/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) RFCMatchBoundsTRSProof [EQUIVALENT, 29 ms] (2) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: active(f(f(a))) -> mark(c(f(g(f(a))))) active(f(X)) -> f(active(X)) active(g(X)) -> g(active(X)) f(mark(X)) -> mark(f(X)) g(mark(X)) -> mark(g(X)) proper(f(X)) -> f(proper(X)) proper(a) -> ok(a) proper(c(X)) -> c(proper(X)) proper(g(X)) -> g(proper(X)) f(ok(X)) -> ok(f(X)) c(ok(X)) -> ok(c(X)) g(ok(X)) -> ok(g(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Q is empty. ---------------------------------------- (1) RFCMatchBoundsTRSProof (EQUIVALENT) Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 7. This implies Q-termination of R. The following rules were used to construct the certificate: active(f(f(a))) -> mark(c(f(g(f(a))))) active(f(X)) -> f(active(X)) active(g(X)) -> g(active(X)) f(mark(X)) -> mark(f(X)) g(mark(X)) -> mark(g(X)) proper(f(X)) -> f(proper(X)) proper(a) -> ok(a) proper(c(X)) -> c(proper(X)) proper(g(X)) -> g(proper(X)) f(ok(X)) -> ok(f(X)) c(ok(X)) -> ok(c(X)) g(ok(X)) -> ok(g(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The certificate found is represented by the following graph. The certificate consists of the following enumerated nodes: 1, 3, 4, 5, 6, 7, 8, 9, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 34, 37, 44, 45, 46, 47, 48, 49, 55, 56, 57, 62, 64, 65, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 88, 92, 94, 97, 98, 99, 100, 101, 102, 103 Node 1 is start node and node 3 is final node. Those nodes are connected through the following edges: * 1 to 4 labelled mark_1(0), ok_1(0)* 1 to 9 labelled f_1(0), g_1(0), c_1(0), top_1(0)* 1 to 8 labelled ok_1(0)* 1 to 24 labelled mark_1(1)* 1 to 25 labelled top_1(1)* 1 to 21 labelled ok_1(1)* 1 to 55 labelled top_1(2)* 1 to 97 labelled top_1(3)* 3 to 3 labelled #_1(0)* 4 to 5 labelled c_1(0)* 4 to 3 labelled f_1(0), g_1(0), c_1(0)* 4 to 17 labelled mark_1(1), ok_1(1)* 5 to 6 labelled f_1(0)* 6 to 7 labelled g_1(0)* 7 to 8 labelled f_1(0)* 8 to 3 labelled a(0)* 9 to 3 labelled active_1(0), proper_1(0)* 9 to 18 labelled mark_1(1)* 9 to 23 labelled f_1(1), g_1(1), c_1(1)* 9 to 22 labelled ok_1(1)* 9 to 26 labelled mark_1(2)* 9 to 27 labelled ok_1(2)* 17 to 3 labelled f_1(1), g_1(1), c_1(1)* 17 to 17 labelled mark_1(1), ok_1(1)* 18 to 19 labelled c_1(1)* 19 to 20 labelled f_1(1)* 20 to 21 labelled g_1(1)* 21 to 22 labelled f_1(1), g_1(1), c_1(1)* 21 to 27 labelled f_1(1), g_1(1), c_1(1)* 22 to 3 labelled a(1)* 23 to 3 labelled active_1(1), proper_1(1)* 23 to 18 labelled mark_1(1)* 23 to 23 labelled f_1(1), g_1(1), c_1(1)* 23 to 22 labelled ok_1(1)* 23 to 26 labelled mark_1(2)* 23 to 27 labelled ok_1(2)* 24 to 18 labelled f_1(1), g_1(1)* 24 to 26 labelled f_1(1), g_1(1)* 25 to 18 labelled proper_1(1)* 25 to 22 labelled active_1(1)* 25 to 28 labelled c_1(2)* 25 to 26 labelled proper_1(1)* 25 to 27 labelled active_1(1)* 25 to 37 labelled f_1(2), g_1(2)* 25 to 45 labelled mark_1(2)* 25 to 64 labelled mark_1(3)* 25 to 77 labelled ok_1(3)* 26 to 18 labelled f_1(2), g_1(2)* 26 to 26 labelled f_1(2), g_1(2)* 27 to 22 labelled f_1(2), g_1(2), c_1(2)* 27 to 27 labelled f_1(2), g_1(2), c_1(2)* 28 to 19 labelled proper_1(2)* 28 to 34 labelled f_1(2)* 28 to 74 labelled ok_1(3)* 34 to 20 labelled proper_1(2)* 34 to 44 labelled g_1(2)* 34 to 70 labelled ok_1(3)* 37 to 18 labelled proper_1(2)* 37 to 26 labelled proper_1(2)* 37 to 22 labelled active_1(2)* 37 to 27 labelled active_1(2)* 37 to 28 labelled c_1(2)* 37 to 56 labelled f_1(3), g_1(3)* 37 to 45 labelled mark_1(2)* 37 to 64 labelled mark_1(3)* 37 to 71 labelled mark_1(4)* 37 to 77 labelled ok_1(3)* 37 to 80 labelled ok_1(4)* 44 to 21 labelled proper_1(2)* 44 to 57 labelled f_1(2), g_1(2), c_1(2)* 44 to 68 labelled ok_1(3)* 44 to 70 labelled ok_1(3)* 45 to 46 labelled c_1(2)* 