109.43/28.68 YES 109.43/28.72 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 109.43/28.72 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 109.43/28.72 109.43/28.72 109.43/28.72 Termination w.r.t. Q of the given QTRS could be proven: 109.43/28.72 109.43/28.72 (0) QTRS 109.43/28.72 (1) QTRS Reverse [EQUIVALENT, 0 ms] 109.43/28.72 (2) QTRS 109.43/28.72 (3) DependencyPairsProof [EQUIVALENT, 486 ms] 109.43/28.72 (4) QDP 109.43/28.72 (5) DependencyGraphProof [EQUIVALENT, 1 ms] 109.43/28.72 (6) AND 109.43/28.72 (7) QDP 109.43/28.72 (8) UsableRulesProof [EQUIVALENT, 0 ms] 109.43/28.72 (9) QDP 109.43/28.72 (10) QDPSizeChangeProof [EQUIVALENT, 0 ms] 109.43/28.72 (11) YES 109.43/28.72 (12) QDP 109.43/28.72 (13) QDPOrderProof [EQUIVALENT, 224 ms] 109.43/28.72 (14) QDP 109.43/28.72 (15) DependencyGraphProof [EQUIVALENT, 0 ms] 109.43/28.72 (16) QDP 109.43/28.72 (17) QDPOrderProof [EQUIVALENT, 84 ms] 109.43/28.72 (18) QDP 109.43/28.72 (19) DependencyGraphProof [EQUIVALENT, 0 ms] 109.43/28.72 (20) QDP 109.43/28.72 (21) QDPOrderProof [EQUIVALENT, 78 ms] 109.43/28.72 (22) QDP 109.43/28.72 (23) QDPOrderProof [EQUIVALENT, 76 ms] 109.43/28.72 (24) QDP 109.43/28.72 (25) DependencyGraphProof [EQUIVALENT, 0 ms] 109.43/28.72 (26) QDP 109.43/28.72 (27) QDPOrderProof [EQUIVALENT, 53 ms] 109.43/28.72 (28) QDP 109.43/28.72 (29) QDPOrderProof [EQUIVALENT, 45 ms] 109.43/28.72 (30) QDP 109.43/28.72 (31) DependencyGraphProof [EQUIVALENT, 0 ms] 109.43/28.72 (32) AND 109.43/28.72 (33) QDP 109.43/28.72 (34) QDPOrderProof [EQUIVALENT, 3338 ms] 109.43/28.72 (35) QDP 109.43/28.72 (36) DependencyGraphProof [EQUIVALENT, 0 ms] 109.43/28.72 (37) QDP 109.43/28.72 (38) UsableRulesProof [EQUIVALENT, 0 ms] 109.43/28.72 (39) QDP 109.43/28.72 (40) QDPSizeChangeProof [EQUIVALENT, 0 ms] 109.43/28.72 (41) YES 109.43/28.72 (42) QDP 109.43/28.72 (43) QDPOrderProof [EQUIVALENT, 33 ms] 109.43/28.72 (44) QDP 109.43/28.72 (45) PisEmptyProof [EQUIVALENT, 0 ms] 109.43/28.72 (46) YES 109.43/28.72 109.43/28.72 109.43/28.72 ---------------------------------------- 109.43/28.72 109.43/28.72 (0) 109.43/28.72 Obligation: 109.43/28.72 Q restricted rewrite system: 109.43/28.72 The TRS R consists of the following rules: 109.43/28.72 109.43/28.72 0(1(2(x1))) -> 0(0(2(1(x1)))) 109.43/28.72 0(1(2(x1))) -> 0(2(1(3(x1)))) 109.43/28.72 0(1(2(x1))) -> 0(2(3(1(x1)))) 109.43/28.72 0(1(4(x1))) -> 0(4(1(1(0(0(x1)))))) 109.43/28.72 0(3(2(x1))) -> 0(0(2(3(x1)))) 109.43/28.72 0(3(2(x1))) -> 0(2(3(1(x1)))) 109.43/28.72 0(3(4(x1))) -> 0(0(4(3(x1)))) 109.43/28.72 0(4(5(x1))) -> 0(0(4(1(5(x1))))) 109.43/28.72 2(0(1(x1))) -> 0(2(1(1(x1)))) 109.43/28.72 2(4(1(x1))) -> 0(4(2(3(1(x1))))) 109.43/28.72 4(3(2(x1))) -> 4(2(3(1(x1)))) 109.43/28.72 0(1(0(1(x1)))) -> 0(0(3(1(1(x1))))) 109.43/28.72 0(1(0(2(x1)))) -> 0(0(2(1(1(1(x1)))))) 109.43/28.72 0(1(4(2(x1)))) -> 0(4(2(1(1(x1))))) 109.43/28.72 0(3(2(4(x1)))) -> 0(4(2(3(0(x1))))) 109.43/28.72 0(3(4(2(x1)))) -> 0(4(2(3(1(x1))))) 109.43/28.72 0(3(5(2(x1)))) -> 0(2(1(3(5(x1))))) 109.43/28.72 0(4(5(3(x1)))) -> 0(0(4(3(5(x1))))) 109.43/28.72 0(5(3(4(x1)))) -> 0(4(3(3(5(x1))))) 109.43/28.72 0(5(4(2(x1)))) -> 0(4(2(1(5(x1))))) 109.43/28.72 2(0(3(1(x1)))) -> 0(3(2(1(1(x1))))) 109.43/28.72 2(2(4(1(x1)))) -> 4(2(2(3(1(x1))))) 109.43/28.72 2(3(4(3(x1)))) -> 2(3(0(4(3(3(x1)))))) 109.43/28.72 2(4(1(1(x1)))) -> 2(0(4(1(1(x1))))) 109.43/28.72 2(4(1(3(x1)))) -> 0(4(3(2(1(x1))))) 109.43/28.72 2(4(3(2(x1)))) -> 2(0(4(2(3(x1))))) 109.43/28.72 2(5(4(3(x1)))) -> 0(4(2(3(5(x1))))) 109.43/28.72 4(2(0(3(x1)))) -> 2(3(0(4(3(x1))))) 109.43/28.72 4(3(4(3(x1)))) -> 4(3(0(4(3(x1))))) 109.43/28.72 0(1(5(3(2(x1))))) -> 0(5(0(3(2(1(x1)))))) 109.43/28.72 0(1(5(4(2(x1))))) -> 5(2(1(0(4(3(x1)))))) 109.43/28.72 0(1(5(4(5(x1))))) -> 0(4(1(2(5(5(x1)))))) 109.43/28.72 0(2(2(4(3(x1))))) -> 2(0(4(3(0(2(x1)))))) 109.43/28.72 0(3(0(4(5(x1))))) -> 5(0(0(4(3(2(x1)))))) 109.43/28.72 0(3(2(4(3(x1))))) -> 0(4(3(3(4(2(x1)))))) 109.43/28.72 0(4(2(5(4(x1))))) -> 0(4(2(1(5(4(x1)))))) 109.43/28.72 0(5(3(2(1(x1))))) -> 0(2(3(1(3(5(x1)))))) 109.43/28.72 0(5(4(1(4(x1))))) -> 4(0(4(1(5(0(x1)))))) 109.43/28.72 2(4(1(5(3(x1))))) -> 0(4(1(3(5(2(x1)))))) 109.43/28.72 2(4(2(0(1(x1))))) -> 2(1(1(2(0(4(x1)))))) 109.43/28.72 2(5(3(4(1(x1))))) -> 5(1(0(4(3(2(x1)))))) 109.43/28.72 4(0(1(5(4(x1))))) -> 4(0(0(4(1(5(x1)))))) 109.43/28.72 4(3(0(2(3(x1))))) -> 0(4(2(3(1(3(x1)))))) 109.43/28.72 4(4(1(2(3(x1))))) -> 0(4(4(2(3(1(x1)))))) 109.43/28.72 4(5(1(0(2(x1))))) -> 1(0(4(2(1(5(x1)))))) 109.43/28.72 109.43/28.72 Q is empty. 109.43/28.72 109.43/28.72 ---------------------------------------- 109.43/28.72 109.43/28.72 (1) QTRS Reverse (EQUIVALENT) 109.43/28.72 We applied the QTRS Reverse Processor [REVERSE]. 109.43/28.72 ---------------------------------------- 109.43/28.72 109.43/28.72 (2) 109.43/28.72 Obligation: 109.43/28.72 Q restricted rewrite system: 109.43/28.72 The TRS R consists of the following rules: 109.43/28.72 109.43/28.72 2(1(0(x1))) -> 1(2(0(0(x1)))) 109.43/28.72 2(1(0(x1))) -> 3(1(2(0(x1)))) 109.43/28.72 2(1(0(x1))) -> 1(3(2(0(x1)))) 109.43/28.72 4(1(0(x1))) -> 0(0(1(1(4(0(x1)))))) 109.43/28.72 2(3(0(x1))) -> 3(2(0(0(x1)))) 109.43/28.72 2(3(0(x1))) -> 1(3(2(0(x1)))) 109.43/28.72 4(3(0(x1))) -> 3(4(0(0(x1)))) 109.43/28.72 5(4(0(x1))) -> 5(1(4(0(0(x1))))) 109.43/28.72 1(0(2(x1))) -> 1(1(2(0(x1)))) 109.43/28.72 1(4(2(x1))) -> 1(3(2(4(0(x1))))) 109.43/28.72 2(3(4(x1))) -> 1(3(2(4(x1)))) 109.43/28.72 1(0(1(0(x1)))) -> 1(1(3(0(0(x1))))) 109.43/28.72 2(0(1(0(x1)))) -> 1(1(1(2(0(0(x1)))))) 109.43/28.72 2(4(1(0(x1)))) -> 1(1(2(4(0(x1))))) 109.43/28.72 4(2(3(0(x1)))) -> 0(3(2(4(0(x1))))) 109.43/28.72 2(4(3(0(x1)))) -> 1(3(2(4(0(x1))))) 109.43/28.72 2(5(3(0(x1)))) -> 5(3(1(2(0(x1))))) 109.43/28.72 3(5(4(0(x1)))) -> 5(3(4(0(0(x1))))) 109.43/28.72 4(3(5(0(x1)))) -> 5(3(3(4(0(x1))))) 109.43/28.72 2(4(5(0(x1)))) -> 5(1(2(4(0(x1))))) 109.43/28.72 1(3(0(2(x1)))) -> 1(1(2(3(0(x1))))) 109.43/28.72 1(4(2(2(x1)))) -> 1(3(2(2(4(x1))))) 109.43/28.72 3(4(3(2(x1)))) -> 3(3(4(0(3(2(x1)))))) 109.43/28.72 1(1(4(2(x1)))) -> 1(1(4(0(2(x1))))) 109.43/28.72 3(1(4(2(x1)))) -> 1(2(3(4(0(x1))))) 109.43/28.72 2(3(4(2(x1)))) -> 3(2(4(0(2(x1))))) 109.43/28.72 3(4(5(2(x1)))) -> 5(3(2(4(0(x1))))) 109.43/28.72 3(0(2(4(x1)))) -> 3(4(0(3(2(x1))))) 109.43/28.72 3(4(3(4(x1)))) -> 3(4(0(3(4(x1))))) 109.43/28.72 2(3(5(1(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 109.43/28.72 2(4(5(1(0(x1))))) -> 3(4(0(1(2(5(x1)))))) 109.43/28.72 5(4(5(1(0(x1))))) -> 5(5(2(1(4(0(x1)))))) 109.43/28.72 3(4(2(2(0(x1))))) -> 2(0(3(4(0(2(x1)))))) 109.43/28.72 5(4(0(3(0(x1))))) -> 2(3(4(0(0(5(x1)))))) 109.43/28.72 3(4(2(3(0(x1))))) -> 2(4(3(3(4(0(x1)))))) 109.43/28.72 4(5(2(4(0(x1))))) -> 4(5(1(2(4(0(x1)))))) 109.43/28.72 1(2(3(5(0(x1))))) -> 5(3(1(3(2(0(x1)))))) 109.43/28.72 4(1(4(5(0(x1))))) -> 0(5(1(4(0(4(x1)))))) 109.43/28.72 3(5(1(4(2(x1))))) -> 2(5(3(1(4(0(x1)))))) 109.43/28.72 1(0(2(4(2(x1))))) -> 4(0(2(1(1(2(x1)))))) 109.43/28.72 1(4(3(5(2(x1))))) -> 2(3(4(0(1(5(x1)))))) 109.43/28.72 4(5(1(0(4(x1))))) -> 5(1(4(0(0(4(x1)))))) 109.43/28.72 3(2(0(3(4(x1))))) -> 3(1(3(2(4(0(x1)))))) 109.43/28.72 3(2(1(4(4(x1))))) -> 1(3(2(4(4(0(x1)))))) 109.43/28.72 2(0(1(5(4(x1))))) -> 5(1(2(4(0(1(x1)))))) 109.43/28.72 109.43/28.72 Q is empty. 109.43/28.72 109.43/28.72 ---------------------------------------- 109.43/28.72 109.43/28.72 (3) DependencyPairsProof (EQUIVALENT) 109.43/28.72 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 109.43/28.72 ---------------------------------------- 109.43/28.72 109.43/28.72 (4) 109.43/28.72 Obligation: 109.43/28.72 Q DP problem: 109.43/28.72 The TRS P consists of the following rules: 109.43/28.72 109.43/28.72 2^1(1(0(x1))) -> 1^1(2(0(0(x1)))) 109.43/28.72 2^1(1(0(x1))) -> 2^1(0(0(x1))) 109.43/28.72 2^1(1(0(x1))) -> 3^1(1(2(0(x1)))) 109.43/28.72 2^1(1(0(x1))) -> 1^1(2(0(x1))) 109.43/28.72 2^1(1(0(x1))) -> 2^1(0(x1)) 109.43/28.72 2^1(1(0(x1))) -> 1^1(3(2(0(x1)))) 109.43/28.72 2^1(1(0(x1))) -> 3^1(2(0(x1))) 109.43/28.72 4^1(1(0(x1))) -> 1^1(1(4(0(x1)))) 109.43/28.72 4^1(1(0(x1))) -> 1^1(4(0(x1))) 109.43/28.72 4^1(1(0(x1))) -> 4^1(0(x1)) 109.43/28.72 2^1(3(0(x1))) -> 3^1(2(0(0(x1)))) 109.43/28.72 2^1(3(0(x1))) -> 2^1(0(0(x1))) 109.43/28.72 2^1(3(0(x1))) -> 1^1(3(2(0(x1)))) 109.43/28.72 2^1(3(0(x1))) -> 3^1(2(0(x1))) 109.43/28.72 2^1(3(0(x1))) -> 2^1(0(x1)) 109.43/28.72 4^1(3(0(x1))) -> 3^1(4(0(0(x1)))) 109.43/28.72 4^1(3(0(x1))) -> 4^1(0(0(x1))) 109.43/28.72 5^1(4(0(x1))) -> 5^1(1(4(0(0(x1))))) 109.43/28.72 5^1(4(0(x1))) -> 1^1(4(0(0(x1)))) 109.43/28.72 5^1(4(0(x1))) -> 4^1(0(0(x1))) 109.43/28.72 1^1(0(2(x1))) -> 1^1(1(2(0(x1)))) 109.43/28.72 1^1(0(2(x1))) -> 1^1(2(0(x1))) 109.43/28.72 1^1(0(2(x1))) -> 2^1(0(x1)) 109.43/28.72 1^1(4(2(x1))) -> 1^1(3(2(4(0(x1))))) 109.43/28.72 1^1(4(2(x1))) -> 3^1(2(4(0(x1)))) 109.43/28.72 1^1(4(2(x1))) -> 2^1(4(0(x1))) 109.43/28.72 1^1(4(2(x1))) -> 4^1(0(x1)) 109.43/28.72 2^1(3(4(x1))) -> 1^1(3(2(4(x1)))) 109.43/28.72 2^1(3(4(x1))) -> 3^1(2(4(x1))) 109.43/28.72 2^1(3(4(x1))) -> 2^1(4(x1)) 109.43/28.72 1^1(0(1(0(x1)))) -> 1^1(1(3(0(0(x1))))) 109.43/28.72 1^1(0(1(0(x1)))) -> 1^1(3(0(0(x1)))) 109.43/28.72 1^1(0(1(0(x1)))) -> 3^1(0(0(x1))) 109.43/28.72 2^1(0(1(0(x1)))) -> 1^1(1(1(2(0(0(x1)))))) 109.43/28.72 2^1(0(1(0(x1)))) -> 1^1(1(2(0(0(x1))))) 109.43/28.72 2^1(0(1(0(x1)))) -> 1^1(2(0(0(x1)))) 109.43/28.72 2^1(0(1(0(x1)))) -> 2^1(0(0(x1))) 109.43/28.72 2^1(4(1(0(x1)))) -> 1^1(1(2(4(0(x1))))) 109.43/28.72 2^1(4(1(0(x1)))) -> 1^1(2(4(0(x1)))) 109.43/28.72 2^1(4(1(0(x1)))) -> 2^1(4(0(x1))) 109.43/28.72 2^1(4(1(0(x1)))) -> 4^1(0(x1)) 109.43/28.72 4^1(2(3(0(x1)))) -> 3^1(2(4(0(x1)))) 109.43/28.72 4^1(2(3(0(x1)))) -> 2^1(4(0(x1))) 109.43/28.72 4^1(2(3(0(x1)))) -> 4^1(0(x1)) 109.43/28.72 2^1(4(3(0(x1)))) -> 1^1(3(2(4(0(x1))))) 109.43/28.72 2^1(4(3(0(x1)))) -> 3^1(2(4(0(x1)))) 109.43/28.72 2^1(4(3(0(x1)))) -> 2^1(4(0(x1))) 109.43/28.72 2^1(4(3(0(x1)))) -> 4^1(0(x1)) 109.43/28.72 2^1(5(3(0(x1)))) -> 5^1(3(1(2(0(x1))))) 109.43/28.72 2^1(5(3(0(x1)))) -> 3^1(1(2(0(x1)))) 109.43/28.72 2^1(5(3(0(x1)))) -> 1^1(2(0(x1))) 109.43/28.72 2^1(5(3(0(x1)))) -> 2^1(0(x1)) 109.43/28.72 3^1(5(4(0(x1)))) -> 5^1(3(4(0(0(x1))))) 109.43/28.72 3^1(5(4(0(x1)))) -> 3^1(4(0(0(x1)))) 109.43/28.72 3^1(5(4(0(x1)))) -> 4^1(0(0(x1))) 109.43/28.72 4^1(3(5(0(x1)))) -> 5^1(3(3(4(0(x1))))) 109.43/28.72 4^1(3(5(0(x1)))) -> 3^1(3(4(0(x1)))) 109.43/28.72 4^1(3(5(0(x1)))) -> 3^1(4(0(x1))) 109.43/28.72 4^1(3(5(0(x1)))) -> 4^1(0(x1)) 109.43/28.73 2^1(4(5(0(x1)))) -> 5^1(1(2(4(0(x1))))) 109.43/28.73 2^1(4(5(0(x1)))) -> 1^1(2(4(0(x1)))) 109.43/28.73 2^1(4(5(0(x1)))) -> 2^1(4(0(x1))) 109.43/28.73 2^1(4(5(0(x1)))) -> 4^1(0(x1)) 109.43/28.73 1^1(3(0(2(x1)))) -> 1^1(1(2(3(0(x1))))) 109.43/28.73 1^1(3(0(2(x1)))) -> 1^1(2(3(0(x1)))) 109.43/28.73 1^1(3(0(2(x1)))) -> 2^1(3(0(x1))) 109.43/28.73 1^1(3(0(2(x1)))) -> 3^1(0(x1)) 109.43/28.73 1^1(4(2(2(x1)))) -> 1^1(3(2(2(4(x1))))) 109.43/28.73 1^1(4(2(2(x1)))) -> 3^1(2(2(4(x1)))) 109.43/28.73 1^1(4(2(2(x1)))) -> 2^1(2(4(x1))) 109.43/28.73 1^1(4(2(2(x1)))) -> 2^1(4(x1)) 109.43/28.73 1^1(4(2(2(x1)))) -> 4^1(x1) 109.43/28.73 3^1(4(3(2(x1)))) -> 3^1(3(4(0(3(2(x1)))))) 109.43/28.73 3^1(4(3(2(x1)))) -> 3^1(4(0(3(2(x1))))) 109.43/28.73 3^1(4(3(2(x1)))) -> 4^1(0(3(2(x1)))) 109.43/28.73 1^1(1(4(2(x1)))) -> 1^1(1(4(0(2(x1))))) 109.43/28.73 1^1(1(4(2(x1)))) -> 1^1(4(0(2(x1)))) 109.43/28.73 1^1(1(4(2(x1)))) -> 4^1(0(2(x1))) 109.43/28.73 3^1(1(4(2(x1)))) -> 1^1(2(3(4(0(x1))))) 109.43/28.73 3^1(1(4(2(x1)))) -> 2^1(3(4(0(x1)))) 109.43/28.73 3^1(1(4(2(x1)))) -> 3^1(4(0(x1))) 109.43/28.73 3^1(1(4(2(x1)))) -> 4^1(0(x1)) 109.43/28.73 2^1(3(4(2(x1)))) -> 3^1(2(4(0(2(x1))))) 109.43/28.73 2^1(3(4(2(x1)))) -> 2^1(4(0(2(x1)))) 109.43/28.73 2^1(3(4(2(x1)))) -> 4^1(0(2(x1))) 109.43/28.73 3^1(4(5(2(x1)))) -> 5^1(3(2(4(0(x1))))) 109.43/28.73 3^1(4(5(2(x1)))) -> 3^1(2(4(0(x1)))) 109.43/28.73 3^1(4(5(2(x1)))) -> 2^1(4(0(x1))) 109.43/28.73 3^1(4(5(2(x1)))) -> 4^1(0(x1)) 109.43/28.73 3^1(0(2(4(x1)))) -> 3^1(4(0(3(2(x1))))) 109.43/28.73 3^1(0(2(4(x1)))) -> 4^1(0(3(2(x1)))) 109.43/28.73 3^1(0(2(4(x1)))) -> 3^1(2(x1)) 109.43/28.73 3^1(0(2(4(x1)))) -> 2^1(x1) 109.43/28.73 3^1(4(3(4(x1)))) -> 3^1(4(0(3(4(x1))))) 109.43/28.73 3^1(4(3(4(x1)))) -> 4^1(0(3(4(x1)))) 109.43/28.73 2^1(3(5(1(0(x1))))) -> 1^1(2(3(0(5(0(x1)))))) 109.43/28.73 2^1(3(5(1(0(x1))))) -> 2^1(3(0(5(0(x1))))) 109.43/28.73 2^1(3(5(1(0(x1))))) -> 3^1(0(5(0(x1)))) 109.43/28.73 2^1(3(5(1(0(x1))))) -> 5^1(0(x1)) 