30.05/8.72 YES 30.05/8.73 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 30.05/8.73 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 30.05/8.73 30.05/8.73 30.05/8.73 Termination w.r.t. Q of the given QTRS could be proven: 30.05/8.73 30.05/8.73 (0) QTRS 30.05/8.73 (1) QTRS Reverse [EQUIVALENT, 0 ms] 30.05/8.73 (2) QTRS 30.05/8.73 (3) FlatCCProof [EQUIVALENT, 0 ms] 30.05/8.73 (4) QTRS 30.05/8.73 (5) RootLabelingProof [EQUIVALENT, 0 ms] 30.05/8.73 (6) QTRS 30.05/8.73 (7) QTRSRRRProof [EQUIVALENT, 51 ms] 30.05/8.73 (8) QTRS 30.05/8.73 (9) DependencyPairsProof [EQUIVALENT, 22 ms] 30.05/8.73 (10) QDP 30.05/8.73 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 30.05/8.73 (12) QDP 30.05/8.73 (13) QDPOrderProof [EQUIVALENT, 97 ms] 30.05/8.73 (14) QDP 30.05/8.73 (15) DependencyGraphProof [EQUIVALENT, 0 ms] 30.05/8.73 (16) AND 30.05/8.73 (17) QDP 30.05/8.73 (18) UsableRulesProof [EQUIVALENT, 0 ms] 30.05/8.73 (19) QDP 30.05/8.73 (20) QDPSizeChangeProof [EQUIVALENT, 0 ms] 30.05/8.73 (21) YES 30.05/8.73 (22) QDP 30.05/8.73 (23) UsableRulesProof [EQUIVALENT, 0 ms] 30.05/8.73 (24) QDP 30.05/8.73 (25) QDPSizeChangeProof [EQUIVALENT, 0 ms] 30.05/8.73 (26) YES 30.05/8.73 30.05/8.73 30.05/8.73 ---------------------------------------- 30.05/8.73 30.05/8.73 (0) 30.05/8.73 Obligation: 30.05/8.73 Q restricted rewrite system: 30.05/8.73 The TRS R consists of the following rules: 30.05/8.73 30.05/8.73 a(b(x1)) -> x1 30.05/8.73 a(c(x1)) -> b(c(a(b(c(a(x1)))))) 30.05/8.73 b(c(x1)) -> x1 30.05/8.73 30.05/8.73 Q is empty. 30.05/8.73 30.05/8.73 ---------------------------------------- 30.05/8.73 30.05/8.73 (1) QTRS Reverse (EQUIVALENT) 30.05/8.73 We applied the QTRS Reverse Processor [REVERSE]. 30.05/8.73 ---------------------------------------- 30.05/8.73 30.05/8.73 (2) 30.05/8.73 Obligation: 30.05/8.73 Q restricted rewrite system: 30.05/8.73 The TRS R consists of the following rules: 30.05/8.73 30.05/8.73 b(a(x1)) -> x1 30.05/8.73 c(a(x1)) -> a(c(b(a(c(b(x1)))))) 30.05/8.73 c(b(x1)) -> x1 30.05/8.73 30.05/8.73 Q is empty. 30.05/8.73 30.05/8.73 ---------------------------------------- 30.05/8.73 30.05/8.73 (3) FlatCCProof (EQUIVALENT) 30.05/8.73 We used flat context closure [ROOTLAB] 30.05/8.73 As Q is empty the flat context closure was sound AND complete. 30.05/8.73 30.05/8.73 ---------------------------------------- 30.05/8.73 30.05/8.73 (4) 30.05/8.73 Obligation: 30.05/8.73 Q restricted rewrite system: 30.05/8.73 The TRS R consists of the following rules: 30.05/8.73 30.05/8.73 b(b(a(x1))) -> b(x1) 30.05/8.73 a(b(a(x1))) -> a(x1) 30.05/8.73 c(b(a(x1))) -> c(x1) 30.05/8.73 b(c(a(x1))) -> b(a(c(b(a(c(b(x1))))))) 30.05/8.73 a(c(a(x1))) -> a(a(c(b(a(c(b(x1))))))) 30.05/8.73 c(c(a(x1))) -> c(a(c(b(a(c(b(x1))))))) 30.05/8.73 b(c(b(x1))) -> b(x1) 30.05/8.73 a(c(b(x1))) -> a(x1) 30.05/8.73 c(c(b(x1))) -> c(x1) 30.05/8.73 30.05/8.73 Q is empty. 30.05/8.73 30.05/8.73 ---------------------------------------- 30.05/8.73 30.05/8.73 (5) RootLabelingProof (EQUIVALENT) 30.05/8.73 We used plain root labeling [ROOTLAB] with the following heuristic: 30.05/8.73 LabelAll: All function symbols get labeled 30.05/8.73 30.05/8.73 As Q is empty the root labeling was sound AND complete. 30.05/8.73 30.05/8.73 ---------------------------------------- 30.05/8.73 30.05/8.73 (6) 30.05/8.73 Obligation: 30.05/8.73 Q restricted rewrite system: 30.05/8.73 The TRS R consists of the following rules: 30.05/8.73 30.05/8.73 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{b_1}(x1) 30.05/8.73 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(x1) 30.05/8.73 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(x1) 30.05/8.73 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{b_1}(x1) 30.05/8.73 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(x1) 30.05/8.73 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(x1) 30.05/8.73 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{b_1}(x1) 30.05/8.73 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(x1) 30.05/8.73 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(x1) 30.05/8.73 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) 30.05/8.73 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) 30.05/8.73 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) 30.05/8.73 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) 30.05/8.73 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) 30.05/8.73 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) 30.05/8.73 c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) 30.05/8.73 c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) 30.05/8.73 c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) 30.05/8.73 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{b_1}(x1) 30.05/8.73 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(x1) 30.05/8.73 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{c_1}(x1) 30.05/8.73 a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{b_1}(x1) 30.05/8.73 a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1) 30.05/8.73 a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{c_1}(x1) 30.05/8.73 c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{b_1}(x1) 30.05/8.73 c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(x1) 30.05/8.73 c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{c_1}(x1) 30.05/8.73 30.05/8.73 Q is empty. 