721.60/196.06 YES 722.82/196.39 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 722.82/196.39 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 722.82/196.39 722.82/196.39 722.82/196.39 Termination w.r.t. Q of the given QTRS could be proven: 722.82/196.39 722.82/196.39 (0) QTRS 722.82/196.39 (1) RootLabelingProof [EQUIVALENT, 0 ms] 722.82/196.39 (2) QTRS 722.82/196.39 (3) DependencyPairsProof [EQUIVALENT, 56 ms] 722.82/196.39 (4) QDP 722.82/196.39 (5) DependencyGraphProof [EQUIVALENT, 3 ms] 722.82/196.39 (6) AND 722.82/196.39 (7) QDP 722.82/196.39 (8) QDPSizeChangeProof [EQUIVALENT, 0 ms] 722.82/196.39 (9) YES 722.82/196.39 (10) QDP 722.82/196.39 (11) QDPOrderProof [EQUIVALENT, 7311 ms] 722.82/196.39 (12) QDP 722.82/196.39 (13) QDPOrderProof [EQUIVALENT, 7579 ms] 722.82/196.39 (14) QDP 722.82/196.39 (15) PisEmptyProof [EQUIVALENT, 0 ms] 722.82/196.39 (16) YES 722.82/196.39 (17) QDP 722.82/196.39 (18) SplitQDPProof [EQUIVALENT, 0 ms] 722.82/196.39 (19) AND 722.82/196.39 (20) QDP 722.82/196.39 (21) SemLabProof [SOUND, 0 ms] 722.82/196.39 (22) QDP 722.82/196.39 (23) DependencyGraphProof [EQUIVALENT, 0 ms] 722.82/196.39 (24) QDP 722.82/196.39 (25) QDPOrderProof [EQUIVALENT, 16 ms] 722.82/196.39 (26) QDP 722.82/196.39 (27) QDPOrderProof [EQUIVALENT, 17 ms] 722.82/196.39 (28) QDP 722.82/196.39 (29) PisEmptyProof [SOUND, 0 ms] 722.82/196.39 (30) TRUE 722.82/196.39 (31) QDP 722.82/196.39 (32) QDPOrderProof [EQUIVALENT, 8630 ms] 722.82/196.39 (33) QDP 722.82/196.39 (34) PisEmptyProof [EQUIVALENT, 0 ms] 722.82/196.39 (35) YES 722.82/196.39 722.82/196.39 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (0) 722.82/196.39 Obligation: 722.82/196.39 Q restricted rewrite system: 722.82/196.39 The TRS R consists of the following rules: 722.82/196.39 722.82/196.39 b(b(a(a(b(x1))))) -> b(a(a(b(b(x1))))) 722.82/196.39 b(a(a(a(b(b(b(x1))))))) -> b(b(b(a(a(a(b(x1))))))) 722.82/196.39 b(a(b(a(a(b(x1)))))) -> b(a(b(b(a(a(a(b(x1)))))))) 722.82/196.39 b(a(a(a(b(a(b(x1))))))) -> b(a(a(b(a(b(x1)))))) 722.82/196.39 722.82/196.39 Q is empty. 722.82/196.39 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (1) RootLabelingProof (EQUIVALENT) 722.82/196.39 We used plain root labeling [ROOTLAB] with the following heuristic: 722.82/196.39 LabelAll: All function symbols get labeled 722.82/196.39 722.82/196.39 As Q is empty the root labeling was sound AND complete. 722.82/196.39 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (2) 722.82/196.39 Obligation: 722.82/196.39 Q restricted rewrite system: 722.82/196.39 The TRS R consists of the following rules: 722.82/196.39 722.82/196.39 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 722.82/196.39 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))) 722.82/196.39 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 722.82/196.39 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))) 722.82/196.39 722.82/196.39 Q is empty. 722.82/196.39 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (3) DependencyPairsProof (EQUIVALENT) 722.82/196.39 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (4) 722.82/196.39 Obligation: 722.82/196.39 Q DP problem: 722.82/196.39 The TRS P consists of the following rules: 722.82/196.39 722.82/196.39 B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 722.82/196.39 B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 722.82/196.39 B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 722.82/196.39 B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 722.82/196.39 B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 722.82/196.39 B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) 722.82/196.39 B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 722.82/196.39 B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))) 722.82/196.39 B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) 722.82/196.39 B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 722.82/196.39 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 722.82/196.39 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) 722.82/196.39 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 722.82/196.39 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 722.82/196.39 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) 722.82/196.39 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 722.82/196.39 B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))) 722.82/196.39 B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))) 722.82/196.39 722.82/196.39 The TRS R consists of the following rules: 722.82/196.39 722.82/196.39 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 722.82/196.39 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))) 722.82/196.39 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 722.82/196.39 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))) 722.82/196.39 722.82/196.39 Q is empty. 722.82/196.39 We have to consider all minimal (P,Q,R)-chains. 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (5) DependencyGraphProof (EQUIVALENT) 722.82/196.39 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 12 less nodes. 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (6) 722.82/196.39 Complex Obligation (AND) 722.82/196.39 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (7) 722.82/196.39 Obligation: 722.82/196.39 Q DP problem: 722.82/196.39 The TRS P consists of the following rules: 722.82/196.39 722.82/196.39 B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 722.82/196.39 B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 722.82/196.39 722.82/196.39 The TRS R consists of the following rules: 722.82/196.39 722.82/196.39 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 722.82/196.39 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))) 722.82/196.39 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 722.82/196.39 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))) 722.82/196.39 722.82/196.39 Q is empty. 