/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [SOUND, 0 ms] (2) QTRS (3) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms] (4) YES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(X)) -> mark(g(h(f(X)))) active(f(X)) -> f(active(X)) active(h(X)) -> h(active(X)) f(mark(X)) -> mark(f(X)) h(mark(X)) -> mark(h(X)) proper(f(X)) -> f(proper(X)) proper(g(X)) -> g(proper(X)) proper(h(X)) -> h(proper(X)) f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) h(ok(X)) -> ok(h(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(h(x0)) f(mark(x0)) h(mark(x0)) proper(f(x0)) proper(g(x0)) proper(h(x0)) f(ok(x0)) g(ok(x0)) h(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: f(active(X)) -> f(h(g(mark(X)))) f(active(X)) -> active(f(X)) h(active(X)) -> active(h(X)) mark(f(X)) -> f(mark(X)) mark(h(X)) -> h(mark(X)) f(proper(X)) -> proper(f(X)) g(proper(X)) -> proper(g(X)) h(proper(X)) -> proper(h(X)) ok(f(X)) -> f(ok(X)) ok(g(X)) -> g(ok(X)) ok(h(X)) -> h(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 4. This implies Q-termination of R. The following rules were used to construct the certificate: f(active(X)) -> f(h(g(mark(X)))) f(active(X)) -> active(f(X)) h(active(X)) -> active(h(X)) mark(f(X)) -> f(mark(X)) mark(h(X)) -> h(mark(X)) f(proper(X)) -> proper(f(X)) g(proper(X)) -> proper(g(X)) h(proper(X)) -> proper(h(X)) ok(f(X)) -> f(ok(X)) ok(g(X)) -> g(ok(X)) ok(h(X)) -> h(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: 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 78, 79, 80, 81, 82, 83, 84, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 98, 99 Node 57 is start node and node 58 is final node. Those nodes are connected through the following edges: * 57 to 59 labelled f_1(0), g_1(0), h_1(0)* 57 to 62 labelled active_1(0), proper_1(0)* 57 to 61 labelled f_1(0), h_1(0)* 57 to 71 labelled f_1(1)* 57 to 74 labelled active_1(1), proper_1(1)* 57 to 93 labelled proper_1(2)* 58 to 58 labelled #_1(0)* 59 to 60 labelled h_1(0)* 59 to 58 labelled ok_1(0)* 59 to 64 labelled f_1(1), g_1(1), h_1(1)* 59 to 65 labelled active_1(1)* 59 to 78 labelled proper_1(1)* 59 to 79 labelled f_1(2)* 59 to 82 labelled active_1(2)* 59 to 96 labelled proper_1(3)* 59 to 98 labelled proper_1(2)* 60 to 61 labelled g_1(0)* 60 to 75 labelled proper_1(1)* 61 to 58 labelled mark_1(0)* 61 to 66 labelled f_1(1), h_1(1)* 61 to 65 labelled proper_1(1)* 61 to 82 labelled proper_1(2)* 62 to 58 labelled f_1(0), h_1(0), g_1(0), top_1(0)* 62 to 67 labelled f_1(1)* 62 to 70 labelled active_1(1), proper_1(1)* 62 to 88 labelled proper_1(2)* 64 to 58 labelled ok_1(1)* 64 to 64 labelled f_1(1), g_1(1), h_1(1)* 64 to 65 labelled active_1(1)* 64 to 79 labelled f_1(2)* 64 to 82 labelled active_1(2)* 64 to 96 labelled proper_1(3)* 64 to 98 labelled proper_1(2)* 65 to 58 labelled top_1(1)* 66 to 58 labelled mark_1(1)* 66 to 66 labelled f_1(1), h_1(1)* 66 to 65 labelled proper_1(1)* 66 to 82 labelled proper_1(2)* 67 to 68 labelled h_1(1)* 67 to 86 labelled proper_1(2)* 68 to 69 labelled g_1(1)* 68 to 83 labelled proper_1(2)* 69 to 58 labelled mark_1(1)* 69 to 66 labelled f_1(1), h_1(1)* 69 to 65 labelled proper_1(1)* 69 to 82 labelled proper_1(2)* 70 to 58 labelled f_1(1), h_1(1), g_1(1)* 70 to 67 labelled f_1(1)* 70 to 70 labelled active_1(1), proper_1(1)* 70 to 88 labelled proper_1(2)* 71 to 72 labelled h_1(1)* 71 to 89 labelled proper_1(2)* 72 to 73 labelled g_1(1)* 72 to 87 labelled proper_1(2)* 73 to 65 labelled mark_1(1)* 73 to 84 labelled proper_1(2)* 73 to 82 labelled mark_1(1)* 73 to 81 labelled f_1(2), h_1(2)* 73 to 94 labelled proper_1(3)* 74 to 65 labelled f_1(1), h_1(1)* 74 to 78 labelled f_1(1), g_1(1), h_1(1)* 74 to 82 labelled f_1(1), h_1(1)* 74 to 96 labelled f_1(1), g_1(1), h_1(1)* 74 to 98 labelled f_1(1), g_1(1), h_1(1)* 75 to 65 labelled g_1(1)* 75 to 82 labelled g_1(1)* 78 to 75 labelled h_1(1)* 79 to 80 labelled h_1(2)* 79 to 95 labelled proper_1(3)* 80 to 81 labelled g_1(2)* 80 to 92 labelled proper_1(3)* 81 to 65 labelled mark_1(2)* 81 to 82 labelled mark_1(2)* 81 to 84 labelled proper_1(2)* 81 to 91 labelled f_1(3), h_1(3)* 81 to 94 labelled proper_1(3)* 81 to 99 labelled proper_1(4)* 82 to 65 labelled f_1(2), h_1(2)* 82 to 82 labelled f_1(2), h_1(2)* 83 to 65 labelled g_1(2)* 83 to 82 labelled g_1(2)* 84 to 58 labelled top_1(2)* 86 to 83 labelled h_1(2)* 87 to 84 labelled g_1(2)* 87 to 94 labelled g_1(2)* 88 to 86 labelled f_1(2)* 89 to 87 labelled h_1(2)* 91 to 65 labelled mark_1(3)* 91 to 82 labelled mark_1(3)* 91 to 84 labelled proper_1(2)* 91 to 91 labelled f_1(3), h_1(3)* 91 to 94 labelled proper_1(3)* 91 to 99 labelled proper_1(4)* 92 to 84 labelled g_1(3)* 92 to 94 labelled g_1(3)* 92 to 99 labelled g_1(3)* 93 to 89 labelled f_1(2)* 94 to 84 labelled f_1(3), h_1(3)* 94 to 94 labelled f_1(3), h_1(3)* 94 to 99 labelled f_1(3), h_1(3)* 95 to 92 labelled h_1(3)* 96 to 95 labelled f_1(3)* 98 to 96 labelled f_1(2), g_1(2), h_1(2)* 98 to 98 labelled f_1(2), g_1(2), h_1(2)* 99 to 94 labelled f_1(4), h_1(4)* 99 to 99 labelled f_1(4), h_1(4) ---------------------------------------- (4) YES