/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [SOUND, 0 ms] (2) QTRS (3) RFCMatchBoundsTRSProof [EQUIVALENT, 9 ms]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: 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 32, 33, 34, 35, 36, 39, 40, 41, 42, 43, 44, 45, 46, 47, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65 Node 16 is start node and node 17 is final node. Those nodes are connected through the following edges: * 16 to 18 labelled f_1(0), g_1(0), h_1(0)* 16 to 21 labelled active_1(0), proper_1(0)* 16 to 20 labelled f_1(0), h_1(0)* 16 to 32 labelled f_1(1)* 16 to 35 labelled active_1(1), proper_1(1)* 16 to 60 labelled proper_1(2)* 17 to 17 labelled #_1(0)* 18 to 19 labelled h_1(0)* 18 to 17 labelled ok_1(0)* 18 to 22 labelled f_1(1), g_1(1), h_1(1)* 18 to 23 labelled active_1(1)* 18 to 39 labelled proper_1(1)* 18 to 40 labelled f_1(2)* 18 to 43 labelled active_1(2)* 18 to 63 labelled proper_1(3)* 18 to 64 labelled proper_1(2)* 19 to 20 labelled g_1(0)* 19 to 36 labelled proper_1(1)* 20 to 17 labelled mark_1(0)* 20 to 24 labelled f_1(1), h_1(1)* 20 to 23 labelled proper_1(1)* 20 to 43 labelled proper_1(2)* 21 to 17 labelled f_1(0), h_1(0), g_1(0), top_1(0)* 21 to 25 labelled f_1(1)* 21 to 28 labelled active_1(1), proper_1(1)* 21 to 56 labelled proper_1(2)* 22 to 17 labelled ok_1(1)* 22 to 22 labelled f_1(1), g_1(1), h_1(1)* 22 to 23 labelled active_1(1)* 22 to 40 labelled f_1(2)* 22 to 43 labelled active_1(2)* 22 to 63 labelled proper_1(3)* 22 to 64 labelled proper_1(2)* 23 to 17 labelled top_1(1)* 24 to 17 labelled mark_1(1)* 24 to 24 labelled f_1(1), h_1(1)* 24 to 23 labelled proper_1(1)* 24 to 43 labelled proper_1(2)* 25 to 26 labelled h_1(1)* 25 to 46 labelled proper_1(2)* 26 to 27 labelled g_1(1)* 26 to 44 labelled proper_1(2)* 27 to 17 labelled mark_1(1)* 27 to 24 labelled f_1(1), h_1(1)* 27 to 23 labelled proper_1(1)* 27 to 43 labelled proper_1(2)* 28 to 17 labelled f_1(1), h_1(1), g_1(1)* 28 to 25 labelled f_1(1)* 28 to 28 labelled active_1(1), proper_1(1)* 28 to 56 labelled proper_1(2)* 32 to 33 labelled h_1(1)* 32 to 57 labelled proper_1(2)* 33 to 34 labelled g_1(1)* 33 to 47 labelled proper_1(2)* 34 to 23 labelled mark_1(1)* 34 to 45 labelled proper_1(2)* 34 to 43 labelled mark_1(1)* 34 to 42 labelled f_1(2), h_1(2)* 34 to 61 labelled proper_1(3)* 35 to 23 labelled f_1(1), h_1(1)* 35 to 39 labelled f_1(1), g_1(1), h_1(1)* 35 to 43 labelled f_1(1), h_1(1)* 35 to 63 labelled f_1(1), g_1(1), h_1(1)* 35 to 64 labelled f_1(1), g_1(1), h_1(1)* 36 to 23 labelled g_1(1)* 36 to 43 labelled g_1(1)* 39 to 36 labelled h_1(1)* 40 to 41 labelled h_1(2)* 40 to 62 labelled proper_1(3)* 41 to 42 labelled g_1(2)* 41 to 59 labelled proper_1(3)* 42 to 23 labelled mark_1(2)* 42 to 43 labelled mark_1(2)* 42 to 45 labelled proper_1(2)* 42 to 58 labelled f_1(3), h_1(3)* 42 to 61 labelled proper_1(3)* 42 to 65 labelled proper_1(4)* 43 to 23 labelled f_1(2), h_1(2)* 43 to 43 labelled f_1(2), h_1(2)* 44 to 23 labelled g_1(2)* 44 to 43 labelled g_1(2)* 45 to 17 labelled top_1(2)* 46 to 44 labelled h_1(2)* 47 to 45 labelled g_1(2)* 47 to 61 labelled g_1(2)* 56 to 46 labelled f_1(2)* 57 to 47 labelled h_1(2)* 58 to 23 labelled mark_1(3)* 58 to 43 labelled mark_1(3)* 58 to 45 labelled proper_1(2)* 58 to 58 labelled f_1(3), h_1(3)* 58 to 61 labelled proper_1(3)* 58 to 65 labelled proper_1(4)* 59 to 45 labelled g_1(3)* 59 to 61 labelled g_1(3)* 59 to 65 labelled g_1(3)* 60 to 57 labelled f_1(2)* 61 to 45 labelled f_1(3), h_1(3)* 61 to 61 labelled f_1(3), h_1(3)* 61 to 65 labelled f_1(3), h_1(3)* 62 to 59 labelled h_1(3)* 63 to 62 labelled f_1(3)* 64 to 63 labelled f_1(2), g_1(2), h_1(2)* 64 to 64 labelled f_1(2), g_1(2), h_1(2)* 65 to 61 labelled f_1(4), h_1(4)* 65 to 65 labelled f_1(4), h_1(4) ---------------------------------------- (4) YES