722.82/196.39 We have to consider all minimal (P,Q,R)-chains. 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (8) QDPSizeChangeProof (EQUIVALENT) 722.82/196.39 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. 722.82/196.39 722.82/196.39 From the DPs we obtained the following set of size-change graphs: 722.82/196.39 *B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 722.82/196.39 The graph contains the following edges 1 > 1 722.82/196.39 722.82/196.39 722.82/196.39 *B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 722.82/196.39 The graph contains the following edges 1 > 1 722.82/196.39 722.82/196.39 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (9) 722.82/196.39 YES 722.82/196.39 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (10) 722.82/196.39 Obligation: 722.82/196.39 Q DP problem: 722.82/196.39 The TRS P consists of the following rules: 722.82/196.39 722.82/196.39 B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 722.82/196.39 B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 722.82/196.39 722.82/196.39 The TRS R consists of the following rules: 722.82/196.39 722.82/196.39 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 722.82/196.39 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))) 722.82/196.39 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 722.82/196.39 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))) 722.82/196.39 722.82/196.39 Q is empty. 722.82/196.39 We have to consider all minimal (P,Q,R)-chains. 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (11) QDPOrderProof (EQUIVALENT) 722.82/196.39 We use the reduction pair processor [LPAR04,JAR06]. 722.82/196.39 722.82/196.39 722.82/196.39 The following pairs can be oriented strictly and are deleted. 722.82/196.39 722.82/196.39 B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 722.82/196.39 The remaining pairs can at least be oriented weakly. 722.82/196.39 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 722.82/196.39 722.82/196.39 <<< 722.82/196.39 POL(B_{A_1}(x_1)) = [[0A]] + [[0A, 0A, -I]] * x_1 722.82/196.39 >>> 722.82/196.39 722.82/196.39 <<< 722.82/196.39 POL(a_{a_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [0A, -I, 0A], [-I, 0A, -I]] * x_1 722.82/196.39 >>> 722.82/196.39 722.82/196.39 <<< 722.82/196.39 POL(a_{b_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [-I, -I, 0A], [0A, -I, -I]] * x_1 722.82/196.39 >>> 722.82/196.39 722.82/196.39 <<< 722.82/196.39 POL(b_{b_1}(x_1)) = [[-I], [0A], [0A]] + [[0A, -I, 0A], [0A, 0A, 0A], [1A, -I, -I]] * x_1 722.82/196.39 >>> 722.82/196.39 722.82/196.39 <<< 722.82/196.39 POL(b_{a_1}(x_1)) = [[-I], [0A], [-I]] + [[0A, -I, -I], [0A, -I, -I], [-I, -I, 0A]] * x_1 722.82/196.39 >>> 722.82/196.39 722.82/196.39 722.82/196.39 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 722.82/196.39 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))) 722.82/196.39 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 722.82/196.39 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))) 722.82/196.39 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 722.82/196.39 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 722.82/196.39 722.82/196.39 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (12) 722.82/196.39 Obligation: 722.82/196.39 Q DP problem: 722.82/196.39 The TRS P consists of the following rules: 722.82/196.39 722.82/196.39 B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 722.82/196.39 722.82/196.39 The TRS R consists of the following rules: 722.82/196.39 722.82/196.39 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 722.82/196.39 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))) 722.82/196.39 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 722.82/196.39 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))) 722.82/196.39 722.82/196.39 Q is empty. 722.82/196.39 We have to consider all minimal (P,Q,R)-chains. 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (13) QDPOrderProof (EQUIVALENT) 722.82/196.39 We use the reduction pair processor [LPAR04,JAR06]. 722.82/196.39 722.82/196.39 722.82/196.39 The following pairs can be oriented strictly and are deleted. 722.82/196.39 722.82/196.39 B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 722.82/196.39 The remaining pairs can at least be oriented weakly. 722.82/196.39 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 722.82/196.39 722.82/196.39 <<< 722.82/196.39 POL(B_{A_1}(x_1)) = [[-I]] + [[0A, 0A, -I]] * x_1 722.82/196.39 >>> 722.82/196.39 722.82/196.39 <<< 722.82/196.39 POL(a_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 722.82/196.39 >>> 722.82/196.39 722.82/196.39 <<< 722.82/196.39 POL(a_{b_1}(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, -I], [0A, 0A, 0A], [-I, -I, -I]] * x_1 722.82/196.39 >>> 722.82/196.39 722.82/196.39 <<< 722.82/196.39 POL(b_{b_1}(x_1)) = [[0A], [0A], [-I]] + [[0A, 0A, 0A], [0A, -I, 1A], [0A, 1A, 0A]] * x_1 722.82/196.39 >>> 722.82/196.39 722.82/196.39 <<< 722.82/196.39 POL(b_{a_1}(x_1)) = [[0A], [-I], [0A]] + [[0A, -I, -I], [-I, -I, 0A], [0A, -I, 0A]] * x_1 722.82/196.39 >>> 722.82/196.39 722.82/196.39 722.82/196.39 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 722.82/196.39 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))) 722.82/196.39 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 722.82/196.39 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))) 722.82/196.39 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 722.82/196.39 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 722.82/196.39 722.82/196.39 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (14) 722.82/196.39 Obligation: 722.82/196.39 Q DP problem: 722.82/196.39 P is empty. 