30.05/8.73 30.05/8.73 ---------------------------------------- 30.05/8.73 30.05/8.73 (7) QTRSRRRProof (EQUIVALENT) 30.05/8.73 Used ordering: 30.05/8.73 Polynomial interpretation [POLO]: 30.05/8.73 30.05/8.73 POL(a_{a_1}(x_1)) = x_1 30.05/8.73 POL(a_{b_1}(x_1)) = 1 + x_1 30.05/8.73 POL(a_{c_1}(x_1)) = x_1 30.05/8.73 POL(b_{a_1}(x_1)) = x_1 30.05/8.73 POL(b_{b_1}(x_1)) = 1 + x_1 30.05/8.73 POL(b_{c_1}(x_1)) = x_1 30.05/8.73 POL(c_{a_1}(x_1)) = x_1 30.05/8.73 POL(c_{b_1}(x_1)) = x_1 30.05/8.73 POL(c_{c_1}(x_1)) = x_1 30.05/8.73 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 30.05/8.73 30.05/8.73 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{b_1}(x1) 30.05/8.73 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(x1) 30.05/8.73 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(x1) 30.05/8.73 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{b_1}(x1) 30.05/8.73 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(x1) 30.05/8.73 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(x1) 30.05/8.73 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{b_1}(x1) 30.05/8.73 c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{b_1}(x1) 30.05/8.73 30.05/8.73 30.05/8.73 30.05/8.73 30.05/8.73 ---------------------------------------- 30.05/8.73 30.05/8.73 (8) 30.05/8.73 Obligation: 30.05/8.73 Q restricted rewrite system: 30.05/8.73 The TRS R consists of the following rules: 30.05/8.73 30.05/8.73 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(x1) 30.05/8.73 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(x1) 30.05/8.73 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) 30.05/8.73 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) 30.05/8.73 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) 30.05/8.73 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) 30.05/8.73 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) 30.05/8.73 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) 30.05/8.73 c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) 30.05/8.73 c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) 30.05/8.73 c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) 30.05/8.73 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{b_1}(x1) 30.05/8.73 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(x1) 30.05/8.73 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{c_1}(x1) 30.05/8.73 a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{b_1}(x1) 30.05/8.73 a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1) 30.05/8.73 a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{c_1}(x1) 30.05/8.73 c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(x1) 30.05/8.73 c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{c_1}(x1) 30.05/8.73 30.05/8.73 Q is empty. 30.05/8.73 30.05/8.73 ---------------------------------------- 30.05/8.73 30.05/8.73 (9) DependencyPairsProof (EQUIVALENT) 30.05/8.73 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 30.05/8.73 ---------------------------------------- 30.05/8.73 30.05/8.73 (10) 30.05/8.73 Obligation: 30.05/8.73 Q DP problem: 30.05/8.73 The TRS P consists of the following rules: 30.05/8.73 30.05/8.73 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{C_1}(x1) 30.05/8.73 B_{C_1}(c_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))))) 30.05/8.73 B_{C_1}(c_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))) 30.05/8.73 B_{C_1}(c_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(x1))) 30.05/8.73 B_{C_1}(c_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{b_1}(x1)) 30.05/8.73 B_{C_1}(c_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))))) 30.05/8.73 B_{C_1}(c_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))) 30.05/8.73 B_{C_1}(c_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(x1))) 30.05/8.73 B_{C_1}(c_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(x1)) 30.05/8.73 B_{C_1}(c_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))))) 30.05/8.73 B_{C_1}(c_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))) 30.05/8.73 B_{C_1}(c_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{c_1}(x1))) 30.05/8.73 B_{C_1}(c_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{c_1}(x1)) 30.05/8.73 B_{C_1}(c_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1) 30.05/8.73 A_{C_1}(c_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))))) 30.05/8.73 A_{C_1}(c_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))) 30.05/8.73 A_{C_1}(c_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(x1))) 30.05/8.73 A_{C_1}(c_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{b_1}(x1)) 30.05/8.73 A_{C_1}(c_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))))) 30.05/8.73 A_{C_1}(c_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))) 30.05/8.73 A_{C_1}(c_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(x1))) 30.05/8.73 A_{C_1}(c_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(x1)) 30.05/8.73 