/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) RFCMatchBoundsTRSProof [EQUIVALENT, 20 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)) 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))))) 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: 3, 4, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 37, 38, 39, 40, 41, 42, 46, 47, 48, 49, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 75, 76, 81, 82, 83, 84 Node 3 is start node and node 4 is final node. Those nodes are connected through the following edges: * 3 to 21 labelled a'_1(0), f_1(0), g_1(0)* 3 to 26 labelled active_1(0), proper_1(0)* 3 to 24 labelled f_1(0)* 3 to 31 labelled active_1(1), proper_1(1)* 3 to 39 labelled a'_1(1)* 3 to 60 labelled a'_1(2)* 4 to 4 labelled #_1(0)* 21 to 22 labelled f_1(0)* 21 to 4 labelled ok_1(0)* 21 to 27 labelled f_1(1), g_1(1)* 21 to 28 labelled active_1(1)* 21 to 37 labelled active_1(2)* 21 to 46 labelled proper_1(1)* 22 to 23 labelled g_1(0)* 22 to 38 labelled proper_1(1)* 23 to 24 labelled f_1(0)* 23 to 31 labelled proper_1(1)* 24 to 4 labelled mark_1(0)* 24 to 29 labelled f_1(1)* 24 to 28 labelled proper_1(1)* 24 to 37 labelled proper_1(2)* 26 to 4 labelled f_1(0), g_1(0), top_1(0)* 26 to 30 labelled active_1(1), proper_1(1)* 27 to 4 labelled ok_1(1)* 27 to 27 labelled f_1(1), g_1(1)* 27 to 28 labelled active_1(1)* 27 to 37 labelled active_1(2)* 28 to 4 labelled top_1(1)* 29 to 4 labelled mark_1(1)* 29 to 29 labelled f_1(1)* 29 to 28 labelled proper_1(1)* 29 to 37 labelled proper_1(2)* 30 to 4 labelled f_1(1), g_1(1)* 30 to 30 labelled active_1(1), proper_1(1)* 31 to 28 labelled f_1(1)* 31 to 37 labelled f_1(1)* 31 to 46 labelled f_1(1), g_1(1)* 37 to 28 labelled f_1(2)* 37 to 37 labelled f_1(2)* 38 to 31 labelled g_1(1)* 39 to 40 labelled f_1(1)* 39 to 46 labelled ok_1(1)* 39 to 54 labelled f_1(2)* 39 to 57 labelled proper_1(2)* 40 to 41 labelled g_1(1)* 40 to 55 labelled proper_1(2)* 41 to 42 labelled f_1(1)* 41 to 48 labelled proper_1(2)* 42 to 28 labelled mark_1(1)* 42 to 47 labelled proper_1(2)* 42 to 37 labelled mark_1(1)* 42 to 49 labelled f_1(2)* 42 to 58 labelled proper_1(3)* 46 to 38 labelled f_1(1)* 47 to 4 labelled top_1(2)* 48 to 47 labelled f_1(2)* 48 to 58 labelled f_1(2)* 49 to 28 labelled mark_1(2)* 49 to 37 labelled mark_1(2)* 49 to 47 labelled proper_1(2)* 49 to 56 labelled f_1(3)* 49 to 58 labelled proper_1(3)* 49 to 63 labelled proper_1(4)* 54 to 38 labelled ok_1(2)* 54 to 59 labelled g_1(2)* 55 to 48 labelled g_1(2)* 56 to 28 labelled mark_1(3)* 56 to 37 labelled mark_1(3)* 56 to 47 labelled proper_1(2)* 56 to 56 labelled f_1(3)* 56 to 58 labelled proper_1(3)* 56 to 63 labelled proper_1(4)* 57 to 55 labelled f_1(2)* 58 to 47 labelled f_1(3)* 58 to 58 labelled f_1(3)* 58 to 63 labelled f_1(3)* 59 to 31 labelled ok_1(2)* 59 to 61 labelled f_1(2), g_1(2)* 59 to 58 labelled active_1(3)* 60 to 57 labelled ok_1(2)* 60 to 64 labelled f_1(3)* 61 to 28 labelled ok_1(2)* 61 to 37 labelled ok_1(2)* 61 to 46 labelled ok_1(2)* 61 to 54 labelled f_1(2)* 61 to 47 labelled active_1(2)* 61 to 62 labelled f_1(3)* 61 to 58 labelled active_1(3)* 61 to 63 labelled active_1(4)* 62 to 28 labelled ok_1(3)* 62 to 37 labelled ok_1(3)* 62 to 47 labelled active_1(2)* 62 to 62 labelled f_1(3)* 62 to 58 labelled active_1(3)* 62 to 63 labelled active_1(4)* 63 to 58 labelled f_1(4)* 63 to 63 labelled f_1(4)* 64 to 55 labelled ok_1(3)* 64 to 65 labelled g_1(3)* 65 to 48 labelled ok_1(3)* 65 to 66 labelled f_1(3)* 65 to 81 labelled active_1(4)* 66 to 47 labelled ok_1(3)* 66 to 58 labelled ok_1(3)* 66 to 75 labelled active_1(3)* 66 to 76 labelled f_1(4)* 66 to 81 labelled active_1(4)* 66 to 83 labelled active_1(5)* 75 to 4 labelled top_1(3)* 76 to 47 labelled ok_1(4)* 76 to 58 labelled ok_1(4)* 76 to 63 labelled ok_1(4)* 76 to 75 labelled active_1(3)* 76 to 76 labelled f_1(4)* 76 to 82 labelled f_1(5)* 76 to 81 labelled active_1(4)* 76 to 83 labelled active_1(5)* 76 to 84 labelled active_1(6)* 81 to 75 labelled f_1(4)* 81 to 81 labelled f_1(4)* 81 to 83 labelled f_1(4)* 82 to 58 labelled ok_1(5)* 82 to 63 labelled ok_1(5)* 82 to 76 labelled f_1(4)* 82 to 82 labelled f_1(5)* 82 to 81 labelled active_1(4)* 82 to 83 labelled active_1(5)* 82 to 84 labelled active_1(6)* 83 to 81 labelled f_1(5)* 83 to 83 labelled f_1(5)* 83 to 84 labelled f_1(5)* 84 to 83 labelled f_1(6)* 84 to 84 labelled f_1(6) ---------------------------------------- (4) YES