722.82/196.39 The TRS R consists of the following rules: 722.82/196.39 722.82/196.39 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 722.82/196.39 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))) 722.82/196.39 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 722.82/196.39 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))) 722.82/196.39 722.82/196.39 Q is empty. 722.82/196.39 We have to consider all minimal (P,Q,R)-chains. 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (15) PisEmptyProof (EQUIVALENT) 722.82/196.39 The TRS P is empty. Hence, there is no (P,Q,R) chain. 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (16) 722.82/196.39 YES 722.82/196.39 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (17) 722.82/196.39 Obligation: 722.82/196.39 Q DP problem: 722.82/196.39 The TRS P consists of the following rules: 722.82/196.39 722.82/196.39 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 722.82/196.39 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 722.82/196.39 722.82/196.39 The TRS R consists of the following rules: 722.82/196.39 722.82/196.39 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 722.82/196.39 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))) 722.82/196.39 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 722.82/196.39 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))) 722.82/196.39 722.82/196.39 Q is empty. 722.82/196.39 We have to consider all minimal (P,Q,R)-chains. 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (18) SplitQDPProof (EQUIVALENT) 722.82/196.39 We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem 722.82/196.39 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (19) 722.82/196.39 Complex Obligation (AND) 722.82/196.39 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (20) 722.82/196.39 Obligation: 722.82/196.39 Q DP problem: 722.82/196.39 The TRS P consists of the following rules: 722.82/196.39 722.82/196.39 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 722.82/196.39 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 722.82/196.39 722.82/196.39 The TRS R consists of the following rules: 722.82/196.39 722.82/196.39 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 722.82/196.39 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))) 722.82/196.39 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 722.82/196.39 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))) 722.82/196.39 722.82/196.39 Q is empty. 722.82/196.39 We have to consider all minimal (P,Q,R)-chains. 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (21) SemLabProof (SOUND) 722.82/196.39 We found the following model for the rules of the TRSs R and P. 722.82/196.39 Interpretation over the domain with elements from 0 to 1. 722.82/196.39 b_{b_1}: 0 722.82/196.39 b_{a_1}: 0 722.82/196.39 B_{A_1}: 0 722.82/196.39 a_{b_1}: 1 722.82/196.39 a_{a_1}: 0 722.82/196.39 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (22) 722.82/196.39 Obligation: 722.82/196.39 Q DP problem: 722.82/196.39 The TRS P consists of the following rules: 722.82/196.39 722.82/196.39 B_{A_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1)))))) -> B_{A_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1)))))))) 722.82/196.39 B_{A_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1)))))) -> B_{A_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1)))))))) 722.82/196.39 B_{A_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1)))))) -> B_{A_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1)))))))) 722.82/196.39 B_{A_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1)))))) -> B_{A_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1)))))))) 722.82/196.39 722.82/196.39 The TRS R consists of the following rules: 722.82/196.39 722.82/196.39 b_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(x1))))) 722.82/196.39 b_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.1(x1))))) 722.82/196.39 b_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(x1))))) 722.82/196.39 b_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.1(x1))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(b_{b_1}.0(x1))))))) -> b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1))))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(b_{b_1}.1(x1))))))) -> b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1))))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(x1))))))) -> b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1))))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(b_{a_1}.1(x1))))))) -> b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1))))))) 722.82/196.39 b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1)))))) -> b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1)))))))) 722.82/196.39 b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1)))))) -> b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1)))))))) 722.82/196.39 b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1)))))) -> b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1)))))))) 722.82/196.39 b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1)))))) -> b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1)))))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1))))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1)))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1))))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1)))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1))))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1)))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1))))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1)))))) 722.82/196.39 722.82/196.39 Q is empty. 