253.81/65.47 YES 254.12/65.56 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 254.12/65.56 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 254.12/65.56 254.12/65.56 254.12/65.56 Termination w.r.t. Q of the given QTRS could be proven: 254.12/65.56 254.12/65.56 (0) QTRS 254.12/65.56 (1) QTRS Reverse [EQUIVALENT, 0 ms] 254.12/65.56 (2) QTRS 254.12/65.56 (3) RootLabelingProof [EQUIVALENT, 0 ms] 254.12/65.56 (4) QTRS 254.12/65.56 (5) DependencyPairsProof [EQUIVALENT, 26 ms] 254.12/65.56 (6) QDP 254.12/65.56 (7) DependencyGraphProof [EQUIVALENT, 0 ms] 254.12/65.56 (8) QDP 254.12/65.56 (9) QDPOrderProof [EQUIVALENT, 7456 ms] 254.12/65.56 (10) QDP 254.12/65.56 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 254.12/65.56 (12) TRUE 254.12/65.56 254.12/65.56 254.12/65.56 ---------------------------------------- 254.12/65.56 254.12/65.56 (0) 254.12/65.56 Obligation: 254.12/65.56 Q restricted rewrite system: 254.12/65.56 The TRS R consists of the following rules: 254.12/65.56 254.12/65.56 b(b(b(x1))) -> b(b(a(b(a(b(a(b(x1)))))))) 254.12/65.56 b(a(b(b(x1)))) -> b(a(b(a(b(a(a(b(x1)))))))) 254.12/65.56 b(b(a(b(a(b(a(a(b(x1))))))))) -> b(b(b(b(a(b(x1)))))) 254.12/65.56 254.12/65.56 Q is empty. 254.12/65.56 254.12/65.56 ---------------------------------------- 254.12/65.56 254.12/65.56 (1) QTRS Reverse (EQUIVALENT) 254.12/65.56 We applied the QTRS Reverse Processor [REVERSE]. 254.12/65.56 ---------------------------------------- 254.12/65.56 254.12/65.56 (2) 254.12/65.56 Obligation: 254.12/65.56 Q restricted rewrite system: 254.12/65.56 The TRS R consists of the following rules: 254.12/65.56 254.12/65.56 b(b(b(x1))) -> b(a(b(a(b(a(b(b(x1)))))))) 254.12/65.56 b(b(a(b(x1)))) -> b(a(a(b(a(b(a(b(x1)))))))) 254.12/65.56 b(a(a(b(a(b(a(b(b(x1))))))))) -> b(a(b(b(b(b(x1)))))) 254.12/65.56 254.12/65.56 Q is empty. 254.12/65.56 254.12/65.56 ---------------------------------------- 254.12/65.56 254.12/65.56 (3) RootLabelingProof (EQUIVALENT) 254.12/65.56 We used plain root labeling [ROOTLAB] with the following heuristic: 254.12/65.56 LabelAll: All function symbols get labeled 254.12/65.56 254.12/65.56 As Q is empty the root labeling was sound AND complete. 254.12/65.56 254.12/65.56 ---------------------------------------- 254.12/65.56 254.12/65.56 (4) 254.12/65.56 Obligation: 254.12/65.56 Q restricted rewrite system: 254.12/65.56 The TRS R consists of the following rules: 254.12/65.56 254.12/65.56 b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 254.12/65.56 b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 254.12/65.56 b_{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_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 254.12/65.56 b_{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}(a_{b_1}(b_{a_1}(x1)))))))) 254.12/65.56 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 254.12/65.56 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 254.12/65.56 254.12/65.56 Q is empty. 254.12/65.56 254.12/65.56 ---------------------------------------- 254.12/65.56 254.12/65.56 (5) DependencyPairsProof (EQUIVALENT) 254.12/65.56 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 254.12/65.56 ---------------------------------------- 254.12/65.56 254.12/65.56 (6) 254.12/65.56 Obligation: 254.12/65.56 Q DP problem: 254.12/65.56 The TRS P consists of the following rules: 254.12/65.56 254.12/65.56 B_{B_1}(b_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 254.12/65.56 B_{B_1}(b_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 254.12/65.56 B_{B_1}(b_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 254.12/65.56 B_{B_1}(b_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 254.12/65.56 B_{B_1}(b_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 254.12/65.56 B_{B_1}(b_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 254.12/65.56 B_{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_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 254.12/65.56 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 254.12/65.56 B_{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}(a_{b_1}(b_{a_1}(x1)))))))) 254.12/65.56 B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) 254.12/65.56 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 254.12/65.56 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 254.12/65.56 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 254.12/65.56 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 254.12/65.56 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 254.12/65.56 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 254.12/65.56 254.12/65.56 The TRS R consists of the following rules: 254.12/65.56 254.12/65.56 b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 254.12/65.56 b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 254.12/65.56 b_{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_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 254.12/65.56 b_{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}(a_{b_1}(b_{a_1}(x1)))))))) 254.12/65.56 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 254.12/65.56 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 254.12/65.56 254.12/65.56 Q is empty. 