109.43/28.73 2^1(4(5(1(0(x1))))) -> 3^1(4(0(1(2(5(x1)))))) 109.43/28.73 2^1(4(5(1(0(x1))))) -> 4^1(0(1(2(5(x1))))) 109.43/28.73 2^1(4(5(1(0(x1))))) -> 1^1(2(5(x1))) 109.43/28.73 2^1(4(5(1(0(x1))))) -> 2^1(5(x1)) 109.43/28.73 2^1(4(5(1(0(x1))))) -> 5^1(x1) 109.43/28.73 5^1(4(5(1(0(x1))))) -> 5^1(5(2(1(4(0(x1)))))) 109.43/28.73 5^1(4(5(1(0(x1))))) -> 5^1(2(1(4(0(x1))))) 109.43/28.73 5^1(4(5(1(0(x1))))) -> 2^1(1(4(0(x1)))) 109.43/28.73 5^1(4(5(1(0(x1))))) -> 1^1(4(0(x1))) 109.43/28.73 5^1(4(5(1(0(x1))))) -> 4^1(0(x1)) 109.43/28.73 3^1(4(2(2(0(x1))))) -> 2^1(0(3(4(0(2(x1)))))) 109.43/28.73 3^1(4(2(2(0(x1))))) -> 3^1(4(0(2(x1)))) 109.43/28.73 3^1(4(2(2(0(x1))))) -> 4^1(0(2(x1))) 109.43/28.73 3^1(4(2(2(0(x1))))) -> 2^1(x1) 109.43/28.73 5^1(4(0(3(0(x1))))) -> 2^1(3(4(0(0(5(x1)))))) 109.43/28.73 5^1(4(0(3(0(x1))))) -> 3^1(4(0(0(5(x1))))) 109.43/28.73 5^1(4(0(3(0(x1))))) -> 4^1(0(0(5(x1)))) 109.43/28.73 5^1(4(0(3(0(x1))))) -> 5^1(x1) 109.43/28.73 3^1(4(2(3(0(x1))))) -> 2^1(4(3(3(4(0(x1)))))) 109.43/28.73 3^1(4(2(3(0(x1))))) -> 4^1(3(3(4(0(x1))))) 109.43/28.73 3^1(4(2(3(0(x1))))) -> 3^1(3(4(0(x1)))) 109.43/28.73 3^1(4(2(3(0(x1))))) -> 3^1(4(0(x1))) 109.43/28.73 3^1(4(2(3(0(x1))))) -> 4^1(0(x1)) 109.43/28.73 4^1(5(2(4(0(x1))))) -> 4^1(5(1(2(4(0(x1)))))) 109.43/28.73 4^1(5(2(4(0(x1))))) -> 5^1(1(2(4(0(x1))))) 109.43/28.73 4^1(5(2(4(0(x1))))) -> 1^1(2(4(0(x1)))) 109.43/28.73 1^1(2(3(5(0(x1))))) -> 5^1(3(1(3(2(0(x1)))))) 109.43/28.73 1^1(2(3(5(0(x1))))) -> 3^1(1(3(2(0(x1))))) 109.43/28.73 1^1(2(3(5(0(x1))))) -> 1^1(3(2(0(x1)))) 109.43/28.73 1^1(2(3(5(0(x1))))) -> 3^1(2(0(x1))) 109.43/28.73 1^1(2(3(5(0(x1))))) -> 2^1(0(x1)) 109.43/28.73 4^1(1(4(5(0(x1))))) -> 5^1(1(4(0(4(x1))))) 109.43/28.73 4^1(1(4(5(0(x1))))) -> 1^1(4(0(4(x1)))) 109.43/28.73 4^1(1(4(5(0(x1))))) -> 4^1(0(4(x1))) 109.43/28.73 4^1(1(4(5(0(x1))))) -> 4^1(x1) 109.43/28.73 3^1(5(1(4(2(x1))))) -> 2^1(5(3(1(4(0(x1)))))) 109.43/28.73 3^1(5(1(4(2(x1))))) -> 5^1(3(1(4(0(x1))))) 109.43/28.73 3^1(5(1(4(2(x1))))) -> 3^1(1(4(0(x1)))) 109.43/28.73 3^1(5(1(4(2(x1))))) -> 1^1(4(0(x1))) 109.43/28.73 3^1(5(1(4(2(x1))))) -> 4^1(0(x1)) 109.43/28.73 1^1(0(2(4(2(x1))))) -> 4^1(0(2(1(1(2(x1)))))) 109.43/28.73 1^1(0(2(4(2(x1))))) -> 2^1(1(1(2(x1)))) 109.43/28.73 1^1(0(2(4(2(x1))))) -> 1^1(1(2(x1))) 109.43/28.73 1^1(0(2(4(2(x1))))) -> 1^1(2(x1)) 109.43/28.73 1^1(4(3(5(2(x1))))) -> 2^1(3(4(0(1(5(x1)))))) 109.43/28.73 1^1(4(3(5(2(x1))))) -> 3^1(4(0(1(5(x1))))) 109.43/28.73 1^1(4(3(5(2(x1))))) -> 4^1(0(1(5(x1)))) 109.43/28.73 1^1(4(3(5(2(x1))))) -> 1^1(5(x1)) 109.43/28.73 1^1(4(3(5(2(x1))))) -> 5^1(x1) 109.43/28.73 4^1(5(1(0(4(x1))))) -> 5^1(1(4(0(0(4(x1)))))) 109.43/28.73 4^1(5(1(0(4(x1))))) -> 1^1(4(0(0(4(x1))))) 109.43/28.73 4^1(5(1(0(4(x1))))) -> 4^1(0(0(4(x1)))) 109.43/28.73 3^1(2(0(3(4(x1))))) -> 3^1(1(3(2(4(0(x1)))))) 109.43/28.73 3^1(2(0(3(4(x1))))) -> 1^1(3(2(4(0(x1))))) 109.43/28.73 3^1(2(0(3(4(x1))))) -> 3^1(2(4(0(x1)))) 109.43/28.73 3^1(2(0(3(4(x1))))) -> 2^1(4(0(x1))) 109.43/28.73 3^1(2(0(3(4(x1))))) -> 4^1(0(x1)) 109.43/28.73 3^1(2(1(4(4(x1))))) -> 1^1(3(2(4(4(0(x1)))))) 109.43/28.73 3^1(2(1(4(4(x1))))) -> 3^1(2(4(4(0(x1))))) 109.43/28.73 3^1(2(1(4(4(x1))))) -> 2^1(4(4(0(x1)))) 109.43/28.73 3^1(2(1(4(4(x1))))) -> 4^1(4(0(x1))) 109.43/28.73 3^1(2(1(4(4(x1))))) -> 4^1(0(x1)) 109.43/28.73 2^1(0(1(5(4(x1))))) -> 5^1(1(2(4(0(1(x1)))))) 109.43/28.73 2^1(0(1(5(4(x1))))) -> 1^1(2(4(0(1(x1))))) 109.43/28.73 2^1(0(1(5(4(x1))))) -> 2^1(4(0(1(x1)))) 109.43/28.73 2^1(0(1(5(4(x1))))) -> 4^1(0(1(x1))) 109.43/28.73 2^1(0(1(5(4(x1))))) -> 1^1(x1) 109.43/28.73 109.43/28.73 The TRS R consists of the following rules: 109.43/28.73 109.43/28.73 2(1(0(x1))) -> 1(2(0(0(x1)))) 109.43/28.73 2(1(0(x1))) -> 3(1(2(0(x1)))) 109.43/28.73 2(1(0(x1))) -> 1(3(2(0(x1)))) 109.43/28.73 4(1(0(x1))) -> 0(0(1(1(4(0(x1)))))) 109.43/28.73 2(3(0(x1))) -> 3(2(0(0(x1)))) 109.43/28.73 2(3(0(x1))) -> 1(3(2(0(x1)))) 109.43/28.73 4(3(0(x1))) -> 3(4(0(0(x1)))) 109.43/28.73 5(4(0(x1))) -> 5(1(4(0(0(x1))))) 109.43/28.73 1(0(2(x1))) -> 1(1(2(0(x1)))) 109.43/28.73 1(4(2(x1))) -> 1(3(2(4(0(x1))))) 109.43/28.73 2(3(4(x1))) -> 1(3(2(4(x1)))) 109.43/28.73 1(0(1(0(x1)))) -> 1(1(3(0(0(x1))))) 109.43/28.73 2(0(1(0(x1)))) -> 1(1(1(2(0(0(x1)))))) 109.43/28.73 2(4(1(0(x1)))) -> 1(1(2(4(0(x1))))) 109.43/28.73 4(2(3(0(x1)))) -> 0(3(2(4(0(x1))))) 109.43/28.73 2(4(3(0(x1)))) -> 1(3(2(4(0(x1))))) 109.43/28.73 2(5(3(0(x1)))) -> 5(3(1(2(0(x1))))) 109.43/28.73 3(5(4(0(x1)))) -> 5(3(4(0(0(x1))))) 109.43/28.73 4(3(5(0(x1)))) -> 5(3(3(4(0(x1))))) 109.43/28.73 2(4(5(0(x1)))) -> 5(1(2(4(0(x1))))) 109.43/28.73 1(3(0(2(x1)))) -> 1(1(2(3(0(x1))))) 109.43/28.73 1(4(2(2(x1)))) -> 1(3(2(2(4(x1))))) 109.43/28.73 3(4(3(2(x1)))) -> 3(3(4(0(3(2(x1)))))) 109.43/28.73 1(1(4(2(x1)))) -> 1(1(4(0(2(x1))))) 109.43/28.73 3(1(4(2(x1)))) -> 1(2(3(4(0(x1))))) 109.43/28.73 2(3(4(2(x1)))) -> 3(2(4(0(2(x1))))) 109.43/28.73 3(4(5(2(x1)))) -> 5(3(2(4(0(x1))))) 109.43/28.73 3(0(2(4(x1)))) -> 3(4(0(3(2(x1))))) 109.43/28.73 3(4(3(4(x1)))) -> 3(4(0(3(4(x1))))) 109.43/28.73 2(3(5(1(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 109.43/28.73 2(4(5(1(0(x1))))) -> 3(4(0(1(2(5(x1)))))) 109.43/28.73 5(4(5(1(0(x1))))) -> 5(5(2(1(4(0(x1)))))) 109.43/28.73 3(4(2(2(0(x1))))) -> 2(0(3(4(0(2(x1)))))) 109.43/28.73 5(4(0(3(0(x1))))) -> 2(3(4(0(0(5(x1)))))) 109.43/28.73 3(4(2(3(0(x1))))) -> 2(4(3(3(4(0(x1)))))) 109.43/28.73 4(5(2(4(0(x1))))) -> 4(5(1(2(4(0(x1)))))) 109.43/28.73 1(2(3(5(0(x1))))) -> 5(3(1(3(2(0(x1)))))) 109.43/28.73 4(1(4(5(0(x1))))) -> 0(5(1(4(0(4(x1)))))) 109.43/28.73 3(5(1(4(2(x1))))) -> 2(5(3(1(4(0(x1)))))) 109.43/28.73 1(0(2(4(2(x1))))) -> 4(0(2(1(1(2(x1)))))) 109.43/28.73 1(4(3(5(2(x1))))) -> 2(3(4(0(1(5(x1)))))) 109.43/28.73 4(5(1(0(4(x1))))) -> 5(1(4(0(0(4(x1)))))) 109.43/28.73 3(2(0(3(4(x1))))) -> 3(1(3(2(4(0(x1)))))) 109.43/28.73 3(2(1(4(4(x1))))) -> 1(3(2(4(4(0(x1)))))) 109.43/28.73 2(0(1(5(4(x1))))) -> 5(1(2(4(0(1(x1)))))) 109.43/28.73 109.43/28.73 Q is empty. 109.43/28.73 We have to consider all minimal (P,Q,R)-chains. 109.43/28.73 ---------------------------------------- 109.43/28.73 109.43/28.73 (5) DependencyGraphProof (EQUIVALENT) 109.43/28.73 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 115 less nodes. 109.43/28.73 ---------------------------------------- 109.43/28.73 109.43/28.73 (6) 109.43/28.73 Complex Obligation (AND) 109.43/28.73 109.43/28.73 ---------------------------------------- 109.43/28.73 109.43/28.73 (7) 109.43/28.73 Obligation: 109.43/28.73 Q DP problem: 109.43/28.73 The TRS P consists of the following rules: 109.43/28.73 109.43/28.73 4^1(1(4(5(0(x1))))) -> 4^1(x1) 109.43/28.73 109.43/28.73 The TRS R consists of the following rules: 109.43/28.73 109.43/28.73 2(1(0(x1))) -> 1(2(0(0(x1)))) 109.43/28.73 2(1(0(x1))) -> 3(1(2(0(x1)))) 109.43/28.73 2(1(0(x1))) -> 1(3(2(0(x1)))) 109.43/28.73 4(1(0(x1))) -> 0(0(1(1(4(0(x1)))))) 109.43/28.73 2(3(0(x1))) -> 3(2(0(0(x1)))) 109.43/28.73 2(3(0(x1))) -> 1(3(2(0(x1)))) 109.43/28.73 4(3(0(x1))) -> 3(4(0(0(x1)))) 109.43/28.73 5(4(0(x1))) -> 5(1(4(0(0(x1))))) 109.43/28.73 1(0(2(x1))) -> 1(1(2(0(x1)))) 109.43/28.73 1(4(2(x1))) -> 1(3(2(4(0(x1))))) 109.43/28.73 2(3(4(x1))) -> 1(3(2(4(x1)))) 109.43/28.73 1(0(1(0(x1)))) -> 1(1(3(0(0(x1))))) 109.43/28.73 2(0(1(0(x1)))) -> 1(1(1(2(0(0(x1)))))) 109.43/28.73 2(4(1(0(x1)))) -> 1(1(2(4(0(x1))))) 109.43/28.73 4(2(3(0(x1)))) -> 0(3(2(4(0(x1))))) 109.43/28.73 2(4(3(0(x1)))) -> 1(3(2(4(0(x1))))) 109.43/28.73 2(5(3(0(x1)))) -> 5(3(1(2(0(x1))))) 109.43/28.73 3(5(4(0(x1)))) -> 5(3(4(0(0(x1))))) 109.43/28.73 4(3(5(0(x1)))) -> 5(3(3(4(0(x1))))) 109.43/28.73 2(4(5(0(x1)))) -> 5(1(2(4(0(x1))))) 109.43/28.73 1(3(0(2(x1)))) -> 1(1(2(3(0(x1))))) 109.43/28.73 1(4(2(2(x1)))) -> 1(3(2(2(4(x1))))) 109.43/28.73 3(4(3(2(x1)))) -> 3(3(4(0(3(2(x1)))))) 109.43/28.73 1(1(4(2(x1)))) -> 1(1(4(0(2(x1))))) 109.43/28.73 3(1(4(2(x1)))) -> 1(2(3(4(0(x1))))) 109.43/28.73 2(3(4(2(x1)))) -> 3(2(4(0(2(x1))))) 109.43/28.73 3(4(5(2(x1)))) -> 5(3(2(4(0(x1))))) 109.43/28.73 3(0(2(4(x1)))) -> 3(4(0(3(2(x1))))) 109.43/28.73 3(4(3(4(x1)))) -> 3(4(0(3(4(x1))))) 109.43/28.73 2(3(5(1(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 109.43/28.73 2(4(5(1(0(x1))))) -> 3(4(0(1(2(5(x1)))))) 109.43/28.73 5(4(5(1(0(x1))))) -> 5(5(2(1(4(0(x1)))))) 109.43/28.73 3(4(2(2(0(x1))))) -> 2(0(3(4(0(2(x1)))))) 109.43/28.73 5(4(0(3(0(x1))))) -> 2(3(4(0(0(5(x1)))))) 109.43/28.73 3(4(2(3(0(x1))))) -> 2(4(3(3(4(0(x1)))))) 109.43/28.73 4(5(2(4(0(x1))))) -> 4(5(1(2(4(0(x1)))))) 109.43/28.73 1(2(3(5(0(x1))))) -> 5(3(1(3(2(0(x1)))))) 109.43/28.73 4(1(4(5(0(x1))))) -> 0(5(1(4(0(4(x1)))))) 109.43/28.73 3(5(1(4(2(x1))))) -> 2(5(3(1(4(0(x1)))))) 109.43/28.73 1(0(2(4(2(x1))))) -> 4(0(2(1(1(2(x1)))))) 109.43/28.73 1(4(3(5(2(x1))))) -> 2(3(4(0(1(5(x1)))))) 109.43/28.73 4(5(1(0(4(x1))))) -> 5(1(4(0(0(4(x1)))))) 109.43/28.73 3(2(0(3(4(x1))))) -> 3(1(3(2(4(0(x1)))))) 109.43/28.73 3(2(1(4(4(x1))))) -> 1(3(2(4(4(0(x1)))))) 109.43/28.73 2(0(1(5(4(x1))))) -> 5(1(2(4(0(1(x1)))))) 109.43/28.73 109.43/28.73 Q is empty. 109.43/28.73 We have to consider all minimal (P,Q,R)-chains. 109.43/28.73 ---------------------------------------- 109.43/28.73 109.43/28.73 (8) UsableRulesProof (EQUIVALENT) 109.43/28.73 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. 109.43/28.73 ---------------------------------------- 109.43/28.73 109.43/28.73 (9) 109.43/28.73 Obligation: 109.43/28.73 Q DP problem: 109.43/28.73 The TRS P consists of the following rules: 109.43/28.73 109.43/28.73 4^1(1(4(5(0(x1))))) -> 4^1(x1) 109.43/28.73 109.43/28.73 R is empty. 109.43/28.73 Q is empty. 109.43/28.73 We have to consider all minimal (P,Q,R)-chains. 109.43/28.73 ---------------------------------------- 109.43/28.73 109.43/28.73 (10) QDPSizeChangeProof (EQUIVALENT) 109.43/28.73 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. 109.43/28.73 109.43/28.73 From the DPs we obtained the following set of size-change graphs: 109.43/28.73 *4^1(1(4(5(0(x1))))) -> 4^1(x1) 109.43/28.73 The graph contains the following edges 1 > 1 109.43/28.73 109.43/28.73 109.43/28.73 ---------------------------------------- 109.43/28.73 109.43/28.73 (11) 109.43/28.73 YES 109.43/28.73 109.43/28.73 ---------------------------------------- 109.43/28.73 109.43/28.73 (12) 109.43/28.73 Obligation: 109.43/28.73 Q DP problem: 109.43/28.73 The TRS P consists of the following rules: 109.43/28.73 109.43/28.73 2^1(1(0(x1))) -> 3^1(1(2(0(x1)))) 109.43/28.73 3^1(1(4(2(x1)))) -> 1^1(2(3(4(0(x1))))) 109.43/28.73 1^1(0(2(x1))) -> 1^1(1(2(0(x1)))) 109.43/28.73 1^1(0(2(x1))) -> 1^1(2(0(x1))) 109.43/28.73 1^1(0(2(x1))) -> 2^1(0(x1)) 109.43/28.73 2^1(0(1(5(4(x1))))) -> 1^1(x1) 109.43/28.73 1^1(3(0(2(x1)))) -> 1^1(1(2(3(0(x1))))) 109.43/28.73 1^1(3(0(2(x1)))) -> 1^1(2(3(0(x1)))) 109.43/28.73 1^1(3(0(2(x1)))) -> 2^1(3(0(x1))) 109.43/28.73 2^1(1(0(x1))) -> 1^1(2(0(x1))) 109.43/28.73 1^1(3(0(2(x1)))) -> 3^1(0(x1)) 109.43/28.73 3^1(0(2(4(x1)))) -> 3^1(2(x1)) 109.43/28.73 3^1(1(4(2(x1)))) -> 2^1(3(4(0(x1)))) 109.43/28.73 2^1(3(4(x1))) -> 1^1(3(2(4(x1)))) 109.43/28.73 1^1(4(2(2(x1)))) -> 1^1(3(2(2(4(x1))))) 109.43/28.73 1^1(4(2(2(x1)))) -> 3^1(2(2(4(x1)))) 109.43/28.73 3^1(0(2(4(x1)))) -> 2^1(x1) 109.43/28.73 2^1(1(0(x1))) -> 2^1(0(x1)) 109.43/28.73 2^1(1(0(x1))) -> 1^1(3(2(0(x1)))) 109.43/28.73 1^1(4(2(2(x1)))) -> 2^1(2(4(x1))) 109.43/28.73 2^1(1(0(x1))) -> 3^1(2(0(x1))) 109.43/28.73 3^1(4(2(2(0(x1))))) -> 2^1(x1) 109.43/28.73 2^1(3(0(x1))) -> 1^1(3(2(0(x1)))) 109.43/28.73 1^1(4(2(2(x1)))) -> 2^1(4(x1)) 109.43/28.73 2^1(3(0(x1))) -> 3^1(2(0(x1))) 109.43/28.73 2^1(3(0(x1))) -> 2^1(0(x1)) 109.43/28.73 2^1(3(4(x1))) -> 3^1(2(4(x1))) 109.43/28.73 2^1(3(4(x1))) -> 2^1(4(x1)) 109.43/28.73 2^1(5(3(0(x1)))) -> 5^1(3(1(2(0(x1))))) 109.43/28.73 5^1(4(0(3(0(x1))))) -> 2^1(3(4(0(0(5(x1)))))) 109.43/28.73 5^1(4(0(3(0(x1))))) -> 5^1(x1) 109.43/28.73 2^1(5(3(0(x1)))) -> 3^1(1(2(0(x1)))) 109.43/28.73 2^1(5(3(0(x1)))) -> 1^1(2(0(x1))) 109.43/28.73 1^1(2(3(5(0(x1))))) -> 5^1(3(1(3(2(0(x1)))))) 109.43/28.73 1^1(2(3(5(0(x1))))) -> 3^1(1(3(2(0(x1))))) 109.43/28.73 1^1(2(3(5(0(x1))))) -> 1^1(3(2(0(x1)))) 109.43/28.73 1^1(2(3(5(0(x1))))) -> 3^1(2(0(x1))) 109.43/28.73 1^1(2(3(5(0(x1))))) -> 2^1(0(x1)) 109.43/28.73 1^1(0(2(4(2(x1))))) -> 2^1(1(1(2(x1)))) 109.43/28.73 2^1(5(3(0(x1)))) -> 2^1(0(x1)) 109.43/28.73 2^1(3(5(1(0(x1))))) -> 1^1(2(3(0(5(0(x1)))))) 109.43/28.73 1^1(0(2(4(2(x1))))) -> 1^1(1(2(x1))) 109.43/28.73 1^1(0(2(4(2(x1))))) -> 1^1(2(x1)) 109.43/28.73 