722.82/196.39 We have to consider all minimal (P,Q,R)-chains. 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (23) DependencyGraphProof (EQUIVALENT) 722.82/196.39 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (24) 722.82/196.39 Obligation: 722.82/196.39 Q DP problem: 722.82/196.39 The TRS P consists of the following rules: 722.82/196.39 722.82/196.39 B_{A_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1)))))) -> B_{A_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1)))))))) 722.82/196.39 B_{A_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1)))))) -> B_{A_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1)))))))) 722.82/196.39 B_{A_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1)))))) -> B_{A_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1)))))))) 722.82/196.39 722.82/196.39 The TRS R consists of the following rules: 722.82/196.39 722.82/196.39 b_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(x1))))) 722.82/196.39 b_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.1(x1))))) 722.82/196.39 b_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(x1))))) 722.82/196.39 b_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.1(x1))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(b_{b_1}.0(x1))))))) -> b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1))))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(b_{b_1}.1(x1))))))) -> b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1))))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(x1))))))) -> b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1))))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(b_{a_1}.1(x1))))))) -> b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1))))))) 722.82/196.39 b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1)))))) -> b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1)))))))) 722.82/196.39 b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1)))))) -> b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1)))))))) 722.82/196.39 b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1)))))) -> b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1)))))))) 722.82/196.39 b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1)))))) -> b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1)))))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1))))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1)))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1))))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1)))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1))))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1)))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1))))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1)))))) 722.82/196.39 722.82/196.39 Q is empty. 722.82/196.39 We have to consider all minimal (P,Q,R)-chains. 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (25) QDPOrderProof (EQUIVALENT) 722.82/196.39 We use the reduction pair processor [LPAR04,JAR06]. 722.82/196.39 722.82/196.39 722.82/196.39 The following pairs can be oriented strictly and are deleted. 722.82/196.39 722.82/196.39 B_{A_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1)))))) -> B_{A_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1)))))))) 722.82/196.39 The remaining pairs can at least be oriented weakly. 722.82/196.39 Used ordering: Polynomial interpretation [POLO]: 722.82/196.39 722.82/196.39 POL(B_{A_1}.1(x_1)) = x_1 722.82/196.39 POL(a_{a_1}.0(x_1)) = 1 722.82/196.39 POL(a_{a_1}.1(x_1)) = 1 + x_1 722.82/196.39 POL(a_{b_1}.0(x_1)) = x_1 722.82/196.39 POL(b_{a_1}.0(x_1)) = 1 + x_1 722.82/196.39 POL(b_{a_1}.1(x_1)) = 0 722.82/196.39 POL(b_{b_1}.0(x_1)) = x_1 722.82/196.39 POL(b_{b_1}.1(x_1)) = x_1 722.82/196.39 722.82/196.39 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 722.82/196.39 722.82/196.39 b_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(x1))))) 722.82/196.39 b_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.1(x1))))) 722.82/196.39 b_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(x1))))) 722.82/196.39 b_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.1(x1))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(b_{b_1}.0(x1))))))) -> b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1))))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(b_{b_1}.1(x1))))))) -> b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1))))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(x1))))))) -> b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1))))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(b_{a_1}.1(x1))))))) -> b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1))))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1))))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1)))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1))))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1)))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1))))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1)))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1))))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1)))))) 722.82/196.39 b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1)))))) -> b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1)))))))) 722.82/196.39 b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1)))))) -> b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1)))))))) 722.82/196.39 