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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [SOUND, 0 ms] (2) QTRS (3) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms]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: 39, 40, 41, 42, 43, 44, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80 Node 39 is start node and node 40 is final node. Those nodes are connected through the following edges: * 39 to 41 labelled f_1(0), g_1(0), h_1(0)* 39 to 44 labelled active_1(0), proper_1(0)* 39 to 43 labelled f_1(0), h_1(0)* 39 to 55 labelled f_1(1)* 39 to 58 labelled active_1(1), proper_1(1)* 39 to 74 labelled proper_1(2)* 40 to 40 labelled #_1(0)* 41 to 42 labelled h_1(0)* 41 to 40 labelled ok_1(0)* 41 to 48 labelled f_1(1), g_1(1), h_1(1)* 41 to 49 labelled active_1(1)* 41 to 60 labelled proper_1(1)* 41 to 61 labelled f_1(2)* 41 to 64 labelled active_1(2)* 41 to 78 labelled proper_1(3)* 41 to 79 labelled proper_1(2)* 42 to 43 labelled g_1(0)* 42 to 59 labelled proper_1(1)* 43 to 40 labelled mark_1(0)* 43 to 50 labelled f_1(1), h_1(1)* 43 to 49 labelled proper_1(1)* 43 to 64 labelled proper_1(2)* 44 to 40 labelled f_1(0), h_1(0), g_1(0), top_1(0)* 44 to 51 labelled f_1(1)* 44 to 54 labelled active_1(1), proper_1(1)* 44 to 70 labelled proper_1(2)* 48 to 40 labelled ok_1(1)* 48 to 48 labelled f_1(1), g_1(1), h_1(1)* 48 to 49 labelled active_1(1)* 48 to 61 labelled f_1(2)* 48 to 64 labelled active_1(2)* 48 to 78 labelled proper_1(3)* 48 to 79 labelled proper_1(2)* 49 to 40 labelled top_1(1)* 50 to 40 labelled mark_1(1)* 50 to 50 labelled f_1(1), h_1(1)* 50 to 49 labelled proper_1(1)* 50 to 64 labelled proper_1(2)* 51 to 52 labelled h_1(1)* 51 to 67 labelled proper_1(2)* 52 to 53 labelled g_1(1)* 52 to 65 labelled proper_1(2)* 53 to 40 labelled mark_1(1)* 53 to 50 labelled f_1(1), h_1(1)* 53 to 49 labelled proper_1(1)* 53 to 64 labelled proper_1(2)* 54 to 40 labelled f_1(1), h_1(1), g_1(1)* 54 to 51 labelled f_1(1)* 54 to 54 labelled active_1(1), proper_1(1)* 54 to 70 labelled proper_1(2)* 55 to 56 labelled h_1(1)* 55 to 71 labelled proper_1(2)* 56 to 57 labelled g_1(1)* 56 to 68 labelled proper_1(2)* 57 to 49 labelled mark_1(1)* 57 to 66 labelled proper_1(2)* 57 to 64 labelled mark_1(1)* 57 to 63 labelled f_1(2), h_1(2)* 57 to 75 labelled proper_1(3)* 58 to 49 labelled f_1(1), h_1(1)* 58 to 60 labelled f_1(1), g_1(1), h_1(1)* 58 to 64 labelled f_1(1), h_1(1)* 58 to 78 labelled f_1(1), g_1(1), h_1(1)* 58 to 79 labelled f_1(1), g_1(1), h_1(1)* 59 to 49 labelled g_1(1)* 59 to 64 labelled g_1(1)* 60 to 59 labelled h_1(1)* 61 to 62 labelled h_1(2)* 61 to 76 labelled proper_1(3)* 62 to 63 labelled g_1(2)* 62 to 73 labelled proper_1(3)* 63 to 49 labelled mark_1(2)* 63 to 64 labelled mark_1(2)* 63 to 66 labelled proper_1(2)* 63 to 72 labelled f_1(3), h_1(3)* 63 to 75 labelled proper_1(3)* 63 to 80 labelled proper_1(4)* 64 to 49 labelled f_1(2), h_1(2)* 64 to 64 labelled f_1(2), h_1(2)* 65 to 49 labelled g_1(2)* 65 to 64 labelled g_1(2)* 66 to 40 labelled top_1(2)* 67 to 65 labelled h_1(2)* 68 to 66 labelled g_1(2)* 68 to 75 labelled g_1(2)* 70 to 67 labelled f_1(2)* 71 to 68 labelled h_1(2)* 72 to 49 labelled mark_1(3)* 72 to 64 labelled mark_1(3)* 72 to 66 labelled proper_1(2)* 72 to 72 labelled f_1(3), h_1(3)* 72 to 75 labelled proper_1(3)* 72 to 80 labelled proper_1(4)* 73 to 66 labelled g_1(3)* 73 to 75 labelled g_1(3)* 73 to 80 labelled g_1(3)* 74 to 71 labelled f_1(2)* 75 to 66 labelled f_1(3), h_1(3)* 75 to 75 labelled f_1(3), h_1(3)* 75 to 80 labelled f_1(3), h_1(3)* 76 to 73 labelled h_1(3)* 78 to 76 labelled f_1(3)* 79 to 78 labelled f_1(2), g_1(2), h_1(2)* 79 to 79 labelled f_1(2), g_1(2), h_1(2)* 80 to 75 labelled f_1(4), h_1(4)* 80 to 80 labelled f_1(4), h_1(4) ---------------------------------------- (4) YES