1^1(4(3(5(2(x1))))) -> 2^1(3(4(0(1(5(x1)))))) 109.43/28.73 1^1(4(3(5(2(x1))))) -> 1^1(5(x1)) 109.43/28.73 1^1(4(3(5(2(x1))))) -> 5^1(x1) 109.43/28.73 2^1(3(5(1(0(x1))))) -> 2^1(3(0(5(0(x1))))) 109.43/28.73 2^1(4(5(1(0(x1))))) -> 1^1(2(5(x1))) 109.43/28.73 2^1(4(5(1(0(x1))))) -> 2^1(5(x1)) 109.43/28.73 2^1(4(5(1(0(x1))))) -> 5^1(x1) 109.43/28.73 109.43/28.73 The TRS R consists of the following rules: 109.43/28.73 109.43/28.73 2(1(0(x1))) -> 1(2(0(0(x1)))) 109.43/28.73 2(1(0(x1))) -> 3(1(2(0(x1)))) 109.43/28.73 2(1(0(x1))) -> 1(3(2(0(x1)))) 109.43/28.73 4(1(0(x1))) -> 0(0(1(1(4(0(x1)))))) 109.43/28.73 2(3(0(x1))) -> 3(2(0(0(x1)))) 109.43/28.73 2(3(0(x1))) -> 1(3(2(0(x1)))) 109.43/28.73 4(3(0(x1))) -> 3(4(0(0(x1)))) 109.43/28.73 5(4(0(x1))) -> 5(1(4(0(0(x1))))) 109.43/28.73 1(0(2(x1))) -> 1(1(2(0(x1)))) 109.43/28.73 1(4(2(x1))) -> 1(3(2(4(0(x1))))) 109.43/28.73 2(3(4(x1))) -> 1(3(2(4(x1)))) 109.43/28.73 1(0(1(0(x1)))) -> 1(1(3(0(0(x1))))) 109.43/28.73 2(0(1(0(x1)))) -> 1(1(1(2(0(0(x1)))))) 109.43/28.73 2(4(1(0(x1)))) -> 1(1(2(4(0(x1))))) 109.43/28.73 4(2(3(0(x1)))) -> 0(3(2(4(0(x1))))) 109.43/28.73 2(4(3(0(x1)))) -> 1(3(2(4(0(x1))))) 109.43/28.73 2(5(3(0(x1)))) -> 5(3(1(2(0(x1))))) 109.43/28.73 3(5(4(0(x1)))) -> 5(3(4(0(0(x1))))) 109.43/28.73 4(3(5(0(x1)))) -> 5(3(3(4(0(x1))))) 109.43/28.73 2(4(5(0(x1)))) -> 5(1(2(4(0(x1))))) 109.43/28.73 1(3(0(2(x1)))) -> 1(1(2(3(0(x1))))) 109.43/28.73 1(4(2(2(x1)))) -> 1(3(2(2(4(x1))))) 109.43/28.73 3(4(3(2(x1)))) -> 3(3(4(0(3(2(x1)))))) 109.43/28.73 1(1(4(2(x1)))) -> 1(1(4(0(2(x1))))) 109.43/28.73 3(1(4(2(x1)))) -> 1(2(3(4(0(x1))))) 109.43/28.73 2(3(4(2(x1)))) -> 3(2(4(0(2(x1))))) 109.43/28.73 3(4(5(2(x1)))) -> 5(3(2(4(0(x1))))) 109.43/28.73 3(0(2(4(x1)))) -> 3(4(0(3(2(x1))))) 109.43/28.73 3(4(3(4(x1)))) -> 3(4(0(3(4(x1))))) 109.43/28.73 2(3(5(1(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 109.43/28.73 2(4(5(1(0(x1))))) -> 3(4(0(1(2(5(x1)))))) 109.43/28.73 5(4(5(1(0(x1))))) -> 5(5(2(1(4(0(x1)))))) 109.43/28.73 3(4(2(2(0(x1))))) -> 2(0(3(4(0(2(x1)))))) 109.43/28.73 5(4(0(3(0(x1))))) -> 2(3(4(0(0(5(x1)))))) 109.43/28.73 3(4(2(3(0(x1))))) -> 2(4(3(3(4(0(x1)))))) 109.43/28.73 4(5(2(4(0(x1))))) -> 4(5(1(2(4(0(x1)))))) 109.43/28.73 1(2(3(5(0(x1))))) -> 5(3(1(3(2(0(x1)))))) 109.43/28.73 4(1(4(5(0(x1))))) -> 0(5(1(4(0(4(x1)))))) 109.43/28.73 3(5(1(4(2(x1))))) -> 2(5(3(1(4(0(x1)))))) 109.43/28.73 1(0(2(4(2(x1))))) -> 4(0(2(1(1(2(x1)))))) 109.43/28.73 1(4(3(5(2(x1))))) -> 2(3(4(0(1(5(x1)))))) 109.43/28.73 4(5(1(0(4(x1))))) -> 5(1(4(0(0(4(x1)))))) 109.43/28.73 3(2(0(3(4(x1))))) -> 3(1(3(2(4(0(x1)))))) 109.43/28.73 3(2(1(4(4(x1))))) -> 1(3(2(4(4(0(x1)))))) 109.43/28.73 2(0(1(5(4(x1))))) -> 5(1(2(4(0(1(x1)))))) 109.43/28.73 109.43/28.73 Q is empty. 109.43/28.73 We have to consider all minimal (P,Q,R)-chains. 109.43/28.73 ---------------------------------------- 109.43/28.73 109.43/28.73 (13) QDPOrderProof (EQUIVALENT) 109.43/28.73 We use the reduction pair processor [LPAR04,JAR06]. 109.43/28.73 109.43/28.73 109.43/28.73 The following pairs can be oriented strictly and are deleted. 109.43/28.73 109.43/28.73 2^1(1(0(x1))) -> 3^1(1(2(0(x1)))) 109.43/28.73 3^1(0(2(4(x1)))) -> 3^1(2(x1)) 109.43/28.73 1^1(4(2(2(x1)))) -> 3^1(2(2(4(x1)))) 109.43/28.73 2^1(1(0(x1))) -> 3^1(2(0(x1))) 109.43/28.73 2^1(3(0(x1))) -> 3^1(2(0(x1))) 109.43/28.73 2^1(3(4(x1))) -> 3^1(2(4(x1))) 109.43/28.73 2^1(5(3(0(x1)))) -> 3^1(1(2(0(x1)))) 109.43/28.73 1^1(2(3(5(0(x1))))) -> 3^1(1(3(2(0(x1))))) 109.43/28.73 1^1(2(3(5(0(x1))))) -> 3^1(2(0(x1))) 109.43/28.73 The remaining pairs can at least be oriented weakly. 109.43/28.73 Used ordering: Polynomial interpretation [POLO]: 109.43/28.73 109.43/28.73 POL(0(x_1)) = 1 109.43/28.73 POL(1(x_1)) = x_1 109.43/28.73 POL(1^1(x_1)) = 1 109.43/28.73 POL(2(x_1)) = 0 109.43/28.73 POL(2^1(x_1)) = 1 109.43/28.73 POL(3(x_1)) = 0 109.43/28.73 POL(3^1(x_1)) = x_1 109.43/28.73 POL(4(x_1)) = 1 109.43/28.73 POL(5(x_1)) = 0 109.43/28.73 POL(5^1(x_1)) = 1 109.43/28.73 109.43/28.73 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 109.43/28.73 109.43/28.73 2(0(1(0(x1)))) -> 1(1(1(2(0(0(x1)))))) 109.43/28.73 2(0(1(5(4(x1))))) -> 5(1(2(4(0(1(x1)))))) 109.43/28.73 5(4(0(3(0(x1))))) -> 2(3(4(0(0(5(x1)))))) 109.43/28.73 2(3(4(x1))) -> 1(3(2(4(x1)))) 109.43/28.73 1(0(2(x1))) -> 1(1(2(0(x1)))) 109.43/28.73 1(3(0(2(x1)))) -> 1(1(2(3(0(x1))))) 109.43/28.73 1(4(2(2(x1)))) -> 1(3(2(2(4(x1))))) 109.43/28.73 1(2(3(5(0(x1))))) -> 5(3(1(3(2(0(x1)))))) 109.43/28.73 1(4(3(5(2(x1))))) -> 2(3(4(0(1(5(x1)))))) 109.43/28.73 1(4(2(x1))) -> 1(3(2(4(0(x1))))) 109.43/28.73 1(0(1(0(x1)))) -> 1(1(3(0(0(x1))))) 109.43/28.73 1(1(4(2(x1)))) -> 1(1(4(0(2(x1))))) 109.43/28.73 1(0(2(4(2(x1))))) -> 4(0(2(1(1(2(x1)))))) 109.43/28.73 3(0(2(4(x1)))) -> 3(4(0(3(2(x1))))) 109.43/28.73 2(1(0(x1))) -> 1(2(0(0(x1)))) 109.43/28.73 2(1(0(x1))) -> 3(1(2(0(x1)))) 109.43/28.73 2(1(0(x1))) -> 1(3(2(0(x1)))) 109.43/28.73 2(3(0(x1))) -> 3(2(0(0(x1)))) 109.43/28.73 2(3(0(x1))) -> 1(3(2(0(x1)))) 109.43/28.73 2(4(1(0(x1)))) -> 1(1(2(4(0(x1))))) 109.43/28.73 2(4(3(0(x1)))) -> 1(3(2(4(0(x1))))) 109.43/28.73 2(5(3(0(x1)))) -> 5(3(1(2(0(x1))))) 109.43/28.73 2(4(5(0(x1)))) -> 5(1(2(4(0(x1))))) 109.43/28.73 2(3(4(2(x1)))) -> 3(2(4(0(2(x1))))) 109.43/28.73 2(3(5(1(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 109.43/28.73 2(4(5(1(0(x1))))) -> 3(4(0(1(2(5(x1)))))) 109.43/28.73 3(5(4(0(x1)))) -> 5(3(4(0(0(x1))))) 109.43/28.73 3(4(3(2(x1)))) -> 3(3(4(0(3(2(x1)))))) 109.43/28.73 3(1(4(2(x1)))) -> 1(2(3(4(0(x1))))) 109.43/28.73 3(4(5(2(x1)))) -> 5(3(2(4(0(x1))))) 109.43/28.73 3(4(3(4(x1)))) -> 3(4(0(3(4(x1))))) 109.43/28.73 3(4(2(2(0(x1))))) -> 2(0(3(4(0(2(x1)))))) 109.43/28.73 3(4(2(3(0(x1))))) -> 2(4(3(3(4(0(x1)))))) 109.43/28.73 3(5(1(4(2(x1))))) -> 2(5(3(1(4(0(x1)))))) 109.43/28.73 3(2(0(3(4(x1))))) -> 3(1(3(2(4(0(x1)))))) 109.43/28.73 3(2(1(4(4(x1))))) -> 1(3(2(4(4(0(x1)))))) 109.43/28.73 5(4(0(x1))) -> 5(1(4(0(0(x1))))) 109.43/28.73 5(4(5(1(0(x1))))) -> 5(5(2(1(4(0(x1)))))) 109.43/28.73 109.43/28.73 109.43/28.73 ---------------------------------------- 109.43/28.73 109.43/28.73 (14) 109.43/28.73 Obligation: 109.43/28.73 Q DP problem: 109.43/28.73 The TRS P consists of the following rules: 109.43/28.73 109.43/28.73 3^1(1(4(2(x1)))) -> 1^1(2(3(4(0(x1))))) 109.43/28.73 1^1(0(2(x1))) -> 1^1(1(2(0(x1)))) 109.43/28.73 1^1(0(2(x1))) -> 1^1(2(0(x1))) 109.43/28.73 1^1(0(2(x1))) -> 2^1(0(x1)) 109.43/28.73 2^1(0(1(5(4(x1))))) -> 1^1(x1) 109.43/28.73 1^1(3(0(2(x1)))) -> 1^1(1(2(3(0(x1))))) 109.76/28.74 1^1(3(0(2(x1)))) -> 1^1(2(3(0(x1)))) 109.76/28.74 1^1(3(0(2(x1)))) -> 2^1(3(0(x1))) 109.76/28.74 2^1(1(0(x1))) -> 1^1(2(0(x1))) 109.76/28.74 1^1(3(0(2(x1)))) -> 3^1(0(x1)) 109.76/28.74 3^1(1(4(2(x1)))) -> 2^1(3(4(0(x1)))) 109.76/28.74 2^1(3(4(x1))) -> 1^1(3(2(4(x1)))) 109.76/28.74 1^1(4(2(2(x1)))) -> 1^1(3(2(2(4(x1))))) 109.76/28.74 3^1(0(2(4(x1)))) -> 2^1(x1) 109.76/28.74 2^1(1(0(x1))) -> 2^1(0(x1)) 109.76/28.74 2^1(1(0(x1))) -> 1^1(3(2(0(x1)))) 109.76/28.74 1^1(4(2(2(x1)))) -> 2^1(2(4(x1))) 109.76/28.74 3^1(4(2(2(0(x1))))) -> 2^1(x1) 109.76/28.74 2^1(3(0(x1))) -> 1^1(3(2(0(x1)))) 109.76/28.74 1^1(4(2(2(x1)))) -> 2^1(4(x1)) 109.76/28.74 2^1(3(0(x1))) -> 2^1(0(x1)) 109.76/28.74 2^1(3(4(x1))) -> 2^1(4(x1)) 109.76/28.74 2^1(5(3(0(x1)))) -> 5^1(3(1(2(0(x1))))) 109.76/28.74 5^1(4(0(3(0(x1))))) -> 2^1(3(4(0(0(5(x1)))))) 109.76/28.74 5^1(4(0(3(0(x1))))) -> 5^1(x1) 109.76/28.74 2^1(5(3(0(x1)))) -> 1^1(2(0(x1))) 109.76/28.74 1^1(2(3(5(0(x1))))) -> 5^1(3(1(3(2(0(x1)))))) 109.76/28.74 1^1(2(3(5(0(x1))))) -> 1^1(3(2(0(x1)))) 109.76/28.74 1^1(2(3(5(0(x1))))) -> 2^1(0(x1)) 109.76/28.74 1^1(0(2(4(2(x1))))) -> 2^1(1(1(2(x1)))) 109.76/28.74 2^1(5(3(0(x1)))) -> 2^1(0(x1)) 109.76/28.74 2^1(3(5(1(0(x1))))) -> 1^1(2(3(0(5(0(x1)))))) 109.76/28.74 1^1(0(2(4(2(x1))))) -> 1^1(1(2(x1))) 109.76/28.74 1^1(0(2(4(2(x1))))) -> 1^1(2(x1)) 109.76/28.74 1^1(4(3(5(2(x1))))) -> 2^1(3(4(0(1(5(x1)))))) 109.76/28.74 1^1(4(3(5(2(x1))))) -> 1^1(5(x1)) 109.76/28.74 1^1(4(3(5(2(x1))))) -> 5^1(x1) 109.76/28.74 2^1(3(5(1(0(x1))))) -> 2^1(3(0(5(0(x1))))) 109.76/28.74 2^1(4(5(1(0(x1))))) -> 1^1(2(5(x1))) 109.76/28.74 2^1(4(5(1(0(x1))))) -> 2^1(5(x1)) 109.76/28.74 2^1(4(5(1(0(x1))))) -> 5^1(x1) 109.76/28.74 109.76/28.74 The TRS R consists of the following rules: 109.76/28.74 109.76/28.74 2(1(0(x1))) -> 1(2(0(0(x1)))) 109.76/28.74 2(1(0(x1))) -> 3(1(2(0(x1)))) 109.76/28.74 2(1(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.74 4(1(0(x1))) -> 0(0(1(1(4(0(x1)))))) 109.76/28.74 2(3(0(x1))) -> 3(2(0(0(x1)))) 109.76/28.74 2(3(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.74 4(3(0(x1))) -> 3(4(0(0(x1)))) 109.76/28.74 5(4(0(x1))) -> 5(1(4(0(0(x1))))) 109.76/28.74 1(0(2(x1))) -> 1(1(2(0(x1)))) 109.76/28.74 1(4(2(x1))) -> 1(3(2(4(0(x1))))) 109.76/28.74 2(3(4(x1))) -> 1(3(2(4(x1)))) 109.76/28.74 1(0(1(0(x1)))) -> 1(1(3(0(0(x1))))) 109.76/28.74 2(0(1(0(x1)))) -> 1(1(1(2(0(0(x1)))))) 109.76/28.74 2(4(1(0(x1)))) -> 1(1(2(4(0(x1))))) 109.76/28.74 4(2(3(0(x1)))) -> 0(3(2(4(0(x1))))) 109.76/28.74 2(4(3(0(x1)))) -> 1(3(2(4(0(x1))))) 109.76/28.74 2(5(3(0(x1)))) -> 5(3(1(2(0(x1))))) 109.76/28.74 3(5(4(0(x1)))) -> 5(3(4(0(0(x1))))) 109.76/28.74 4(3(5(0(x1)))) -> 5(3(3(4(0(x1))))) 109.76/28.74 2(4(5(0(x1)))) -> 5(1(2(4(0(x1))))) 109.76/28.74 1(3(0(2(x1)))) -> 1(1(2(3(0(x1))))) 109.76/28.74 1(4(2(2(x1)))) -> 1(3(2(2(4(x1))))) 109.76/28.74 3(4(3(2(x1)))) -> 3(3(4(0(3(2(x1)))))) 109.76/28.74 1(1(4(2(x1)))) -> 1(1(4(0(2(x1))))) 109.76/28.74 3(1(4(2(x1)))) -> 1(2(3(4(0(x1))))) 109.76/28.74 2(3(4(2(x1)))) -> 3(2(4(0(2(x1))))) 109.76/28.74 3(4(5(2(x1)))) -> 5(3(2(4(0(x1))))) 109.76/28.74 3(0(2(4(x1)))) -> 3(4(0(3(2(x1))))) 109.76/28.74 3(4(3(4(x1)))) -> 3(4(0(3(4(x1))))) 109.76/28.74 2(3(5(1(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 109.76/28.74 2(4(5(1(0(x1))))) -> 3(4(0(1(2(5(x1)))))) 109.76/28.74 5(4(5(1(0(x1))))) -> 5(5(2(1(4(0(x1)))))) 109.76/28.74 3(4(2(2(0(x1))))) -> 2(0(3(4(0(2(x1)))))) 109.76/28.74 5(4(0(3(0(x1))))) -> 2(3(4(0(0(5(x1)))))) 109.76/28.74 3(4(2(3(0(x1))))) -> 2(4(3(3(4(0(x1)))))) 109.76/28.74 4(5(2(4(0(x1))))) -> 4(5(1(2(4(0(x1)))))) 109.76/28.74 1(2(3(5(0(x1))))) -> 5(3(1(3(2(0(x1)))))) 109.76/28.74 4(1(4(5(0(x1))))) -> 0(5(1(4(0(4(x1)))))) 109.76/28.74 3(5(1(4(2(x1))))) -> 2(5(3(1(4(0(x1)))))) 109.76/28.74 1(0(2(4(2(x1))))) -> 4(0(2(1(1(2(x1)))))) 109.76/28.74 1(4(3(5(2(x1))))) -> 2(3(4(0(1(5(x1)))))) 109.76/28.74 4(5(1(0(4(x1))))) -> 5(1(4(0(0(4(x1)))))) 109.76/28.74 3(2(0(3(4(x1))))) -> 3(1(3(2(4(0(x1)))))) 109.76/28.74 3(2(1(4(4(x1))))) -> 1(3(2(4(4(0(x1)))))) 109.76/28.74 2(0(1(5(4(x1))))) -> 5(1(2(4(0(1(x1)))))) 109.76/28.74 109.76/28.74 Q is empty. 109.76/28.74 We have to consider all minimal (P,Q,R)-chains. 