/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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 [SOUND, 0 ms]
(2) QTRS
(3) RFCMatchBoundsTRSProof [EQUIVALENT, 14 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))

The set Q consists of the following terms:

   active(c(x0))
   active(h(x0))
   active(f(x0))
   f(mark(x0))
   h(mark(x0))
   proper(f(x0))
   proper(c(x0))
   proper(g(x0))
   proper(d(x0))
   proper(h(x0))
   f(ok(x0))
   c(ok(x0))
   g(ok(x0))
   d(ok(x0))
   h(ok(x0))
   top(mark(x0))
   top(ok(x0))


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(1) QTRS Reverse (SOUND)
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:
24, 25, 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, 64, 65, 66, 67, 71, 72, 73, 74, 88, 89, 90, 91, 92, 93, 94

Node 24 is start node and node 25 is final node.

Those nodes are connected through the following edges:

* 24 to 28 labelled f_1(0), c_1(0), g_1(0), d_1(0), h_1(0)* 24 to 31 labelled d_1(0), f_1(0), h_1(0)* 24 to 30 labelled d_1(0)* 24 to 32 labelled active_1(0), proper_1(0)* 24 to 41 labelled active_1(1), proper_1(1)* 24 to 42 labelled d_1(1)* 24 to 43 labelled d_1(1)* 24 to 45 labelled proper_1(1)* 24 to 55 labelled f_1(1)* 24 to 64 labelled proper_1(2)* 25 to 25 labelled #_1(0)* 28 to 29 labelled g_1(0)* 28 to 25 labelled ok_1(0)* 28 to 33 labelled f_1(1), c_1(1), g_1(1), d_1(1), h_1(1)* 28 to 34 labelled active_1(1)* 28 to 48 labelled active_1(2)* 28 to 49 labelled d_1(2)* 28 to 50 labelled d_1(2)* 28 to 59 labelled f_1(2)* 28 to 65 labelled proper_1(1)* 28 to 71 labelled proper_1(3)* 28 to 88 labelled proper_1(2)* 29 to 30 labelled f_1(0)* 29 to 47 labelled proper_1(1)* 30 to 31 labelled c_1(0)* 30 to 46 labelled proper_1(1)* 31 to 25 labelled mark_1(0)* 31 to 35 labelled f_1(1), h_1(1)* 31 to 34 labelled proper_1(1)* 31 to 48 labelled proper_1(2)* 32 to 25 labelled f_1(0), h_1(0), c_1(0), g_1(0), d_1(0), top_1(0)* 32 to 36 labelled f_1(1)* 32 to 40 labelled active_1(1), proper_1(1)* 32 to 38 labelled d_1(1)* 32 to 35 labelled d_1(1)* 32 to 52 labelled proper_1(2)* 33 to 25 labelled ok_1(1)* 33 to 33 labelled f_1(1), c_1(1), g_1(1), d_1(1), h_1(1)* 33 to 34 labelled active_1(1)* 33 to 48 labelled active_1(2)* 33 to 49 labelled d_1(2)* 33 to 50 labelled d_1(2)* 33 to 59 labelled f_1(2)* 33 to 71 labelled proper_1(3)* 33 to 88 labelled proper_1(2)* 34 to 25 labelled top_1(1)* 35 to 25 labelled mark_1(1)* 35 to 35 labelled f_1(1), h_1(1)* 35 to 34 labelled proper_1(1)* 35 to 48 labelled proper_1(2)* 36 to 37 labelled g_1(1)* 36 to 72 labelled proper_1(2)* 37 to 38 labelled f_1(1)* 37 to 66 labelled proper_1(2)* 38 to 39 labelled c_1(1)* 38 to 53 labelled proper_1(2)* 39 to 25 labelled mark_1(1)* 39 to 35 labelled f_1(1), h_1(1)* 39 to 34 labelled proper_1(1)* 39 to 48 labelled proper_1(2)* 40 to 25 labelled f_1(1), h_1(1), c_1(1), g_1(1), d_1(1)* 40 to 36 labelled f_1(1)* 40 to 40 labelled active_1(1), proper_1(1)* 40 to 38 labelled d_1(1)* 40 to 35 labelled d_1(1)* 40 to 52 labelled proper_1(2)* 41 to 34 labelled f_1(1), h_1(1)* 41 to 46 labelled d_1(1)* 41 to 48 labelled