))) 66.19/17.80 2(4(4(3(0(x1))))) -> 2(2(4(3(0(4(x1)))))) 66.19/17.80 3(5(2(4(0(x1))))) -> 2(1(5(3(4(0(x1)))))) 66.19/17.80 0(2(1(5(0(x1))))) -> 2(0(5(1(0(3(x1)))))) 66.19/17.80 0(2(2(4(4(x1))))) -> 4(2(2(0(1(4(x1)))))) 66.19/17.80 0(2(1(5(4(x1))))) -> 2(2(1(4(0(5(x1)))))) 66.19/17.80 2(3(2(5(4(x1))))) -> 2(2(5(3(4(5(x1)))))) 66.19/17.80 2(3(0(1(5(x1))))) -> 2(0(1(3(0(5(x1)))))) 66.19/17.80 3(5(0(1(5(x1))))) -> 1(3(1(0(5(5(x1)))))) 66.19/17.80 0(0(3(1(5(x1))))) -> 0(2(1(0(5(3(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 2(5(1(2(0(3(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 2(3(0(1(0(5(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 3(2(1(1(0(5(x1)))))) 66.19/17.80 0(2(3(1(5(x1))))) -> 0(2(5(1(3(5(x1)))))) 66.19/17.80 3(2(3(1(5(x1))))) -> 2(1(5(3(4(3(x1)))))) 66.19/17.80 2(5(4(1(5(x1))))) -> 2(5(1(4(1(5(x1)))))) 66.19/17.80 2(3(1(5(5(x1))))) -> 2(1(4(5(5(3(x1)))))) 66.19/17.80 66.19/17.80 Q is empty. 66.19/17.80 We have to consider all minimal (P,Q,R)-chains. 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (32) QDPOrderProof (EQUIVALENT) 66.19/17.80 We use the reduction pair processor [LPAR04,JAR06]. 66.19/17.80 66.19/17.80 66.19/17.80 The following pairs can be oriented strictly and are deleted. 66.19/17.80 66.19/17.80 2^1(0(3(1(5(x1))))) -> 2^1(0(3(x1))) 66.19/17.80 The remaining pairs can at least be oriented weakly. 66.19/17.80 Used ordering: Polynomial interpretation [POLO]: 66.19/17.80 66.19/17.80 POL(0(x_1)) = x_1 66.19/17.80 POL(1(x_1)) = x_1 66.19/17.80 POL(2(x_1)) = 0 66.19/17.80 POL(2^1(x_1)) = x_1 66.19/17.80 POL(3(x_1)) = x_1 66.19/17.80 POL(4(x_1)) = 0 66.19/17.80 POL(5(x_1)) = 1 + x_1 66.19/17.80 66.19/17.80 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 66.19/17.80 66.19/17.80 3(1(4(0(x1)))) -> 2(2(0(3(4(1(x1)))))) 66.19/17.80 3(2(4(0(x1)))) -> 2(0(3(4(5(x1))))) 66.19/17.80 3(1(5(0(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(3(4(3(x1))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(5(3(4(x1))))) 66.19/17.80 3(1(4(4(x1)))) -> 2(2(1(4(3(4(x1)))))) 66.19/17.80 3(1(4(0(0(x1))))) -> 3(2(0(1(0(4(x1)))))) 66.19/17.80 3(5(2(4(0(x1))))) -> 2(1(5(3(4(0(x1)))))) 66.19/17.80 3(5(0(1(5(x1))))) -> 1(3(1(0(5(5(x1)))))) 66.19/17.80 3(2(3(1(5(x1))))) -> 2(1(5(3(4(3(x1)))))) 66.19/17.80 0(0(0(x1))) -> 2(0(1(0(0(x1))))) 66.19/17.80 0(3(5(0(x1)))) -> 0(3(4(1(0(5(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 2(0(5(1(2(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 4(2(2(0(1(5(x1)))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(5(x1))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(2(5(x1)))))) 