109.76/28.74 ---------------------------------------- 109.76/28.74 109.76/28.74 (15) DependencyGraphProof (EQUIVALENT) 109.76/28.74 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 109.76/28.74 ---------------------------------------- 109.76/28.74 109.76/28.74 (16) 109.76/28.74 Obligation: 109.76/28.74 Q DP problem: 109.76/28.74 The TRS P consists of the following rules: 109.76/28.74 109.76/28.74 1^1(0(2(x1))) -> 1^1(2(0(x1))) 109.76/28.74 1^1(0(2(x1))) -> 1^1(1(2(0(x1)))) 109.76/28.74 1^1(0(2(x1))) -> 2^1(0(x1)) 109.76/28.74 2^1(0(1(5(4(x1))))) -> 1^1(x1) 109.76/28.74 1^1(3(0(2(x1)))) -> 1^1(1(2(3(0(x1))))) 109.76/28.74 1^1(3(0(2(x1)))) -> 1^1(2(3(0(x1)))) 109.76/28.74 1^1(3(0(2(x1)))) -> 2^1(3(0(x1))) 109.76/28.74 2^1(1(0(x1))) -> 1^1(2(0(x1))) 109.76/28.74 1^1(3(0(2(x1)))) -> 3^1(0(x1)) 109.76/28.74 3^1(0(2(4(x1)))) -> 2^1(x1) 109.76/28.74 2^1(1(0(x1))) -> 2^1(0(x1)) 109.76/28.74 2^1(1(0(x1))) -> 1^1(3(2(0(x1)))) 109.76/28.74 1^1(4(2(2(x1)))) -> 1^1(3(2(2(4(x1))))) 109.76/28.74 1^1(4(2(2(x1)))) -> 2^1(2(4(x1))) 109.76/28.74 2^1(3(0(x1))) -> 1^1(3(2(0(x1)))) 109.76/28.74 1^1(4(2(2(x1)))) -> 2^1(4(x1)) 109.76/28.74 2^1(3(0(x1))) -> 2^1(0(x1)) 109.76/28.74 2^1(3(4(x1))) -> 1^1(3(2(4(x1)))) 109.76/28.74 1^1(2(3(5(0(x1))))) -> 5^1(3(1(3(2(0(x1)))))) 109.76/28.74 5^1(4(0(3(0(x1))))) -> 2^1(3(4(0(0(5(x1)))))) 109.76/28.74 2^1(3(4(x1))) -> 2^1(4(x1)) 109.76/28.74 2^1(5(3(0(x1)))) -> 5^1(3(1(2(0(x1))))) 109.76/28.74 5^1(4(0(3(0(x1))))) -> 5^1(x1) 109.76/28.74 2^1(5(3(0(x1)))) -> 1^1(2(0(x1))) 109.76/28.74 1^1(2(3(5(0(x1))))) -> 1^1(3(2(0(x1)))) 109.76/28.74 1^1(2(3(5(0(x1))))) -> 2^1(0(x1)) 109.76/28.74 1^1(0(2(4(2(x1))))) -> 2^1(1(1(2(x1)))) 109.76/28.74 2^1(5(3(0(x1)))) -> 2^1(0(x1)) 109.76/28.74 2^1(3(5(1(0(x1))))) -> 1^1(2(3(0(5(0(x1)))))) 109.76/28.74 1^1(0(2(4(2(x1))))) -> 1^1(1(2(x1))) 109.76/28.74 1^1(0(2(4(2(x1))))) -> 1^1(2(x1)) 109.76/28.74 1^1(4(3(5(2(x1))))) -> 2^1(3(4(0(1(5(x1)))))) 109.76/28.74 1^1(4(3(5(2(x1))))) -> 1^1(5(x1)) 109.76/28.74 1^1(4(3(5(2(x1))))) -> 5^1(x1) 109.76/28.74 2^1(3(5(1(0(x1))))) -> 2^1(3(0(5(0(x1))))) 109.76/28.74 2^1(4(5(1(0(x1))))) -> 1^1(2(5(x1))) 109.76/28.74 2^1(4(5(1(0(x1))))) -> 2^1(5(x1)) 109.76/28.74 2^1(4(5(1(0(x1))))) -> 5^1(x1) 109.76/28.74 109.76/28.74 The TRS R consists of the following rules: 109.76/28.74 109.76/28.74 2(1(0(x1))) -> 1(2(0(0(x1)))) 109.76/28.74 2(1(0(x1))) -> 3(1(2(0(x1)))) 109.76/28.74 2(1(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.74 4(1(0(x1))) -> 0(0(1(1(4(0(x1)))))) 109.76/28.74 2(3(0(x1))) -> 3(2(0(0(x1)))) 109.76/28.74 2(3(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.74 4(3(0(x1))) -> 3(4(0(0(x1)))) 109.76/28.74 5(4(0(x1))) -> 5(1(4(0(0(x1))))) 109.76/28.74 1(0(2(x1))) -> 1(1(2(0(x1)))) 109.76/28.74 1(4(2(x1))) -> 1(3(2(4(0(x1))))) 109.76/28.74 2(3(4(x1))) -> 1(3(2(4(x1)))) 109.76/28.74 1(0(1(0(x1)))) -> 1(1(3(0(0(x1))))) 109.76/28.74 2(0(1(0(x1)))) -> 1(1(1(2(0(0(x1)))))) 109.76/28.74 2(4(1(0(x1)))) -> 1(1(2(4(0(x1))))) 109.76/28.74 4(2(3(0(x1)))) -> 0(3(2(4(0(x1))))) 109.76/28.74 2(4(3(0(x1)))) -> 1(3(2(4(0(x1))))) 109.76/28.74 2(5(3(0(x1)))) -> 5(3(1(2(0(x1))))) 109.76/28.74 3(5(4(0(x1)))) -> 5(3(4(0(0(x1))))) 109.76/28.74 4(3(5(0(x1)))) -> 5(3(3(4(0(x1))))) 109.76/28.74 2(4(5(0(x1)))) -> 5(1(2(4(0(x1))))) 109.76/28.74 1(3(0(2(x1)))) -> 1(1(2(3(0(x1))))) 109.76/28.74 1(4(2(2(x1)))) -> 1(3(2(2(4(x1))))) 109.76/28.74 3(4(3(2(x1)))) -> 3(3(4(0(3(2(x1)))))) 109.76/28.74 1(1(4(2(x1)))) -> 1(1(4(0(2(x1))))) 109.76/28.74 3(1(4(2(x1)))) -> 1(2(3(4(0(x1))))) 109.76/28.74 2(3(4(2(x1)))) -> 3(2(4(0(2(x1))))) 109.76/28.74 3(4(5(2(x1)))) -> 5(3(2(4(0(x1))))) 109.76/28.74 3(0(2(4(x1)))) -> 3(4(0(3(2(x1))))) 109.76/28.74 3(4(3(4(x1)))) -> 3(4(0(3(4(x1))))) 109.76/28.74 2(3(5(1(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 109.76/28.74 2(4(5(1(0(x1))))) -> 3(4(0(1(2(5(x1)))))) 109.76/28.74 5(4(5(1(0(x1))))) -> 5(5(2(1(4(0(x1)))))) 109.76/28.74 3(4(2(2(0(x1))))) -> 2(0(3(4(0(2(x1)))))) 109.76/28.74 5(4(0(3(0(x1))))) -> 2(3(4(0(0(5(x1)))))) 109.76/28.74 3(4(2(3(0(x1))))) -> 2(4(3(3(4(0(x1)))))) 109.76/28.74 4(5(2(4(0(x1))))) -> 4(5(1(2(4(0(x1)))))) 109.76/28.74 1(2(3(5(0(x1))))) -> 5(3(1(3(2(0(x1)))))) 109.76/28.74 4(1(4(5(0(x1))))) -> 0(5(1(4(0(4(x1)))))) 109.76/28.74 3(5(1(4(2(x1))))) -> 2(5(3(1(4(0(x1)))))) 109.76/28.74 1(0(2(4(2(x1))))) -> 4(0(2(1(1(2(x1)))))) 109.76/28.74 1(4(3(5(2(x1))))) -> 2(3(4(0(1(5(x1)))))) 109.76/28.74 4(5(1(0(4(x1))))) -> 5(1(4(0(0(4(x1)))))) 109.76/28.74 3(2(0(3(4(x1))))) -> 3(1(3(2(4(0(x1)))))) 109.76/28.74 3(2(1(4(4(x1))))) -> 1(3(2(4(4(0(x1)))))) 109.76/28.74 2(0(1(5(4(x1))))) -> 5(1(2(4(0(1(x1)))))) 109.76/28.74 109.76/28.74 Q is empty. 109.76/28.74 We have to consider all minimal (P,Q,R)-chains. 109.76/28.74 ---------------------------------------- 109.76/28.74 109.76/28.74 (17) QDPOrderProof (EQUIVALENT) 109.76/28.74 We use the reduction pair processor [LPAR04,JAR06]. 109.76/28.74 109.76/28.74 109.76/28.74 The following pairs can be oriented strictly and are deleted. 109.76/28.74 109.76/28.74 2^1(0(1(5(4(x1))))) -> 1^1(x1) 109.76/28.74 2^1(5(3(0(x1)))) -> 1^1(2(0(x1))) 109.76/28.74 1^1(2(3(5(0(x1))))) -> 1^1(3(2(0(x1)))) 109.76/28.74 1^1(2(3(5(0(x1))))) -> 2^1(0(x1)) 109.76/28.74 2^1(5(3(0(x1)))) -> 2^1(0(x1)) 109.76/28.74 The remaining pairs can at least be oriented weakly. 109.76/28.74 Used ordering: Polynomial interpretation [POLO]: 109.76/28.74 109.76/28.74 POL(0(x_1)) = x_1 109.76/28.74 POL(1(x_1)) = x_1 109.76/28.74 POL(1^1(x_1)) = x_1 109.76/28.74 POL(2(x_1)) = x_1 109.76/28.74 POL(2^1(x_1)) = x_1 109.76/28.74 POL(3(x_1)) = x_1 109.76/28.74 POL(3^1(x_1)) = x_1 109.76/28.74 POL(4(x_1)) = x_1 109.76/28.74 POL(5(x_1)) = 1 + x_1 109.76/28.74 POL(5^1(x_1)) = 1 + x_1 109.76/28.74 109.76/28.74 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 109.76/28.74 109.76/28.74 2(0(1(0(x1)))) -> 1(1(1(2(0(0(x1)))))) 109.76/28.74 2(0(1(5(4(x1))))) -> 5(1(2(4(0(1(x1)))))) 109.76/28.74 5(4(0(3(0(x1))))) -> 2(3(4(0(0(5(x1)))))) 109.76/28.74 2(3(4(x1))) -> 1(3(2(4(x1)))) 109.76/28.74 1(0(2(x1))) -> 1(1(2(0(x1)))) 109.76/28.74 1(3(0(2(x1)))) -> 1(1(2(3(0(x1))))) 109.76/28.74 1(4(2(2(x1)))) -> 1(3(2(2(4(x1))))) 109.76/28.74 1(2(3(5(0(x1))))) -> 5(3(1(3(2(0(x1)))))) 109.76/28.74 1(4(3(5(2(x1))))) -> 2(3(4(0(1(5(x1)))))) 109.76/28.74 1(4(2(x1))) -> 1(3(2(4(0(x1))))) 109.76/28.74 1(0(1(0(x1)))) -> 1(1(3(0(0(x1))))) 109.76/28.75 1(1(4(2(x1)))) -> 1(1(4(0(2(x1))))) 109.76/28.75 1(0(2(4(2(x1))))) -> 4(0(2(1(1(2(x1)))))) 109.76/28.75 3(0(2(4(x1)))) -> 3(4(0(3(2(x1))))) 109.76/28.75 2(1(0(x1))) -> 1(2(0(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 3(1(2(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 2(3(0(x1))) -> 3(2(0(0(x1)))) 109.76/28.75 2(3(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 2(4(1(0(x1)))) -> 1(1(2(4(0(x1))))) 109.76/28.75 2(4(3(0(x1)))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(5(3(0(x1)))) -> 5(3(1(2(0(x1))))) 109.76/28.75 2(4(5(0(x1)))) -> 5(1(2(4(0(x1))))) 109.76/28.75 2(3(4(2(x1)))) -> 3(2(4(0(2(x1))))) 109.76/28.75 2(3(5(1(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 109.76/28.75 2(4(5(1(0(x1))))) -> 3(4(0(1(2(5(x1)))))) 109.76/28.75 3(5(4(0(x1)))) -> 5(3(4(0(0(x1))))) 109.76/28.75 3(4(3(2(x1)))) -> 3(3(4(0(3(2(x1)))))) 109.76/28.75 3(1(4(2(x1)))) -> 1(2(3(4(0(x1))))) 109.76/28.75 3(4(5(2(x1)))) -> 5(3(2(4(0(x1))))) 109.76/28.75 3(4(3(4(x1)))) -> 3(4(0(3(4(x1))))) 109.76/28.75 3(4(2(2(0(x1))))) -> 2(0(3(4(0(2(x1)))))) 109.76/28.75 3(4(2(3(0(x1))))) -> 2(4(3(3(4(0(x1)))))) 109.76/28.75 3(5(1(4(2(x1))))) -> 2(5(3(1(4(0(x1)))))) 109.76/28.75 3(2(0(3(4(x1))))) -> 3(1(3(2(4(0(x1)))))) 109.76/28.75 3(2(1(4(4(x1))))) -> 1(3(2(4(4(0(x1)))))) 109.76/28.75 4(1(0(x1))) -> 0(0(1(1(4(0(x1)))))) 109.76/28.75 4(3(0(x1))) -> 3(4(0(0(x1)))) 109.76/28.75 4(2(3(0(x1)))) -> 0(3(2(4(0(x1))))) 109.76/28.75 4(3(5(0(x1)))) -> 5(3(3(4(0(x1))))) 109.76/28.75 4(5(2(4(0(x1))))) -> 4(5(1(2(4(0(x1)))))) 109.76/28.75 4(1(4(5(0(x1))))) -> 0(5(1(4(0(4(x1)))))) 109.76/28.75 4(5(1(0(4(x1))))) -> 5(1(4(0(0(4(x1)))))) 109.76/28.75 5(4(0(x1))) -> 5(1(4(0(0(x1))))) 109.76/28.75 5(4(5(1(0(x1))))) -> 5(5(2(1(4(0(x1)))))) 109.76/28.75 109.76/28.75 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (18) 109.76/28.75 Obligation: 109.76/28.75 Q DP problem: 109.76/28.75 The TRS P consists of the following rules: 109.76/28.75 109.76/28.75 1^1(0(2(x1))) -> 1^1(2(0(x1))) 109.76/28.75 1^1(0(2(x1))) -> 1^1(1(2(0(x1)))) 109.76/28.75 1^1(0(2(x1))) -> 2^1(0(x1)) 109.76/28.75 1^1(3(0(2(x1)))) -> 1^1(1(2(3(0(x1))))) 109.76/28.75 1^1(3(0(2(x1)))) -> 1^1(2(3(0(x1)))) 109.76/28.75 1^1(3(0(2(x1)))) -> 2^1(3(0(x1))) 109.76/28.75 2^1(1(0(x1))) -> 1^1(2(0(x1))) 109.76/28.75 1^1(3(0(2(x1)))) -> 3^1(0(x1)) 109.76/28.75 3^1(0(2(4(x1)))) -> 2^1(x1) 109.76/28.75 2^1(1(0(x1))) -> 2^1(0(x1)) 109.76/28.75 2^1(1(0(x1))) -> 1^1(3(2(0(x1)))) 109.76/28.75 1^1(4(2(2(x1)))) -> 1^1(3(2(2(4(x1))))) 109.76/28.75 1^1(4(2(2(x1)))) -> 2^1(2(4(x1))) 109.76/28.75 2^1(3(0(x1))) -> 1^1(3(2(0(x1)))) 109.76/28.75 1^1(4(2(2(x1)))) -> 2^1(4(x1)) 109.76/28.75 2^1(3(0(x1))) -> 2^1(0(x1)) 109.76/28.75 2^1(3(4(x1))) -> 1^1(3(2(4(x1)))) 109.76/28.75 1^1(2(3(5(0(x1))))) -> 5^1(3(1(3(2(0(x1)))))) 109.76/28.75 5^1(4(0(3(0(x1))))) -> 2^1(3(4(0(0(5(x1)))))) 109.76/28.75 2^1(3(4(x1))) -> 2^1(4(x1)) 109.76/28.75 2^1(5(3(0(x1)))) -> 5^1(3(1(2(0(x1))))) 109.76/28.75 5^1(4(0(3(0(x1))))) -> 5^1(x1) 109.76/28.75 1^1(0(2(4(2(x1))))) -> 2^1(1(1(2(x1)))) 109.76/28.75 2^1(3(5(1(0(x1))))) -> 1^1(2(3(0(5(0(x1)))))) 109.76/28.75 1^1(0(2(4(2(x1))))) -> 1^1(1(2(x1))) 109.76/28.75 1^1(0(2(4(2(x1))))) -> 1^1(2(x1)) 109.76/28.75 1^1(4(3(5(2(x1))))) -> 2^1(3(4(0(1(5(x1)))))) 109.76/28.75 1^1(4(3(5(2(x1))))) -> 1^1(5(x1)) 109.76/28.75 1^1(4(3(5(2(x1))))) -> 5^1(x1) 109.76/28.75 2^1(3(5(1(0(x1))))) -> 2^1(3(0(5(0(x1))))) 109.76/28.75 2^1(4(5(1(0(x1))))) -> 1^1(2(5(x1))) 109.76/28.75 2^1(4(5(1(0(x1))))) -> 2^1(5(x1)) 109.76/28.75 2^1(4(5(1(0(x1))))) -> 5^1(x1) 109.76/28.75 109.76/28.75 The TRS R consists of the following rules: 109.76/28.75 109.76/28.75 2(1(0(x1))) -> 1(2(0(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 3(1(2(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 4(1(0(x1))) -> 0(0(1(1(4(0(x1)))))) 109.76/28.75 2(3(0(x1))) -> 3(2(0(0(x1)))) 109.76/28.75 2(3(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 4(3(0(x1))) -> 3(4(0(0(x1)))) 109.76/28.75 5(4(0(x1))) -> 5(1(4(0(0(x1))))) 109.76/28.75 1(0(2(x1))) -> 1(1(2(0(x1)))) 109.76/28.75 1(4(2(x1))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(3(4(x1))) -> 1(3(2(4(x1)))) 109.76/28.75 1(0(1(0(x1)))) -> 1(1(3(0(0(x1))))) 109.76/28.75 2(0(1(0(x1)))) -> 1(1(1(2(0(0(x1)))))) 109.76/28.75 2(4(1(0(x1)))) -> 1(1(2(4(0(x1))))) 109.76/28.75 4(2(3(0(x1)))) -> 0(3(2(4(0(x1))))) 109.76/28.75 2(4(3(0(x1)))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(5(3(0(x1)))) -> 5(3(1(2(0(x1))))) 109.76/28.75 3(5(4(0(x1)))) -> 5(3(4(0(0(x1))))) 109.76/28.75 4(3(5(0(x1)))) -> 5(3(3(4(0(x1))))) 109.76/28.75 2(4(5(0(x1)))) -> 5(1(2(4(0(x1))))) 109.76/28.75 1(3(0(2(x1)))) -> 1(1(2(3(0(x1))))) 109.76/28.75 1(4(2(2(x1)))) -> 1(3(2(2(4(x1))))) 109.76/28.75 3(4(3(2(x1)))) -> 3(3(4(0(3(2(x1)))))) 109.76/28.75 1(1(4(2(x1)))) -> 1(1(4(0(2(x1))))) 109.76/28.75 3(1(4(2(x1)))) -> 1(2(3(4(0(x1))))) 109.76/28.75 2(3(4(2(x1)))) -> 3(2(4(0(2(x1))))) 109.76/28.75 3(4(5(2(x1)))) -> 5(3(2(4(0(x1))))) 109.76/28.75 3(0(2(4(x1)))) -> 3(4(0(3(2(x1))))) 109.76/28.75 3(4(3(4(x1)))) -> 3(4(0(3(4(x1))))) 109.76/28.75 2(3(5(1(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 109.76/28.75 2(4(5(1(0(x1))))) -> 3(4(0(1(2(5(x1)))))) 109.76/28.75 5(4(5(1(0(x1))))) -> 5(5(2(1(4(0(x1)))))) 109.76/28.75 3(4(2(2(0(x1))))) -> 2(0(3(4(0(2(x1)))))) 109.76/28.75 5(4(0(3(0(x1))))) -> 2(3(4(0(0(5(x1)))))) 109.76/28.75 3(4(2(3(0(x1))))) -> 2(4(3(3(4(0(x1)))))) 109.76/28.75 4(5(2(4(0(x1))))) -> 4(5(1(2(4(0(x1)))))) 109.76/28.75 1(2(3(5(0(x1))))) -> 5(3(1(3(2(0(x1)))))) 109.76/28.75 4(1(4(5(0(x1))))) -> 0(5(1(4(0(4(x1)))))) 109.76/28.75 3(5(1(4(2(x1))))) -> 2(5(3(1(4(0(x1)))))) 109.76/28.75 1(0(2(4(2(x1))))) -> 4(0(2(1(1(2(x1)))))) 109.76/28.75 1(4(3(5(2(x1))))) -> 2(3(4(0(1(5(x1)))))) 109.76/28.75 4(5(1(0(4(x1))))) -> 5(1(4(0(0(4(x1)))))) 109.76/28.75 3(2(0(3(4(x1))))) -> 3(1(3(2(4(0(x1)))))) 109.76/28.75 3(2(1(4(4(x1))))) -> 1(3(2(4(4(0(x1)))))) 109.76/28.75 2(0(1(5(4(x1))))) -> 5(1(2(4(0(1(x1)))))) 109.76/28.75 109.76/28.75 Q is empty. 109.76/28.75 We have to consider all minimal (P,Q,R)-chains. 