/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 0 ms]
(4) YES


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(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   active(f(f(X))) -> mark(c(f(g(f(X)))))
   active(c(X)) -> mark(d(X))
   active(h(X)) -> mark(c(d(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(c(X)) -> c(proper(X))
   proper(g(X)) -> g(proper(X))
   proper(d(X)) -> d(proper(X))
   proper(h(X)) -> h(proper(X))
   f(ok(X)) -> ok(f(X))
   c(ok(X)) -> ok(c(X))
   g(ok(X)) -> ok(g(X))
   d(ok(X)) -> ok(d(X))
   h(ok(X)) -> ok(h(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

Q is empty.

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(1) QTRS Reverse (EQUIVALENT)
We applied the QTRS Reverse Processor [REVERSE].
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(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f(f(active(X))) -> f(g(f(c(mark(X)))))
   c(active(X)) -> d(mark(X))
   h(active(X)) -> d(c(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))
   c(proper(X)) -> proper(c(X))
   g(proper(X)) -> proper(g(X))
   d(proper(X)) -> proper(d(X))
   h(proper(X)) -> proper(h(X))
   ok(f(X)) -> f(ok(X))
   ok(c(X)) -> c(ok(X))
   ok(g(X)) -> g(ok(X))
   ok(d(X)) -> d(ok(X))
   ok(h(X)) -> h(ok(X))
   mark(top(X)) -> proper(top(X))
   ok(top(X)) -> active(top(X))

Q is empty.

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(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(f(active(X))) -> f(g(f(c(mark(X)))))
   c(active(X)) -> d(mark(X))
   h(active(X)) -> d(c(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))
   c(proper(X)) -> proper(c(X))
   g(proper(X)) -> proper(g(X))
   d(proper(X)) -> proper(d(X))
   h(proper(X)) -> proper(h(X))
   ok(f(X)) -> f(ok(X))
   ok(c(X)) -> c(ok(X))
   ok(g(X)) -> g(ok(X))
   ok(d(X)) -> d(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:
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, 64, 65, 66, 67, 68, 69, 70, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84

Node 30 is start node and node 31 is final node.

Those nodes are connected through the following edges:

