/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) RFCMatchBoundsTRSProof [EQUIVALENT, 40 ms] (4) 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) QTRS Reverse (EQUIVALENT) 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(c(mark(x)))))) f(active(X)) -> active(f(X)) g(active(X)) -> active(g(X)) mark(f(X)) -> f(mark(X)) mark(g(X)) -> g(mark(X)) f(proper(X)) -> proper(f(X)) a'(proper(x)) -> a'(ok(x)) c(proper(X)) -> proper(c(X)) g(proper(X)) -> proper(g(X)) ok(f(X)) -> f(ok(X)) ok(c(X)) -> c(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(c(mark(x)))))) f(active(X)) -> active(f(X)) g(active(X)) -> active(g(X)) mark(f(X)) -> f(mark(X)) mark(g(X)) -> g(mark(X)) f(proper(X)) -> proper(f(X)) a'(proper(x)) -> a'(ok(x)) c(proper(X)) -> proper(c(X)) g(proper(X)) -> proper(g(X)) ok(f(X)) -> f(ok(X)) ok(c(X)) -> c(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: 18, 19, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 67, 68, 72, 73, 74, 78, 79, 80, 81, 82, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99 Node 18 is start node and node 19 is final node. Those nodes are connected through the following edges: * 18 to 34 labelled a'_1(0), f_1(0), c_1(0), g_1(0)* 18 to 39 labelled active_1(0), proper_1(0)* 18 to 38 labelled f_1(0), g_1(0)* 18 to 44 labelled active_1(1), proper_1(1)* 18 to 51 labelled a'_1(1)* 18 to 80 labelled a'_1(2)* 19 to 19 labelled #_1(0)* 34 to 35 labelled f_1(0)* 34 to 19 labelled ok_1(0)* 34 to 40 labelled f_1(1), c_1(1), g_1(1)* 34 to 41 labelled active_1(1)* 34 to 49 labelled active_1(2)* 34 to 58 labelled proper_1(1)* 35 to 36 labelled g_1(0)* 35 to 56 labelled proper_1(1)* 36 to 37 labelled f_1(0)* 36 to 50 labelled proper_1(1)* 37 to 38 labelled c_1(0)* 37 to 45 labelled proper_1(1)* 38 to 19 labelled mark_1(0)* 38 to 42 labelled f_1(1), g_1(1)* 38 to 41 labelled proper_1(1)* 38 to 49 labelled proper_1(2)* 39 to 19 labelled f_1(0), g_1(0), c_1(0), top_1(0)* 39 to 43 labelled active_1(1), proper_1(1)* 40 to 19 labelled ok_1(1)* 40 to 40 labelled f_1(1), c_1(1), g_1(1)* 40 to 41 labelled active_1(1)* 40 to 49 labelled active_1(2)* 41 to 19 labelled top_1(1)* 42 to 19 labelled mark_1(1)* 42 to 42 labelled f_1(1), g_1(1)* 42 to 41 labelled proper_1(1)* 42 to 49 labelled proper_1(2)* 43 to 19 labelled f_1(1), g_1(1), c_1(1)* 43 to 43 labelled active_1(1), proper_1(1)* 44 to 41 labelled f_1(1), g_1(1)* 44 to 49 labelled f_1(1), g_1(1)* 44 to 58 labelled f_1(1), c_1(1), g_1(1)* 45 to 41 labelled c_1(1)* 45 to 49 labelled c_1(1)* 49 to 41 labelled f_1(2), g_1(2)* 49 to 49 labelled f_1(2), g_1(2)* 50 to 45 labelled f_1(1)* 51 to 52 labelled f_1(1)* 51 to 58 labelled ok_1(1)* 51 to 72 labelled