/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [SOUND, 0 ms] (2) QTRS (3) RFCMatchBoundsTRSProof [EQUIVALENT, 9 ms] (4) YES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: 32, 33, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 83, 84, 85, 87, 88, 89, 90, 92, 93, 95, 96, 97, 98, 99, 102 Node 32 is start node and node 33 is final node. Those nodes are connected through the following edges: * 32 to 49 labelled a'_1(0), f_1(0), c_1(0), g_1(0)* 32 to 54 labelled active_1(0), proper_1(0)* 32 to 53 labelled f_1(0), g_1(0)* 32 to 59 labelled active_1(1), proper_1(1)* 32 to 63 labelled a'_1(1)* 32 to 83 labelled a'_1(2)* 33 to 33 labelled #_1(0)* 49 to 50 labelled f_1(0)* 49 to 33 labelled ok_1(0)* 49 to 55 labelled f_1(1), c_1(1), g_1(1)* 49 to 56 labelled active_1(1)* 49 to 61 labelled active_1(2)* 49 to 70 labelled proper_1(1)* 50 to 51 labelled g_1(0)* 50 to 68 labelled proper_1(1)* 51 to 52 labelled f_1(0)* 51 to 62 labelled proper_1(1)* 52 to 53 labelled c_1(0)* 52 to 60 labelled proper_1(1)* 53 to 33 labelled mark_1(0)* 53 to 57 labelled f_1(1), g_1(1)* 53 to 56 labelled proper_1(1)* 53 to 61 labelled proper_1(2)* 54 to 33 labelled f_1(0), g_1(0), c_1(0), top_1(0)* 54 to 58 labelled active_1(1), proper_1(1)* 55 to 33 labelled ok_1(1)* 55 to 55 labelled f_1(1), c_1(1), g_1(1)* 55 to 56 labelled active_1(1)* 55 to 61 labelled active_1(2)* 56 to 33 labelled top_1(1)* 57 to 33 labelled mark_1(1)* 57 to 57 labelled f_1(1), g_1(1)* 57 to 56 labelled proper_1(1)* 57 to 61 labelled proper_1(2)* 58 to 33 labelled f_1(1), g_1(1), c_1(1)* 58 to 58 labelled active_1(1), proper_1(1)* 59 to 56 labelled f_1(1), g_1(1)* 59 to 61 labelled f_1(1), g_1(1)* 59 to 70 labelled f_1(1), c_1(1), g_1(1)* 60 to 56 labelled c_1(1)* 60 to 61 labelled c_1(1)* 61 to 56 labelled f_1(2), g_1(2)* 61 to 61 labelled f_1(2), g_1(2)* 62 to 60 labelled f_1(1)* 63 to 64 labelled f_1(1)* 63 to 70 labelled ok_1(1)* 63 to 75 labelled f_1(2)* 63 to 78 labelled proper_1(2)* 64 to 65 labelled g_1(1)* 64 to 76 labelled proper_1(2)* 65 to 66 labelled f_1(1)* 65 to 73 labelled proper_1(2)* 66 to 67 labelled c_1(1)* 66 to 71 labelled proper_1(2)* 67 to 56 labelled mark_1(1)* 67 to 69 labelled proper_1(2)* 67 to 61 labelled mark_1(1)* 67 to 72 labelled f_1(2), g_1(2)* 67 to 77 labelled proper_1(3)* 68 to 62 labelled g_1(1)* 69 to 33 labelled top_1(2)* 70 to 68 labelled f_1(1)* 71 to 69 labelled c_1(2)* 71 to 77 labelled c_1(2)* 72 to 56 labelled mark_1(2)* 72 to 61 labelled mark_1(2)* 72 to 69 labelled proper_1(2)* 72 to 74 labelled f_1(3), g_1(3)* 72 to 77 labelled proper_1(3)* 72 to 84 labelled proper_1(4)* 73 to 71 labelled f_1(2)* 74 to 56 labelled mark_1(3)* 74 to 61 labelled mark_1(3)* 74 to 69 labelled proper_1(2)* 74 to 74 labelled f_1(3), g_1(3)* 74 to 77 labelled proper_1(3)* 74 to 84 labelled proper_1(4)* 75 to 68 labelled ok_1(2)* 75 to 79 labelled g_1(2)* 76 to 73 labelled g_1(2)* 77 to 69 labelled f_1(3), g_1(3)* 77 to 77 labelled f_1(3), g_1(3)* 77 to 84 labelled f_1(3), g_1(3)* 78 to 76 labelled f_1(2)* 79 to 62 labelled ok_1(2)* 79 to 85 labelled f_1(2)* 83 to 78 labelled ok_1(2)* 83 to 87 labelled f_1(3)* 84 to 77 labelled f_1(4), g_1(4)* 84 to 84 labelled f_1(4), g_1(4)* 85 to 60 labelled ok_1(2)* 85 to 88 labelled c_1(2)* 87 to 76 labelled ok_1(3)* 87 to 90 labelled g_1(3)* 88 to 56 labelled ok_1(2)* 88 to 61 labelled ok_1(2)* 88 to 69 labelled active_1(2)* 88 to 89 labelled f_1(3), g_1(3)* 88 to 77 labelled active_1(3)* 88 to 84 labelled active_1(4)* 89 to 56 labelled ok_1(3)* 89 to 61 labelled ok_1(3)* 89 to 69 labelled active_1(2)* 89 to 89 labelled f_1(3), g_1(3)* 89 to 77 labelled active_1(3)* 89 to 84 labelled active_1(4)* 90 to 73 labelled ok_1(3)* 90 to 92 labelled f_1(3)* 92 to 71 labelled ok_1(3)* 92 to 93 labelled c_1(3)* 93 to 69 labelled ok_1(3)* 93 to 77 labelled ok_1(3)* 93 to 95 labelled active_1(3)* 93 to 96 labelled f_1(4), g_1(4)* 93 to 98 labelled active_1(4)* 93 to 99 labelled active_1(5)* 95 to 33 labelled top_1(3)* 96 to 69 labelled ok_1(4)* 96 to 77 labelled ok_1(4)* 96 to 84 labelled ok_1(4)* 96 to 95 labelled active_1(3)* 96 to 96 labelled f_1(4), g_1(4)* 96 to 97 labelled f_1(5), g_1(5)* 96 to 98 labelled active_1(4)* 96 to 99 labelled active_1(5)* 96 to 102 labelled active_1(6)* 97 to 77 labelled ok_1(5)* 97 to 84 labelled ok_1(5)* 97 to 96 labelled f_1(4), g_1(4)* 97 to 97 labelled f_1(5), g_1(5)* 97 to 98 labelled active_1(4)* 97 to 99 labelled active_1(5)* 97 to 102 labelled active_1(6)* 98 to 95 labelled f_1(4), g_1(4)* 99 to 98 labelled f_1(5), g_1(5)* 99 to 99 labelled f_1(5), g_1(5)* 99 to 102 labelled f_1(5), g_1(5)* 102 to 99 labelled f_1(6), g_1(6)* 102 to 102 labelled f_1(6), g_1(6) ---------------------------------------- (4) YES