/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) RFCMatchBoundsTRSProof [EQUIVALENT, 31 ms] (2) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: active(f(f(a))) -> mark(f(g(f(a)))) active(f(X)) -> f(active(X)) f(mark(X)) -> mark(f(X)) proper(f(X)) -> f(proper(X)) proper(a) -> ok(a) proper(g(X)) -> g(proper(X)) f(ok(X)) -> ok(f(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(f(g(f(a)))) active(f(X)) -> f(active(X)) f(mark(X)) -> mark(f(X)) proper(f(X)) -> f(proper(X)) proper(a) -> ok(a) proper(g(X)) -> g(proper(X)) f(ok(X)) -> ok(f(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, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50 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 7 labelled f_1(0), g_1(0), top_1(0)* 1 to 6 labelled ok_1(0)* 1 to 14 labelled mark_1(1)* 1 to 15 labelled top_1(1)* 1 to 11 labelled ok_1(1)* 1 to 25 labelled top_1(2)* 1 to 44 labelled top_1(3)* 2 to 2 labelled #_1(0)* 3 to 4 labelled f_1(0)* 3 to 2 labelled f_1(0), g_1(0)* 3 to 8 labelled mark_1(1), ok_1(1)* 4 to 5 labelled g_1(0)* 5 to 6 labelled f_1(0)* 6 to 2 labelled a(0)* 7 to 2 labelled active_1(0), proper_1(0)* 7 to 9 labelled mark_1(1)* 7 to 13 labelled f_1(1), g_1(1)* 7 to 12 labelled ok_1(1)* 7 to 16 labelled mark_1(2)* 7 to 17 labelled ok_1(2)* 8 to 2 labelled f_1(1), g_1(1)* 8 to 8 labelled mark_1(1), ok_1(1)* 9 to 10 labelled f_1(1)* 10 to 11 labelled g_1(1)* 11 to 12 labelled f_1(1), g_1(1)* 11 to 17 labelled f_1(1), g_1(1)* 12 to 2 labelled a(1)* 13 to 2 labelled active_1(1), proper_1(1)* 13 to 9 labelled mark_1(1)* 13 to 13 labelled f_1(1), g_1(1)* 13 to 12 labelled ok_1(1)* 13 to 16 labelled mark_1(2)* 13 to 17 labelled ok_1(2)* 14 to 9 labelled f_1(1)* 14 to 16 labelled f_1(1)* 15 to 9 labelled proper_1(1)* 15 to 12 labelled active_1(1)* 15 to 18 labelled f_1(2)* 15 to 16 labelled proper_1(1)* 15 to 17 labelled active_1(1)* 15 to 21 labelled mark_1(2)* 15 to 28 labelled mark_1(3)* 15 to 33 labelled ok_1(3)* 16 to 9 labelled f_1(2)* 16 to 16 labelled f_1(2)* 17 to 12 labelled f_1(2), g_1(2)* 17 to 17 labelled f_1(2), g_1(2)* 18 to 10 labelled proper_1(2)* 18 to 19 labelled g_1(2)* 18 to 9 labelled proper_1(2)* 18 to 16 labelled proper_1(2)* 18 to 12 labelled active_1(2)* 18 to 17 labelled active_1(2)* 18 to 18 labelled f_1(2)* 18 to 26 labelled f_1(3)* 18 to 21 labelled mark_1(2)* 18 to 28 labelled mark_1(3)* 18 to 31 labelled ok_1(3)* 18 to 33 labelled ok_1(3)* 18 to 34 labelled mark_1(4)* 18 to 37 labelled ok_1(4)* 19 to 11 labelled proper_1(2)* 19 to 20 labelled f_1(2), g_1(2)* 19 to 29 labelled ok_1(3)* 19 to 31 labelled ok_1(3)* 20 to 12 labelled proper_1(2)* 20 to 17 labelled proper_1(2)* 20 to 27 labelled f_1(3), g_1(3)* 20 to 24 labelled ok_1(2)* 20 to 29 labelled ok_1(3)* 20 to 35 labelled ok_1(4)* 21 to 22 labelled f_1(2)* 22 to 23 labelled g_1(2)* 23 to 24 labelled f_1(2)* 24 to 2 labelled a(2)* 25 to 21 labelled proper_1(2)* 25 to 30 labelled f_1(3)* 25 to 28 labelled proper_1(2)* 25 to 33 labelled active_1(2)* 25 to 43 labelled ok_1(4)* 