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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) RFCMatchBoundsTRSProof [EQUIVALENT, 58 ms] (2) YES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)) The set Q consists of the following terms: active(f(f(a))) active(g(x0)) g(mark(x0)) proper(f(x0)) proper(a) proper(g(x0)) f(ok(x0)) g(ok(x0)) top(mark(x0)) top(ok(x0)) ---------------------------------------- (1) RFCMatchBoundsTRSProof (EQUIVALENT) Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 7. This implies Q-termination of R. The following rules were used to construct the certificate: 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)) The certificate found is represented by the following graph. The certificate consists of the following enumerated nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 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 Node 1 is start node and node 2 is final node. Those nodes are connected through the following edges: * 1 to 3 labelled mark_1(0), ok_1(0)* 1 to 7 labelled g_1(0), top_1(0)* 1 to 8 labelled f_1(0), g_1(0), top_1(0)* 1 to 6 labelled ok_1(0)* 1 to 17 labelled mark_1(1)* 1 to 18 labelled top_1(1)* 1 to 13 labelled ok_1(1)* 1 to 35 labelled top_1(2)* 1 to 57 labelled top_1(3)* 2 to 2 labelled #_1(0)* 3 to 4 labelled f_1(0)* 3 to 2 labelled g_1(0)* 3 to 9 labelled mark_1(1), ok_1(1)* 4 to 5 labelled g_1(0)* 4 to 20 labelled ok_1(1)* 5 to 6 labelled f_1(0)* 5 to 19 labelled ok_1(1)* 6 to 2 labelled a(0), f_1(0)* 6 to 10 labelled ok_1(1)* 7 to 2 labelled active_1(0)* 7 to 11 labelled mark_1(1)* 7 to 15 labelled g_1(1)* 7 to 21 labelled mark_1(2)* 8 to 2 labelled proper_1(0)* 8 to 16 labelled f_1(1), g_1(1)* 8 to 14 labelled ok_1(1)* 8 to 22 labelled ok_1(2)* 9 to 2 labelled g_1(1)* 9 to 9 labelled mark_1(1), ok_1(1)* 9 to 20 labelled f_1(1)* 9 to 27 labelled ok_1(2)* 10 to 2 labelled f_1(1)* 10 to 10 labelled ok_1(1)* 11 to 12 labelled f_1(1)* 12 to 13 labelled g_1(1)* 13 to 14 labelled f_1(1), g_1(1)* 13 to 22 labelled f_1(1), g_1(1)* 14 to 2 labelled a(1)* 15 to 2 labelled active_1(1)* 15 to 11 labelled mark_1(1)* 15 to 15 labelled g_1(1)* 15 to 21 labelled mark_1(2)* 16 to 2 labelled proper_1(1)* 16 to 16 labelled f_1(1), g_1(1)* 16 to 14 labelled ok_1(1)* 16 to 22 labelled ok_1(2)* 17 to 11 labelled g_1(1)* 17 to 21 labelled g_1(1)* 18 to 11 labelled proper_1(1)* 18 to 14 labelled active_1(1)* 18 to 23 labelled f_1(2)* 18 to 21 labelled proper_1(1)* 18 to 22 labelled active_1(1)* 18 to 28 labelled g_1(2)* 18 to 31 labelled mark_1(2)* 18 to 39 labelled mark_1(3)* 18 to 45 labelled ok_1(3)* 19 to 10 labelled f_1(1)* 19 to 24 labelled ok_1(2)* 20 to 19 labelled g_1(1)* 20 to 25 labelled ok_1(2)* 21 to 11 labelled g_1(2)* 21 to 21 labelled g_1(2)* 22 to 14 labelled f_1(2), g_1(2)* 22 to 22 labelled f_1(2), g_1(2)* 23 to 12 labelled proper_1(2)* 23 to 26 labelled g_1(2)* 23 to 43 labelled ok_1(3)* 24 to 10 labelled f_1(2)* 24 to 24 labelled ok_1(2)* 25 to 24 labelled g_1(2)* 25 to 29 labelled ok_1(3)* 26 to 13 labelled proper_1(2)* 26 to 30 labelled f_1(2), g_1(2)* 26 to 40 labelled ok_1(3)* 26 to 43 labelled ok_1(3)* 27 to 25 labelled f_1(2)* 27 to 38 labelled ok_1(3)* 28 to 11 labelled proper_1(2)* 28 to 21 labelled proper_1(2)* 28 to 14 labelled active_1(2)* 28 to 22 labelled active_1(2)* 28 to 23 labelled f_1(2)* 28 to 36 labelled g_1(3)* 28 to 31 labelled mark_1(2)* 28 to 39 labelled mark_1(3)* 28 to 45 labelled ok_1(3)* 28 to 46 labelled mark_1(4)* 28 to 50 labelled ok_1(4)* 