A_{C_1}(c_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))))) 30.05/8.73 A_{C_1}(c_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))) 30.05/8.73 A_{C_1}(c_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{c_1}(x1))) 30.05/8.73 A_{C_1}(c_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{c_1}(x1)) 30.05/8.73 A_{C_1}(c_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1) 30.05/8.73 C_{C_1}(c_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))))) 30.05/8.73 C_{C_1}(c_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))) 30.05/8.73 C_{C_1}(c_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{b_1}(x1))) 30.05/8.73 C_{C_1}(c_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{b_1}(x1)) 30.05/8.73 C_{C_1}(c_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))))) 30.05/8.73 C_{C_1}(c_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))) 30.05/8.73 C_{C_1}(c_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(x1))) 30.05/8.73 C_{C_1}(c_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(x1)) 30.05/8.73 C_{C_1}(c_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))))) 30.05/8.73 C_{C_1}(c_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))) 30.05/8.73 C_{C_1}(c_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{c_1}(x1))) 30.05/8.73 C_{C_1}(c_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{c_1}(x1)) 30.05/8.73 C_{C_1}(c_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1) 30.05/8.73 A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1) 30.05/8.73 C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1) 30.05/8.73 30.05/8.73 The TRS R consists of the following rules: 30.05/8.73 30.05/8.73 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(x1) 30.05/8.73 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(x1) 30.05/8.73 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) 30.05/8.73 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) 30.05/8.73 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) 30.05/8.73 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) 30.05/8.73 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) 30.05/8.73 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) 30.05/8.73 c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) 30.05/8.73 c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) 30.05/8.73 c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) 30.05/8.73 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{b_1}(x1) 30.05/8.73 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(x1) 30.05/8.73 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{c_1}(x1) 30.05/8.73 a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{b_1}(x1) 30.05/8.73 a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1) 30.05/8.73 a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{c_1}(x1) 30.05/8.73 c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(x1) 30.05/8.73 c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{c_1}(x1) 30.05/8.73 30.05/8.73 Q is empty. 30.05/8.73 We have to consider all minimal (P,Q,R)-chains. 30.05/8.73 ---------------------------------------- 30.05/8.73 30.05/8.73 (11) DependencyGraphProof (EQUIVALENT) 30.05/8.73 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes. 30.05/8.73 ---------------------------------------- 30.05/8.73 30.05/8.73 (12) 30.05/8.73 Obligation: 30.05/8.73 Q DP problem: 30.05/8.73 The TRS P consists of the following rules: 30.05/8.73 30.05/8.73 C_{C_1}(c_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))))) 30.05/8.73 A_{C_1}(c_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))))) 30.05/8.73 A_{C_1}(c_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))) 30.05/8.73 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{C_1}(x1) 30.05/8.73 C_{C_1}(c_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))) 30.05/8.73 C_{C_1}(c_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))))) 30.05/8.73 A_{C_1}(c_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))))) 30.05/8.73 A_{C_1}(c_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))) 30.05/8.73 A_{C_1}(c_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(x1))) 30.05/8.73 A_{C_1}(c_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(x1)) 30.05/8.73 A_{C_1}(c_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))))) 30.05/8.73 A_{C_1}(c_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))) 30.05/8.73 A_{C_1}(c_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{c_1}(x1))) 30.05/8.73 A_{C_1}(c_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{c_1}(x1)) 30.05/8.73 A_{C_1}(c_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1) 30.05/8.73 B_{C_1}(c_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))))) 30.05/8.73 A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1) 30.05/8.73 B_{C_1}(c_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))) 30.05/8.73 B_{C_1}(c_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))))) 30.05/8.73 B_{C_1}(c_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))) 30.05/8.73 B_{C_1}(c_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(x1))) 30.05/8.73 B_{C_1}(c_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(x1)) 