b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1)))))) -> b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1)))))))) 722.82/196.39 b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1)))))) -> b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1)))))))) 722.82/196.39 722.82/196.39 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (26) 722.82/196.39 Obligation: 722.82/196.39 Q DP problem: 722.82/196.39 The TRS P consists of the following rules: 722.82/196.39 722.82/196.39 B_{A_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1)))))) -> B_{A_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1)))))))) 722.82/196.39 B_{A_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1)))))) -> B_{A_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1)))))))) 722.82/196.39 722.82/196.39 The TRS R consists of the following rules: 722.82/196.39 722.82/196.39 b_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(x1))))) 722.82/196.39 b_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.1(x1))))) 722.82/196.39 b_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(x1))))) 722.82/196.39 b_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.1(x1))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(b_{b_1}.0(x1))))))) -> b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1))))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(b_{b_1}.1(x1))))))) -> b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1))))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(x1))))))) -> b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1))))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(b_{a_1}.1(x1))))))) -> b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1))))))) 722.82/196.39 b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1)))))) -> b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1)))))))) 722.82/196.39 b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1)))))) -> b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1)))))))) 722.82/196.39 b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1)))))) -> b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1)))))))) 722.82/196.39 b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1)))))) -> b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1)))))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1))))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1)))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1))))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1)))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1))))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1)))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1))))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1)))))) 722.82/196.39 722.82/196.39 Q is empty. 722.82/196.39 We have to consider all minimal (P,Q,R)-chains. 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (27) QDPOrderProof (EQUIVALENT) 722.82/196.39 We use the reduction pair processor [LPAR04,JAR06]. 722.82/196.39 722.82/196.39 722.82/196.39 The following pairs can be oriented strictly and are deleted. 722.82/196.39 722.82/196.39 B_{A_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1)))))) -> B_{A_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1)))))))) 722.82/196.39 The remaining pairs can at least be oriented weakly. 722.82/196.39 Used ordering: Polynomial interpretation [POLO]: 722.82/196.39 722.82/196.39 POL(B_{A_1}.1(x_1)) = x_1 722.82/196.39 POL(a_{a_1}.0(x_1)) = 1 722.82/196.39 POL(a_{a_1}.1(x_1)) = x_1 722.82/196.39 POL(a_{b_1}.0(x_1)) = x_1 722.82/196.39 POL(b_{a_1}.0(x_1)) = x_1 722.82/196.39 POL(b_{a_1}.1(x_1)) = 1 722.82/196.39 POL(b_{b_1}.0(x_1)) = 0 722.82/196.39 POL(b_{b_1}.1(x_1)) = x_1 722.82/196.39 722.82/196.39 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 722.82/196.39 722.82/196.39 b_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(x1))))) 722.82/196.39 b_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.1(x1))))) 722.82/196.39 b_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(x1))))) 722.82/196.39 b_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.1(x1))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(b_{b_1}.0(x1))))))) -> b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1))))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(x1))))))) -> b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1))))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(b_{a_1}.1(x1))))))) -> b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1))))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1))))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1)))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1))))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1)))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1))))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1)))))) 722.82/196.39 b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1)))))) -> b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1)))))))) 722.82/196.39 b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1)))))) -> b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1)))))))) 722.82/196.39 b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1)))))) -> b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1)))))))) 722.82/196.39 b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1)))))) -> b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1)))))))) 722.82/196.39 722.82/196.39 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (28) 722.82/196.39 Obligation: 722.82/196.39 Q DP problem: 722.82/196.39 The TRS P consists of the following rules: 722.82/196.39 722.82/196.39 B_{A_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1)))))) -> B_{A_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1)))))))) 