/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [SOUND, 0 ms] (2) QTRS (3) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms] (4) YES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(g(X)) -> g(active(X)) g(mark(X)) -> mark(g(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 set Q consists of the following terms: active(f(f(a))) active(g(x0)) g(mark(x0)) proper(f(x0)) proper(a) proper(g(x0)) f(ok(x0)) g(ok(x0)) top(mark(x0)) top(ok(x0)) ---------------------------------------- (1) QTRS Reverse (SOUND) We applied the QTRS Reverse Processor [REVERSE]. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a'(f(f(active(x)))) -> a'(f(g(f(mark(x))))) g(active(X)) -> active(g(X)) mark(g(X)) -> g(mark(X)) f(proper(X)) -> proper(f(X)) a'(proper(x)) -> a'(ok(x)) g(proper(X)) -> proper(g(X)) ok(f(X)) -> f(ok(X)) ok(g(X)) -> g(ok(X)) mark(top(X)) -> proper(top(X)) ok(top(X)) -> active(top(X)) Q is empty. ---------------------------------------- (3) RFCMatchBoundsTRSProof (EQUIVALENT) Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 6. This implies Q-termination of R. The following rules were used to construct the certificate: a'(f(f(active(x)))) -> a'(f(g(f(mark(x))))) g(active(X)) -> active(g(X)) mark(g(X)) -> g(mark(X)) f(proper(X)) -> proper(f(X)) a'(proper(x)) -> a'(ok(x)) g(proper(X)) -> proper(g(X)) ok(f(X)) -> f(ok(X)) ok(g(X)) -> g(ok(X)) mark(top(X)) -> proper(top(X)) ok(top(X)) -> active(top(X)) The certificate found is represented by the following graph. The certificate consists of the following enumerated nodes: 27, 28, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 85, 86, 87, 93 Node 27 is start node and node 28 is final node. Those nodes are connected through the following edges: * 27 to 37 labelled a'_1(0), f_1(0), g_1(0)* 27 to 41 labelled active_1(0), proper_1(0)* 27 to 40 labelled g_1(0)* 27 to 42 labelled proper_1(0)* 27 to 48 labelled active_1(1), proper_1(1)* 27 to 52 labelled a'_1(1)* 27 to 74 labelled a'_1(2)* 28 to 28 labelled #_1(0)* 37 to 38 labelled f_1(0)* 37 to 28 labelled ok_1(0)* 37 to 43 labelled f_1(1), g_1(1)* 37 to 44 labelled active_1(1)* 37 to 50 labelled active_1(2)* 37 to 56 labelled proper_1(1)* 38 to 39 labelled g_1(0)* 38 to 51 labelled proper_1(1)* 39 to 40 labelled f_1(0)* 39 to 49 labelled proper_1(1)* 40 to 28 labelled mark_1(0)* 40 to 45 labelled g_1(1)* 40 to 44 labelled proper_1(1)* 40 to 50 labelled proper_1(2)* 41 to 28 labelled g_1(0), top_1(0)* 41 to 46 labelled active_1(1), proper_1(1)* 42 to 28 labelled f_1(0)* 42 to 47 labelled proper_1(1)* 43 to 28 labelled ok_1(1)* 43 to 43 labelled f_1(1), g_1(1)* 43 to 44 labelled active_1(1)* 43 to 50 labelled active_1(2)* 44 to 28 labelled top_1(1)* 45 to 28 labelled mark_1(1)* 45 to 45 labelled g_1(1)* 45 to 44 labelled proper_1(1)* 45 to 50 labelled proper_1(2)* 