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (19) DependencyGraphProof (EQUIVALENT) 109.76/28.75 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (20) 109.76/28.75 Obligation: 109.76/28.75 Q DP problem: 109.76/28.75 The TRS P consists of the following rules: 109.76/28.75 109.76/28.75 1^1(0(2(x1))) -> 1^1(1(2(0(x1)))) 109.76/28.75 1^1(0(2(x1))) -> 1^1(2(0(x1))) 109.76/28.75 1^1(3(0(2(x1)))) -> 1^1(1(2(3(0(x1))))) 109.76/28.75 1^1(3(0(2(x1)))) -> 1^1(2(3(0(x1)))) 109.76/28.75 1^1(3(0(2(x1)))) -> 2^1(3(0(x1))) 109.76/28.75 2^1(1(0(x1))) -> 1^1(2(0(x1))) 109.76/28.75 1^1(3(0(2(x1)))) -> 3^1(0(x1)) 109.76/28.75 3^1(0(2(4(x1)))) -> 2^1(x1) 109.76/28.75 2^1(1(0(x1))) -> 1^1(3(2(0(x1)))) 109.76/28.75 1^1(4(2(2(x1)))) -> 1^1(3(2(2(4(x1))))) 109.76/28.75 1^1(4(2(2(x1)))) -> 2^1(2(4(x1))) 109.76/28.75 2^1(3(0(x1))) -> 1^1(3(2(0(x1)))) 109.76/28.75 1^1(4(2(2(x1)))) -> 2^1(4(x1)) 109.76/28.75 2^1(3(4(x1))) -> 1^1(3(2(4(x1)))) 109.76/28.75 1^1(2(3(5(0(x1))))) -> 5^1(3(1(3(2(0(x1)))))) 109.76/28.75 5^1(4(0(3(0(x1))))) -> 2^1(3(4(0(0(5(x1)))))) 109.76/28.75 2^1(3(4(x1))) -> 2^1(4(x1)) 109.76/28.75 2^1(5(3(0(x1)))) -> 5^1(3(1(2(0(x1))))) 109.76/28.75 5^1(4(0(3(0(x1))))) -> 5^1(x1) 109.76/28.75 2^1(3(5(1(0(x1))))) -> 1^1(2(3(0(5(0(x1)))))) 109.76/28.75 1^1(0(2(4(2(x1))))) -> 2^1(1(1(2(x1)))) 109.76/28.75 2^1(3(5(1(0(x1))))) -> 2^1(3(0(5(0(x1))))) 109.76/28.75 2^1(4(5(1(0(x1))))) -> 1^1(2(5(x1))) 109.76/28.75 1^1(0(2(4(2(x1))))) -> 1^1(1(2(x1))) 109.76/28.75 1^1(0(2(4(2(x1))))) -> 1^1(2(x1)) 109.76/28.75 1^1(4(3(5(2(x1))))) -> 2^1(3(4(0(1(5(x1)))))) 109.76/28.75 1^1(4(3(5(2(x1))))) -> 1^1(5(x1)) 109.76/28.75 1^1(4(3(5(2(x1))))) -> 5^1(x1) 109.76/28.75 2^1(4(5(1(0(x1))))) -> 2^1(5(x1)) 109.76/28.75 2^1(4(5(1(0(x1))))) -> 5^1(x1) 109.76/28.75 109.76/28.75 The TRS R consists of the following rules: 109.76/28.75 109.76/28.75 2(1(0(x1))) -> 1(2(0(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 3(1(2(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 4(1(0(x1))) -> 0(0(1(1(4(0(x1)))))) 109.76/28.75 2(3(0(x1))) -> 3(2(0(0(x1)))) 109.76/28.75 2(3(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 4(3(0(x1))) -> 3(4(0(0(x1)))) 109.76/28.75 5(4(0(x1))) -> 5(1(4(0(0(x1))))) 109.76/28.75 1(0(2(x1))) -> 1(1(2(0(x1)))) 109.76/28.75 1(4(2(x1))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(3(4(x1))) -> 1(3(2(4(x1)))) 109.76/28.75 1(0(1(0(x1)))) -> 1(1(3(0(0(x1))))) 109.76/28.75 2(0(1(0(x1)))) -> 1(1(1(2(0(0(x1)))))) 109.76/28.75 2(4(1(0(x1)))) -> 1(1(2(4(0(x1))))) 109.76/28.75 4(2(3(0(x1)))) -> 0(3(2(4(0(x1))))) 109.76/28.75 2(4(3(0(x1)))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(5(3(0(x1)))) -> 5(3(1(2(0(x1))))) 109.76/28.75 3(5(4(0(x1)))) -> 5(3(4(0(0(x1))))) 109.76/28.75 4(3(5(0(x1)))) -> 5(3(3(4(0(x1))))) 109.76/28.75 2(4(5(0(x1)))) -> 5(1(2(4(0(x1))))) 109.76/28.75 1(3(0(2(x1)))) -> 1(1(2(3(0(x1))))) 109.76/28.75 1(4(2(2(x1)))) -> 1(3(2(2(4(x1))))) 109.76/28.75 3(4(3(2(x1)))) -> 3(3(4(0(3(2(x1)))))) 109.76/28.75 1(1(4(2(x1)))) -> 1(1(4(0(2(x1))))) 109.76/28.75 3(1(4(2(x1)))) -> 1(2(3(4(0(x1))))) 109.76/28.75 2(3(4(2(x1)))) -> 3(2(4(0(2(x1))))) 109.76/28.75 3(4(5(2(x1)))) -> 5(3(2(4(0(x1))))) 109.76/28.75 3(0(2(4(x1)))) -> 3(4(0(3(2(x1))))) 109.76/28.75 3(4(3(4(x1)))) -> 3(4(0(3(4(x1))))) 109.76/28.75 2(3(5(1(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 109.76/28.75 2(4(5(1(0(x1))))) -> 3(4(0(1(2(5(x1)))))) 109.76/28.75 5(4(5(1(0(x1))))) -> 5(5(2(1(4(0(x1)))))) 109.76/28.75 3(4(2(2(0(x1))))) -> 2(0(3(4(0(2(x1)))))) 109.76/28.75 5(4(0(3(0(x1))))) -> 2(3(4(0(0(5(x1)))))) 109.76/28.75 3(4(2(3(0(x1))))) -> 2(4(3(3(4(0(x1)))))) 109.76/28.75 4(5(2(4(0(x1))))) -> 4(5(1(2(4(0(x1)))))) 109.76/28.75 1(2(3(5(0(x1))))) -> 5(3(1(3(2(0(x1)))))) 109.76/28.75 4(1(4(5(0(x1))))) -> 0(5(1(4(0(4(x1)))))) 109.76/28.75 3(5(1(4(2(x1))))) -> 2(5(3(1(4(0(x1)))))) 109.76/28.75 1(0(2(4(2(x1))))) -> 4(0(2(1(1(2(x1)))))) 109.76/28.75 1(4(3(5(2(x1))))) -> 2(3(4(0(1(5(x1)))))) 109.76/28.75 4(5(1(0(4(x1))))) -> 5(1(4(0(0(4(x1)))))) 109.76/28.75 3(2(0(3(4(x1))))) -> 3(1(3(2(4(0(x1)))))) 109.76/28.75 3(2(1(4(4(x1))))) -> 1(3(2(4(4(0(x1)))))) 109.76/28.75 2(0(1(5(4(x1))))) -> 5(1(2(4(0(1(x1)))))) 109.76/28.75 109.76/28.75 Q is empty. 109.76/28.75 We have to consider all minimal (P,Q,R)-chains. 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (21) QDPOrderProof (EQUIVALENT) 109.76/28.75 We use the reduction pair processor [LPAR04,JAR06]. 109.76/28.75 109.76/28.75 109.76/28.75 The following pairs can be oriented strictly and are deleted. 109.76/28.75 109.76/28.75 1^1(0(2(x1))) -> 1^1(1(2(0(x1)))) 109.76/28.75 1^1(0(2(x1))) -> 1^1(2(0(x1))) 109.76/28.75 1^1(4(2(2(x1)))) -> 1^1(3(2(2(4(x1))))) 109.76/28.75 1^1(4(2(2(x1)))) -> 2^1(2(4(x1))) 109.76/28.75 1^1(4(2(2(x1)))) -> 2^1(4(x1)) 109.76/28.75 1^1(0(2(4(2(x1))))) -> 2^1(1(1(2(x1)))) 109.76/28.75 1^1(0(2(4(2(x1))))) -> 1^1(1(2(x1))) 109.76/28.75 1^1(0(2(4(2(x1))))) -> 1^1(2(x1)) 109.76/28.75 1^1(4(3(5(2(x1))))) -> 2^1(3(4(0(1(5(x1)))))) 109.76/28.75 1^1(4(3(5(2(x1))))) -> 1^1(5(x1)) 109.76/28.75 1^1(4(3(5(2(x1))))) -> 5^1(x1) 109.76/28.75 The remaining pairs can at least be oriented weakly. 109.76/28.75 Used ordering: Polynomial interpretation [POLO]: 109.76/28.75 109.76/28.75 POL(0(x_1)) = 1 109.76/28.75 POL(1(x_1)) = x_1 109.76/28.75 POL(1^1(x_1)) = x_1 109.76/28.75 POL(2(x_1)) = 0 109.76/28.75 POL(2^1(x_1)) = 0 109.76/28.75 POL(3(x_1)) = 0 109.76/28.75 POL(3^1(x_1)) = 0 109.76/28.75 POL(4(x_1)) = 1 109.76/28.75 POL(5(x_1)) = 0 109.76/28.75 POL(5^1(x_1)) = 0 109.76/28.75 109.76/28.75 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 109.76/28.75 109.76/28.75 2(0(1(0(x1)))) -> 1(1(1(2(0(0(x1)))))) 109.76/28.75 2(0(1(5(4(x1))))) -> 5(1(2(4(0(1(x1)))))) 109.76/28.75 5(4(0(3(0(x1))))) -> 2(3(4(0(0(5(x1)))))) 109.76/28.75 2(3(4(x1))) -> 1(3(2(4(x1)))) 109.76/28.75 1(0(2(x1))) -> 1(1(2(0(x1)))) 109.76/28.75 1(3(0(2(x1)))) -> 1(1(2(3(0(x1))))) 109.76/28.75 1(4(2(2(x1)))) -> 1(3(2(2(4(x1))))) 109.76/28.75 1(2(3(5(0(x1))))) -> 5(3(1(3(2(0(x1)))))) 109.76/28.75 1(4(3(5(2(x1))))) -> 2(3(4(0(1(5(x1)))))) 109.76/28.75 1(4(2(x1))) -> 1(3(2(4(0(x1))))) 109.76/28.75 1(0(1(0(x1)))) -> 1(1(3(0(0(x1))))) 109.76/28.75 1(1(4(2(x1)))) -> 1(1(4(0(2(x1))))) 109.76/28.75 1(0(2(4(2(x1))))) -> 4(0(2(1(1(2(x1)))))) 109.76/28.75 3(0(2(4(x1)))) -> 3(4(0(3(2(x1))))) 109.76/28.75 2(1(0(x1))) -> 1(2(0(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 3(1(2(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 2(3(0(x1))) -> 3(2(0(0(x1)))) 109.76/28.75 2(3(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 2(4(1(0(x1)))) -> 1(1(2(4(0(x1))))) 109.76/28.75 2(4(3(0(x1)))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(5(3(0(x1)))) -> 5(3(1(2(0(x1))))) 109.76/28.75 2(4(5(0(x1)))) -> 5(1(2(4(0(x1))))) 109.76/28.75 2(3(4(2(x1)))) -> 3(2(4(0(2(x1))))) 109.76/28.75 2(3(5(1(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 109.76/28.75 2(4(5(1(0(x1))))) -> 3(4(0(1(2(5(x1)))))) 109.76/28.75 3(5(4(0(x1)))) -> 5(3(4(0(0(x1))))) 109.76/28.75 3(4(3(2(x1)))) -> 3(3(4(0(3(2(x1)))))) 109.76/28.75 3(1(4(2(x1)))) -> 1(2(3(4(0(x1))))) 109.76/28.75 3(4(5(2(x1)))) -> 5(3(2(4(0(x1))))) 109.76/28.75 3(4(3(4(x1)))) -> 3(4(0(3(4(x1))))) 109.76/28.75 3(4(2(2(0(x1))))) -> 2(0(3(4(0(2(x1)))))) 109.76/28.75 3(4(2(3(0(x1))))) -> 2(4(3(3(4(0(x1)))))) 109.76/28.75 3(5(1(4(2(x1))))) -> 2(5(3(1(4(0(x1)))))) 109.76/28.75 3(2(0(3(4(x1))))) -> 3(1(3(2(4(0(x1)))))) 109.76/28.75 3(2(1(4(4(x1))))) -> 1(3(2(4(4(0(x1)))))) 109.76/28.75 5(4(0(x1))) -> 5(1(4(0(0(x1))))) 109.76/28.75 5(4(5(1(0(x1))))) -> 5(5(2(1(4(0(x1)))))) 109.76/28.75 109.76/28.75 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (22) 109.76/28.75 Obligation: 109.76/28.75 Q DP problem: 109.76/28.75 The TRS P consists of the following rules: 109.76/28.75 109.76/28.75 1^1(3(0(2(x1)))) -> 1^1(1(2(3(0(x1))))) 109.76/28.75 1^1(3(0(2(x1)))) -> 1^1(2(3(0(x1)))) 109.76/28.75 1^1(3(0(2(x1)))) -> 2^1(3(0(x1))) 109.76/28.75 2^1(1(0(x1))) -> 1^1(2(0(x1))) 109.76/28.75 1^1(3(0(2(x1)))) -> 3^1(0(x1)) 109.76/28.75 3^1(0(2(4(x1)))) -> 2^1(x1) 109.76/28.75 2^1(1(0(x1))) -> 1^1(3(2(0(x1)))) 109.76/28.75 2^1(3(0(x1))) -> 1^1(3(2(0(x1)))) 109.76/28.75 2^1(3(4(x1))) -> 1^1(3(2(4(x1)))) 109.76/28.75 1^1(2(3(5(0(x1))))) -> 5^1(3(1(3(2(0(x1)))))) 109.76/28.75 5^1(4(0(3(0(x1))))) -> 2^1(3(4(0(0(5(x1)))))) 109.76/28.75 2^1(3(4(x1))) -> 2^1(4(x1)) 109.76/28.75 2^1(5(3(0(x1)))) -> 5^1(3(1(2(0(x1))))) 109.76/28.75 5^1(4(0(3(0(x1))))) -> 5^1(x1) 109.76/28.75 2^1(3(5(1(0(x1))))) -> 1^1(2(3(0(5(0(x1)))))) 109.76/28.75 2^1(3(5(1(0(x1))))) -> 2^1(3(0(5(0(x1))))) 109.76/28.75 2^1(4(5(1(0(x1))))) -> 1^1(2(5(x1))) 109.76/28.75 2^1(4(5(1(0(x1))))) -> 2^1(5(x1)) 109.76/28.75 2^1(4(5(1(0(x1))))) -> 5^1(x1) 109.76/28.75 109.76/28.75 The TRS R consists of the following rules: 109.76/28.75 109.76/28.75 2(1(0(x1))) -> 1(2(0(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 3(1(2(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 4(1(0(x1))) -> 0(0(1(1(4(0(x1)))))) 109.76/28.75 2(3(0(x1))) -> 3(2(0(0(x1)))) 109.76/28.75 2(3(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 4(3(0(x1))) -> 3(4(0(0(x1)))) 109.76/28.75 5(4(0(x1))) -> 5(1(4(0(0(x1))))) 109.76/28.75 1(0(2(x1))) -> 1(1(2(0(x1)))) 109.76/28.75 1(4(2(x1))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(3(4(x1))) -> 1(3(2(4(x1)))) 109.76/28.75 1(0(1(0(x1)))) -> 1(1(3(0(0(x1))))) 109.76/28.75 2(0(1(0(x1)))) -> 1(1(1(2(0(0(x1)))))) 109.76/28.75 2(4(1(0(x1)))) -> 1(1(2(4(0(x1))))) 109.76/28.75 4(2(3(0(x1)))) -> 0(3(2(4(0(x1))))) 109.76/28.75 2(4(3(0(x1)))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(5(3(0(x1)))) -> 5(3(1(2(0(x1))))) 109.76/28.75 3(5(4(0(x1)))) -> 5(3(4(0(0(x1))))) 109.76/28.75 4(3(5(0(x1)))) -> 5(3(3(4(0(x1))))) 109.76/28.75 2(4(5(0(x1)))) -> 5(1(2(4(0(x1))))) 109.76/28.75 1(3(0(2(x1)))) -> 1(1(2(3(0(x1))))) 109.76/28.75 1(4(2(2(x1)))) -> 1(3(2(2(4(x1))))) 109.76/28.75 3(4(3(2(x1)))) -> 3(3(4(0(3(2(x1)))))) 109.76/28.75 1(1(4(2(x1)))) -> 1(1(4(0(2(x1))))) 109.76/28.75 3(1(4(2(x1)))) -> 1(2(3(4(0(x1))))) 109.76/28.75 2(3(4(2(x1)))) -> 3(2(4(0(2(x1))))) 109.76/28.75 3(4(5(2(x1)))) -> 5(3(2(4(0(x1))))) 109.76/28.75 3(0(2(4(x1)))) -> 3(4(0(3(2(x1))))) 109.76/28.75 3(4(3(4(x1)))) -> 3(4(0(3(4(x1))))) 109.76/28.75 2(3(5(1(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 109.76/28.75 2(4(5(1(0(x1))))) -> 3(4(0(1(2(5(x1)))))) 109.76/28.75 5(4(5(1(0(x1))))) -> 5(5(2(1(4(0(x1)))))) 109.76/28.75 3(4(2(2(0(x1))))) -> 2(0(3(4(0(2(x1)))))) 109.76/28.75 5(4(0(3(0(x1))))) -> 2(3(4(0(0(5(x1)))))) 109.76/28.75 3(4(2(3(0(x1))))) -> 2(4(3(3(4(0(x1)))))) 109.76/28.75 4(5(2(4(0(x1))))) -> 4(5(1(2(4(0(x1)))))) 109.76/28.75 1(2(3(5(0(x1))))) -> 5(3(1(3(2(0(x1)))))) 109.76/28.75 4(1(4(5(0(x1))))) -> 0(5(1(4(0(4(x1)))))) 109.76/28.75 3(5(1(4(2(x1))))) -> 2(5(3(1(4(0(x1)))))) 109.76/28.75 1(0(2(4(2(x1))))) -> 4(0(2(1(1(2(x1)))))) 109.76/28.75 1(4(3(5(2(x1))))) -> 2(3(4(0(1(5(x1)))))) 109.76/28.75 4(5(1(0(4(x1))))) -> 5(1(4(0(0(4(x1)))))) 109.76/28.75 3(2(0(3(4(x1))))) -> 3(1(3(2(4(0(x1)))))) 109.76/28.75 3(2(1(4(4(x1))))) -> 1(3(2(4(4(0(x1)))))) 109.76/28.75 2(0(1(5(4(x1))))) -> 5(1(2(4(0(1(x1)))))) 109.76/28.75 109.76/28.75 Q is empty. 109.76/28.75 We have to consider all minimal (P,Q,R)-chains. 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (23) QDPOrderProof (EQUIVALENT) 109.76/28.75 We use the reduction pair processor [LPAR04,JAR06]. 109.76/28.75 109.76/28.75 109.76/28.75 The following pairs can be oriented strictly and are deleted. 109.76/28.75 109.76/28.75 1^1(3(0(2(x1)))) -> 1^1(1(2(3(0(x1))))) 109.76/28.75 1^1(3(0(2(x1)))) -> 1^1(2(3(0(x1)))) 109.76/28.75 1^1(3(0(2(x1)))) -> 2^1(3(0(x1))) 109.76/28.75 1^1(3(0(2(x1)))) -> 3^1(0(x1)) 109.76/28.75 The remaining pairs can at least be oriented weakly. 109.76/28.75 Used ordering: Polynomial interpretation [POLO]: 109.76/28.75 109.76/28.75 POL(0(x_1)) = 1 109.76/28.75 POL(1(x_1)) = 0 109.76/28.75 POL(1^1(x_1)) = x_1 109.76/28.75 POL(2(x_1)) = 0 109.76/28.75 POL(2^1(x_1)) = 0 109.76/28.75 POL(3(x_1)) = x_1 109.76/28.75 POL(3^1(x_1)) = 0 109.76/28.75 POL(4(x_1)) = 0 109.76/28.75 POL(5(x_1)) = 0 109.76/28.75 POL(5^1(x_1)) = 0 109.76/28.75 109.76/28.75 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 109.76/28.75 109.76/28.75 3(0(2(4(x1)))) -> 3(4(0(3(2(x1))))) 109.76/28.75 2(1(0(x1))) -> 1(2(0(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 3(1(2(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 2(3(0(x1))) -> 3(2(0(0(x1)))) 109.76/28.75 2(3(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 5(4(0(3(0(x1))))) -> 2(3(4(0(0(5(x1)))))) 109.76/28.75 2(3(4(x1))) -> 1(3(2(4(x1)))) 109.76/28.75 1(0(2(x1))) -> 1(1(2(0(x1)))) 109.76/28.75 1(3(0(2(x1)))) -> 1(1(2(3(0(x1))))) 109.76/28.75 1(4(2(2(x1)))) -> 1(3(2(2(4(x1))))) 109.76/28.75 1(2(3(5(0(x1))))) -> 5(3(1(3(2(0(x1)))))) 109.76/28.75 1(4(3(5(2(x1))))) -> 2(3(4(0(1(5(x1)))))) 109.76/28.75 2(0(1(0(x1)))) -> 1(1(1(2(0(0(x1)))))) 109.76/28.75 2(4(1(0(x1)))) -> 1(1(2(4(0(x1))))) 109.76/28.75 2(4(3(0(x1)))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(5(3(0(x1)))) -> 5(3(1(2(0(x1))))) 109.76/28.75 2(4(5(0(x1)))) -> 5(1(2(4(0(x1))))) 109.76/28.75 2(3(4(2(x1)))) -> 3(2(4(0(2(x1))))) 109.76/28.75 2(3(5(1(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 109.76/28.75 2(4(5(1(0(x1))))) -> 3(4(0(1(2(5(x1)))))) 109.76/28.75 2(0(1(5(4(x1))))) -> 5(1(2(4(0(1(x1)))))) 109.76/28.75 1(4(2(x1))) -> 1(3(2(4(0(x1))))) 109.76/28.75 1(0(1(0(x1)))) -> 1(1(3(0(0(x1))))) 109.76/28.75 1(1(4(2(x1)))) -> 1(1(4(0(2(x1))))) 109.76/28.75 1(0(2(4(2(x1))))) -> 4(0(2(1(1(2(x1)))))) 109.76/28.75 3(5(4(0(x1)))) -> 5(3(4(0(0(x1))))) 109.76/28.75 3(4(3(2(x1)))) -> 3(3(4(0(3(2(x1)))))) 109.76/28.75 3(1(4(2(x1)))) -> 1(2(3(4(0(x1))))) 109.76/28.75 3(4(5(2(x1)))) -> 5(3(2(4(0(x1))))) 109.76/28.75 3(4(3(4(x1)))) -> 3(4(0(3(4(x1))))) 109.76/28.75 3(4(2(2(0(x1))))) -> 2(0(3(4(0(2(x1)))))) 109.76/28.75 3(4(2(3(0(x1))))) -> 2(4(3(3(4(0(x1)))))) 109.76/28.75 3(5(1(4(2(x1))))) -> 2(5(3(1(4(0(x1)))))) 109.76/28.75 3(2(0(3(4(x1))))) -> 3(1(3(2(4(0(x1)))))) 109.76/28.75 3(2(1(4(4(x1))))) -> 1(3(2(4(4(0(x1)))))) 109.76/28.75 