254.12/65.56 We have to consider all minimal (P,Q,R)-chains. 254.12/65.56 ---------------------------------------- 254.12/65.56 254.12/65.56 (7) DependencyGraphProof (EQUIVALENT) 254.12/65.56 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 10 less nodes. 254.12/65.56 ---------------------------------------- 254.12/65.56 254.12/65.56 (8) 254.12/65.56 Obligation: 254.12/65.56 Q DP problem: 254.12/65.56 The TRS P consists of the following rules: 254.12/65.56 254.12/65.56 B_{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_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 254.12/65.56 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 254.12/65.56 B_{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}(a_{b_1}(b_{a_1}(x1)))))))) 254.12/65.56 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 254.12/65.56 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 254.12/65.56 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 254.12/65.56 254.12/65.56 The TRS R consists of the following rules: 254.12/65.56 254.12/65.56 b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 254.12/65.56 b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 254.12/65.56 b_{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_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 254.12/65.56 b_{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}(a_{b_1}(b_{a_1}(x1)))))))) 254.12/65.56 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 254.12/65.56 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 254.12/65.56 254.12/65.56 Q is empty. 254.12/65.56 We have to consider all minimal (P,Q,R)-chains. 254.12/65.56 ---------------------------------------- 254.12/65.56 254.12/65.56 (9) QDPOrderProof (EQUIVALENT) 254.12/65.56 We use the reduction pair processor [LPAR04,JAR06]. 254.12/65.56 254.12/65.56 254.12/65.56 The following pairs can be oriented strictly and are deleted. 254.12/65.56 254.12/65.56 B_{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_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 254.12/65.56 B_{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}(a_{b_1}(b_{a_1}(x1)))))))) 254.12/65.56 The remaining pairs can at least be oriented weakly. 254.12/65.56 Used ordering: Matrix interpretation [MATRO] to (N^3, +, *, >=, >) : 254.12/65.56 254.12/65.56 <<< 254.12/65.56 POL(B_{B_1}(x_1)) = [[1]] + [[1, 0, 0]] * x_1 254.12/65.56 >>> 254.12/65.56 254.12/65.56 <<< 254.12/65.56 POL(b_{a_1}(x_1)) = [[0], [0], [0]] + [[0, 0, 1], [0, 1, 0], [0, 0, 0]] * x_1 254.12/65.56 >>> 254.12/65.56 254.12/65.56 <<< 254.12/65.56 POL(a_{b_1}(x_1)) = [[0], [0], [0]] + [[0, 1, 0], [1, 0, 0], [0, 0, 1]] * x_1 254.12/65.56 >>> 254.12/65.56 254.12/65.56 <<< 254.12/65.56 POL(b_{b_1}(x_1)) = [[0], [0], [1]] + [[1, 0, 0], [1, 1, 0], [1, 0, 0]] * x_1 254.12/65.56 >>> 254.12/65.56 254.12/65.56 <<< 254.12/65.56 POL(B_{A_1}(x_1)) = [[0]] + [[0, 0, 1]] * x_1 254.12/65.56 >>> 254.12/65.56 254.12/65.56 <<< 254.12/65.56 POL(a_{a_1}(x_1)) = [[0], [0], [0]] + [[0, 0, 0], [1, 0, 0], [1, 0, 0]] * x_1 254.12/65.56 >>> 254.12/65.56 254.12/65.56 254.12/65.56 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 254.12/65.56 254.12/65.56 b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 254.12/65.56 b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 254.12/65.56 b_{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_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 254.12/65.56 b_{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}(a_{b_1}(b_{a_1}(x1)))))))) 254.12/65.56 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 254.12/65.56 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 254.12/65.56 254.12/65.56 254.12/65.56 ---------------------------------------- 254.12/65.56 254.12/65.56 (10) 254.12/65.56 Obligation: 254.12/65.56 Q DP problem: 254.12/65.56 The TRS P consists of the following rules: 254.12/65.56 254.12/65.56 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 254.12/65.56 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 254.12/65.56 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 254.12/65.56 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 254.12/65.56 254.12/65.56 The TRS R consists of the following rules: 254.12/65.56 254.12/65.56 b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 254.12/65.56 b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 254.12/65.56 b_{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_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 254.12/65.56 b_{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}(a_{b_1}(b_{a_1}(x1)))))))) 254.12/65.56 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 254.12/65.56 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 254.12/65.56 254.12/65.56 Q is empty. 254.12/65.56 We have to consider all minimal (P,Q,R)-chains. 254.12/65.56 ---------------------------------------- 254.12/65.56 254.12/65.56 (11) DependencyGraphProof (EQUIVALENT) 254.12/65.56 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes. 254.12/65.56 ---------------------------------------- 254.12/65.56 254.12/65.56 (12) 254.12/65.56 TRUE 254.58/65.68 EOF