30.05/8.73 B_{C_1}(c_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))))) 30.05/8.73 B_{C_1}(c_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))) 30.05/8.73 B_{C_1}(c_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{c_1}(x1))) 30.05/8.73 B_{C_1}(c_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{c_1}(x1)) 30.05/8.73 B_{C_1}(c_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1) 30.05/8.73 C_{C_1}(c_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))) 30.05/8.73 C_{C_1}(c_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(x1))) 30.05/8.73 C_{C_1}(c_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(x1)) 30.05/8.73 C_{C_1}(c_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))))) 30.05/8.73 C_{C_1}(c_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))) 30.05/8.73 C_{C_1}(c_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{c_1}(x1))) 30.05/8.73 C_{C_1}(c_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{c_1}(x1)) 30.05/8.73 C_{C_1}(c_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1) 30.05/8.73 C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1) 30.05/8.73 30.05/8.73 The TRS R consists of the following rules: 30.05/8.73 30.05/8.73 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(x1) 30.05/8.73 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(x1) 30.05/8.73 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) 30.05/8.73 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) 30.05/8.73 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) 30.05/8.73 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) 30.05/8.73 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) 30.05/8.73 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) 30.05/8.73 c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) 30.05/8.73 c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) 30.05/8.73 c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) 30.05/8.73 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{b_1}(x1) 30.05/8.73 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(x1) 30.05/8.73 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{c_1}(x1) 30.05/8.73 a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{b_1}(x1) 30.05/8.73 a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1) 30.05/8.73 a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{c_1}(x1) 30.05/8.73 c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(x1) 30.05/8.73 c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{c_1}(x1) 30.05/8.73 30.05/8.73 Q is empty. 30.05/8.73 We have to consider all minimal (P,Q,R)-chains. 30.05/8.73 ---------------------------------------- 30.05/8.73 30.05/8.73 (13) QDPOrderProof (EQUIVALENT) 30.05/8.73 We use the reduction pair processor [LPAR04,JAR06]. 30.05/8.73 30.05/8.73 30.05/8.73 The following pairs can be oriented strictly and are deleted. 30.05/8.73 30.05/8.73 C_{C_1}(c_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))))) 30.05/8.73 A_{C_1}(c_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))))) 30.05/8.73 A_{C_1}(c_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))) 30.05/8.73 C_{C_1}(c_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))) 30.05/8.73 C_{C_1}(c_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))))) 30.05/8.73 A_{C_1}(c_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))))) 30.05/8.73 A_{C_1}(c_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))) 30.05/8.73 A_{C_1}(c_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(x1))) 30.05/8.73 A_{C_1}(c_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(x1)) 30.05/8.73 A_{C_1}(c_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))))) 30.05/8.73 A_{C_1}(c_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))) 30.05/8.73 A_{C_1}(c_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{c_1}(x1))) 30.05/8.73 A_{C_1}(c_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{c_1}(x1)) 30.05/8.73 A_{C_1}(c_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1) 30.05/8.73 B_{C_1}(c_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))))) 30.05/8.73 B_{C_1}(c_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))) 30.05/8.73 B_{C_1}(c_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))))) 30.05/8.73 B_{C_1}(c_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))) 30.05/8.73 B_{C_1}(c_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(x1))) 30.05/8.73 B_{C_1}(c_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(x1)) 30.05/8.73 B_{C_1}(c_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))))) 30.05/8.73 B_{C_1}(c_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))) 30.05/8.73 B_{C_1}(c_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{c_1}(x1))) 30.05/8.73 B_{C_1}(c_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{c_1}(x1)) 30.05/8.73 B_{C_1}(c_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1) 30.05/8.73 C_{C_1}(c_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))) 30.05/8.73 C_{C_1}(c_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(x1))) 30.05/8.73 C_{C_1}(c_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(x1)) 30.05/8.73 C_{C_1}(c_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))))) 30.05/8.73 C_{C_1}(c_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))) 30.05/8.73 C_{C_1}(c_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{c_1}(x1))) 30.05/8.73 C_{C_1}(c_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{c_1}(x1)) 30.05/8.73 C_{C_1}(c_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1) 30.05/8.73 The remaining pairs can at least be oriented weakly. 