5(4(0(x1))) -> 5(1(4(0(0(x1))))) 109.76/28.75 5(4(5(1(0(x1))))) -> 5(5(2(1(4(0(x1)))))) 109.76/28.75 109.76/28.75 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (24) 109.76/28.75 Obligation: 109.76/28.75 Q DP problem: 109.76/28.75 The TRS P consists of the following rules: 109.76/28.75 109.76/28.75 2^1(1(0(x1))) -> 1^1(2(0(x1))) 109.76/28.75 3^1(0(2(4(x1)))) -> 2^1(x1) 109.76/28.75 2^1(1(0(x1))) -> 1^1(3(2(0(x1)))) 109.76/28.75 2^1(3(0(x1))) -> 1^1(3(2(0(x1)))) 109.76/28.75 2^1(3(4(x1))) -> 1^1(3(2(4(x1)))) 109.76/28.75 1^1(2(3(5(0(x1))))) -> 5^1(3(1(3(2(0(x1)))))) 109.76/28.75 5^1(4(0(3(0(x1))))) -> 2^1(3(4(0(0(5(x1)))))) 109.76/28.75 2^1(3(4(x1))) -> 2^1(4(x1)) 109.76/28.75 2^1(5(3(0(x1)))) -> 5^1(3(1(2(0(x1))))) 109.76/28.75 5^1(4(0(3(0(x1))))) -> 5^1(x1) 109.76/28.75 2^1(3(5(1(0(x1))))) -> 1^1(2(3(0(5(0(x1)))))) 109.76/28.75 2^1(3(5(1(0(x1))))) -> 2^1(3(0(5(0(x1))))) 109.76/28.75 2^1(4(5(1(0(x1))))) -> 1^1(2(5(x1))) 109.76/28.75 2^1(4(5(1(0(x1))))) -> 2^1(5(x1)) 109.76/28.75 2^1(4(5(1(0(x1))))) -> 5^1(x1) 109.76/28.75 109.76/28.75 The TRS R consists of the following rules: 109.76/28.75 109.76/28.75 2(1(0(x1))) -> 1(2(0(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 3(1(2(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 4(1(0(x1))) -> 0(0(1(1(4(0(x1)))))) 109.76/28.75 2(3(0(x1))) -> 3(2(0(0(x1)))) 109.76/28.75 2(3(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 4(3(0(x1))) -> 3(4(0(0(x1)))) 109.76/28.75 5(4(0(x1))) -> 5(1(4(0(0(x1))))) 109.76/28.75 1(0(2(x1))) -> 1(1(2(0(x1)))) 109.76/28.75 1(4(2(x1))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(3(4(x1))) -> 1(3(2(4(x1)))) 109.76/28.75 1(0(1(0(x1)))) -> 1(1(3(0(0(x1))))) 109.76/28.75 2(0(1(0(x1)))) -> 1(1(1(2(0(0(x1)))))) 109.76/28.75 2(4(1(0(x1)))) -> 1(1(2(4(0(x1))))) 109.76/28.75 4(2(3(0(x1)))) -> 0(3(2(4(0(x1))))) 109.76/28.75 2(4(3(0(x1)))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(5(3(0(x1)))) -> 5(3(1(2(0(x1))))) 109.76/28.75 3(5(4(0(x1)))) -> 5(3(4(0(0(x1))))) 109.76/28.75 4(3(5(0(x1)))) -> 5(3(3(4(0(x1))))) 109.76/28.75 2(4(5(0(x1)))) -> 5(1(2(4(0(x1))))) 109.76/28.75 1(3(0(2(x1)))) -> 1(1(2(3(0(x1))))) 109.76/28.75 1(4(2(2(x1)))) -> 1(3(2(2(4(x1))))) 109.76/28.75 3(4(3(2(x1)))) -> 3(3(4(0(3(2(x1)))))) 109.76/28.75 1(1(4(2(x1)))) -> 1(1(4(0(2(x1))))) 109.76/28.75 3(1(4(2(x1)))) -> 1(2(3(4(0(x1))))) 109.76/28.75 2(3(4(2(x1)))) -> 3(2(4(0(2(x1))))) 109.76/28.75 3(4(5(2(x1)))) -> 5(3(2(4(0(x1))))) 109.76/28.75 3(0(2(4(x1)))) -> 3(4(0(3(2(x1))))) 109.76/28.75 3(4(3(4(x1)))) -> 3(4(0(3(4(x1))))) 109.76/28.75 2(3(5(1(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 109.76/28.75 2(4(5(1(0(x1))))) -> 3(4(0(1(2(5(x1)))))) 109.76/28.75 5(4(5(1(0(x1))))) -> 5(5(2(1(4(0(x1)))))) 109.76/28.75 3(4(2(2(0(x1))))) -> 2(0(3(4(0(2(x1)))))) 109.76/28.75 5(4(0(3(0(x1))))) -> 2(3(4(0(0(5(x1)))))) 109.76/28.75 3(4(2(3(0(x1))))) -> 2(4(3(3(4(0(x1)))))) 109.76/28.75 4(5(2(4(0(x1))))) -> 4(5(1(2(4(0(x1)))))) 109.76/28.75 1(2(3(5(0(x1))))) -> 5(3(1(3(2(0(x1)))))) 109.76/28.75 4(1(4(5(0(x1))))) -> 0(5(1(4(0(4(x1)))))) 109.76/28.75 3(5(1(4(2(x1))))) -> 2(5(3(1(4(0(x1)))))) 109.76/28.75 1(0(2(4(2(x1))))) -> 4(0(2(1(1(2(x1)))))) 109.76/28.75 1(4(3(5(2(x1))))) -> 2(3(4(0(1(5(x1)))))) 109.76/28.75 4(5(1(0(4(x1))))) -> 5(1(4(0(0(4(x1)))))) 109.76/28.75 3(2(0(3(4(x1))))) -> 3(1(3(2(4(0(x1)))))) 109.76/28.75 3(2(1(4(4(x1))))) -> 1(3(2(4(4(0(x1)))))) 109.76/28.75 2(0(1(5(4(x1))))) -> 5(1(2(4(0(1(x1)))))) 109.76/28.75 109.76/28.75 Q is empty. 109.76/28.75 We have to consider all minimal (P,Q,R)-chains. 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (25) DependencyGraphProof (EQUIVALENT) 109.76/28.75 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (26) 109.76/28.75 Obligation: 109.76/28.75 Q DP problem: 109.76/28.75 The TRS P consists of the following rules: 109.76/28.75 109.76/28.75 1^1(2(3(5(0(x1))))) -> 5^1(3(1(3(2(0(x1)))))) 109.76/28.75 5^1(4(0(3(0(x1))))) -> 2^1(3(4(0(0(5(x1)))))) 109.76/28.75 2^1(3(4(x1))) -> 1^1(3(2(4(x1)))) 109.76/28.75 2^1(3(4(x1))) -> 2^1(4(x1)) 109.76/28.75 2^1(1(0(x1))) -> 1^1(2(0(x1))) 109.76/28.75 2^1(1(0(x1))) -> 1^1(3(2(0(x1)))) 109.76/28.75 2^1(3(0(x1))) -> 1^1(3(2(0(x1)))) 109.76/28.75 2^1(5(3(0(x1)))) -> 5^1(3(1(2(0(x1))))) 109.76/28.75 5^1(4(0(3(0(x1))))) -> 5^1(x1) 109.76/28.75 2^1(3(5(1(0(x1))))) -> 1^1(2(3(0(5(0(x1)))))) 109.76/28.75 2^1(3(5(1(0(x1))))) -> 2^1(3(0(5(0(x1))))) 109.76/28.75 2^1(4(5(1(0(x1))))) -> 1^1(2(5(x1))) 109.76/28.75 2^1(4(5(1(0(x1))))) -> 2^1(5(x1)) 109.76/28.75 2^1(4(5(1(0(x1))))) -> 5^1(x1) 109.76/28.75 109.76/28.75 The TRS R consists of the following rules: 109.76/28.75 109.76/28.75 2(1(0(x1))) -> 1(2(0(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 3(1(2(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 4(1(0(x1))) -> 0(0(1(1(4(0(x1)))))) 109.76/28.75 2(3(0(x1))) -> 3(2(0(0(x1)))) 109.76/28.75 2(3(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 4(3(0(x1))) -> 3(4(0(0(x1)))) 109.76/28.75 5(4(0(x1))) -> 5(1(4(0(0(x1))))) 109.76/28.75 1(0(2(x1))) -> 1(1(2(0(x1)))) 109.76/28.75 1(4(2(x1))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(3(4(x1))) -> 1(3(2(4(x1)))) 109.76/28.75 1(0(1(0(x1)))) -> 1(1(3(0(0(x1))))) 109.76/28.75 2(0(1(0(x1)))) -> 1(1(1(2(0(0(x1)))))) 109.76/28.75 2(4(1(0(x1)))) -> 1(1(2(4(0(x1))))) 109.76/28.75 4(2(3(0(x1)))) -> 0(3(2(4(0(x1))))) 109.76/28.75 2(4(3(0(x1)))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(5(3(0(x1)))) -> 5(3(1(2(0(x1))))) 109.76/28.75 3(5(4(0(x1)))) -> 5(3(4(0(0(x1))))) 109.76/28.75 4(3(5(0(x1)))) -> 5(3(3(4(0(x1))))) 109.76/28.75 2(4(5(0(x1)))) -> 5(1(2(4(0(x1))))) 109.76/28.75 1(3(0(2(x1)))) -> 1(1(2(3(0(x1))))) 109.76/28.75 1(4(2(2(x1)))) -> 1(3(2(2(4(x1))))) 109.76/28.75 3(4(3(2(x1)))) -> 3(3(4(0(3(2(x1)))))) 109.76/28.75 1(1(4(2(x1)))) -> 1(1(4(0(2(x1))))) 109.76/28.75 3(1(4(2(x1)))) -> 1(2(3(4(0(x1))))) 109.76/28.75 2(3(4(2(x1)))) -> 3(2(4(0(2(x1))))) 109.76/28.75 3(4(5(2(x1)))) -> 5(3(2(4(0(x1))))) 109.76/28.75 3(0(2(4(x1)))) -> 3(4(0(3(2(x1))))) 109.76/28.75 3(4(3(4(x1)))) -> 3(4(0(3(4(x1))))) 109.76/28.75 2(3(5(1(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 109.76/28.75 2(4(5(1(0(x1))))) -> 3(4(0(1(2(5(x1)))))) 109.76/28.75 5(4(5(1(0(x1))))) -> 5(5(2(1(4(0(x1)))))) 109.76/28.75 3(4(2(2(0(x1))))) -> 2(0(3(4(0(2(x1)))))) 109.76/28.75 5(4(0(3(0(x1))))) -> 2(3(4(0(0(5(x1)))))) 109.76/28.75 3(4(2(3(0(x1))))) -> 2(4(3(3(4(0(x1)))))) 109.76/28.75 4(5(2(4(0(x1))))) -> 4(5(1(2(4(0(x1)))))) 109.76/28.75 1(2(3(5(0(x1))))) -> 5(3(1(3(2(0(x1)))))) 109.76/28.75 4(1(4(5(0(x1))))) -> 0(5(1(4(0(4(x1)))))) 109.76/28.75 3(5(1(4(2(x1))))) -> 2(5(3(1(4(0(x1)))))) 109.76/28.75 1(0(2(4(2(x1))))) -> 4(0(2(1(1(2(x1)))))) 109.76/28.75 1(4(3(5(2(x1))))) -> 2(3(4(0(1(5(x1)))))) 109.76/28.75 4(5(1(0(4(x1))))) -> 5(1(4(0(0(4(x1)))))) 109.76/28.75 3(2(0(3(4(x1))))) -> 3(1(3(2(4(0(x1)))))) 109.76/28.75 3(2(1(4(4(x1))))) -> 1(3(2(4(4(0(x1)))))) 109.76/28.75 2(0(1(5(4(x1))))) -> 5(1(2(4(0(1(x1)))))) 109.76/28.75 109.76/28.75 Q is empty. 109.76/28.75 We have to consider all minimal (P,Q,R)-chains. 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (27) QDPOrderProof (EQUIVALENT) 109.76/28.75 We use the reduction pair processor [LPAR04,JAR06]. 109.76/28.75 109.76/28.75 109.76/28.75 The following pairs can be oriented strictly and are deleted. 109.76/28.75 109.76/28.75 2^1(1(0(x1))) -> 1^1(2(0(x1))) 109.76/28.75 2^1(1(0(x1))) -> 1^1(3(2(0(x1)))) 109.76/28.75 2^1(5(3(0(x1)))) -> 5^1(3(1(2(0(x1))))) 109.76/28.75 2^1(3(5(1(0(x1))))) -> 1^1(2(3(0(5(0(x1)))))) 109.76/28.75 2^1(3(5(1(0(x1))))) -> 2^1(3(0(5(0(x1))))) 109.76/28.75 2^1(4(5(1(0(x1))))) -> 1^1(2(5(x1))) 109.76/28.75 2^1(4(5(1(0(x1))))) -> 5^1(x1) 109.76/28.75 The remaining pairs can at least be oriented weakly. 109.76/28.75 Used ordering: Polynomial interpretation [POLO]: 109.76/28.75 109.76/28.75 POL(0(x_1)) = 0 109.76/28.75 POL(1(x_1)) = 1 109.76/28.75 POL(1^1(x_1)) = 0 109.76/28.75 POL(2(x_1)) = 1 109.76/28.75 POL(2^1(x_1)) = x_1 109.76/28.75 POL(3(x_1)) = x_1 109.76/28.75 POL(4(x_1)) = x_1 109.76/28.75 POL(5(x_1)) = 1 109.76/28.75 POL(5^1(x_1)) = 0 109.76/28.75 109.76/28.75 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 109.76/28.75 109.76/28.75 5(4(0(3(0(x1))))) -> 2(3(4(0(0(5(x1)))))) 109.76/28.75 2(3(4(x1))) -> 1(3(2(4(x1)))) 109.76/28.75 1(0(2(x1))) -> 1(1(2(0(x1)))) 109.76/28.75 1(3(0(2(x1)))) -> 1(1(2(3(0(x1))))) 109.76/28.75 1(4(2(2(x1)))) -> 1(3(2(2(4(x1))))) 109.76/28.75 1(2(3(5(0(x1))))) -> 5(3(1(3(2(0(x1)))))) 109.76/28.75 1(4(3(5(2(x1))))) -> 2(3(4(0(1(5(x1)))))) 109.76/28.75 1(4(2(x1))) -> 1(3(2(4(0(x1))))) 109.76/28.75 1(0(1(0(x1)))) -> 1(1(3(0(0(x1))))) 109.76/28.75 1(1(4(2(x1)))) -> 1(1(4(0(2(x1))))) 109.76/28.75 1(0(2(4(2(x1))))) -> 4(0(2(1(1(2(x1)))))) 109.76/28.75 5(4(0(x1))) -> 5(1(4(0(0(x1))))) 109.76/28.75 5(4(5(1(0(x1))))) -> 5(5(2(1(4(0(x1)))))) 109.76/28.75 4(1(0(x1))) -> 0(0(1(1(4(0(x1)))))) 109.76/28.75 4(3(0(x1))) -> 3(4(0(0(x1)))) 109.76/28.75 4(2(3(0(x1)))) -> 0(3(2(4(0(x1))))) 109.76/28.75 4(3(5(0(x1)))) -> 5(3(3(4(0(x1))))) 109.76/28.75 4(5(2(4(0(x1))))) -> 4(5(1(2(4(0(x1)))))) 109.76/28.75 4(1(4(5(0(x1))))) -> 0(5(1(4(0(4(x1)))))) 109.76/28.75 4(5(1(0(4(x1))))) -> 5(1(4(0(0(4(x1)))))) 109.76/28.75 109.76/28.75 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (28) 109.76/28.75 Obligation: 109.76/28.75 Q DP problem: 109.76/28.75 The TRS P consists of the following rules: 109.76/28.75 109.76/28.75 1^1(2(3(5(0(x1))))) -> 5^1(3(1(3(2(0(x1)))))) 109.76/28.75 5^1(4(0(3(0(x1))))) -> 2^1(3(4(0(0(5(x1)))))) 109.76/28.75 2^1(3(4(x1))) -> 1^1(3(2(4(x1)))) 109.76/28.75 2^1(3(4(x1))) -> 2^1(4(x1)) 109.76/28.75 2^1(3(0(x1))) -> 1^1(3(2(0(x1)))) 109.76/28.75 5^1(4(0(3(0(x1))))) -> 5^1(x1) 109.76/28.75 2^1(4(5(1(0(x1))))) -> 2^1(5(x1)) 109.76/28.75 109.76/28.75 The TRS R consists of the following rules: 109.76/28.75 109.76/28.75 2(1(0(x1))) -> 1(2(0(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 3(1(2(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 4(1(0(x1))) -> 0(0(1(1(4(0(x1)))))) 109.76/28.75 2(3(0(x1))) -> 3(2(0(0(x1)))) 109.76/28.75 2(3(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 4(3(0(x1))) -> 3(4(0(0(x1)))) 109.76/28.75 5(4(0(x1))) -> 5(1(4(0(0(x1))))) 109.76/28.75 1(0(2(x1))) -> 1(1(2(0(x1)))) 109.76/28.75 1(4(2(x1))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(3(4(x1))) -> 1(3(2(4(x1)))) 109.76/28.75 1(0(1(0(x1)))) -> 1(1(3(0(0(x1))))) 109.76/28.75 2(0(1(0(x1)))) -> 1(1(1(2(0(0(x1)))))) 109.76/28.75 2(4(1(0(x1)))) -> 1(1(2(4(0(x1))))) 109.76/28.75 4(2(3(0(x1)))) -> 0(3(2(4(0(x1))))) 109.76/28.75 2(4(3(0(x1)))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(5(3(0(x1)))) -> 5(3(1(2(0(x1))))) 109.76/28.75 3(5(4(0(x1)))) -> 5(3(4(0(0(x1))))) 109.76/28.75 4(3(5(0(x1)))) -> 5(3(3(4(0(x1))))) 109.76/28.75 2(4(5(0(x1)))) -> 5(1(2(4(0(x1))))) 109.76/28.75 1(3(0(2(x1)))) -> 1(1(2(3(0(x1))))) 109.76/28.75 1(4(2(2(x1)))) -> 1(3(2(2(4(x1))))) 109.76/28.75 3(4(3(2(x1)))) -> 3(3(4(0(3(2(x1)))))) 109.76/28.75 1(1(4(2(x1)))) -> 1(1(4(0(2(x1))))) 109.76/28.75 3(1(4(2(x1)))) -> 1(2(3(4(0(x1))))) 109.76/28.75 2(3(4(2(x1)))) -> 3(2(4(0(2(x1))))) 109.76/28.75 3(4(5(2(x1)))) -> 5(3(2(4(0(x1))))) 109.76/28.75 3(0(2(4(x1)))) -> 3(4(0(3(2(x1))))) 109.76/28.75 3(4(3(4(x1)))) -> 3(4(0(3(4(x1))))) 109.76/28.75 2(3(5(1(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 109.76/28.75 2(4(5(1(0(x1))))) -> 3(4(0(1(2(5(x1)))))) 109.76/28.75 5(4(5(1(0(x1))))) -> 5(5(2(1(4(0(x1)))))) 109.76/28.75 3(4(2(2(0(x1))))) -> 2(0(3(4(0(2(x1)))))) 109.76/28.75 5(4(0(3(0(x1))))) -> 2(3(4(0(0(5(x1)))))) 109.76/28.75 3(4(2(3(0(x1))))) -> 2(4(3(3(4(0(x1)))))) 109.76/28.75 4(5(2(4(0(x1))))) -> 4(5(1(2(4(0(x1)))))) 109.76/28.75 1(2(3(5(0(x1))))) -> 5(3(1(3(2(0(x1)))))) 109.76/28.75 4(1(4(5(0(x1))))) -> 0(5(1(4(0(4(x1)))))) 109.76/28.75 3(5(1(4(2(x1))))) -> 2(5(3(1(4(0(x1)))))) 109.76/28.75 1(0(2(4(2(x1))))) -> 4(0(2(1(1(2(x1)))))) 109.76/28.75 1(4(3(5(2(x1))))) -> 2(3(4(0(1(5(x1)))))) 109.76/28.75 4(5(1(0(4(x1))))) -> 5(1(4(0(0(4(x1)))))) 109.76/28.75 3(2(0(3(4(x1))))) -> 3(1(3(2(4(0(x1)))))) 109.76/28.75 3(2(1(4(4(x1))))) -> 1(3(2(4(4(0(x1)))))) 109.76/28.75 2(0(1(5(4(x1))))) -> 5(1(2(4(0(1(x1)))))) 109.76/28.75 109.76/28.75 Q is empty. 109.76/28.75 We have to consider all minimal (P,Q,R)-chains. 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (29) QDPOrderProof (EQUIVALENT) 109.76/28.75 We use the reduction pair processor [LPAR04,JAR06]. 