f_1(2)* 51 to 78 labelled proper_1(2)* 52 to 53 labelled g_1(1)* 52 to 73 labelled proper_1(2)* 53 to 54 labelled f_1(1)* 53 to 67 labelled proper_1(2)* 54 to 55 labelled c_1(1)* 54 to 59 labelled proper_1(2)* 55 to 41 labelled mark_1(1)* 55 to 57 labelled proper_1(2)* 55 to 49 labelled mark_1(1)* 55 to 60 labelled f_1(2), g_1(2)* 55 to 74 labelled proper_1(3)* 56 to 50 labelled g_1(1)* 57 to 19 labelled top_1(2)* 58 to 56 labelled f_1(1)* 59 to 57 labelled c_1(2)* 59 to 74 labelled c_1(2)* 60 to 41 labelled mark_1(2)* 60 to 49 labelled mark_1(2)* 60 to 57 labelled proper_1(2)* 60 to 68 labelled f_1(3), g_1(3)* 60 to 74 labelled proper_1(3)* 60 to 81 labelled proper_1(4)* 67 to 59 labelled f_1(2)* 68 to 41 labelled mark_1(3)* 68 to 49 labelled mark_1(3)* 68 to 57 labelled proper_1(2)* 68 to 68 labelled f_1(3), g_1(3)* 68 to 74 labelled proper_1(3)* 68 to 81 labelled proper_1(4)* 72 to 56 labelled ok_1(2)* 72 to 79 labelled g_1(2)* 73 to 67 labelled g_1(2)* 74 to 57 labelled f_1(3), g_1(3)* 74 to 74 labelled f_1(3), g_1(3)* 74 to 81 labelled f_1(3), g_1(3)* 78 to 73 labelled f_1(2)* 79 to 50 labelled ok_1(2)* 79 to 82 labelled f_1(2)* 80 to 78 labelled ok_1(2)* 80 to 87 labelled f_1(3)* 81 to 74 labelled f_1(4), g_1(4)* 81 to 81 labelled f_1(4), g_1(4)* 82 to 45 labelled ok_1(2)* 82 to 88 labelled c_1(2)* 87 to 73 labelled ok_1(3)* 87 to 90 labelled g_1(3)* 88 to 41 labelled ok_1(2)* 88 to 49 labelled ok_1(2)* 88 to 57 labelled active_1(2)* 88 to 89 labelled f_1(3), g_1(3)* 88 to 74 labelled active_1(3)* 88 to 81 labelled active_1(4)* 89 to 41 labelled ok_1(3)* 89 to 49 labelled ok_1(3)* 89 to 57 labelled active_1(2)* 89 to 89 labelled f_1(3), g_1(3)* 89 to 74 labelled active_1(3)* 89 to 81 labelled active_1(4)* 90 to 67 labelled ok_1(3)* 90 to 91 labelled f_1(3)* 91 to 59 labelled ok_1(3)* 91 to 92 labelled c_1(3)* 92 to 57 labelled ok_1(3)* 92 to 74 labelled ok_1(3)* 92 to 93 labelled active_1(3)* 92 to 94 labelled f_1(4), g_1(4)* 92 to 97 labelled active_1(4)* 92 to 98 labelled active_1(5)* 93 to 19 labelled top_1(3)* 94 to 57 labelled ok_1(4)* 94 to 74 labelled ok_1(4)* 94 to 81 labelled ok_1(4)* 94 to 93 labelled active_1(3)* 94 to 94 labelled f_1(4), g_1(4)* 94 to 95 labelled f_1(5), g_1(5)* 94 to 97 labelled active_1(4)* 94 to 98 labelled active_1(5)* 94 to 99 labelled active_1(6)* 95 to 74 labelled ok_1(5)* 95 to 81 labelled ok_1(5)* 95 to 94 labelled f_1(4), g_1(4)* 95 to 95 labelled f_1(5), g_1(5)* 95 to 97 labelled active_1(4)* 95 to 98 labelled active_1(5)* 95 to 99 labelled active_1(6)* 97 to 93 labelled f_1(4), g_1(4)* 98 to 97 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) ---------------------------------------- (4) YES