66.19/17.80 0(3(1(5(x1)))) -> 3(1(2(0(5(x1))))) 66.19/17.80 0(0(3(5(x1)))) -> 2(0(3(4(0(5(x1)))))) 66.19/17.80 0(4(1(3(0(x1))))) -> 3(0(1(0(1(4(x1)))))) 66.19/17.80 0(0(2(3(0(x1))))) -> 3(2(0(1(0(0(x1)))))) 66.19/17.80 0(2(1(5(0(x1))))) -> 2(0(5(1(0(3(x1)))))) 66.19/17.80 0(2(2(4(4(x1))))) -> 4(2(2(0(1(4(x1)))))) 66.19/17.80 0(2(1(5(4(x1))))) -> 2(2(1(4(0(5(x1)))))) 66.19/17.80 0(0(3(1(5(x1))))) -> 0(2(1(0(5(3(x1)))))) 66.19/17.80 0(2(3(1(5(x1))))) -> 0(2(5(1(3(5(x1)))))) 66.19/17.80 66.19/17.80 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (33) 66.19/17.80 Obligation: 66.19/17.80 Q DP problem: 66.19/17.80 P is empty. 66.19/17.80 The TRS R consists of the following rules: 66.19/17.80 66.19/17.80 0(0(0(x1))) -> 2(0(1(0(0(x1))))) 66.19/17.80 2(3(0(x1))) -> 2(0(3(4(x1)))) 66.19/17.80 2(4(0(0(x1)))) -> 2(0(1(4(0(x1))))) 66.19/17.80 2(5(0(0(x1)))) -> 0(3(2(0(5(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(0(1(3(0(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(0(1(1(3(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(1(3(4(1(0(x1)))))) 66.19/17.80 3(1(4(0(x1)))) -> 2(2(0(3(4(1(x1)))))) 66.19/17.80 3(2(4(0(x1)))) -> 2(0(3(4(5(x1))))) 66.19/17.80 2(5(4(0(x1)))) -> 2(4(2(2(0(5(x1)))))) 66.19/17.80 3(1(5(0(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 0(3(5(0(x1)))) -> 0(3(4(1(0(5(x1)))))) 66.19/17.80 2(3(5(0(x1)))) -> 3(2(0(5(1(5(x1)))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(3(4(3(x1))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(5(3(4(x1))))) 66.19/17.80 3(1(4(4(x1)))) -> 2(2(1(4(3(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 2(0(5(1(2(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 4(2(2(0(1(5(x1)))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(5(x1))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(2(5(x1)))))) 66.19/17.80 0(3(1(5(x1)))) -> 3(1(2(0(5(x1))))) 66.19/17.80 2(3(1(5(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 2(3(1(5(x1)))) -> 2(2(5(1(1(3(x1)))))) 66.19/17.80 0(0(3(5(x1)))) -> 2(0(3(4(0(5(x1)))))) 66.19/17.80 3(1(4(0(0(x1))))) -> 3(2(0(1(0(4(x1)))))) 66.19/17.80 2(5(4(0(0(x1))))) -> 4(2(0(1(0(5(x1)))))) 66.19/17.80 2(3(5(0(0(x1))))) -> 3(2(0(5(1(0(x1)))))) 66.19/17.80 2(5(0(1(0(x1))))) -> 0(1(5(2(0(1(x1)))))) 66.19/17.80 2(5(4(1(0(x1))))) -> 4(2(0(5(1(2(x1)))))) 66.19/17.80 0(4(1(3(0(x1))))) -> 3(0(1(0(1(4(x1)))))) 66.19/17.80 0(0(2(3(0(x1))))) -> 3(2(0(1(0(0(x1)))))) 66.19/17.80 2(0(4(3(0(x1))))) -> 0(1(2(0(3(4(x1)))))) 66.19/17.80 2(0(4(3(0(x1))))) -> 0(3(2(0(3(4(x1)))))) 66.19/17.80 2(4(4(3(0(x1))))) -> 2(2(4(3(0(4(x1)))))) 