26 to 9 labelled proper_1(3)* 26 to 16 labelled proper_1(3)* 26 to 12 labelled active_1(3)* 26 to 17 labelled active_1(3)* 26 to 18 labelled f_1(2)* 26 to 26 labelled f_1(3)* 26 to 21 labelled mark_1(2)* 26 to 28 labelled mark_1(3)* 26 to 33 labelled ok_1(3)* 26 to 34 labelled mark_1(4)* 26 to 37 labelled ok_1(4)* 27 to 12 labelled proper_1(3)* 27 to 17 labelled proper_1(3)* 27 to 27 labelled f_1(3), g_1(3)* 27 to 24 labelled ok_1(2)* 27 to 29 labelled ok_1(3)* 27 to 35 labelled ok_1(4)* 28 to 21 labelled f_1(3)* 28 to 28 labelled f_1(3)* 28 to 34 labelled f_1(3)* 29 to 24 labelled f_1(3), g_1(3)* 29 to 29 labelled f_1(3)* 29 to 35 labelled f_1(3), g_1(3)* 30 to 22 labelled proper_1(3)* 30 to 32 labelled g_1(3)* 30 to 21 labelled proper_1(3)* 30 to 28 labelled proper_1(3)* 30 to 30 labelled f_1(3)* 30 to 38 labelled f_1(4)* 30 to 34 labelled proper_1(3)* 30 to 31 labelled active_1(3)* 30 to 33 labelled active_1(3)* 30 to 37 labelled active_1(3)* 30 to 41 labelled ok_1(4)* 30 to 43 labelled ok_1(4)* 30 to 45 labelled ok_1(5)* 31 to 29 labelled g_1(3)* 31 to 31 labelled g_1(3)* 32 to 23 labelled proper_1(3)* 32 to 36 labelled f_1(3)* 32 to 40 labelled ok_1(4)* 33 to 31 labelled f_1(3)* 33 to 33 labelled f_1(3)* 33 to 37 labelled f_1(3)* 34 to 28 labelled f_1(4)* 34 to 34 labelled f_1(4)* 35 to 29 labelled f_1(4), g_1(4)* 35 to 35 labelled f_1(4), g_1(4)* 36 to 24 labelled proper_1(3)* 36 to 39 labelled ok_1(3)* 37 to 33 labelled f_1(4)* 37 to 37 labelled f_1(4)* 38 to 21 labelled proper_1(4)* 38 to 28 labelled proper_1(4)* 38 to 30 labelled f_1(3)* 38 to 38 labelled f_1(4)* 38 to 34 labelled proper_1(4)* 38 to 31 labelled active_1(4)* 38 to 33 labelled active_1(4)* 38 to 37 labelled active_1(4)* 38 to 42 labelled f_1(5)* 38 to 43 labelled ok_1(4)* 38 to 45 labelled ok_1(5)* 38 to 46 labelled ok_1(6)* 39 to 2 labelled a(3)* 40 to 39 labelled f_1(4)* 41 to 40 labelled g_1(4)* 42 to 28 labelled proper_1(5)* 42 to 34 labelled proper_1(5)* 42 to 38 labelled f_1(4)* 42 to 42 labelled f_1(5)* 42 to 33 labelled active_1(5)* 42 to 37 labelled active_1(5)* 42 to 45 labelled ok_1(5)* 42 to 46 labelled ok_1(6)* 43 to 41 labelled f_1(4)* 43 to 43 labelled f_1(4)* 43 to 45 labelled f_1(4)* 44 to 43 labelled active_1(3)* 44 to 47 labelled f_1(4)* 45 to 43 labelled f_1(5)* 45 to 45 labelled f_1(5)* 45 to 46 labelled f_1(5)* 46 to 45 labelled f_1(6)* 46 to 46 labelled f_1(6)* 47 to 41 labelled active_1(4)* 47 to 43 labelled active_1(4)* 47 to 48 labelled f_1(5)* 47 to 45 labelled active_1(4)* 48 to 41 labelled active_1(5)* 48 to 43 labelled active_1(5)* 48 to 45 labelled active_1(5)* 48 to 48 labelled f_1(5)* 48 to 49 labelled f_1(6)* 48 to 46 labelled active_1(5)* 49 to 43 labelled active_1(6)* 49 to 45 labelled active_1(6)* 49 to 46 labelled active_1(6)* 49 to 48 labelled f_1(5)* 49 to 49 labelled f_1(6)* 49 to 50 labelled f_1(7)* 50 to 45 labelled active_1(7)* 50 to 46 labelled active_1(7)* 50 to 49 labelled f_1(6)* 50 to 50 labelled f_1(7) ---------------------------------------- (2) YES