/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) DependencyPairsProof [EQUIVALENT, 0 ms] (2) QDP (3) DependencyGraphProof [EQUIVALENT, 0 ms] (4) QDP (5) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] (6) QDP (7) RFCMatchBoundsDPProof [EQUIVALENT, 0 ms] (8) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f(a, f(a, f(b, f(a, f(a, f(b, f(a, x))))))) -> f(a, f(b, f(a, f(a, f(b, f(a, f(a, f(a, f(b, x))))))))) Q is empty. ---------------------------------------- (1) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (2) Obligation: Q DP problem: The TRS P consists of the following rules: F(a, f(a, f(b, f(a, f(a, f(b, f(a, x))))))) -> F(a, f(b, f(a, f(a, f(b, f(a, f(a, f(a, f(b, x))))))))) F(a, f(a, f(b, f(a, f(a, f(b, f(a, x))))))) -> F(b, f(a, f(a, f(b, f(a, f(a, f(a, f(b, x)))))))) F(a, f(a, f(b, f(a, f(a, f(b, f(a, x))))))) -> F(a, f(a, f(b, f(a, f(a, f(a, f(b, x))))))) F(a, f(a, f(b, f(a, f(a, f(b, f(a, x))))))) -> F(a, f(b, f(a, f(a, f(a, f(b, x)))))) F(a, f(a, f(b, f(a, f(a, f(b, f(a, x))))))) -> F(b, f(a, f(a, f(a, f(b, x))))) F(a, f(a, f(b, f(a, f(a, f(b, f(a, x))))))) -> F(a, f(a, f(a, f(b, x)))) F(a, f(a, f(b, f(a, f(a, f(b, f(a, x))))))) -> F(a, f(a, f(b, x))) F(a, f(a, f(b, f(a, f(a, f(b, f(a, x))))))) -> F(a, f(b, x)) F(a, f(a, f(b, f(a, f(a, f(b, f(a, x))))))) -> F(b, x) The TRS R consists of the following rules: f(a, f(a, f(b, f(a, f(a, f(b, f(a, x))))))) -> f(a, f(b, f(a, f(a, f(b, f(a, f(a, f(a, f(b, x))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (3) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes. ---------------------------------------- (4) Obligation: Q DP problem: The TRS P consists of the following rules: F(a, f(a, f(b, f(a, f(a, f(b, f(a, x))))))) -> F(a, f(a, f(a, f(b, x)))) F(a, f(a, f(b, f(a, f(a, f(b, f(a, x))))))) -> F(a, f(a, f(b, f(a, f(a, f(a, f(b, x))))))) F(a, f(a, f(b, f(a, f(a, f(b, f(a, x))))))) -> F(a, f(a, f(b, x))) The TRS R consists of the following rules: f(a, f(a, f(b, f(a, f(a, f(b, f(a, x))))))) -> f(a, f(b, f(a, f(a, f(b, f(a, f(a, f(a, f(b, x))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (5) UsableRulesReductionPairsProof (EQUIVALENT) First, we A-transformed [FROCOS05] the QDP-Problem. Then we obtain the following A-transformed DP problem. The pairs P are: a1(a(b(a(a(b(a(x))))))) -> a1(a(a(b(x)))) a1(a(b(a(a(b(a(x))))))) -> a1(a(b(a(a(a(b(x))))))) a1(a(b(a(a(b(a(x))))))) -> a1(a(b(x))) and the Q and R are: Q restricted rewrite system: The TRS R consists of the following rules: a(a(b(a(a(b(a(x))))))) -> a(b(a(a(b(a(a(a(b(x))))))))) Q is empty. By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. No dependency pairs are removed. The following rules are removed from R: f(a, f(a, f(b, f(a, f(a, f(b, f(a, x))))))) -> f(a, f(b, f(a, f(a, f(b, f(a, f(a, f(a, f(b, x))))))))) Used ordering: POLO with Polynomial interpretation [POLO]: POL(a(x_1)) = x_1 POL(a1(x_1)) = 2*x_1 POL(b(x_1)) = x_1 ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: a1(a(b(a(a(b(a(x))))))) -> a1(a(a(b(x)))) a1(a(b(a(a(b(a(x))))))) -> a1(a(b(a(a(a(b(x))))))) a1(a(b(a(a(b(a(x))))))) -> a1(a(b(x))) The TRS R consists of the following rules: a(a(b(a(a(b(a(x))))))) -> a(b(a(a(b(a(a(a(b(x))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (7) RFCMatchBoundsDPProof (EQUIVALENT) Finiteness of the DP problem can be shown by a matchbound of 2. As the DP problem is minimal we only have to initialize the certificate graph by the