YES proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [SOUND, 0 ms] (2) QTRS (3) RFCMatchBoundsTRSProof [EQUIVALENT, 16 ms]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(f(X)) -> f(active(X)) f(mark(X)) -> mark(f(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(x0)) f(mark(x0)) proper(f(x0)) proper(a) proper(g(x0)) f(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(mark(x))))) f(active(X)) -> active(f(X)) mark(f(X)) -> f(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))))) f(active(X)) -> active(f(X)) mark(f(X)) -> f(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: 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, 70, 71, 72, 73, 74, 76, 77, 83, 84, 85, 86 Node 40 is start node and node 41 is final node. Those nodes are connected through the following edges: * 40 to 42 labelled a'_1(0), f_1(0), g_1(0)* 40 to 46 labelled active_1(0), proper_1(0)* 40 to 45 labelled f_1(0)* 40 to 51 labelled active_1(1), proper_1(1)* 40 to 54 labelled a'_1(1)* 40 to 68 labelled a'_1(2)* 41 to 41 labelled #_1(0)* 42 to 43 labelled f_1(0)* 42 to 41 labelled ok_1(0)* 42 to 47 labelled f_1(1), g_1(1)* 42 to 48 labelled active_1(1)* 42 to 52 labelled active_1(2)* 42 to 58 labelled proper_1(1)* 43 to 44 labelled g_1(0)* 43 to 53 labelled proper_1(1)* 44 to 45 labelled f_1(0)* 44 to 51 labelled proper_1(1)* 45 to 41 labelled mark_1(0)* 45 to 49 labelled f_1(1)* 45 to 48 labelled proper_1(1)* 45 to 52 labelled proper_1(2)* 46 to 41 labelled f_1(0), g_1(0), top_1(0)* 46 to 50 labelled active_1(1), proper_1(1)* 47 to 41 labelled ok_1(1)* 47 to 47 labelled f_1(1), g_1(1)* 47 to 48 labelled active_1(1)* 47 to 52 labelled active_1(2)* 48 to 41 labelled top_1(1)* 49 to 41 labelled mark_1(1)* 49 to 49 labelled f_1(1)* 49 to 48 labelled proper_1(1)* 49 to 52 labelled proper_1(2)* 50 to 41 labelled f_1(1), g_1(1)* 50 to 50 labelled active_1(1), proper_1(1)* 51 to 48 labelled f_1(1)* 51 to 52 labelled f_1(1)* 51 to 58 labelled f_1(1), g_1(1)* 52 to 48 labelled f_1(2)* 52 to 52 labelled f_1(2)* 53 to 51 labelled g_1(1)* 54 to 55 labelled f_1(1)* 54 to 58 labelled ok_1(1)* 54 to 62 labelled f_1(2)* 54 to 65 labelled proper_1(2)* 55 to 56 labelled g_1(1)* 55 to 63 labelled proper_1(2)* 56 to 57 labelled f_1(1)* 56 to 60 labelled proper_1(2)* 57 to 48 labelled mark_1(1)* 57 to 59 labelled proper_1(2)* 57 to 52 labelled mark_1(1)* 57 to 61 labelled f_1(2)* 57 to 66 labelled proper_1(3)* 58 to 53 labelled f_1(1)* 59 to 41 labelled top_1(2)* 60 to 59 labelled f_1(2)* 60 to 66 labelled f_1(2)* 61 to 48 labelled mark_1(2)* 61 to 52 labelled mark_1(2)* 61 to 59 labelled proper_1(2)* 61 to 64 labelled f_1(3)* 61 to 66 labelled proper_1(3)* 61 to 71 labelled proper_1(4)* 62 to 53 labelled ok_1(2)* 62 to 67 labelled g_1(2)* 63 to 60 labelled g_1(2)* 64 to 48 labelled mark_1(3)* 64 to 52 labelled mark_1(3)* 64 to 59 labelled proper_1(2)* 64 to 64 labelled f_1(3)* 64 to 66 labelled proper_1(3)* 64 to 71 labelled proper_1(4)* 65 to 63 labelled f_1(2)* 66 to 59 labelled f_1(3)* 66 to 66 labelled f_1(3)* 66 to 71 labelled f_1(3)* 67 to 51 labelled ok_1(2)* 67 to 69 labelled f_1(2), g_1(2)* 67 to 66 labelled active_1(3)* 68 to 65 labelled ok_1(2)* 68 to 72 labelled f_1(3)* 69 to 48 labelled ok_1(2)* 69 to 52 labelled ok_1(2)* 69 to 58 labelled ok_1(2)* 69 to 62 labelled f_1(2)* 69 to 59 labelled active_1(2)* 69 to 70 labelled f_1(3)* 69 to 66 labelled active_1(3)* 69 to 71 labelled active_1(4)* 70 to 48 labelled ok_1(3)* 70 to 52 labelled ok_1(3)* 70 to 59 labelled active_1(2)* 70 to 70 labelled f_1(3)* 70 to 66 labelled active_1(3)* 70 to 71 labelled active_1(4)* 71 to 66 labelled f_1(4)* 71 to 71 labelled f_1(4)* 72 to 63 labelled ok_1(3)* 72 to 73 labelled g_1(3)* 73 to 60 labelled ok_1(3)* 73 to 74 labelled f_1(3)* 73 to 83 labelled active_1(4)* 74 to 59 labelled ok_1(3)* 74 to 66 labelled ok_1(3)* 74 to 76 labelled active_1(3)* 74 to 77 labelled f_1(4)* 74 to 83 labelled active_1(4)* 74 to 85 labelled active_1(5)* 76 to 41 labelled top_1(3)* 77 to 59 labelled ok_1(4)* 77 to 66 labelled ok_1(4)* 77 to 71 labelled ok_1(4)* 77 to 76 labelled active_1(3)* 77 to 77 labelled f_1(4)* 77 to 84 labelled f_1(5)* 77 to 83 labelled active_1(4)* 77 to 85 labelled active_1(5)* 77 to 86 labelled active_1(6)* 83 to 76 labelled f_1(4)* 83 to 83 labelled f_1(4)* 83 to 85 labelled f_1(4)* 84 to 66 labelled ok_1(5)* 84 to 71 labelled ok_1(5)* 84 to 77 labelled f_1(4)* 84 to 84 labelled f_1(5)* 84 to 83 labelled active_1(4)* 84 to 85 labelled active_1(5)* 84 to 86 labelled active_1(6)* 85 to 83 labelled f_1(5)* 85 to 85 labelled f_1(5)* 85 to 86 labelled f_1(5)* 86 to 85 labelled f_1(6)* 86 to 86 labelled f_1(6) ---------------------------------------- (4) YES