109.76/28.75 109.76/28.75 109.76/28.75 The following pairs can be oriented strictly and are deleted. 109.76/28.75 109.76/28.75 2^1(3(4(x1))) -> 2^1(4(x1)) 109.76/28.75 The remaining pairs can at least be oriented weakly. 109.76/28.75 Used ordering: Polynomial interpretation [POLO]: 109.76/28.75 109.76/28.75 POL(0(x_1)) = 0 109.76/28.75 POL(1(x_1)) = 0 109.76/28.75 POL(1^1(x_1)) = 1 109.76/28.75 POL(2(x_1)) = 0 109.76/28.75 POL(2^1(x_1)) = x_1 109.76/28.75 POL(3(x_1)) = 1 + x_1 109.76/28.75 POL(4(x_1)) = x_1 109.76/28.75 POL(5(x_1)) = 0 109.76/28.75 POL(5^1(x_1)) = 1 109.76/28.75 109.76/28.75 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 109.76/28.75 109.76/28.75 5(4(0(3(0(x1))))) -> 2(3(4(0(0(5(x1)))))) 109.76/28.75 2(3(4(x1))) -> 1(3(2(4(x1)))) 109.76/28.75 1(0(2(x1))) -> 1(1(2(0(x1)))) 109.76/28.75 1(3(0(2(x1)))) -> 1(1(2(3(0(x1))))) 109.76/28.75 1(4(2(2(x1)))) -> 1(3(2(2(4(x1))))) 109.76/28.75 1(2(3(5(0(x1))))) -> 5(3(1(3(2(0(x1)))))) 109.76/28.75 1(4(3(5(2(x1))))) -> 2(3(4(0(1(5(x1)))))) 109.76/28.75 1(4(2(x1))) -> 1(3(2(4(0(x1))))) 109.76/28.75 1(0(1(0(x1)))) -> 1(1(3(0(0(x1))))) 109.76/28.75 1(1(4(2(x1)))) -> 1(1(4(0(2(x1))))) 109.76/28.75 1(0(2(4(2(x1))))) -> 4(0(2(1(1(2(x1)))))) 109.76/28.75 5(4(0(x1))) -> 5(1(4(0(0(x1))))) 109.76/28.75 5(4(5(1(0(x1))))) -> 5(5(2(1(4(0(x1)))))) 109.76/28.75 4(1(0(x1))) -> 0(0(1(1(4(0(x1)))))) 109.76/28.75 4(3(0(x1))) -> 3(4(0(0(x1)))) 109.76/28.75 4(2(3(0(x1)))) -> 0(3(2(4(0(x1))))) 109.76/28.75 4(3(5(0(x1)))) -> 5(3(3(4(0(x1))))) 109.76/28.75 4(5(2(4(0(x1))))) -> 4(5(1(2(4(0(x1)))))) 109.76/28.75 4(1(4(5(0(x1))))) -> 0(5(1(4(0(4(x1)))))) 109.76/28.75 4(5(1(0(4(x1))))) -> 5(1(4(0(0(4(x1)))))) 109.76/28.75 109.76/28.75 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (30) 109.76/28.75 Obligation: 109.76/28.75 Q DP problem: 109.76/28.75 The TRS P consists of the following rules: 109.76/28.75 109.76/28.75 1^1(2(3(5(0(x1))))) -> 5^1(3(1(3(2(0(x1)))))) 109.76/28.75 5^1(4(0(3(0(x1))))) -> 2^1(3(4(0(0(5(x1)))))) 109.76/28.75 2^1(3(4(x1))) -> 1^1(3(2(4(x1)))) 109.76/28.75 2^1(3(0(x1))) -> 1^1(3(2(0(x1)))) 109.76/28.75 5^1(4(0(3(0(x1))))) -> 5^1(x1) 109.76/28.75 2^1(4(5(1(0(x1))))) -> 2^1(5(x1)) 109.76/28.75 109.76/28.75 The TRS R consists of the following rules: 109.76/28.75 109.76/28.75 2(1(0(x1))) -> 1(2(0(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 3(1(2(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 4(1(0(x1))) -> 0(0(1(1(4(0(x1)))))) 109.76/28.75 2(3(0(x1))) -> 3(2(0(0(x1)))) 109.76/28.75 2(3(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 4(3(0(x1))) -> 3(4(0(0(x1)))) 109.76/28.75 5(4(0(x1))) -> 5(1(4(0(0(x1))))) 109.76/28.75 1(0(2(x1))) -> 1(1(2(0(x1)))) 109.76/28.75 1(4(2(x1))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(3(4(x1))) -> 1(3(2(4(x1)))) 109.76/28.75 1(0(1(0(x1)))) -> 1(1(3(0(0(x1))))) 109.76/28.75 2(0(1(0(x1)))) -> 1(1(1(2(0(0(x1)))))) 109.76/28.75 2(4(1(0(x1)))) -> 1(1(2(4(0(x1))))) 109.76/28.75 4(2(3(0(x1)))) -> 0(3(2(4(0(x1))))) 109.76/28.75 2(4(3(0(x1)))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(5(3(0(x1)))) -> 5(3(1(2(0(x1))))) 109.76/28.75 3(5(4(0(x1)))) -> 5(3(4(0(0(x1))))) 109.76/28.75 4(3(5(0(x1)))) -> 5(3(3(4(0(x1))))) 109.76/28.75 2(4(5(0(x1)))) -> 5(1(2(4(0(x1))))) 109.76/28.75 1(3(0(2(x1)))) -> 1(1(2(3(0(x1))))) 109.76/28.75 1(4(2(2(x1)))) -> 1(3(2(2(4(x1))))) 109.76/28.75 3(4(3(2(x1)))) -> 3(3(4(0(3(2(x1)))))) 109.76/28.75 1(1(4(2(x1)))) -> 1(1(4(0(2(x1))))) 109.76/28.75 3(1(4(2(x1)))) -> 1(2(3(4(0(x1))))) 109.76/28.75 2(3(4(2(x1)))) -> 3(2(4(0(2(x1))))) 109.76/28.75 3(4(5(2(x1)))) -> 5(3(2(4(0(x1))))) 109.76/28.75 3(0(2(4(x1)))) -> 3(4(0(3(2(x1))))) 109.76/28.75 3(4(3(4(x1)))) -> 3(4(0(3(4(x1))))) 109.76/28.75 2(3(5(1(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 109.76/28.75 2(4(5(1(0(x1))))) -> 3(4(0(1(2(5(x1)))))) 109.76/28.75 5(4(5(1(0(x1))))) -> 5(5(2(1(4(0(x1)))))) 109.76/28.75 3(4(2(2(0(x1))))) -> 2(0(3(4(0(2(x1)))))) 109.76/28.75 5(4(0(3(0(x1))))) -> 2(3(4(0(0(5(x1)))))) 109.76/28.75 3(4(2(3(0(x1))))) -> 2(4(3(3(4(0(x1)))))) 109.76/28.75 4(5(2(4(0(x1))))) -> 4(5(1(2(4(0(x1)))))) 109.76/28.75 1(2(3(5(0(x1))))) -> 5(3(1(3(2(0(x1)))))) 109.76/28.75 4(1(4(5(0(x1))))) -> 0(5(1(4(0(4(x1)))))) 109.76/28.75 3(5(1(4(2(x1))))) -> 2(5(3(1(4(0(x1)))))) 109.76/28.75 1(0(2(4(2(x1))))) -> 4(0(2(1(1(2(x1)))))) 109.76/28.75 1(4(3(5(2(x1))))) -> 2(3(4(0(1(5(x1)))))) 109.76/28.75 4(5(1(0(4(x1))))) -> 5(1(4(0(0(4(x1)))))) 109.76/28.75 3(2(0(3(4(x1))))) -> 3(1(3(2(4(0(x1)))))) 109.76/28.75 3(2(1(4(4(x1))))) -> 1(3(2(4(4(0(x1)))))) 109.76/28.75 2(0(1(5(4(x1))))) -> 5(1(2(4(0(1(x1)))))) 109.76/28.75 109.76/28.75 Q is empty. 109.76/28.75 We have to consider all minimal (P,Q,R)-chains. 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (31) DependencyGraphProof (EQUIVALENT) 109.76/28.75 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (32) 109.76/28.75 Complex Obligation (AND) 109.76/28.75 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (33) 109.76/28.75 Obligation: 109.76/28.75 Q DP problem: 109.76/28.75 The TRS P consists of the following rules: 109.76/28.75 109.76/28.75 5^1(4(0(3(0(x1))))) -> 2^1(3(4(0(0(5(x1)))))) 109.76/28.75 2^1(3(4(x1))) -> 1^1(3(2(4(x1)))) 109.76/28.75 1^1(2(3(5(0(x1))))) -> 5^1(3(1(3(2(0(x1)))))) 109.76/28.75 5^1(4(0(3(0(x1))))) -> 5^1(x1) 109.76/28.75 109.76/28.75 The TRS R consists of the following rules: 109.76/28.75 109.76/28.75 2(1(0(x1))) -> 1(2(0(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 3(1(2(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 4(1(0(x1))) -> 0(0(1(1(4(0(x1)))))) 109.76/28.75 2(3(0(x1))) -> 3(2(0(0(x1)))) 109.76/28.75 2(3(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 4(3(0(x1))) -> 3(4(0(0(x1)))) 109.76/28.75 5(4(0(x1))) -> 5(1(4(0(0(x1))))) 109.76/28.75 1(0(2(x1))) -> 1(1(2(0(x1)))) 109.76/28.75 1(4(2(x1))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(3(4(x1))) -> 1(3(2(4(x1)))) 109.76/28.75 1(0(1(0(x1)))) -> 1(1(3(0(0(x1))))) 109.76/28.75 2(0(1(0(x1)))) -> 1(1(1(2(0(0(x1)))))) 109.76/28.75 2(4(1(0(x1)))) -> 1(1(2(4(0(x1))))) 109.76/28.75 4(2(3(0(x1)))) -> 0(3(2(4(0(x1))))) 109.76/28.75 2(4(3(0(x1)))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(5(3(0(x1)))) -> 5(3(1(2(0(x1))))) 109.76/28.75 3(5(4(0(x1)))) -> 5(3(4(0(0(x1))))) 109.76/28.75 4(3(5(0(x1)))) -> 5(3(3(4(0(x1))))) 109.76/28.75 2(4(5(0(x1)))) -> 5(1(2(4(0(x1))))) 109.76/28.75 1(3(0(2(x1)))) -> 1(1(2(3(0(x1))))) 109.76/28.75 1(4(2(2(x1)))) -> 1(3(2(2(4(x1))))) 109.76/28.75 3(4(3(2(x1)))) -> 3(3(4(0(3(2(x1)))))) 109.76/28.75 1(1(4(2(x1)))) -> 1(1(4(0(2(x1))))) 109.76/28.75 3(1(4(2(x1)))) -> 1(2(3(4(0(x1))))) 109.76/28.75 2(3(4(2(x1)))) -> 3(2(4(0(2(x1))))) 109.76/28.75 3(4(5(2(x1)))) -> 5(3(2(4(0(x1))))) 109.76/28.75 3(0(2(4(x1)))) -> 3(4(0(3(2(x1))))) 109.76/28.75 3(4(3(4(x1)))) -> 3(4(0(3(4(x1))))) 109.76/28.75 2(3(5(1(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 109.76/28.75 2(4(5(1(0(x1))))) -> 3(4(0(1(2(5(x1)))))) 109.76/28.75 5(4(5(1(0(x1))))) -> 5(5(2(1(4(0(x1)))))) 109.76/28.75 3(4(2(2(0(x1))))) -> 2(0(3(4(0(2(x1)))))) 109.76/28.75 5(4(0(3(0(x1))))) -> 2(3(4(0(0(5(x1)))))) 109.76/28.75 3(4(2(3(0(x1))))) -> 2(4(3(3(4(0(x1)))))) 109.76/28.75 4(5(2(4(0(x1))))) -> 4(5(1(2(4(0(x1)))))) 109.76/28.75 1(2(3(5(0(x1))))) -> 5(3(1(3(2(0(x1)))))) 109.76/28.75 4(1(4(5(0(x1))))) -> 0(5(1(4(0(4(x1)))))) 109.76/28.75 3(5(1(4(2(x1))))) -> 2(5(3(1(4(0(x1)))))) 109.76/28.75 1(0(2(4(2(x1))))) -> 4(0(2(1(1(2(x1)))))) 109.76/28.75 1(4(3(5(2(x1))))) -> 2(3(4(0(1(5(x1)))))) 109.76/28.75 4(5(1(0(4(x1))))) -> 5(1(4(0(0(4(x1)))))) 109.76/28.75 3(2(0(3(4(x1))))) -> 3(1(3(2(4(0(x1)))))) 109.76/28.75 3(2(1(4(4(x1))))) -> 1(3(2(4(4(0(x1)))))) 109.76/28.75 2(0(1(5(4(x1))))) -> 5(1(2(4(0(1(x1)))))) 109.76/28.75 109.76/28.75 Q is empty. 109.76/28.75 We have to consider all minimal (P,Q,R)-chains. 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (34) QDPOrderProof (EQUIVALENT) 109.76/28.75 We use the reduction pair processor [LPAR04,JAR06]. 109.76/28.75 109.76/28.75 109.76/28.75 The following pairs can be oriented strictly and are deleted. 109.76/28.75 109.76/28.75 1^1(2(3(5(0(x1))))) -> 5^1(3(1(3(2(0(x1)))))) 109.76/28.75 The remaining pairs can at least be oriented weakly. 109.76/28.75 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 109.76/28.75 109.76/28.75 POL( 1^1_1(x_1) ) = max{0, 2x_1 - 2} 109.76/28.75 POL( 5^1_1(x_1) ) = 0 109.76/28.75 POL( 0_1(x_1) ) = max{0, -2} 109.76/28.75 POL( 2^1_1(x_1) ) = 2x_1 109.76/28.75 POL( 3_1(x_1) ) = x_1 109.76/28.75 POL( 4_1(x_1) ) = 2x_1 109.76/28.75 POL( 5_1(x_1) ) = x_1 + 1 109.76/28.75 POL( 1_1(x_1) ) = max{0, x_1 - 1} 109.76/28.75 POL( 2_1(x_1) ) = x_1 + 1 109.76/28.75 109.76/28.75 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 109.76/28.75 109.76/28.75 5(4(0(x1))) -> 5(1(4(0(0(x1))))) 109.76/28.75 5(4(5(1(0(x1))))) -> 5(5(2(1(4(0(x1)))))) 109.76/28.75 5(4(0(3(0(x1))))) -> 2(3(4(0(0(5(x1)))))) 109.76/28.75 2(3(4(x1))) -> 1(3(2(4(x1)))) 109.76/28.75 1(0(2(x1))) -> 1(1(2(0(x1)))) 109.76/28.75 1(3(0(2(x1)))) -> 1(1(2(3(0(x1))))) 109.76/28.75 1(4(2(2(x1)))) -> 1(3(2(2(4(x1))))) 109.76/28.75 1(2(3(5(0(x1))))) -> 5(3(1(3(2(0(x1)))))) 109.76/28.75 1(4(3(5(2(x1))))) -> 2(3(4(0(1(5(x1)))))) 109.76/28.75 4(1(0(x1))) -> 0(0(1(1(4(0(x1)))))) 109.76/28.75 4(3(0(x1))) -> 3(4(0(0(x1)))) 109.76/28.75 4(2(3(0(x1)))) -> 0(3(2(4(0(x1))))) 109.76/28.75 4(3(5(0(x1)))) -> 5(3(3(4(0(x1))))) 109.76/28.75 4(5(2(4(0(x1))))) -> 4(5(1(2(4(0(x1)))))) 109.76/28.75 4(1(4(5(0(x1))))) -> 0(5(1(4(0(4(x1)))))) 109.76/28.75 4(5(1(0(4(x1))))) -> 5(1(4(0(0(4(x1)))))) 109.76/28.75 2(1(0(x1))) -> 1(2(0(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 3(1(2(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 2(3(0(x1))) -> 3(2(0(0(x1)))) 109.76/28.75 2(3(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 2(0(1(0(x1)))) -> 1(1(1(2(0(0(x1)))))) 109.76/28.75 2(4(1(0(x1)))) -> 1(1(2(4(0(x1))))) 109.76/28.75 2(4(3(0(x1)))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(5(3(0(x1)))) -> 5(3(1(2(0(x1))))) 109.76/28.75 2(4(5(0(x1)))) -> 5(1(2(4(0(x1))))) 109.76/28.75 2(3(4(2(x1)))) -> 3(2(4(0(2(x1))))) 109.76/28.75 2(3(5(1(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 109.76/28.75 2(4(5(1(0(x1))))) -> 3(4(0(1(2(5(x1)))))) 109.76/28.75 2(0(1(5(4(x1))))) -> 5(1(2(4(0(1(x1)))))) 109.76/28.75 3(5(4(0(x1)))) -> 5(3(4(0(0(x1))))) 109.76/28.75 3(4(3(2(x1)))) -> 3(3(4(0(3(2(x1)))))) 109.76/28.75 3(1(4(2(x1)))) -> 1(2(3(4(0(x1))))) 109.76/28.75 3(4(5(2(x1)))) -> 5(3(2(4(0(x1))))) 109.76/28.75 3(0(2(4(x1)))) -> 3(4(0(3(2(x1))))) 109.76/28.75 3(4(3(4(x1)))) -> 3(4(0(3(4(x1))))) 109.76/28.75 3(4(2(2(0(x1))))) -> 2(0(3(4(0(2(x1)))))) 109.76/28.75 3(4(2(3(0(x1))))) -> 2(4(3(3(4(0(x1)))))) 109.76/28.75 3(5(1(4(2(x1))))) -> 2(5(3(1(4(0(x1)))))) 109.76/28.75 3(2(0(3(4(x1))))) -> 3(1(3(2(4(0(x1)))))) 109.76/28.75 3(2(1(4(4(x1))))) -> 1(3(2(4(4(0(x1)))))) 109.76/28.75 1(4(2(x1))) -> 1(3(2(4(0(x1))))) 109.76/28.75 1(0(1(0(x1)))) -> 1(1(3(0(0(x1))))) 109.76/28.75 1(1(4(2(x1)))) -> 1(1(4(0(2(x1))))) 109.76/28.75 1(0(2(4(2(x1))))) -> 4(0(2(1(1(2(x1)))))) 109.76/28.75 109.76/28.75 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (35) 109.76/28.75 Obligation: 109.76/28.75 Q DP problem: 109.76/28.75 The TRS P consists of the following rules: 109.76/28.75 109.76/28.75 5^1(4(0(3(0(x1))))) -> 2^1(3(4(0(0(5(x1)))))) 109.76/28.75 2^1(3(4(x1))) -> 1^1(3(2(4(x1)))) 109.76/28.75 5^1(4(0(3(0(x1))))) -> 5^1(x1) 109.76/28.75 109.76/28.75 The TRS R consists of the following rules: 109.76/28.75 109.76/28.75 2(1(0(x1))) -> 1(2(0(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 3(1(2(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 4(1(0(x1))) -> 0(0(1(1(4(0(x1)))))) 109.76/28.75 2(3(0(x1))) -> 3(2(0(0(x1)))) 109.76/28.75 2(3(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 4(3(0(x1))) -> 3(4(0(0(x1)))) 109.76/28.75 5(4(0(x1))) -> 5(1(4(0(0(x1))))) 109.76/28.75 1(0(2(x1))) -> 1(1(2(0(x1)))) 109.76/28.75 1(4(2(x1))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(3(4(x1))) -> 1(3(2(4(x1)))) 109.76/28.75 1(0(1(0(x1)))) -> 1(1(3(0(0(x1))))) 109.76/28.75 2(0(1(0(x1)))) -> 1(1(1(2(0(0(x1)))))) 109.76/28.75 2(4(1(0(x1)))) -> 1(1(2(4(0(x1))))) 109.76/28.75 4(2(3(0(x1)))) -> 0(3(2(4(0(x1))))) 109.76/28.75 2(4(3(0(x1)))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(5(3(0(x1)))) -> 5(3(1(2(0(x1))))) 109.76/28.75 3(5(4(0(x1)))) -> 5(3(4(0(0(x1))))) 109.76/28.75 4(3(5(0(x1)))) -> 5(3(3(4(0(x1))))) 109.76/28.75 2(4(5(0(x1)))) -> 5(1(2(4(0(x1))))) 109.76/28.75 1(3(0(2(x1)))) -> 1(1(2(3(0(x1))))) 109.76/28.75 1(4(2(2(x1)))) -> 1(3(2(2(4(x1))))) 109.76/28.75 3(4(3(2(x1)))) -> 3(3(4(0(3(2(x1)))))) 109.76/28.75 1(1(4(2(x1)))) -> 1(1(4(0(2(x1))))) 109.76/28.75 3(1(4(2(x1)))) -> 1(2(3(4(0(x1))))) 109.76/28.75 2(3(4(2(x1)))) -> 3(2(4(0(2(x1))))) 109.76/28.75 3(4(5(2(x1)))) -> 5(3(2(4(0(x1))))) 109.76/28.75 3(0(2(4(x1)))) -> 3(4(0(3(2(x1))))) 109.76/28.75 3(4(3(4(x1)))) -> 3(4(0(3(4(x1))))) 109.76/28.75 2(3(5(1(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 109.76/28.75 2(4(5(1(0(x1))))) -> 3(4(0(1(2(5(x1)))))) 109.76/28.75 5(4(5(1(0(x1))))) -> 5(5(2(1(4(0(x1)))))) 109.76/28.75 3(4(2(2(0(x1))))) -> 2(0(3(4(0(2(x1)))))) 109.76/28.75 5(4(0(3(0(x1))))) -> 2(3(4(0(0(5(x1)))))) 109.76/28.75 3(4(2(3(0(x1))))) -> 2(4(3(3(4(0(x1)))))) 109.76/28.75 4(5(2(4(0(x1))))) -> 4(5(1(2(4(0(x1)))))) 109.76/28.75 1(2(3(5(0(x1))))) -> 5(3(1(3(2(0(x1)))))) 109.76/28.75 4(1(4(5(0(x1))))) -> 0(5(1(4(0(4(x1)))))) 109.76/28.75 3(5(1(4(2(x1))))) -> 2(5(3(1(4(0(x1)))))) 109.76/28.75 1(0(2(4(2(x1))))) -> 4(0(2(1(1(2(x1)))))) 109.76/28.75 1(4(3(5(2(x1))))) -> 2(3(4(0(1(5(x1)))))) 109.76/28.75 4(5(1(0(4(x1))))) -> 5(1(4(0(0(4(x1)))))) 109.76/28.75 3(2(0(3(4(x1))))) -> 3(1(3(2(4(0(x1)))))) 109.76/28.75 3(2(1(4(4(x1))))) -> 1(3(2(4(4(0(x1)))))) 109.76/28.75 2(0(1(5(4(x1))))) -> 5(1(2(4(0(1(x1)))))) 109.76/28.75 109.76/28.75 Q is empty. 