13.04/4.24 YES 13.68/4.35 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 13.68/4.35 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 13.68/4.35 13.68/4.35 13.68/4.35 Termination w.r.t. Q of the given QTRS could be proven: 13.68/4.35 13.68/4.35 (0) QTRS 13.68/4.35 (1) FlatCCProof [EQUIVALENT, 0 ms] 13.68/4.35 (2) QTRS 13.68/4.35 (3) RootLabelingProof [EQUIVALENT, 0 ms] 13.68/4.35 (4) QTRS 13.68/4.35 (5) DependencyPairsProof [EQUIVALENT, 2 ms] 13.68/4.35 (6) QDP 13.68/4.35 (7) DependencyGraphProof [EQUIVALENT, 0 ms] 13.68/4.35 (8) AND 13.68/4.35 (9) QDP 13.68/4.35 (10) UsableRulesProof [EQUIVALENT, 3 ms] 13.68/4.35 (11) QDP 13.68/4.35 (12) QDPOrderProof [EQUIVALENT, 39 ms] 13.68/4.35 (13) QDP 13.68/4.35 (14) DependencyGraphProof [EQUIVALENT, 0 ms] 13.68/4.35 (15) TRUE 13.68/4.35 (16) QDP 13.68/4.35 (17) UsableRulesProof [EQUIVALENT, 2 ms] 13.68/4.35 (18) QDP 13.68/4.35 (19) QDPSizeChangeProof [EQUIVALENT, 1 ms] 13.68/4.35 (20) YES 13.68/4.35 13.68/4.35 13.68/4.35 ---------------------------------------- 13.68/4.35 13.68/4.35 (0) 13.68/4.35 Obligation: 13.68/4.35 Q restricted rewrite system: 13.68/4.35 The TRS R consists of the following rules: 13.68/4.35 13.68/4.35 a(b(a(a(x1)))) -> a(b(b(a(x1)))) 13.68/4.35 a(b(a(b(x1)))) -> a(b(b(a(x1)))) 13.68/4.35 b(b(a(a(x1)))) -> a(b(a(a(x1)))) 13.68/4.35 b(b(b(b(x1)))) -> a(a(a(a(x1)))) 13.68/4.35 13.68/4.35 Q is empty. 13.68/4.35 13.68/4.35 ---------------------------------------- 13.68/4.35 13.68/4.35 (1) FlatCCProof (EQUIVALENT) 13.68/4.35 We used flat context closure [ROOTLAB] 13.68/4.35 As Q is empty the flat context closure was sound AND complete. 13.68/4.35 13.68/4.35 ---------------------------------------- 13.68/4.35 13.68/4.35 (2) 13.68/4.35 Obligation: 13.68/4.35 Q restricted rewrite system: 13.68/4.35 The TRS R consists of the following rules: 13.68/4.35 13.68/4.35 a(b(a(a(x1)))) -> a(b(b(a(x1)))) 13.68/4.35 a(b(a(b(x1)))) -> a(b(b(a(x1)))) 13.68/4.35 a(b(b(a(a(x1))))) -> a(a(b(a(a(x1))))) 13.68/4.35 b(b(b(a(a(x1))))) -> b(a(b(a(a(x1))))) 13.68/4.35 a(b(b(b(b(x1))))) -> a(a(a(a(a(x1))))) 13.68/4.35 b(b(b(b(b(x1))))) -> b(a(a(a(a(x1))))) 13.68/4.35 13.68/4.35 Q is empty. 13.68/4.35 13.68/4.35 ---------------------------------------- 13.68/4.35 13.68/4.35 (3) RootLabelingProof (EQUIVALENT) 13.68/4.35 We used plain root labeling [ROOTLAB] with the following heuristic: 13.68/4.35 LabelAll: All function symbols get labeled 13.68/4.35 13.68/4.35 As Q is empty the root labeling was sound AND complete. 13.68/4.35 13.68/4.35 ---------------------------------------- 13.68/4.35 13.68/4.35 (4) 13.68/4.35 Obligation: 13.68/4.35 Q restricted rewrite system: 13.68/4.35 The TRS R consists of the following rules: 13.68/4.35 13.68/4.35 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 13.68/4.35 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 13.68/4.35 a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 13.68/4.35 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 13.68/4.35 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 13.68/4.35 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 13.68/4.35 b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 13.68/4.35 b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 13.68/4.35 13.68/4.35 Q is empty. 13.68/4.35 13.68/4.35 ---------------------------------------- 13.68/4.35 13.68/4.35 (5) DependencyPairsProof (EQUIVALENT) 13.68/4.35 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 13.68/4.35 ---------------------------------------- 13.68/4.35 13.68/4.35 (6) 13.68/4.35 Obligation: 13.68/4.35 Q DP problem: 13.68/4.35 The TRS P consists of the following rules: 13.68/4.35 13.68/4.35 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 13.68/4.35 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 13.68/4.35 A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 13.68/4.35 A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) 13.68/4.35 A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 13.68/4.35 A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 13.68/4.35 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 