722.82/196.39 722.82/196.39 The TRS R consists of the following rules: 722.82/196.39 722.82/196.39 b_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(x1))))) 722.82/196.39 b_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.1(x1))))) 722.82/196.39 b_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(x1))))) 722.82/196.39 b_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.1(x1))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(b_{b_1}.0(x1))))))) -> b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1))))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(b_{b_1}.1(x1))))))) -> b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1))))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(x1))))))) -> b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1))))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{b_1}.0(b_{a_1}.1(x1))))))) -> b_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1))))))) 722.82/196.39 b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1)))))) -> b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1)))))))) 722.82/196.39 b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1)))))) -> b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1)))))))) 722.82/196.39 b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1)))))) -> b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1)))))))) 722.82/196.39 b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1)))))) -> b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1)))))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1))))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{b_1}.0(x1)))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1))))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{b_1}.1(x1)))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1))))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{a_1}.0(x1)))))) 722.82/196.39 b_{a_1}.0(a_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1))))))) -> b_{a_1}.0(a_{a_1}.1(a_{b_1}.0(b_{a_1}.1(a_{b_1}.0(b_{a_1}.1(x1)))))) 722.82/196.39 722.82/196.39 Q is empty. 722.82/196.39 We have to consider all minimal (P,Q,R)-chains. 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (29) PisEmptyProof (SOUND) 722.82/196.39 The TRS P is empty. Hence, there is no (P,Q,R) chain. 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (30) 722.82/196.39 TRUE 722.82/196.39 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (31) 722.82/196.39 Obligation: 722.82/196.39 Q DP problem: 722.82/196.39 The TRS P consists of the following rules: 722.82/196.39 722.82/196.39 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 722.82/196.39 722.82/196.39 The TRS R consists of the following rules: 722.82/196.39 722.82/196.39 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 722.82/196.39 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))) 722.82/196.39 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 722.82/196.39 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))) 722.82/196.39 722.82/196.39 Q is empty. 722.82/196.39 We have to consider all minimal (P,Q,R)-chains. 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (32) QDPOrderProof (EQUIVALENT) 722.82/196.39 We use the reduction pair processor [LPAR04,JAR06]. 722.82/196.39 722.82/196.39 722.82/196.39 The following pairs can be oriented strictly and are deleted. 722.82/196.39 722.82/196.39 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 722.82/196.39 The remaining pairs can at least be oriented weakly. 722.82/196.39 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 722.82/196.39 722.82/196.39 <<< 722.82/196.39 POL(B_{A_1}(x_1)) = [[0A]] + [[-I, 0A, -I]] * x_1 722.82/196.39 >>> 722.82/196.39 722.82/196.39 <<< 722.82/196.39 POL(a_{b_1}(x_1)) = [[0A], [0A], [1A]] + [[1A, 0A, 0A], [0A, -I, -I], [0A, 0A, 1A]] * x_1 722.82/196.39 >>> 722.82/196.39 722.82/196.39 <<< 722.82/196.39 POL(b_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, 0A], [0A, 0A, 0A], [-I, 0A, -I]] * x_1 722.82/196.39 >>> 722.82/196.39 722.82/196.39 <<< 722.82/196.39 POL(a_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, -I], [-I, -I, 0A], [-I, 0A, -I]] * x_1 722.82/196.39 >>> 722.82/196.39 722.82/196.39 <<< 722.82/196.39 POL(b_{b_1}(x_1)) = [[0A], [-I], [-I]] + [[-I, -I, 0A], [0A, 0A, -I], [0A, -I, -I]] * x_1 722.82/196.39 >>> 722.82/196.39 722.82/196.39 722.82/196.39 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 722.82/196.39 722.82/196.39 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 722.82/196.39 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))) 722.82/196.39 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 722.82/196.39 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 722.82/196.39 722.82/196.39 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (33) 722.82/196.39 Obligation: 722.82/196.39 Q DP problem: 722.82/196.39 P is empty. 722.82/196.39 The TRS R consists of the following rules: 722.82/196.39 722.82/196.39 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 722.82/196.39 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))) 722.82/196.39 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 722.82/196.39 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))) 722.82/196.39 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))) 722.82/196.39 722.82/196.39 Q is empty. 722.82/196.39 We have to consider all minimal (P,Q,R)-chains. 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (34) PisEmptyProof (EQUIVALENT) 722.82/196.39 The TRS P is empty. Hence, there is no (P,Q,R) chain. 722.82/196.39 ---------------------------------------- 722.82/196.39 722.82/196.39 (35) 722.82/196.39 YES 723.14/196.52 EOF