/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [SOUND, 0 ms] (2) QTRS (3) RFCMatchBoundsTRSProof [EQUIVALENT, 2 ms]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)) The set Q consists of the following terms: active(f(x0)) active(g(x0)) f(mark(x0)) g(mark(x0)) proper(f(x0)) proper(a) proper(c(x0)) proper(g(x0)) f(ok(x0)) c(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(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: 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 39, 40, 41, 42, 43, 44, 45, 46, 47, 52, 53, 54, 55, 56, 57, 58, 63, 64, 65, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 90, 91, 92, 95, 96, 98 Node 22 is start node and node 23 is final node. Those nodes are connected through the following edges: * 22 to 24 labelled a'_1(0), f_1(0), c_1(0), g_1(0)* 22 to 29 labelled active_1(0), proper_1(0)* 22 to 28 labelled f_1(0), g_1(0)* 22 to 39 labelled active_1(1), proper_1(1)* 22 to 43 labelled a'_1(1)* 22 to 70 labelled a'_1(2)* 23 to 23 labelled #_1(0)* 24 to 25 labelled f_1(0)* 24 to 23 labelled ok_1(0)* 24 to 30 labelled f_1(1), c_1(1), g_1(1)* 24 to 31 labelled active_1(1)* 24 to 41 labelled active_1(2)* 24 to 54 labelled proper_1(1)* 25 to 26 labelled g_1(0)* 25 to 52 labelled proper_1(1)* 26 to 27 labelled f_1(0)* 26 to 42 labelled proper_1(1)* 27 to 28 labelled c_1(0)* 27 to 40 labelled proper_1(1)* 28 to 23 labelled mark_1(0)* 28 to 32 labelled f_1(1), g_1(1)* 28 to 31 labelled proper_1(1)* 28 to 41 labelled proper_1(2)* 29 to 23 labelled f_1(0), g_1(0), c_1(0), top_1(0)* 29 to 33 labelled active_1(1), proper_1(1)* 30 to 23 labelled ok_1(1)* 30 to 30 labelled f_1(1), c_1(1), g_1(1)* 30 to 31 labelled active_1(1)* 30 to 41 labelled active_1(2)* 31 to 23 labelled top_1(1)* 32 to 23 labelled mark_1(1)* 32 to 32 labelled f_1(1), g_1(1)* 32 to 31 labelled proper_1(1)* 32 to 41 labelled proper_1(2)* 33 to 23 labelled f_1(1), g_1(1), c_1(1)* 33 to 33 labelled active_1(1), proper_1(1)* 39 to 31 labelled f_1(1), g_1(1)* 39 to 41 labelled f_1(1), g_1(1)* 39 to 54 labelled f_1(1), c_1(1), g_1(1)* 40 to 31 labelled c_1(1)* 40 to 41 labelled c_1(1)* 41 to 31 labelled f_1(2), g_1(2)* 41 to 41 labelled f_1(2), g_1(2)* 42 to 40 labelled f_1(1)* 43 to 44 labelled f_1(1)* 43 to 54 labelled ok_1(1)* 43 to 63 labelled f_1(2)* 43 to 68 labelled proper_1(2)* 44 to 45 labelled g_1(1)* 44 to 64 labelled proper_1(2)* 45 to 46 labelled f_1(1)* 45 to 57 labelled proper_1(2)* 46 to 47 labelled c_1(1)* 46 to 55 labelled proper_1(2)* 47 to 31 labelled mark_1(1)* 47 to 53 labelled proper_1(2)* 47 to 41 labelled mark_1(1)* 47 to 56 labelled f_1(2), g_1(2)* 47 to 65 labelled proper_1(3)* 52 to 42 labelled g_1(1)* 53 to 23 labelled top_1(2)* 54 to 52 labelled f_1(1)* 55 to 53 labelled c_1(2)* 55 to 65 labelled c_1(2)* 56 to 31 labelled mark_1(2)* 56 to 41 labelled mark_1(2)* 56 to 53 labelled proper_1(2)* 56 to 58 labelled f_1(3), g_1(3)* 56 to 65 labelled proper_1(3)* 56 to 71 labelled proper_1(4)* 57 to 55 labelled f_1(2)* 58 to 31 labelled mark_1(3)* 58 to 41 labelled mark_1(3)* 58 to 53 labelled proper_1(2)* 58 to 58 labelled f_1(3), g_1(3)* 58 to 65 labelled proper_1(3)* 58 to 71 labelled proper_1(4)* 63 to 52 labelled ok_1(2)* 63 to 69 labelled g_1(2)* 64 to 57 labelled g_1(2)* 65 to 53 labelled f_1(3), g_1(3)* 65 to 65 labelled f_1(3), g_1(3)* 65 to 71 labelled f_1(3), g_1(3)* 68 to 64 labelled f_1(2)* 69 to 42 labelled ok_1(2)* 69 to 72 labelled f_1(2)* 70 to 68 labelled ok_1(2)* 70 to 73 labelled f_1(3)* 71 to 65 labelled f_1(4), g_1(4)* 71 to 71 labelled f_1(4), g_1(4)* 72 to 40 labelled ok_1(2)* 72 to 74 labelled c_1(2)* 73 to 64 labelled ok_1(3)* 73 to 76 labelled g_1(3)* 74 to 31 labelled ok_1(2)* 74 to 41 labelled ok_1(2)* 74 to 53 labelled active_1(2)* 74 to 75 labelled f_1(3), g_1(3)* 74 to 65 labelled active_1(3)* 74 to 71 labelled active_1(4)* 75 to 31 labelled ok_1(3)* 75 to 41 labelled ok_1(3)* 75 to 53 labelled active_1(2)* 75 to 75 labelled f_1(3), g_1(3)* 75 to 65 labelled active_1(3)* 75 to 71 labelled active_1(4)* 76 to 57 labelled ok_1(3)* 76 to 77 labelled f_1(3)* 77 to 55 labelled ok_1(3)* 77 to 78 labelled c_1(3)* 78 to 53 labelled ok_1(3)* 78 to 65 labelled ok_1(3)* 78 to 90 labelled active_1(3)* 78 to 91 labelled f_1(4), g_1(4)* 78 to 95 labelled active_1(4)* 78 to 96 labelled active_1(5)* 90 to 23 labelled top_1(3)* 91 to 53 labelled ok_1(4)* 91 to 65 labelled ok_1(4)* 91 to 71 labelled ok_1(4)* 91 to 90 labelled active_1(3)* 91 to 91 labelled f_1(4), g_1(4)* 91 to 92 labelled f_1(5), g_1(5)* 91 to 95 labelled active_1(4)* 91 to 96 labelled active_1(5)* 91 to 98 labelled active_1(6)* 92 to 65 labelled ok_1(5)* 92 to 71 labelled ok_1(5)* 92 to 91 labelled f_1(4), g_1(4)* 92 to 92 labelled f_1(5), g_1(5)* 92 to 95 labelled active_1(4)* 92 to 96 labelled active_1(5)* 92 to 98 labelled active_1(6)* 95 to 90 labelled f_1(4), g_1(4)* 96 to 95 labelled f_1(5), g_1(5)* 96 to 96 labelled f_1(5), g_1(5)* 96 to 98 labelled f_1(5), g_1(5)* 98 to 96 labelled f_1(6), g_1(6)* 98 to 98 labelled f_1(6), g_1(6) ---------------------------------------- (4) YES