66.19/17.80 3(5(2(4(0(x1))))) -> 2(1(5(3(4(0(x1)))))) 66.19/17.80 0(2(1(5(0(x1))))) -> 2(0(5(1(0(3(x1)))))) 66.19/17.80 0(2(2(4(4(x1))))) -> 4(2(2(0(1(4(x1)))))) 66.19/17.80 0(2(1(5(4(x1))))) -> 2(2(1(4(0(5(x1)))))) 66.19/17.80 2(3(2(5(4(x1))))) -> 2(2(5(3(4(5(x1)))))) 66.19/17.80 2(3(0(1(5(x1))))) -> 2(0(1(3(0(5(x1)))))) 66.19/17.80 3(5(0(1(5(x1))))) -> 1(3(1(0(5(5(x1)))))) 66.19/17.80 0(0(3(1(5(x1))))) -> 0(2(1(0(5(3(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 2(5(1(2(0(3(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 2(3(0(1(0(5(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 3(2(1(1(0(5(x1)))))) 66.19/17.80 0(2(3(1(5(x1))))) -> 0(2(5(1(3(5(x1)))))) 66.19/17.80 3(2(3(1(5(x1))))) -> 2(1(5(3(4(3(x1)))))) 66.19/17.80 2(5(4(1(5(x1))))) -> 2(5(1(4(1(5(x1)))))) 66.19/17.80 2(3(1(5(5(x1))))) -> 2(1(4(5(5(3(x1)))))) 66.19/17.80 66.19/17.80 Q is empty. 66.19/17.80 We have to consider all minimal (P,Q,R)-chains. 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (34) PisEmptyProof (EQUIVALENT) 66.19/17.80 The TRS P is empty. Hence, there is no (P,Q,R) chain. 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (35) 66.19/17.80 YES 66.19/17.80 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (36) 66.19/17.80 Obligation: 66.19/17.80 Q DP problem: 66.19/17.80 The TRS P consists of the following rules: 66.19/17.80 66.19/17.80 2^1(5(4(1(0(x1))))) -> 2^1(x1) 66.19/17.80 66.19/17.80 The TRS R consists of the following rules: 66.19/17.80 66.19/17.80 0(0(0(x1))) -> 2(0(1(0(0(x1))))) 66.19/17.80 2(3(0(x1))) -> 2(0(3(4(x1)))) 66.19/17.80 2(4(0(0(x1)))) -> 2(0(1(4(0(x1))))) 66.19/17.80 2(5(0(0(x1)))) -> 0(3(2(0(5(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(0(1(3(0(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(0(1(1(3(x1))))) 66.19/17.80 2(3(1(0(x1)))) -> 2(1(3(4(1(0(x1)))))) 66.19/17.80 3(1(4(0(x1)))) -> 2(2(0(3(4(1(x1)))))) 66.19/17.80 3(2(4(0(x1)))) -> 2(0(3(4(5(x1))))) 66.19/17.80 2(5(4(0(x1)))) -> 2(4(2(2(0(5(x1)))))) 66.19/17.80 3(1(5(0(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 0(3(5(0(x1)))) -> 0(3(4(1(0(5(x1)))))) 66.19/17.80 2(3(5(0(x1)))) -> 3(2(0(5(1(5(x1)))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(3(4(3(x1))))) 66.19/17.80 3(2(0(4(x1)))) -> 2(0(5(3(4(x1))))) 66.19/17.80 3(1(4(4(x1)))) -> 2(2(1(4(3(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 2(0(5(1(2(4(x1)))))) 66.19/17.80 0(2(5(4(x1)))) -> 4(2(2(0(1(5(x1)))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(5(x1))))) 66.19/17.80 0(0(1(5(x1)))) -> 0(2(0(1(2(5(x1)))))) 66.19/17.80 0(3(1(5(x1)))) -> 3(1(2(0(5(x1))))) 66.19/17.80 