46 to 47 labelled f_1(2)* 47 to 48 labelled g_1(2)* 48 to 49 labelled f_1(2)* 49 to 3 labelled a(2)* 55 to 45 labelled proper_1(2)* 55 to 65 labelled c_1(3)* 55 to 64 labelled proper_1(2)* 55 to 72 labelled f_1(3), g_1(3)* 55 to 77 labelled active_1(2)* 55 to 94 labelled ok_1(4)* 56 to 18 labelled proper_1(3)* 56 to 26 labelled proper_1(3)* 56 to 22 labelled active_1(3)* 56 to 27 labelled active_1(3)* 56 to 28 labelled c_1(2)* 56 to 56 labelled f_1(3), g_1(3)* 56 to 45 labelled mark_1(2)* 56 to 64 labelled mark_1(3)* 56 to 71 labelled mark_1(4)* 56 to 77 labelled ok_1(3)* 56 to 80 labelled ok_1(4)* 57 to 22 labelled proper_1(2)* 57 to 27 labelled proper_1(2)* 57 to 62 labelled f_1(3), g_1(3), c_1(3)* 57 to 49 labelled ok_1(2)* 57 to 68 labelled ok_1(3)* 57 to 75 labelled ok_1(4)* 62 to 22 labelled proper_1(3)* 62 to 27 labelled proper_1(3)* 62 to 62 labelled f_1(3), g_1(3), c_1(3)* 62 to 49 labelled ok_1(2)* 62 to 68 labelled ok_1(3)* 62 to 75 labelled ok_1(4)* 64 to 45 labelled f_1(3), g_1(3)* 64 to 64 labelled f_1(3), g_1(3)* 64 to 71 labelled f_1(3), g_1(3)* 65 to 46 labelled proper_1(3)* 65 to 69 labelled f_1(3)* 65 to 92 labelled ok_1(4)* 68 to 49 labelled f_1(3), g_1(3), c_1(3)* 68 to 68 labelled f_1(3), c_1(3)* 68 to 75 labelled f_1(3), g_1(3), c_1(3)* 69 to 47 labelled proper_1(3)* 69 to 73 labelled g_1(3)* 69 to 88 labelled ok_1(4)* 70 to 68 labelled g_1(3)* 70 to 70 labelled g_1(3)* 71 to 64 labelled f_1(4), g_1(4)* 71 to 71 labelled f_1(4), g_1(4)* 72 to 45 labelled proper_1(3)* 72 to 65 labelled c_1(3)* 72 to 64 labelled proper_1(3)* 72 to 78 labelled f_1(4), g_1(4)* 72 to 71 labelled proper_1(3)* 72 to 77 labelled active_1(3)* 72 to 80 labelled active_1(3)* 72 to 94 labelled ok_1(4)* 72 to 98 labelled ok_1(5)* 73 to 48 labelled proper_1(3)* 73 to 76 labelled f_1(3)* 73 to 82 labelled ok_1(4)* 74 to 70 labelled f_1(3)* 75 to 68 labelled f_1(4), g_1(4), c_1(4)* 75 to 75 labelled f_1(4), g_1(4), c_1(4)* 76 to 49 labelled proper_1(3)* 76 to 79 labelled ok_1(3)* 77 to 74 labelled c_1(3)* 77 to 77 labelled f_1(3), g_1(3)* 77 to 80 labelled f_1(3), g_1(3)* 78 to 45 labelled proper_1(4)* 78 to 64 labelled proper_1(4)* 78 to 71 labelled proper_1(4)* 78 to 65 labelled c_1(3)* 78 to 78 labelled f_1(4), g_1(4)* 78 to 81 labelled f_1(5), g_1(5)* 78 to 77 labelled active_1(4)* 78 to 80 labelled active_1(4)* 78 to 94 labelled ok_1(4)* 78 to 98 labelled ok_1(5)* 78 to 99 labelled ok_1(6)* 79 to 3 labelled a(3)* 80 to 77 labelled f_1(4), g_1(4)* 80 to 80 labelled f_1(4), g_1(4)* 81 to 64 labelled proper_1(5)* 81 to 71 labelled proper_1(5)* 81 to 78 labelled f_1(4), g_1(4)* 81 to 81 labelled f_1(5), g_1(5)* 81 to 77 labelled active_1(5)* 81 to 80 labelled active_1(5)* 81 to 98 labelled ok_1(5)* 81 to 99 labelled ok_1(6)* 82 to 79 labelled f_1(4)* 88 to 82 labelled g_1(4)* 92 to 88 labelled f_1(4)* 94 to 92 labelled c_1(4)* 94 to 94 labelled f_1(4), g_1(4)* 94 to 98 labelled f_1(4), g_1(4)* 97 to 94 labelled active_1(3)* 97 to 100 labelled f_1(4), g_1(4)* 98 to 94 labelled f_1(5), g_1(5)* 98 to 98 labelled f_1(5), g_1(5)* 98 to 99 labelled f_1(5), g_1(5)* 99 to 98 labelled f_1(6), g_1(6)* 99 to 99 labelled f_1(6), g_1(6)* 100 to 94 labelled active_1(4)* 100 to 101 labelled f_1(5), g_1(5)* 100 to 98 labelled active_1(4)* 101 to 94 labelled active_1(5)* 101 to 98 labelled active_1(5)* 101 to 101 labelled f_1(5), g_1(5)* 101 to 102 labelled f_1(6), g_1(6)* 101 to 99 labelled active_1(5)* 102 to 94 labelled active_1(6)* 102 to 98 labelled active_1(6)* 102 to 99 labelled active_1(6)* 102 to 101 labelled f_1(5), g_1(5)* 102 to 102 labelled f_1(6), g_1(6)* 102 to 103 labelled f_1(7), g_1(7)* 103 to 98 labelled active_1(7)* 103 to 99 labelled active_1(7)* 103 to 102 labelled f_1(6), g_1(6)* 103 to 103 labelled f_1(7), g_1(7) ---------------------------------------- (2) YES