13.68/4.35 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) 13.68/4.35 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1) 13.68/4.35 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 13.68/4.35 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 13.68/4.35 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 13.68/4.35 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 13.68/4.35 A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) 13.68/4.35 B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) 13.68/4.35 13.68/4.35 The TRS R consists of the following rules: 13.68/4.35 13.68/4.35 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 13.68/4.35 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 13.68/4.35 a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 13.68/4.35 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 13.68/4.35 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 13.68/4.35 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 13.68/4.35 b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 13.68/4.35 b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 13.68/4.35 13.68/4.35 Q is empty. 13.68/4.35 We have to consider all minimal (P,Q,R)-chains. 13.68/4.35 ---------------------------------------- 13.68/4.35 13.68/4.35 (7) DependencyGraphProof (EQUIVALENT) 13.68/4.35 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 9 less nodes. 13.68/4.35 ---------------------------------------- 13.68/4.35 13.68/4.35 (8) 13.68/4.35 Complex Obligation (AND) 13.68/4.35 13.68/4.35 ---------------------------------------- 13.68/4.35 13.68/4.35 (9) 13.68/4.35 Obligation: 13.68/4.35 Q DP problem: 13.68/4.35 The TRS P consists of the following rules: 13.68/4.35 13.68/4.35 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 13.68/4.35 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 13.68/4.35 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 13.68/4.35 A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 13.68/4.35 13.68/4.35 The TRS R consists of the following rules: 13.68/4.35 13.68/4.35 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 13.68/4.35 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 13.68/4.35 a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 13.68/4.35 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 13.68/4.35 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 13.68/4.35 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 13.68/4.35 b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 13.68/4.35 b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 13.68/4.35 13.68/4.35 Q is empty. 13.68/4.35 We have to consider all minimal (P,Q,R)-chains. 13.68/4.35 ---------------------------------------- 13.68/4.35 13.68/4.35 (10) UsableRulesProof (EQUIVALENT) 13.68/4.35 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. 13.68/4.35 ---------------------------------------- 13.68/4.35 13.68/4.35 (11) 13.68/4.35 Obligation: 13.68/4.35 Q DP problem: 13.68/4.35 The TRS P consists of the following rules: 13.68/4.35 13.68/4.35 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 13.68/4.35 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 13.68/4.35 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 13.68/4.35 A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 13.68/4.35 13.68/4.35 The TRS R consists of the following rules: 13.68/4.35 13.68/4.35 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 13.68/4.35 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 13.68/4.35 a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 13.68/4.35 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 13.68/4.35 13.68/4.35 Q is empty. 13.68/4.35 We have to consider all minimal (P,Q,R)-chains. 13.68/4.35 ---------------------------------------- 13.68/4.35 13.68/4.35 (12) QDPOrderProof (EQUIVALENT) 13.68/4.35 We use the reduction pair processor [LPAR04,JAR06]. 13.68/4.35 13.68/4.35 13.68/4.35 The following pairs can be oriented strictly and are deleted. 13.68/4.35 13.68/4.35 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 13.68/4.35 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 13.68/4.35 The remaining pairs can at least be oriented weakly. 