109.76/28.75 We have to consider all minimal (P,Q,R)-chains. 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (36) DependencyGraphProof (EQUIVALENT) 109.76/28.75 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (37) 109.76/28.75 Obligation: 109.76/28.75 Q DP problem: 109.76/28.75 The TRS P consists of the following rules: 109.76/28.75 109.76/28.75 5^1(4(0(3(0(x1))))) -> 5^1(x1) 109.76/28.75 109.76/28.75 The TRS R consists of the following rules: 109.76/28.75 109.76/28.75 2(1(0(x1))) -> 1(2(0(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 3(1(2(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 4(1(0(x1))) -> 0(0(1(1(4(0(x1)))))) 109.76/28.75 2(3(0(x1))) -> 3(2(0(0(x1)))) 109.76/28.75 2(3(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 4(3(0(x1))) -> 3(4(0(0(x1)))) 109.76/28.75 5(4(0(x1))) -> 5(1(4(0(0(x1))))) 109.76/28.75 1(0(2(x1))) -> 1(1(2(0(x1)))) 109.76/28.75 1(4(2(x1))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(3(4(x1))) -> 1(3(2(4(x1)))) 109.76/28.75 1(0(1(0(x1)))) -> 1(1(3(0(0(x1))))) 109.76/28.75 2(0(1(0(x1)))) -> 1(1(1(2(0(0(x1)))))) 109.76/28.75 2(4(1(0(x1)))) -> 1(1(2(4(0(x1))))) 109.76/28.75 4(2(3(0(x1)))) -> 0(3(2(4(0(x1))))) 109.76/28.75 2(4(3(0(x1)))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(5(3(0(x1)))) -> 5(3(1(2(0(x1))))) 109.76/28.75 3(5(4(0(x1)))) -> 5(3(4(0(0(x1))))) 109.76/28.75 4(3(5(0(x1)))) -> 5(3(3(4(0(x1))))) 109.76/28.75 2(4(5(0(x1)))) -> 5(1(2(4(0(x1))))) 109.76/28.75 1(3(0(2(x1)))) -> 1(1(2(3(0(x1))))) 109.76/28.75 1(4(2(2(x1)))) -> 1(3(2(2(4(x1))))) 109.76/28.75 3(4(3(2(x1)))) -> 3(3(4(0(3(2(x1)))))) 109.76/28.75 1(1(4(2(x1)))) -> 1(1(4(0(2(x1))))) 109.76/28.75 3(1(4(2(x1)))) -> 1(2(3(4(0(x1))))) 109.76/28.75 2(3(4(2(x1)))) -> 3(2(4(0(2(x1))))) 109.76/28.75 3(4(5(2(x1)))) -> 5(3(2(4(0(x1))))) 109.76/28.75 3(0(2(4(x1)))) -> 3(4(0(3(2(x1))))) 109.76/28.75 3(4(3(4(x1)))) -> 3(4(0(3(4(x1))))) 109.76/28.75 2(3(5(1(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 109.76/28.75 2(4(5(1(0(x1))))) -> 3(4(0(1(2(5(x1)))))) 109.76/28.75 5(4(5(1(0(x1))))) -> 5(5(2(1(4(0(x1)))))) 109.76/28.75 3(4(2(2(0(x1))))) -> 2(0(3(4(0(2(x1)))))) 109.76/28.75 5(4(0(3(0(x1))))) -> 2(3(4(0(0(5(x1)))))) 109.76/28.75 3(4(2(3(0(x1))))) -> 2(4(3(3(4(0(x1)))))) 109.76/28.75 4(5(2(4(0(x1))))) -> 4(5(1(2(4(0(x1)))))) 109.76/28.75 1(2(3(5(0(x1))))) -> 5(3(1(3(2(0(x1)))))) 109.76/28.75 4(1(4(5(0(x1))))) -> 0(5(1(4(0(4(x1)))))) 109.76/28.75 3(5(1(4(2(x1))))) -> 2(5(3(1(4(0(x1)))))) 109.76/28.75 1(0(2(4(2(x1))))) -> 4(0(2(1(1(2(x1)))))) 109.76/28.75 1(4(3(5(2(x1))))) -> 2(3(4(0(1(5(x1)))))) 109.76/28.75 4(5(1(0(4(x1))))) -> 5(1(4(0(0(4(x1)))))) 109.76/28.75 3(2(0(3(4(x1))))) -> 3(1(3(2(4(0(x1)))))) 109.76/28.75 3(2(1(4(4(x1))))) -> 1(3(2(4(4(0(x1)))))) 109.76/28.75 2(0(1(5(4(x1))))) -> 5(1(2(4(0(1(x1)))))) 109.76/28.75 109.76/28.75 Q is empty. 109.76/28.75 We have to consider all minimal (P,Q,R)-chains. 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (38) UsableRulesProof (EQUIVALENT) 109.76/28.75 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. 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (39) 109.76/28.75 Obligation: 109.76/28.75 Q DP problem: 109.76/28.75 The TRS P consists of the following rules: 109.76/28.75 109.76/28.75 5^1(4(0(3(0(x1))))) -> 5^1(x1) 109.76/28.75 109.76/28.75 R is empty. 109.76/28.75 Q is empty. 109.76/28.75 We have to consider all minimal (P,Q,R)-chains. 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (40) QDPSizeChangeProof (EQUIVALENT) 109.76/28.75 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. 109.76/28.75 109.76/28.75 From the DPs we obtained the following set of size-change graphs: 109.76/28.75 *5^1(4(0(3(0(x1))))) -> 5^1(x1) 109.76/28.75 The graph contains the following edges 1 > 1 109.76/28.75 109.76/28.75 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (41) 109.76/28.75 YES 109.76/28.75 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (42) 109.76/28.75 Obligation: 109.76/28.75 Q DP problem: 109.76/28.75 The TRS P consists of the following rules: 109.76/28.75 109.76/28.75 2^1(4(5(1(0(x1))))) -> 2^1(5(x1)) 109.76/28.75 109.76/28.75 The TRS R consists of the following rules: 109.76/28.75 109.76/28.75 2(1(0(x1))) -> 1(2(0(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 3(1(2(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 4(1(0(x1))) -> 0(0(1(1(4(0(x1)))))) 109.76/28.75 2(3(0(x1))) -> 3(2(0(0(x1)))) 109.76/28.75 2(3(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 4(3(0(x1))) -> 3(4(0(0(x1)))) 109.76/28.75 5(4(0(x1))) -> 5(1(4(0(0(x1))))) 109.76/28.75 1(0(2(x1))) -> 1(1(2(0(x1)))) 109.76/28.75 1(4(2(x1))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(3(4(x1))) -> 1(3(2(4(x1)))) 109.76/28.75 1(0(1(0(x1)))) -> 1(1(3(0(0(x1))))) 109.76/28.75 2(0(1(0(x1)))) -> 1(1(1(2(0(0(x1)))))) 109.76/28.75 2(4(1(0(x1)))) -> 1(1(2(4(0(x1))))) 109.76/28.75 4(2(3(0(x1)))) -> 0(3(2(4(0(x1))))) 109.76/28.75 2(4(3(0(x1)))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(5(3(0(x1)))) -> 5(3(1(2(0(x1))))) 109.76/28.75 3(5(4(0(x1)))) -> 5(3(4(0(0(x1))))) 109.76/28.75 4(3(5(0(x1)))) -> 5(3(3(4(0(x1))))) 109.76/28.75 2(4(5(0(x1)))) -> 5(1(2(4(0(x1))))) 109.76/28.75 1(3(0(2(x1)))) -> 1(1(2(3(0(x1))))) 109.76/28.75 1(4(2(2(x1)))) -> 1(3(2(2(4(x1))))) 109.76/28.75 3(4(3(2(x1)))) -> 3(3(4(0(3(2(x1)))))) 109.76/28.75 1(1(4(2(x1)))) -> 1(1(4(0(2(x1))))) 109.76/28.75 3(1(4(2(x1)))) -> 1(2(3(4(0(x1))))) 109.76/28.75 2(3(4(2(x1)))) -> 3(2(4(0(2(x1))))) 109.76/28.75 3(4(5(2(x1)))) -> 5(3(2(4(0(x1))))) 109.76/28.75 3(0(2(4(x1)))) -> 3(4(0(3(2(x1))))) 109.76/28.75 3(4(3(4(x1)))) -> 3(4(0(3(4(x1))))) 109.76/28.75 2(3(5(1(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 109.76/28.75 2(4(5(1(0(x1))))) -> 3(4(0(1(2(5(x1)))))) 109.76/28.75 5(4(5(1(0(x1))))) -> 5(5(2(1(4(0(x1)))))) 109.76/28.75 3(4(2(2(0(x1))))) -> 2(0(3(4(0(2(x1)))))) 109.76/28.75 5(4(0(3(0(x1))))) -> 2(3(4(0(0(5(x1)))))) 109.76/28.75 3(4(2(3(0(x1))))) -> 2(4(3(3(4(0(x1)))))) 109.76/28.75 4(5(2(4(0(x1))))) -> 4(5(1(2(4(0(x1)))))) 109.76/28.75 1(2(3(5(0(x1))))) -> 5(3(1(3(2(0(x1)))))) 109.76/28.75 4(1(4(5(0(x1))))) -> 0(5(1(4(0(4(x1)))))) 109.76/28.75 3(5(1(4(2(x1))))) -> 2(5(3(1(4(0(x1)))))) 109.76/28.75 1(0(2(4(2(x1))))) -> 4(0(2(1(1(2(x1)))))) 109.76/28.75 1(4(3(5(2(x1))))) -> 2(3(4(0(1(5(x1)))))) 109.76/28.75 4(5(1(0(4(x1))))) -> 5(1(4(0(0(4(x1)))))) 109.76/28.75 3(2(0(3(4(x1))))) -> 3(1(3(2(4(0(x1)))))) 109.76/28.75 3(2(1(4(4(x1))))) -> 1(3(2(4(4(0(x1)))))) 109.76/28.75 2(0(1(5(4(x1))))) -> 5(1(2(4(0(1(x1)))))) 109.76/28.75 109.76/28.75 Q is empty. 109.76/28.75 We have to consider all minimal (P,Q,R)-chains. 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (43) QDPOrderProof (EQUIVALENT) 109.76/28.75 We use the reduction pair processor [LPAR04,JAR06]. 109.76/28.75 109.76/28.75 109.76/28.75 The following pairs can be oriented strictly and are deleted. 109.76/28.75 109.76/28.75 2^1(4(5(1(0(x1))))) -> 2^1(5(x1)) 109.76/28.75 The remaining pairs can at least be oriented weakly. 109.76/28.75 Used ordering: Polynomial interpretation [POLO]: 109.76/28.75 109.76/28.75 POL(0(x_1)) = 0 109.76/28.75 POL(1(x_1)) = 1 109.76/28.75 POL(2(x_1)) = x_1 109.76/28.75 POL(2^1(x_1)) = x_1 109.76/28.75 POL(3(x_1)) = 1 109.76/28.75 POL(4(x_1)) = 1 + x_1 109.76/28.75 POL(5(x_1)) = 1 109.76/28.75 109.76/28.75 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 109.76/28.75 109.76/28.75 5(4(0(x1))) -> 5(1(4(0(0(x1))))) 109.76/28.75 5(4(5(1(0(x1))))) -> 5(5(2(1(4(0(x1)))))) 109.76/28.75 5(4(0(3(0(x1))))) -> 2(3(4(0(0(5(x1)))))) 109.76/28.75 2(3(4(x1))) -> 1(3(2(4(x1)))) 109.76/28.75 1(0(2(x1))) -> 1(1(2(0(x1)))) 109.76/28.75 1(3(0(2(x1)))) -> 1(1(2(3(0(x1))))) 109.76/28.75 1(4(2(2(x1)))) -> 1(3(2(2(4(x1))))) 109.76/28.75 1(2(3(5(0(x1))))) -> 5(3(1(3(2(0(x1)))))) 109.76/28.75 1(4(3(5(2(x1))))) -> 2(3(4(0(1(5(x1)))))) 109.76/28.75 1(1(4(2(x1)))) -> 1(1(4(0(2(x1))))) 109.76/28.75 1(0(2(4(2(x1))))) -> 4(0(2(1(1(2(x1)))))) 109.76/28.75 1(4(2(x1))) -> 1(3(2(4(0(x1))))) 109.76/28.75 1(0(1(0(x1)))) -> 1(1(3(0(0(x1))))) 109.76/28.75 109.76/28.75 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (44) 109.76/28.75 Obligation: 109.76/28.75 Q DP problem: 109.76/28.75 P is empty. 109.76/28.75 The TRS R consists of the following rules: 109.76/28.75 109.76/28.75 2(1(0(x1))) -> 1(2(0(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 3(1(2(0(x1)))) 109.76/28.75 2(1(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 4(1(0(x1))) -> 0(0(1(1(4(0(x1)))))) 109.76/28.75 2(3(0(x1))) -> 3(2(0(0(x1)))) 109.76/28.75 2(3(0(x1))) -> 1(3(2(0(x1)))) 109.76/28.75 4(3(0(x1))) -> 3(4(0(0(x1)))) 109.76/28.75 5(4(0(x1))) -> 5(1(4(0(0(x1))))) 109.76/28.75 1(0(2(x1))) -> 1(1(2(0(x1)))) 109.76/28.75 1(4(2(x1))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(3(4(x1))) -> 1(3(2(4(x1)))) 109.76/28.75 1(0(1(0(x1)))) -> 1(1(3(0(0(x1))))) 109.76/28.75 2(0(1(0(x1)))) -> 1(1(1(2(0(0(x1)))))) 109.76/28.75 2(4(1(0(x1)))) -> 1(1(2(4(0(x1))))) 109.76/28.75 4(2(3(0(x1)))) -> 0(3(2(4(0(x1))))) 109.76/28.75 2(4(3(0(x1)))) -> 1(3(2(4(0(x1))))) 109.76/28.75 2(5(3(0(x1)))) -> 5(3(1(2(0(x1))))) 109.76/28.75 3(5(4(0(x1)))) -> 5(3(4(0(0(x1))))) 109.76/28.75 4(3(5(0(x1)))) -> 5(3(3(4(0(x1))))) 109.76/28.75 2(4(5(0(x1)))) -> 5(1(2(4(0(x1))))) 109.76/28.75 1(3(0(2(x1)))) -> 1(1(2(3(0(x1))))) 109.76/28.75 1(4(2(2(x1)))) -> 1(3(2(2(4(x1))))) 109.76/28.75 3(4(3(2(x1)))) -> 3(3(4(0(3(2(x1)))))) 109.76/28.75 1(1(4(2(x1)))) -> 1(1(4(0(2(x1))))) 109.76/28.75 3(1(4(2(x1)))) -> 1(2(3(4(0(x1))))) 109.76/28.75 2(3(4(2(x1)))) -> 3(2(4(0(2(x1))))) 109.76/28.75 3(4(5(2(x1)))) -> 5(3(2(4(0(x1))))) 109.76/28.75 3(0(2(4(x1)))) -> 3(4(0(3(2(x1))))) 109.76/28.75 3(4(3(4(x1)))) -> 3(4(0(3(4(x1))))) 109.76/28.75 2(3(5(1(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 109.76/28.75 2(4(5(1(0(x1))))) -> 3(4(0(1(2(5(x1)))))) 109.76/28.75 5(4(5(1(0(x1))))) -> 5(5(2(1(4(0(x1)))))) 109.76/28.75 3(4(2(2(0(x1))))) -> 2(0(3(4(0(2(x1)))))) 109.76/28.75 5(4(0(3(0(x1))))) -> 2(3(4(0(0(5(x1)))))) 109.76/28.75 3(4(2(3(0(x1))))) -> 2(4(3(3(4(0(x1)))))) 109.76/28.75 4(5(2(4(0(x1))))) -> 4(5(1(2(4(0(x1)))))) 109.76/28.75 1(2(3(5(0(x1))))) -> 5(3(1(3(2(0(x1)))))) 109.76/28.75 4(1(4(5(0(x1))))) -> 0(5(1(4(0(4(x1)))))) 109.76/28.75 3(5(1(4(2(x1))))) -> 2(5(3(1(4(0(x1)))))) 109.76/28.75 1(0(2(4(2(x1))))) -> 4(0(2(1(1(2(x1)))))) 109.76/28.75 1(4(3(5(2(x1))))) -> 2(3(4(0(1(5(x1)))))) 109.76/28.75 4(5(1(0(4(x1))))) -> 5(1(4(0(0(4(x1)))))) 109.76/28.75 3(2(0(3(4(x1))))) -> 3(1(3(2(4(0(x1)))))) 109.76/28.75 3(2(1(4(4(x1))))) -> 1(3(2(4(4(0(x1)))))) 109.76/28.75 2(0(1(5(4(x1))))) -> 5(1(2(4(0(1(x1)))))) 109.76/28.75 109.76/28.75 Q is empty. 109.76/28.75 We have to consider all minimal (P,Q,R)-chains. 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (45) PisEmptyProof (EQUIVALENT) 109.76/28.75 The TRS P is empty. Hence, there is no (P,Q,R) chain. 109.76/28.75 ---------------------------------------- 109.76/28.75 109.76/28.75 (46) 109.76/28.75 YES 109.97/28.88 EOF