f_1(1), h_1(1), d_1(1)* 41 to 65 labelled f_1(1), c_1(1), g_1(1), d_1(1), h_1(1)* 41 to 71 labelled f_1(1), c_1(1), g_1(1), d_1(1), h_1(1)* 41 to 88 labelled f_1(1), c_1(1), g_1(1), d_1(1), h_1(1)* 42 to 34 labelled mark_1(1)* 42 to 54 labelled proper_1(2)* 42 to 48 labelled mark_1(1)* 42 to 49 labelled f_1(2), h_1(2)* 42 to 89 labelled proper_1(3)* 43 to 44 labelled c_1(1)* 43 to 67 labelled proper_1(2)* 44 to 34 labelled mark_1(1)* 44 to 54 labelled proper_1(2)* 44 to 48 labelled mark_1(1)* 44 to 49 labelled f_1(2), h_1(2)* 44 to 89 labelled proper_1(3)* 45 to 34 labelled d_1(1)* 46 to 34 labelled c_1(1)* 46 to 48 labelled c_1(1)* 47 to 46 labelled f_1(1)* 48 to 34 labelled f_1(2), h_1(2)* 48 to 48 labelled f_1(2), h_1(2)* 49 to 34 labelled mark_1(2)* 49 to 48 labelled mark_1(2)* 49 to 54 labelled proper_1(2)* 49 to 73 labelled f_1(3), h_1(3)* 49 to 89 labelled proper_1(3)* 49 to 94 labelled proper_1(4)* 50 to 51 labelled c_1(2)* 50 to 74 labelled proper_1(3)* 51 to 34 labelled mark_1(2)* 51 to 48 labelled mark_1(2)* 51 to 54 labelled proper_1(2)* 51 to 73 labelled f_1(3), h_1(3)* 51 to 89 labelled proper_1(3)* 51 to 94 labelled proper_1(4)* 52 to 34 labelled d_1(2)* 52 to 48 labelled d_1(2)* 52 to 53 labelled d_1(2)* 52 to 72 labelled f_1(2)* 53 to 34 labelled c_1(2)* 53 to 48 labelled c_1(2)* 54 to 25 labelled top_1(2)* 55 to 56 labelled g_1(1)* 55 to 92 labelled proper_1(2)* 56 to 57 labelled f_1(1)* 56 to 90 labelled proper_1(2)* 57 to 58 labelled c_1(1)* 57 to 67 labelled proper_1(2)* 58 to 34 labelled mark_1(1)* 58 to 54 labelled proper_1(2)* 58 to 48 labelled mark_1(1)* 58 to 49 labelled f_1(2), h_1(2)* 58 to 89 labelled proper_1(3)* 59 to 60 labelled g_1(2)* 59 to 93 labelled proper_1(3)* 60 to 61 labelled f_1(2)* 60 to 91 labelled proper_1(3)* 61 to 62 labelled c_1(2)* 61 to 74 labelled proper_1(3)* 62 to 34 labelled mark_1(2)* 62 to 54 labelled proper_1(2)* 62 to 48 labelled mark_1(2)* 62 to 73 labelled f_1(3), h_1(3)* 62 to 89 labelled proper_1(3)* 62 to 94 labelled proper_1(4)* 64 to 54 labelled d_1(2)* 64 to 67 labelled d_1(2)* 64 to 89 labelled d_1(2)* 64 to 92 labelled f_1(2)* 65 to 47 labelled g_1(1)* 66 to 53 labelled f_1(2)* 67 to 54 labelled c_1(2)* 67 to 89 labelled c_1(2)* 71 to 54 labelled d_1(3)* 71 to 74 labelled d_1(3)* 71 to 89 labelled d_1(3)* 71 to 93 labelled f_1(3)* 71 to 94 labelled d_1(3)* 72 to 66 labelled g_1(2)* 73 to 34 labelled mark_1(3)* 73 to 48 labelled mark_1(3)* 73 to 73 labelled f_1(3), h_1(3)* 73 to 54 labelled proper_1(2)* 73 to 89 labelled proper_1(3)* 73 to 94 labelled proper_1(4)* 74 to 54 labelled c_1(3)* 74 to 89 labelled c_1(3)* 74 to 94 labelled c_1(3)* 88 to 71 labelled f_1(2), c_1(2), g_1(2), d_1(2), h_1(2)* 88 to 88 labelled f_1(2), c_1(2), g_1(2), d_1(2), h_1(2)* 89 to 54 labelled f_1(3), h_1(3)* 89 to 89 labelled f_1(3), h_1(3)* 89 to 94 labelled f_1(3), h_1(3)* 90 to 67 labelled f_1(2)* 91 to 74 labelled f_1(3)* 92 to 90 labelled g_1(2)* 93 to 91 labelled g_1(3)* 94 to 89 labelled f_1(4), h_1(4)* 94 to 94 labelled f_1(4), h_1(4)


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(4)
YES