* 30 to 32 labelled f_1(0), c_1(0), g_1(0), d_1(0), h_1(0)* 30 to 35 labelled d_1(0), f_1(0), h_1(0)* 30 to 34 labelled d_1(0)* 30 to 36 labelled active_1(0), proper_1(0)* 30 to 45 labelled active_1(1), proper_1(1)* 30 to 46 labelled d_1(1)* 30 to 47 labelled d_1(1)* 30 to 49 labelled proper_1(1)* 30 to 59 labelled f_1(1)* 30 to 67 labelled proper_1(2)* 31 to 31 labelled #_1(0)* 32 to 33 labelled g_1(0)* 32 to 31 labelled ok_1(0)* 32 to 37 labelled f_1(1), c_1(1), g_1(1), d_1(1), h_1(1)* 32 to 38 labelled active_1(1)* 32 to 52 labelled active_1(2)* 32 to 53 labelled d_1(2)* 32 to 54 labelled d_1(2)* 32 to 63 labelled f_1(2)* 32 to 68 labelled proper_1(1)* 32 to 74 labelled proper_1(3)* 32 to 78 labelled proper_1(2)* 33 to 34 labelled f_1(0)* 33 to 51 labelled proper_1(1)* 34 to 35 labelled c_1(0)* 34 to 50 labelled proper_1(1)* 35 to 31 labelled mark_1(0)* 35 to 39 labelled f_1(1), h_1(1)* 35 to 38 labelled proper_1(1)* 35 to 52 labelled proper_1(2)* 36 to 31 labelled f_1(0), h_1(0), c_1(0), g_1(0), d_1(0), top_1(0)* 36 to 40 labelled f_1(1)* 36 to 44 labelled active_1(1), proper_1(1)* 36 to 42 labelled d_1(1)* 36 to 39 labelled d_1(1)* 36 to 56 labelled proper_1(2)* 37 to 31 labelled ok_1(1)* 37 to 37 labelled f_1(1), c_1(1), g_1(1), d_1(1), h_1(1)* 37 to 38 labelled active_1(1)* 37 to 52 labelled active_1(2)* 37 to 53 labelled d_1(2)* 37 to 54 labelled d_1(2)* 37 to 63 labelled f_1(2)* 37 to 74 labelled proper_1(3)* 37 to 78 labelled proper_1(2)* 38 to 31 labelled top_1(1)* 39 to 31 labelled mark_1(1)* 39 to 39 labelled f_1(1), h_1(1)* 39 to 38 labelled proper_1(1)* 39 to 52 labelled proper_1(2)* 40 to 41 labelled g_1(1)* 40 to 75 labelled proper_1(2)* 41 to 42 labelled f_1(1)* 41 to 69 labelled proper_1(2)* 42 to 43 labelled c_1(1)* 42 to 57 labelled proper_1(2)* 43 to 31 labelled mark_1(1)* 43 to 39 labelled f_1(1), h_1(1)* 43 to 38 labelled proper_1(1)* 43 to 52 labelled proper_1(2)* 44 to 31 labelled f_1(1), h_1(1), c_1(1), g_1(1), d_1(1)* 44 to 40 labelled f_1(1)* 44 to 44 labelled active_1(1), proper_1(1)* 44 to 42 labelled d_1(1)* 44 to 39 labelled d_1(1)* 44 to 56 labelled proper_1(2)* 45 to 38 labelled f_1(1), h_1(1)* 45 to 50 labelled d_1(1)* 45 to 52 labelled f_1(1), h_1(1), d_1(1)* 45 to 68 labelled f_1(1), c_1(1), g_1(1), d_1(1), h_1(1)* 45 to 74 labelled f_1(1), c_1(1), g_1(1), d_1(1), h_1(1)* 45 to 78 labelled f_1(1), c_1(1), g_1(1), d_1(1), h_1(1)* 46 to 38 labelled mark_1(1)* 46 to 58 labelled proper_1(2)* 46 to 52 labelled mark_1(1)* 46 to 53 labelled f_1(2), h_1(2)* 46 to 79 labelled proper_1(3)* 47 to 48 labelled c_1(1)* 47 to 70 labelled proper_1(2)* 48 to 38 labelled mark_1(1)* 48 to 58 labelled proper_1(2)* 48 to 52 labelled mark_1(1)* 48 to 53 labelled f_1(2), h_1(2)* 48 to 79 labelled proper_1(3)* 49 to 38 labelled d_1(1)* 50 to 38 labelled c_1(1)* 50 to 52 labelled c_1(1)* 51 to 50 labelled f_1(1)* 52 to 38 labelled f_1(2), h_1(2)* 52 to 52 labelled f_1(2), h_1(2)* 53 to 38 labelled mark_1(2)* 53 to 52 labelled mark_1(2)* 53 to 58 labelled proper_1(2)* 53 to 76 labelled f_1(3), h_1(3)* 53 to 79 labelled proper_1(3)* 53 to 84 labelled proper_1(4)* 54 to 55 labelled c_1(2)* 54 to 77 labelled proper_1(3)* 55 to 38 labelled mark_1(2)* 55 to 52 labelled mark_1(2)* 55 to 58 labelled proper_1(2)* 55 to 76 labelled f_1(3), h_1(3)* 55 to 79 labelled proper_1(3)* 55 to 84 labelled proper_1(4)* 56 to 38 labelled d_1(2)* 56 to 52 labelled d_1(2)* 56 to 57 labelled d_1(2)* 56 to 75 labelled f_1(2)* 57 to 38 labelled c_1(2)* 57 to 52 labelled c_1(2)* 58 to 31 labelled top_1(2)* 59 to 60 labelled g_1(1)* 59 to 82 labelled proper_1(2)* 60 to 61 labelled f_1(1)* 60 to 80 labelled proper_1(2)* 61 to 62 labelled c_1(1)* 61 to 70 labelled proper_1(2)* 62 to 38 labelled mark_1(1)* 62 to 58 labelled proper_1(2)* 62 to 52 labelled mark_1(1)* 62 to 53 labelled f_1(2), h_1(2)* 62 to 79 labelled proper_1(3)* 63 to 64 labelled g_1(2)* 63 to 83 labelled proper_1(3)* 64 to 65 labelled f_1(2)* 64 to 81 labelled proper_1(3)* 65 to 66 labelled c_1(2)* 65 to 77 labelled proper_1(3)* 66 to 38 labelled mark_1(2)* 66 to 58 labelled proper_1(2)* 66 to 52 labelled mark_1(2)* 66 to 76 labelled f_1(3), h_1(3)* 66 to 79 labelled proper_1(3)* 66 to 84 labelled proper_1(4)* 67 to 58 labelled d_1(2)* 67 to 70 labelled d_1(2)* 67 to 79 labelled d_1(2)* 67 to 82 labelled f_1(2)* 68 to 51 labelled g_1(1)* 69 to 57 labelled f_1(2)* 70 to 58 labelled c_1(2)* 70 to 79 labelled c_1(2)* 74 to 58 labelled d_1(3)* 74 to 77 labelled d_1(3)* 74 to 79 labelled d_1(3)* 74 to 83 labelled f_1(3)* 74 to 84 labelled d_1(3)* 75 to 69 labelled g_1(2)* 76 to 38 labelled mark_1(3)* 76 to 52 labelled mark_1(3)* 76 to 76 labelled f_1(3), h_1(3)* 76 to 58 labelled proper_1(2)* 76 to 79 labelled proper_1(3)* 76 to 84 labelled proper_1(4)* 77 to 58 labelled c_1(3)* 77 to 79 labelled c_1(3)* 77 to 84 labelled c_1(3)* 78 to 74 labelled f_1(2), c_1(2), g_1(2), d_1(2), h_1(2)* 78 to 78 labelled f_1(2), c_1(2), g_1(2), d_1(2), h_1(2)* 79 to 58 labelled f_1(3), h_1(3)* 79 to 79 labelled f_1(3), h_1(3)* 79 to 84 labelled f_1(3), h_1(3)* 80 to 70 labelled f_1(2)* 81 to 77 labelled f_1(3)* 82 to 80 labelled g_1(2)* 83 to 81 labelled g_1(3)* 84 to 79 labelled f_1(4), h_1(4)* 84 to 84 labelled f_1(4), h_1(4)


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(4)
YES