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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) RFCMatchBoundsTRSProof [EQUIVALENT, 19 ms] (4) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: active(f(f(a))) -> mark(f(g(f(a)))) active(g(X)) -> g(active(X)) g(mark(X)) -> mark(g(X)) proper(f(X)) -> f(proper(X)) proper(a) -> ok(a) proper(g(X)) -> g(proper(X)) f(ok(X)) -> ok(f(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(mark(x))))) g(active(X)) -> active(g(X)) mark(g(X)) -> g(mark(X)) f(proper(X)) -> proper(f(X)) a'(proper(x)) -> a'(ok(x)) g(proper(X)) -> proper(g(X)) ok(f(X)) -> f(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(mark(x))))) g(active(X)) -> active(g(X)) mark(g(X)) -> g(mark(X)) f(proper(X)) -> proper(f(X)) a'(proper(x)) -> a'(ok(x)) g(proper(X)) -> proper(g(X)) ok(f(X)) -> f(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: 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 74, 75, 78, 79, 80, 86, 87, 90, 91, 94, 98, 99, 102 Node 39 is start node and node 40 is final node. Those nodes are connected through the following edges: * 39 to 41 labelled a'_1(0), f_1(0), g_1(0)* 39 to 45 labelled active_1(0), proper_1(0)* 39 to 44 labelled g_1(0)* 39 to 46 labelled proper_1(0)* 39 to 52 labelled active_1(1), proper_1(1)* 39 to 56 labelled a'_1(1)* 39 to 74 labelled a'_1(2)* 40 to 40 labelled #_1(0)* 41 to 42 labelled f_1(0)* 41 to 40 labelled ok_1(0)* 41 to 47 labelled f_1(1), g_1(1)* 41 to 48 labelled active_1(1)* 41 to 54 labelled active_1(2)* 41 to 60 labelled proper_1(1)* 42 to 43 labelled g_1(0)* 42 to 55 labelled proper_1(1)* 43 to 44 labelled f_1(0)* 43 to 53 labelled proper_1(1)* 44 to 40 labelled mark_1(0)* 44 to 49 labelled g_1(1)* 44 to 48 labelled proper_1(1)* 44 to 54 labelled proper_1(2)* 45 to 40 labelled g_1(0), top_1(0)* 45 to 50 labelled active_1(1), proper_1(1)* 46 to 40 labelled f_1(0)* 46 to 51 labelled proper_1(1)* 47 to 40 labelled ok_1(1)* 47 to 47 labelled f_1(1), g_1(1)* 47 to 48 labelled active_1(1)* 47 to 54 labelled active_1(2)* 48 to 40 labelled top_1(1)* 49 to 40 labelled mark_1(1)* 49 to 49 labelled g_1(1)* 49 to 48 labelled proper_1(1)* 49 to 54 labelled proper_1(2)* 50 to 40 labelled g_1(1)* 50 to 50 labelled active_1(1), proper_1(1)* 51 to 40 labelled f_1(1)* 51 to 51 labelled proper_1(1)* 52 to 48 labelled g_1(1)* 52 to 54 labelled g_1(1)* 52 to 60 labelled f_1(1), g_1(1)* 53 to 48 labelled f_1(1)* 53 to 54 labelled f_1(1)* 54 to 48 labelled g_1(2)* 54 to 54 labelled g_1(2)* 55 to 53 labelled g_1(1)* 56 to 57 labelled f_1(1)* 56 to 60 labelled ok_1(1)* 56 to 64 labelled f_1(2)* 56 to 67 labelled proper_1(2)* 57 to 58 labelled g_1(1)* 57 to 65 labelled proper_1(2)* 58 to 59 labelled f_1(1)* 58 to 62 labelled proper_1(2)* 59 to 48 labelled mark_1(1)* 59 to 61 labelled proper_1(2)* 59 to 54 labelled mark_1(1)* 59 to 63 labelled g_1(2)* 59 to 68 labelled proper_1(3)* 60 to 55 labelled f_1(1)* 61 to 40 labelled top_1(2)* 62 to 61 labelled f_1(2)* 62 to 68 labelled f_1(2)* 63 to 48 labelled mark_1(2)* 63 to 54 labelled mark_1(2)* 63 to 61 labelled proper_1(2)* 63 to 66 labelled g_1(3)* 63 to 68 labelled proper_1(3)* 63 to 79 labelled proper_1(4)* 64 to 55 labelled ok_1(2)* 64 to 69 labelled g_1(2)* 65 to 62 labelled g_1(2)* 66 to 48 labelled mark_1(3)* 66 to 54 labelled mark_1(3)* 66 to 61 labelled proper_1(2)* 66 to 66 labelled g_1(3)* 66 to 68 labelled proper_1(3)* 66 to 79 labelled proper_1(4)* 67 to 65 labelled f_1(2)* 68 to 61 labelled g_1(3)* 68 to 68 labelled g_1(3)* 68 to 79 labelled g_1(3)* 69 to 53 labelled ok_1(2)* 69 to 75 labelled f_1(2)* 74 to 67 labelled ok_1(2)* 74 to 80 labelled f_1(3)* 75 to 48 labelled ok_1(2)* 75 to 54 labelled ok_1(2)* 75 to 61 labelled active_1(2)* 75 to 78 labelled g_1(3)* 75 to 68 labelled active_1(3)* 75 to 79 labelled active_1(4)* 78 to 48 labelled ok_1(3)* 78 to 54 labelled ok_1(3)* 78 to 61 labelled active_1(2)* 78 to 78 labelled g_1(3)* 78 to 68 labelled active_1(3)* 78 to 79 labelled active_1(4)* 79 to 68 labelled g_1(4)* 79 to 79 labelled g_1(4)* 80 to 65 labelled ok_1(3)* 80 to 86 labelled g_1(3)* 86 to 62 labelled ok_1(3)* 86 to 87 labelled f_1(3)* 87 to 61 labelled ok_1(3)* 87 to 68 labelled ok_1(3)* 87 to 90 labelled active_1(3)* 87 to 91 labelled g_1(4)* 87 to 98 labelled active_1(4)* 87 to 99 labelled active_1(5)* 90 to 40 labelled top_1(3)* 91 to 61 labelled ok_1(4)* 91 to 68 labelled ok_1(4)* 91 to 79 labelled ok_1(4)* 91 to 90 labelled active_1(3)* 91 to 91 labelled g_1(4)* 91 to 94 labelled g_1(5)* 91 to 98 labelled active_1(4)* 91 to 99 labelled active_1(5)* 91 to 102 labelled active_1(6)* 94 to 68 labelled ok_1(5)* 94 to 79 labelled ok_1(5)* 94 to 91 labelled g_1(4)* 94 to 94 labelled g_1(5)* 94 to 98 labelled active_1(4)* 94 to 99 labelled active_1(5)* 94 to 102 labelled active_1(6)* 98 to 90 labelled g_1(4)* 99 to 98 labelled g_1(5)* 99 to 99 labelled g_1(5)* 99 to 102 labelled g_1(5)* 102 to 99 labelled g_1(6)* 102 to 102 labelled g_1(6) ---------------------------------------- (4) YES