2(3(1(5(x1)))) -> 2(1(5(1(0(3(x1)))))) 66.19/17.80 2(3(1(5(x1)))) -> 2(2(5(1(1(3(x1)))))) 66.19/17.80 0(0(3(5(x1)))) -> 2(0(3(4(0(5(x1)))))) 66.19/17.80 3(1(4(0(0(x1))))) -> 3(2(0(1(0(4(x1)))))) 66.19/17.80 2(5(4(0(0(x1))))) -> 4(2(0(1(0(5(x1)))))) 66.19/17.80 2(3(5(0(0(x1))))) -> 3(2(0(5(1(0(x1)))))) 66.19/17.80 2(5(0(1(0(x1))))) -> 0(1(5(2(0(1(x1)))))) 66.19/17.80 2(5(4(1(0(x1))))) -> 4(2(0(5(1(2(x1)))))) 66.19/17.80 0(4(1(3(0(x1))))) -> 3(0(1(0(1(4(x1)))))) 66.19/17.80 0(0(2(3(0(x1))))) -> 3(2(0(1(0(0(x1)))))) 66.19/17.80 2(0(4(3(0(x1))))) -> 0(1(2(0(3(4(x1)))))) 66.19/17.80 2(0(4(3(0(x1))))) -> 0(3(2(0(3(4(x1)))))) 66.19/17.80 2(4(4(3(0(x1))))) -> 2(2(4(3(0(4(x1)))))) 66.19/17.80 3(5(2(4(0(x1))))) -> 2(1(5(3(4(0(x1)))))) 66.19/17.80 0(2(1(5(0(x1))))) -> 2(0(5(1(0(3(x1)))))) 66.19/17.80 0(2(2(4(4(x1))))) -> 4(2(2(0(1(4(x1)))))) 66.19/17.80 0(2(1(5(4(x1))))) -> 2(2(1(4(0(5(x1)))))) 66.19/17.80 2(3(2(5(4(x1))))) -> 2(2(5(3(4(5(x1)))))) 66.19/17.80 2(3(0(1(5(x1))))) -> 2(0(1(3(0(5(x1)))))) 66.19/17.80 3(5(0(1(5(x1))))) -> 1(3(1(0(5(5(x1)))))) 66.19/17.80 0(0(3(1(5(x1))))) -> 0(2(1(0(5(3(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 2(5(1(2(0(3(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 2(3(0(1(0(5(x1)))))) 66.19/17.80 2(0(3(1(5(x1))))) -> 3(2(1(1(0(5(x1)))))) 66.19/17.80 0(2(3(1(5(x1))))) -> 0(2(5(1(3(5(x1)))))) 66.19/17.80 3(2(3(1(5(x1))))) -> 2(1(5(3(4(3(x1)))))) 66.19/17.80 2(5(4(1(5(x1))))) -> 2(5(1(4(1(5(x1)))))) 66.19/17.80 2(3(1(5(5(x1))))) -> 2(1(4(5(5(3(x1)))))) 66.19/17.80 66.19/17.80 Q is empty. 66.19/17.80 We have to consider all minimal (P,Q,R)-chains. 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (37) UsableRulesProof (EQUIVALENT) 66.19/17.80 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. 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (38) 66.19/17.80 Obligation: 66.19/17.80 Q DP problem: 66.19/17.80 The TRS P consists of the following rules: 66.19/17.80 66.19/17.80 2^1(5(4(1(0(x1))))) -> 2^1(x1) 66.19/17.80 66.19/17.80 R is empty. 66.19/17.80 Q is empty. 66.19/17.80 We have to consider all minimal (P,Q,R)-chains. 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (39) QDPSizeChangeProof (EQUIVALENT) 66.19/17.80 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. 66.19/17.80 66.19/17.80 From the DPs we obtained the following set of size-change graphs: 66.19/17.80 *2^1(5(4(1(0(x1))))) -> 2^1(x1) 66.19/17.80 The graph contains the following edges 1 > 1 66.19/17.80 66.19/17.80 66.19/17.80 ---------------------------------------- 66.19/17.80 66.19/17.80 (40) 66.19/17.80 YES 66.49/17.92 EOF