30.05/8.73 Used ordering: Polynomial interpretation [POLO]: 30.05/8.73 30.05/8.73 POL(A_{C_1}(x_1)) = 1 + x_1 30.05/8.73 POL(B_{C_1}(x_1)) = 1 + x_1 30.05/8.73 POL(C_{B_1}(x_1)) = x_1 30.05/8.73 POL(C_{C_1}(x_1)) = 1 + x_1 30.05/8.73 POL(a_{a_1}(x_1)) = 1 + x_1 30.05/8.73 POL(a_{b_1}(x_1)) = 1 + x_1 30.05/8.73 POL(a_{c_1}(x_1)) = 1 + x_1 30.05/8.73 POL(b_{a_1}(x_1)) = x_1 30.05/8.73 POL(b_{b_1}(x_1)) = x_1 30.05/8.73 POL(b_{c_1}(x_1)) = x_1 30.05/8.73 POL(c_{a_1}(x_1)) = 1 + x_1 30.05/8.73 POL(c_{b_1}(x_1)) = x_1 30.05/8.73 POL(c_{c_1}(x_1)) = 1 + x_1 30.05/8.73 30.05/8.73 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 30.05/8.73 30.05/8.73 a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{b_1}(x1) 30.05/8.73 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(x1) 30.05/8.73 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(x1) 30.05/8.73 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) 30.05/8.73 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) 30.05/8.73 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) 30.05/8.73 a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1) 30.05/8.73 a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{c_1}(x1) 30.05/8.73 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) 30.05/8.73 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) 30.05/8.73 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) 30.05/8.73 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{b_1}(x1) 30.05/8.73 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(x1) 30.05/8.73 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{c_1}(x1) 30.05/8.73 c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) 30.05/8.73 c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) 30.05/8.73 c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) 30.05/8.73 c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{c_1}(x1) 30.05/8.73 c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(x1) 30.05/8.73 30.05/8.73 30.05/8.73 ---------------------------------------- 30.05/8.73 30.05/8.73 (14) 30.05/8.73 Obligation: 30.05/8.73 Q DP problem: 30.05/8.73 The TRS P consists of the following rules: 30.05/8.73 30.05/8.73 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{C_1}(x1) 30.05/8.73 A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1) 30.05/8.73 C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1) 30.05/8.73 30.05/8.73 The TRS R consists of the following rules: 30.05/8.73 30.05/8.73 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(x1) 30.05/8.73 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(x1) 30.05/8.73 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) 30.05/8.73 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) 30.05/8.73 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) 30.05/8.73 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) 30.05/8.73 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) 30.05/8.73 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) 30.05/8.73 c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) 30.05/8.73 c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) 30.05/8.73 c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) 30.05/8.73 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{b_1}(x1) 30.05/8.73 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(x1) 30.05/8.73 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{c_1}(x1) 30.05/8.73 a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{b_1}(x1) 30.05/8.73 a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1) 30.05/8.73 a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{c_1}(x1) 30.05/8.73 c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(x1) 30.05/8.73 c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{c_1}(x1) 30.05/8.73 30.05/8.73 Q is empty. 30.05/8.73 We have to consider all minimal (P,Q,R)-chains. 30.05/8.73 ---------------------------------------- 30.05/8.73 30.05/8.73 (15) DependencyGraphProof (EQUIVALENT) 30.05/8.73 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. 