/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) RFCMatchBoundsTRSProof [EQUIVALENT, 37 ms] (2) YES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)) The set Q consists of the following terms: active(f(x0)) active(g(x0)) f(mark(x0)) g(mark(x0)) proper(f(x0)) proper(a) proper(c(x0)) proper(g(x0)) f(ok(x0)) c(ok(x0)) g(ok(x0)) top(mark(x0)) top(ok(x0)) ---------------------------------------- (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, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 51, 52, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 89, 90, 93, 94, 95, 105, 107 Node 1 is start node and node 2 is final node. Those nodes are connected through the following edges: * 1 to 3 labelled mark_1(0), ok_1(0)* 1 to 8 labelled f_1(0), g_1(0), c_1(0), top_1(0)* 1 to 7 labelled ok_1(0)* 1 to 16 labelled mark_1(1)* 1 to 17 labelled top_1(1)* 1 to 13 labelled ok_1(1)* 1 to 29 labelled top_1(2)* 1 to 89 labelled top_1(3)* 2 to 2 labelled #_1(0)* 3 to 4 labelled c_1(0)* 3 to 2 labelled f_1(0), g_1(0), c_1(0)* 3 to 9 labelled mark_1(1), ok_1(1)* 4 to 5 labelled f_1(0)* 5 to 6 labelled g_1(0)* 6 to 7 labelled f_1(0)* 7 to 2 labelled a(0)* 8 to 2 labelled active_1(0), proper_1(0)* 8 to 10 labelled mark_1(1)* 8 to 15 labelled f_1(1), g_1(1), c_1(1)* 8 to 14 labelled ok_1(1)* 8 to 18 labelled mark_1(2)* 8 to 19 labelled ok_1(2)* 9 to 2 labelled f_1(1), g_1(1), c_1(1)* 9 to 9 labelled mark_1(1), ok_1(1)* 10 to 11 labelled c_1(1)* 11 to 12 labelled f_1(1)* 12 to 13 labelled g_1(1)* 13 to 14 labelled f_1(1), g_1(1), c_1(1)* 13 to 19 labelled f_1(1), g_1(1), c_1(1)* 14 to 2 labelled a(1)* 15 to 2 labelled active_1(1), proper_1(1)* 15 to 10 labelled mark_1(1)* 15 to 15 labelled f_1(1), g_1(1), c_1(1)* 15 to 14 labelled ok_1(1)* 15 to 18 labelled mark_1(2)* 15 to 19 labelled ok_1(2)* 16 to 10 labelled f_1(1), g_1(1)* 16 to 18 labelled f_1(1), g_1(1)* 17 to 10 labelled proper_1(1)* 17 to 14 labelled active_1(1)* 17 to 20 labelled c_1(2)* 17 to 18 labelled proper_1(1)* 17 to 19 labelled active_1(1)* 17 to 22 labelled f_1(2), g_1(2)* 17 to 24 labelled mark_1(2)* 17 to 33 labelled mark_1(3)* 17 to 70 labelled ok_1(3)* 18 to 10 labelled f_1(2), g_1(2)* 18 to 18 labelled f_1(2), g_1(2)* 19 to 14 labelled f_1(2), g_1(2), c_1(2)* 19 to 19 labelled f_1(2), g_1(2), c_1(2)* 20 to 11 labelled proper_1(2)* 20 to 21 labelled f_1(2)* 20 to 67 labelled ok_1(3)* 21 to 12 labelled proper_1(2)* 21 to 23 labelled g_1(2)* 21 to 51 labelled ok_1(3)* 22 to 10 labelled proper_1(2)* 22 to 18 labelled proper_1(2)* 22 to 14 labelled active_1(2)* 22 to 19 labelled active_1(2)* 22 to 20 labelled c_1(2)* 22 to 30 labelled f_1(3), g_1(3)* 22 to 24 labelled mark_1(2)* 22 to 33 labelled mark_1(3)* 22 to 52 labelled mark_1(4)* 22 to 70 labelled ok_1(3)* 22 to 73 labelled ok_1(4)* 23 to 13 labelled proper_1(2)* 23 to 31 labelled f_1(2), g_1(2), c_1(2)* 23 to 35 labelled ok_1(3)* 23 to 51 labelled ok_1(3)* 24 to 25 labelled c_1(2)* 25 to 26 labelled f_1(2)* 26 to 27 labelled g_1(2)* 27 to 28 labelled f_1(2)* 28 to 2 labelled a(2)* 29 to 24 labelled proper_1(2)* 29 to 34 labelled c_1(3)* 29 to 33 labelled proper_1(2)* 29 to 65 labelled f_1(3), g_1(3)* 29 to 70 labelled active_1(2)* 29 to 78 labelled ok_1(4)* 30 to 10 labelled proper_1(3)* 30 to 18 labelled proper_1(3)* 30 to 14 labelled active_1(3)* 30 to 19 labelled active_1(3)* 30 to 20 labelled c_1(2)* 