rules of P: a1(a(b(a(a(b(a(x))))))) -> a1(a(a(b(x)))) a1(a(b(a(a(b(a(x))))))) -> a1(a(b(a(a(a(b(x))))))) a1(a(b(a(a(b(a(x))))))) -> a1(a(b(x))) To find matches we regarded all rules of R and P: a(a(b(a(a(b(a(x))))))) -> a(b(a(a(b(a(a(a(b(x))))))))) a1(a(b(a(a(b(a(x))))))) -> a1(a(a(b(x)))) a1(a(b(a(a(b(a(x))))))) -> a1(a(b(a(a(a(b(x))))))) a1(a(b(a(a(b(a(x))))))) -> a1(a(b(x))) The certificate found is represented by the following graph. The certificate consists of the following enumerated nodes: 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104 Node 44 is start node and node 45 is final node. Those nodes are connected through the following edges: * 44 to 46 labelled a1_1(0)* 44 to 49 labelled a1_1(0)* 44 to 47 labelled a1_1(0)* 44 to 55 labelled a1_1(1)* 44 to 58 labelled a1_1(1)* 44 to 56 labelled a1_1(1)* 44 to 80 labelled a1_1(2)* 44 to 83 labelled a1_1(2)* 44 to 81 labelled a1_1(2)* 45 to 45 labelled #_1(0)* 46 to 47 labelled a_1(0)* 46 to 64 labelled a_1(1)* 47 to 48 labelled a_1(0)* 48 to 45 labelled b_1(0)* 49 to 50 labelled a_1(0)* 50 to 51 labelled b_1(0)* 51 to 52 labelled a_1(0)* 51 to 72 labelled a_1(1)* 52 to 53 labelled a_1(0)* 52 to 64 labelled a_1(1)* 53 to 54 labelled a_1(0)* 54 to 45 labelled b_1(0)* 55 to 56 labelled a_1(1)* 55 to 64 labelled a_1(1)* 56 to 57 labelled a_1(1)* 57 to 45 labelled b_1(1)* 57 to 69 labelled b_1(1)* 57 to 66 labelled b_1(1)* 57 to 89 labelled b_1(1)* 57 to 97 labelled b_1(1)* 58 to 59 labelled a_1(1)* 59 to 60 labelled b_1(1)* 60 to 61 labelled a_1(1)* 60 to 89 labelled a_1(2)* 61 to 62 labelled a_1(1)* 61 to 64 labelled a_1(1)* 62 to 63 labelled a_1(1)* 63 to 45 labelled b_1(1)* 63 to 69 labelled b_1(1)* 63 to 66 labelled b_1(1)* 63 to 89 labelled b_1(1)* 63 to 97 labelled b_1(1)* 64 to 65 labelled b_1(1)* 65 to 66 labelled a_1(1)* 65 to 97 labelled a_1(2)* 66 to 67 labelled a_1(1)* 67 to 68 labelled b_1(1)* 68 to 69 labelled a_1(1)* 68 to 89 labelled a_1(2)* 69 to 70 labelled a_1(1)* 69 to 64 labelled a_1(1)* 70 to 71 labelled a_1(1)* 71 to 45 labelled b_1(1)* 72 to 73 labelled b_1(1)* 73 to 74 labelled a_1(1)* 73 to 97 labelled a_1(2)* 74 to 75 labelled a_1(1)* 75 to 76 labelled b_1(1)* 76 to 77 labelled a_1(1)* 76 to 89 labelled a_1(2)* 77 to 78 labelled a_1(1)* 77 to 64 labelled a_1(1)* 78 to 79 labelled a_1(1)* 79 to 69 labelled b_1(1)* 79 to 89 labelled b_1(1)* 80 to 81 labelled a_1(2)* 80 to 64 labelled a_1(1)* 81 to 82 labelled a_1(2)* 82 to 69 labelled b_1(2)* 82 to 66 labelled b_1(2)* 82 to 89 labelled b_1(2)* 82 to 97 labelled b_1(2)* 83 to 84 labelled a_1(2)* 84 to 85 labelled b_1(2)* 85 to 86 labelled a_1(2)* 85 to 89 labelled a_1(2)* 86 to 87 labelled a_1(2)* 86 to 64 labelled a_1(1)* 87 to 88 labelled a_1(2)* 88 to 69 labelled b_1(2)* 88 to 66 labelled b_1(2)* 88 to 89 labelled b_1(2)* 88 to 97 labelled b_1(2)* 89 to 90 labelled b_1(2)* 90 to 91 labelled a_1(2)* 90 to 97 labelled a_1(2)* 91 to 92 labelled a_1(2)* 92 to 93 labelled b_1(2)* 93 to 94 labelled a_1(2)* 93 to 89 labelled a_1(2)* 94 to 95 labelled a_1(2)* 94 to 64 labelled a_1(1)* 95 to 96 labelled a_1(2)* 96 to 69 labelled b_1(2)* 96 to 89 labelled b_1(2)* 97 to 98 labelled b_1(2)* 98 to 99 labelled a_1(2)* 99 to 100 labelled a_1(2)* 100 to 101 labelled b_1(2)* 101 to 102 labelled a_1(2)* 102 to 103 labelled a_1(2)* 103 to 104 labelled a_1(2)* 104 to 66 labelled b_1(2)* 104 to 97 labelled b_1(2) ---------------------------------------- (8) YES