29 to 24 labelled g_1(3)* 29 to 29 labelled ok_1(3)* 30 to 14 labelled proper_1(2)* 30 to 22 labelled proper_1(2)* 30 to 37 labelled f_1(3), g_1(3)* 30 to 34 labelled ok_1(2)* 30 to 40 labelled ok_1(3)* 30 to 47 labelled ok_1(4)* 31 to 32 labelled f_1(2)* 32 to 33 labelled g_1(2)* 33 to 34 labelled f_1(2)* 34 to 2 labelled a(2)* 35 to 31 labelled proper_1(2)* 35 to 41 labelled f_1(3)* 35 to 39 labelled proper_1(2)* 35 to 48 labelled g_1(3)* 35 to 45 labelled active_1(2)* 35 to 56 labelled ok_1(4)* 36 to 11 labelled proper_1(3)* 36 to 21 labelled proper_1(3)* 36 to 14 labelled active_1(3)* 36 to 22 labelled active_1(3)* 36 to 23 labelled f_1(2)* 36 to 36 labelled g_1(3)* 36 to 31 labelled mark_1(2)* 36 to 39 labelled mark_1(3)* 36 to 45 labelled ok_1(3)* 36 to 46 labelled mark_1(4)* 36 to 50 labelled ok_1(4)* 37 to 14 labelled proper_1(3)* 37 to 22 labelled proper_1(3)* 37 to 37 labelled f_1(3), g_1(3)* 37 to 34 labelled ok_1(2)* 37 to 40 labelled ok_1(3)* 37 to 47 labelled ok_1(4)* 38 to 29 labelled f_1(3)* 38 to 42 labelled ok_1(4)* 39 to 31 labelled g_1(3)* 39 to 39 labelled g_1(3)* 39 to 46 labelled g_1(3)* 40 to 34 labelled f_1(3), g_1(3)* 40 to 40 labelled f_1(3)* 40 to 47 labelled f_1(3), g_1(3)* 41 to 32 labelled proper_1(3)* 41 to 44 labelled g_1(3)* 41 to 54 labelled ok_1(4)* 42 to 29 labelled f_1(4)* 42 to 42 labelled ok_1(4)* 43 to 40 labelled g_1(3)* 43 to 43 labelled g_1(3)* 44 to 33 labelled proper_1(3)* 44 to 49 labelled f_1(3)* 44 to 53 labelled ok_1(4)* 45 to 43 labelled f_1(3)* 45 to 45 labelled g_1(3)* 45 to 50 labelled g_1(3)* 46 to 39 labelled g_1(4)* 46 to 46 labelled g_1(4)* 47 to 40 labelled f_1(4), g_1(4)* 47 to 47 labelled f_1(4), g_1(4)* 48 to 31 labelled proper_1(3)* 48 to 41 labelled f_1(3)* 48 to 39 labelled proper_1(3)* 48 to 52 labelled g_1(4)* 48 to 46 labelled proper_1(3)* 48 to 45 labelled active_1(3)* 48 to 50 labelled active_1(3)* 48 to 56 labelled ok_1(4)* 48 to 58 labelled ok_1(5)* 49 to 34 labelled proper_1(3)* 49 to 51 labelled ok_1(3)* 50 to 45 labelled g_1(4)* 50 to 50 labelled g_1(4)* 51 to 2 labelled a(3)* 52 to 31 labelled proper_1(4)* 52 to 39 labelled proper_1(4)* 52 to 46 labelled proper_1(4)* 52 to 41 labelled f_1(3)* 52 to 52 labelled g_1(4)* 52 to 45 labelled active_1(4)* 52 to 50 labelled active_1(4)* 52 to 55 labelled g_1(5)* 52 to 56 labelled ok_1(4)* 52 to 58 labelled ok_1(5)* 52 to 59 labelled ok_1(6)* 53 to 51 labelled f_1(4)* 54 to 53 labelled g_1(4)* 55 to 39 labelled proper_1(5)* 55 to 46 labelled proper_1(5)* 55 to 52 labelled g_1(4)* 55 to 55 labelled g_1(5)* 55 to 45 labelled active_1(5)* 55 to 50 labelled active_1(5)* 55 to 58 labelled ok_1(5)* 55 to 59 labelled ok_1(6)* 56 to 54 labelled f_1(4)* 56 to 56 labelled g_1(4)* 56 to 58 labelled g_1(4)* 57 to 56 labelled active_1(3)* 57 to 60 labelled g_1(4)* 58 to 56 labelled g_1(5)* 58 to 58 labelled g_1(5)* 58 to 59 labelled g_1(5)* 59 to 58 labelled g_1(6)* 59 to 59 labelled g_1(6)* 60 to 56 labelled active_1(4)* 60 to 61 labelled g_1(5)* 60 to 58 labelled active_1(4)* 61 to 56 labelled active_1(5)* 61 to 58 labelled active_1(5)* 61 to 61 labelled g_1(5)* 61 to 62 labelled g_1(6)* 61 to 59 labelled active_1(5)* 62 to 56 labelled active_1(6)* 62 to 58 labelled active_1(6)* 62 to 59 labelled active_1(6)* 62 to 61 labelled g_1(5)* 62 to 62 labelled g_1(6)* 62 to 63 labelled g_1(7)* 63 to 58 labelled active_1(7)* 63 to 59 labelled active_1(7)* 63 to 62 labelled g_1(6)* 63 to 63 labelled g_1(7) ---------------------------------------- (2) YES