46 to 28 labelled g_1(1)* 46 to 46 labelled active_1(1), proper_1(1)* 47 to 28 labelled f_1(1)* 47 to 47 labelled proper_1(1)* 48 to 44 labelled g_1(1)* 48 to 50 labelled g_1(1)* 48 to 56 labelled f_1(1), g_1(1)* 49 to 44 labelled f_1(1)* 49 to 50 labelled f_1(1)* 50 to 44 labelled g_1(2)* 50 to 50 labelled g_1(2)* 51 to 49 labelled g_1(1)* 52 to 53 labelled f_1(1)* 52 to 56 labelled ok_1(1)* 52 to 68 labelled f_1(2)* 52 to 71 labelled proper_1(2)* 53 to 54 labelled g_1(1)* 53 to 69 labelled proper_1(2)* 54 to 55 labelled f_1(1)* 54 to 66 labelled proper_1(2)* 55 to 44 labelled mark_1(1)* 55 to 57 labelled proper_1(2)* 55 to 50 labelled mark_1(1)* 55 to 67 labelled g_1(2)* 55 to 72 labelled proper_1(3)* 56 to 51 labelled f_1(1)* 57 to 28 labelled top_1(2)* 66 to 57 labelled f_1(2)* 66 to 72 labelled f_1(2)* 67 to 44 labelled mark_1(2)* 67 to 50 labelled mark_1(2)* 67 to 57 labelled proper_1(2)* 67 to 70 labelled g_1(3)* 67 to 72 labelled proper_1(3)* 67 to 77 labelled proper_1(4)* 68 to 51 labelled ok_1(2)* 68 to 73 labelled g_1(2)* 69 to 66 labelled g_1(2)* 70 to 44 labelled mark_1(3)* 70 to 50 labelled mark_1(3)* 70 to 57 labelled proper_1(2)* 70 to 70 labelled g_1(3)* 70 to 72 labelled proper_1(3)* 70 to 77 labelled proper_1(4)* 71 to 69 labelled f_1(2)* 72 to 57 labelled g_1(3)* 72 to 72 labelled g_1(3)* 72 to 77 labelled g_1(3)* 73 to 49 labelled ok_1(2)* 73 to 75 labelled f_1(2)* 74 to 71 labelled ok_1(2)* 74 to 78 labelled f_1(3)* 75 to 44 labelled ok_1(2)* 75 to 50 labelled ok_1(2)* 75 to 57 labelled active_1(2)* 75 to 76 labelled g_1(3)* 75 to 72 labelled active_1(3)* 75 to 77 labelled active_1(4)* 76 to 44 labelled ok_1(3)* 76 to 50 labelled ok_1(3)* 76 to 57 labelled active_1(2)* 76 to 76 labelled g_1(3)* 76 to 72 labelled active_1(3)* 76 to 77 labelled active_1(4)* 77 to 72 labelled g_1(4)* 77 to 77 labelled g_1(4)* 78 to 69 labelled ok_1(3)* 78 to 79 labelled g_1(3)* 79 to 66 labelled ok_1(3)* 79 to 80 labelled f_1(3)* 80 to 57 labelled ok_1(3)* 80 to 72 labelled ok_1(3)* 80 to 81 labelled active_1(3)* 80 to 82 labelled g_1(4)* 80 to 86 labelled active_1(4)* 80 to 87 labelled active_1(5)* 81 to 28 labelled top_1(3)* 82 to 57 labelled ok_1(4)* 82 to 72 labelled ok_1(4)* 82 to 77 labelled ok_1(4)* 82 to 81 labelled active_1(3)* 82 to 82 labelled g_1(4)* 82 to 85 labelled g_1(5)* 82 to 86 labelled active_1(4)* 82 to 87 labelled active_1(5)* 82 to 93 labelled active_1(6)* 85 to 72 labelled ok_1(5)* 85 to 77 labelled ok_1(5)* 85 to 82 labelled g_1(4)* 85 to 85 labelled g_1(5)* 85 to 86 labelled active_1(4)* 85 to 87 labelled active_1(5)* 85 to 93 labelled active_1(6)* 86 to 81 labelled g_1(4)* 87 to 86 labelled g_1(5)* 87 to 87 labelled g_1(5)* 87 to 93 labelled g_1(5)* 93 to 87 labelled g_1(6)* 93 to 93 labelled g_1(6) ---------------------------------------- (4) YES