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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms] (4) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: active(c) -> mark(f(g(c))) active(f(g(X))) -> mark(g(X)) proper(c) -> ok(c) proper(f(X)) -> f(proper(X)) 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: c'(active(x)) -> c'(g(f(mark(x)))) g(f(active(X))) -> g(mark(X)) c'(proper(x)) -> c'(ok(x)) f(proper(X)) -> proper(f(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 5. This implies Q-termination of R. The following rules were used to construct the certificate: c'(active(x)) -> c'(g(f(mark(x)))) g(f(active(X))) -> g(mark(X)) c'(proper(x)) -> c'(ok(x)) f(proper(X)) -> proper(f(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: 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94 Node 53 is start node and node 54 is final node. Those nodes are connected through the following edges: * 53 to 55 labelled c'_1(0), f_1(0), g_1(0)* 53 to 57 labelled g_1(0)* 53 to 58 labelled proper_1(0), active_1(0)* 53 to 64 labelled c'_1(1)* 53 to 67 labelled proper_1(1)* 53 to 66 labelled g_1(1)* 53 to 73 labelled proper_1(2)* 53 to 78 labelled c'_1(2)* 53 to 92 labelled c'_1(3)* 54 to 54 labelled #_1(0)* 55 to 56 labelled g_1(0)* 55 to 54 labelled ok_1(0)* 55 to 60 labelled f_1(1), g_1(1)* 55 to 61 labelled active_1(1)* 55 to 69 labelled proper_1(1)* 55 to 72 labelled g_1(2)* 55 to 75 labelled proper_1(3)* 55 to 79 labelled proper_1(2)* 56 to 57 labelled f_1(0)* 56 to 68 labelled proper_1(1)* 57 to 54 labelled mark_1(0)* 57 to 61 labelled proper_1(1)* 58 to 54 labelled f_1(0), g_1(0), top_1(0)* 58 to 62 labelled proper_1(1)* 58 to 63 labelled g_1(1)* 58 to 70 labelled proper_1(2)* 60 to 54 labelled ok_1(1)* 60 to 60 labelled f_1(1), g_1(1)* 60 to 61 labelled active_1(1)* 60 to 72 labelled g_1(2)* 60 to 75 labelled proper_1(3)* 60 to 79 labelled proper_1(2)* 61 to 54 labelled top_1(1)* 62 to 54 labelled f_1(1), g_1(1)* 62 to 62 labelled proper_1(1)* 62 to 63 labelled g_1(1)* 62 to 70 labelled proper_1(2)* 63 to 54 labelled mark_1(1)* 63 to 61 labelled proper_1(1)* 64 to 65 labelled g_1(1)* 64 to 69 labelled ok_1(1)* 64 to 76 labelled proper_1(2)* 64 to 77 labelled g_1(2)* 64 to 75 labelled ok_1(1)* 64 to 79 labelled ok_1(1)* 64 to 82 labelled f_1(2), g_1(2)* 64 to 85 labelled g_1(3)* 64 to 88 labelled proper_1(4)* 65 to 66 labelled f_1(1)* 65 to 74 labelled proper_1(2)* 66 to 61 labelled mark_1(1)* 66 to 71 labelled proper_1(2)* 67 to 61 labelled g_1(1)* 67 to 69 labelled f_1(1), g_1(1)* 67 to 75 labelled f_1(1), g_1(1)* 67 to 79 labelled f_1(1), g_1(1)* 68 to 61 labelled f_1(1)* 69 to 68 labelled g_1(1)* 70 to 61 labelled g_1(2)* 71 to 54 labelled top_1(2)* 72 to 61 labelled mark_1(2)* 72 to 71 labelled proper_1(2)* 73 to 71 labelled g_1(2)* 74 to 71 labelled f_1(2)* 75 to 71 labelled g_1(3)* 76 to 74 labelled g_1(2)* 77 to 68 labelled ok_1(2)* 77 to 80 labelled f_1(2)* 77 to 71 labelled ok_1(2)* 77 to 84 labelled active_1(3)* 78 to 76 labelled ok_1(2)* 78 to 81 labelled g_1(3)* 78 to 89 labelled g_1(4)* 78 to 88 labelled ok_1(2)* 78 to 91 labelled proper_1(5)* 79 to 75 labelled f_1(2), g_1(2)* 79 to 79 labelled f_1(2), g_1(2)* 80 to 61 labelled ok_1(2)* 80 to 71 labelled active_1(2)* 81 to 74 labelled ok_1(3)* 81 to 83 labelled f_1(3)* 81 to 84 labelled ok_1(3)* 81 to 90 labelled active_1(4)* 82 to 75 labelled ok_1(2)* 82 to 79 labelled ok_1(2)* 82 to 83 labelled g_1(3)* 82 to 86 labelled f_1(3), g_1(3)* 83 to 71 labelled ok_1(3)* 83 to 84 labelled active_1(3)* 84 to 54 labelled top_1(3)* 85 to 71 labelled mark_1(3)* 85 to 84 labelled proper_1(3)* 86 to 75 labelled ok_1(3)* 86 to 79 labelled ok_1(3)* 86 to 87 labelled g_1(4)* 86 to 86 labelled f_1(3), g_1(3)* 87 to 71 labelled ok_1(4)* 87 to 84 labelled active_1(3)* 88 to 84 labelled g_1(4)* 89 to 84 labelled mark_1(4)* 89 to 90 labelled proper_1(4)* 90 to 54 labelled top_1(4)* 91 to 90 labelled g_1(5)* 92 to 91 labelled ok_1(3)* 92 to 93 labelled g_1(4)* 93 to 90 labelled ok_1(4)* 93 to 94 labelled active_1(5)* 94 to 54 labelled top_1(5) ---------------------------------------- (4) YES