13.68/4.35 Used ordering: Polynomial interpretation [POLO]: 13.68/4.35 13.68/4.35 POL(A_{B_1}(x_1)) = x_1 13.68/4.35 POL(a_{a_1}(x_1)) = 1 + x_1 13.68/4.35 POL(a_{b_1}(x_1)) = 1 + x_1 13.68/4.35 POL(b_{a_1}(x_1)) = 1 + x_1 13.68/4.35 POL(b_{b_1}(x_1)) = 1 + x_1 13.68/4.35 13.68/4.35 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 13.68/4.35 13.68/4.35 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 13.68/4.35 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 13.68/4.35 a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 13.68/4.35 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 13.68/4.35 13.68/4.35 13.68/4.35 ---------------------------------------- 13.68/4.35 13.68/4.35 (13) 13.68/4.35 Obligation: 13.68/4.35 Q DP problem: 13.68/4.35 The TRS P consists of the following rules: 13.68/4.35 13.68/4.35 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 13.68/4.35 A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 13.68/4.35 13.68/4.35 The TRS R consists of the following rules: 13.68/4.35 13.68/4.35 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 13.68/4.35 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 13.68/4.35 a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 13.68/4.35 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 13.68/4.35 13.68/4.35 Q is empty. 13.68/4.35 We have to consider all minimal (P,Q,R)-chains. 13.68/4.35 ---------------------------------------- 13.68/4.35 13.68/4.35 (14) DependencyGraphProof (EQUIVALENT) 13.68/4.35 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 13.68/4.35 ---------------------------------------- 13.68/4.35 13.68/4.35 (15) 13.68/4.35 TRUE 13.68/4.35 13.68/4.35 ---------------------------------------- 13.68/4.35 13.68/4.35 (16) 13.68/4.35 Obligation: 13.68/4.35 Q DP problem: 13.68/4.35 The TRS P consists of the following rules: 13.68/4.35 13.68/4.35 A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) 13.68/4.35 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1) 13.68/4.35 13.68/4.35 The TRS R consists of the following rules: 13.68/4.35 13.68/4.35 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 13.68/4.35 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 13.68/4.35 a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 13.68/4.35 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 13.68/4.35 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 13.68/4.35 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 13.68/4.35 a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 13.68/4.35 b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 13.68/4.35 b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 13.68/4.35 13.68/4.35 Q is empty. 13.68/4.35 We have to consider all minimal (P,Q,R)-chains. 13.68/4.35 ---------------------------------------- 13.68/4.35 13.68/4.35 (17) UsableRulesProof (EQUIVALENT) 13.68/4.35 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. 13.68/4.35 ---------------------------------------- 13.68/4.35 13.68/4.35 (18) 13.68/4.35 Obligation: 13.68/4.35 Q DP problem: 13.68/4.35 The TRS P consists of the following rules: 13.77/4.35 13.77/4.35 A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) 13.77/4.35 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1) 13.77/4.35 13.77/4.35 R is empty. 13.77/4.35 Q is empty. 13.77/4.35 We have to consider all minimal (P,Q,R)-chains. 13.77/4.35 ---------------------------------------- 13.77/4.35 13.77/4.35 (19) QDPSizeChangeProof (EQUIVALENT) 13.77/4.35 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. 13.77/4.35 13.77/4.35 From the DPs we obtained the following set of size-change graphs: 13.77/4.35 *A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) 13.77/4.35 The graph contains the following edges 1 > 1 13.77/4.35 13.77/4.35 13.77/4.35 *A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1) 13.77/4.35 The graph contains the following edges 1 > 1 13.77/4.35 13.77/4.35 13.77/4.35 ---------------------------------------- 13.77/4.35 13.77/4.35 (20) 13.77/4.35 YES 13.95/4.43 EOF