30 to 30 labelled f_1(3), g_1(3)* 30 to 24 labelled mark_1(2)* 30 to 33 labelled mark_1(3)* 30 to 52 labelled mark_1(4)* 30 to 70 labelled ok_1(3)* 30 to 73 labelled ok_1(4)* 31 to 14 labelled proper_1(2)* 31 to 19 labelled proper_1(2)* 31 to 32 labelled f_1(3), g_1(3), c_1(3)* 31 to 28 labelled ok_1(2)* 31 to 35 labelled ok_1(3)* 31 to 68 labelled ok_1(4)* 32 to 14 labelled proper_1(3)* 32 to 19 labelled proper_1(3)* 32 to 32 labelled f_1(3), g_1(3), c_1(3)* 32 to 28 labelled ok_1(2)* 32 to 35 labelled ok_1(3)* 32 to 68 labelled ok_1(4)* 33 to 24 labelled f_1(3), g_1(3)* 33 to 33 labelled f_1(3), g_1(3)* 33 to 52 labelled f_1(3), g_1(3)* 34 to 25 labelled proper_1(3)* 34 to 36 labelled f_1(3)* 34 to 77 labelled ok_1(4)* 35 to 28 labelled f_1(3), g_1(3), c_1(3)* 35 to 35 labelled f_1(3), c_1(3)* 35 to 68 labelled f_1(3), g_1(3), c_1(3)* 36 to 26 labelled proper_1(3)* 36 to 66 labelled g_1(3)* 36 to 76 labelled ok_1(4)* 51 to 35 labelled g_1(3)* 51 to 51 labelled g_1(3)* 52 to 33 labelled f_1(4), g_1(4)* 52 to 52 labelled f_1(4), g_1(4)* 65 to 24 labelled proper_1(3)* 65 to 34 labelled c_1(3)* 65 to 33 labelled proper_1(3)* 65 to 71 labelled f_1(4), g_1(4)* 65 to 52 labelled proper_1(3)* 65 to 70 labelled active_1(3)* 65 to 73 labelled active_1(3)* 65 to 78 labelled ok_1(4)* 65 to 90 labelled ok_1(5)* 66 to 27 labelled proper_1(3)* 66 to 69 labelled f_1(3)* 66 to 75 labelled ok_1(4)* 67 to 51 labelled f_1(3)* 68 to 35 labelled f_1(4), g_1(4), c_1(4)* 68 to 68 labelled f_1(4), g_1(4), c_1(4)* 69 to 28 labelled proper_1(3)* 69 to 72 labelled ok_1(3)* 70 to 67 labelled c_1(3)* 70 to 70 labelled f_1(3), g_1(3)* 70 to 73 labelled f_1(3), g_1(3)* 71 to 24 labelled proper_1(4)* 71 to 33 labelled proper_1(4)* 71 to 52 labelled proper_1(4)* 71 to 34 labelled c_1(3)* 71 to 71 labelled f_1(4), g_1(4)* 71 to 74 labelled f_1(5), g_1(5)* 71 to 70 labelled active_1(4)* 71 to 73 labelled active_1(4)* 71 to 78 labelled ok_1(4)* 71 to 90 labelled ok_1(5)* 71 to 93 labelled ok_1(6)* 72 to 2 labelled a(3)* 73 to 70 labelled f_1(4), g_1(4)* 73 to 73 labelled f_1(4), g_1(4)* 74 to 33 labelled proper_1(5)* 74 to 52 labelled proper_1(5)* 74 to 71 labelled f_1(4), g_1(4)* 74 to 74 labelled f_1(5), g_1(5)* 74 to 70 labelled active_1(5)* 74 to 73 labelled active_1(5)* 74 to 90 labelled ok_1(5)* 74 to 93 labelled ok_1(6)* 75 to 72 labelled f_1(4)* 76 to 75 labelled g_1(4)* 77 to 76 labelled f_1(4)* 78 to 77 labelled c_1(4)* 78 to 78 labelled f_1(4), g_1(4)* 78 to 90 labelled f_1(4), g_1(4)* 89 to 78 labelled active_1(3)* 89 to 94 labelled f_1(4), g_1(4)* 90 to 78 labelled f_1(5), g_1(5)* 90 to 90 labelled f_1(5), g_1(5)* 90 to 93 labelled f_1(5), g_1(5)* 93 to 90 labelled f_1(6), g_1(6)* 93 to 93 labelled f_1(6), g_1(6)* 94 to 78 labelled active_1(4)* 94 to 95 labelled f_1(5), g_1(5)* 94 to 90 labelled active_1(4)* 95 to 78 labelled active_1(5)* 95 to 90 labelled active_1(5)* 95 to 95 labelled f_1(5), g_1(5)* 95 to 105 labelled f_1(6), g_1(6)* 95 to 93 labelled active_1(5)* 105 to 78 labelled active_1(6)* 105 to 90 labelled active_1(6)* 105 to 93 labelled active_1(6)* 105 to 95 labelled f_1(5), g_1(5)* 105 to 105 labelled f_1(6), g_1(6)* 105 to 107 labelled f_1(7), g_1(7)* 107 to 90 labelled active_1(7)* 107 to 93 labelled active_1(7)* 107 to 105 labelled f_1(6), g_1(6)* 107 to 107 labelled f_1(7), g_1(7) ---------------------------------------- (2) YES