30.05/8.73 ---------------------------------------- 30.05/8.73 30.05/8.73 (16) 30.05/8.73 Complex Obligation (AND) 30.05/8.73 30.05/8.73 ---------------------------------------- 30.05/8.73 30.05/8.73 (17) 30.05/8.73 Obligation: 30.05/8.73 Q DP problem: 30.05/8.73 The TRS P consists of the following rules: 30.05/8.73 30.05/8.73 C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1) 30.05/8.73 30.05/8.73 The TRS R consists of the following rules: 30.05/8.73 30.05/8.73 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(x1) 30.05/8.73 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(x1) 30.05/8.73 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) 30.05/8.73 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) 30.05/8.73 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) 30.05/8.73 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) 30.05/8.73 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) 30.05/8.73 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) 30.05/8.73 c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) 30.05/8.73 c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) 30.05/8.73 c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) 30.05/8.73 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{b_1}(x1) 30.05/8.73 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(x1) 30.05/8.73 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{c_1}(x1) 30.05/8.73 a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{b_1}(x1) 30.05/8.73 a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1) 30.05/8.73 a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{c_1}(x1) 30.05/8.73 c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(x1) 30.05/8.73 c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{c_1}(x1) 30.05/8.73 30.05/8.73 Q is empty. 30.05/8.73 We have to consider all minimal (P,Q,R)-chains. 30.05/8.73 ---------------------------------------- 30.05/8.73 30.05/8.73 (18) UsableRulesProof (EQUIVALENT) 30.05/8.73 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. 30.05/8.73 ---------------------------------------- 30.05/8.73 30.05/8.73 (19) 30.05/8.73 Obligation: 30.05/8.73 Q DP problem: 30.05/8.73 The TRS P consists of the following rules: 30.05/8.73 30.05/8.73 C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1) 30.05/8.73 30.05/8.73 R is empty. 30.05/8.73 Q is empty. 30.05/8.73 We have to consider all minimal (P,Q,R)-chains. 30.05/8.73 ---------------------------------------- 30.05/8.73 30.05/8.73 (20) QDPSizeChangeProof (EQUIVALENT) 30.05/8.73 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. 30.05/8.73 30.05/8.73 From the DPs we obtained the following set of size-change graphs: 30.05/8.73 *C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1) 30.05/8.73 The graph contains the following edges 1 > 1 30.05/8.73 30.05/8.73 30.05/8.73 ---------------------------------------- 30.05/8.73 30.05/8.73 (21) 30.05/8.73 YES 30.05/8.73 30.05/8.73 ---------------------------------------- 30.05/8.73 30.05/8.73 (22) 30.05/8.73 Obligation: 30.05/8.73 Q DP problem: 30.05/8.73 The TRS P consists of the following rules: 30.05/8.73 30.05/8.73 A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1) 30.05/8.73 30.05/8.73 The TRS R consists of the following rules: 30.05/8.73 30.05/8.73 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(x1) 30.05/8.73 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(x1) 30.05/8.73 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) 30.05/8.73 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) 30.05/8.73 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) 30.05/8.73 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) 30.05/8.73 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) 30.05/8.73 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) 30.05/8.73 c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) 30.05/8.73 c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) 30.05/8.73 c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) 30.05/8.73 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{b_1}(x1) 30.05/8.73 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(x1) 30.05/8.73 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{c_1}(x1) 30.05/8.73 a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{b_1}(x1) 30.05/8.73 a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1) 30.05/8.73 a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{c_1}(x1) 30.05/8.73 c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(x1) 30.05/8.73 c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{c_1}(x1) 30.05/8.73 30.05/8.73 Q is empty. 30.05/8.73 We have to consider all minimal (P,Q,R)-chains. 30.05/8.73 ---------------------------------------- 30.05/8.73 30.05/8.73 (23) UsableRulesProof (EQUIVALENT) 30.05/8.73 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. 30.05/8.73 ---------------------------------------- 30.05/8.73 30.05/8.73 (24) 30.05/8.73 Obligation: 30.05/8.73 Q DP problem: 30.05/8.73 The TRS P consists of the following rules: 30.05/8.73 30.05/8.73 A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1) 30.05/8.73 30.05/8.73 R is empty. 30.05/8.73 Q is empty. 30.05/8.73 We have to consider all minimal (P,Q,R)-chains. 30.05/8.73 ---------------------------------------- 30.05/8.73 30.05/8.73 (25) QDPSizeChangeProof (EQUIVALENT) 30.05/8.73 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. 30.05/8.73 30.05/8.73 From the DPs we obtained the following set of size-change graphs: 30.05/8.73 *A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1) 30.05/8.73 The graph contains the following edges 1 > 1 30.05/8.73 30.05/8.73 30.05/8.73 ---------------------------------------- 30.05/8.73 30.05/8.73 (26) 30.05/8.73 YES 30.49/8.77 EOF