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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 4 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:
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, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97

Node 46 is start node and node 47 is final node.

Those nodes are connected through the following edges:

* 46 to 48 labelled f_1(0), c_1(0), g_1(0), d_1(0), h_1(0)* 46 to 51 labelled d_1(0), f_1(0), h_1(0)* 46 to 50 labelled d_1(0)* 46 to 52 labelled active_1(0), proper_1(0)* 46 to 61 labelled active_1(1), proper_1(1)* 46 to 62 labelled d_1(1)* 46 to 63 labelled d_1(1)* 46 to 65 labelled proper_1(1)* 46 to 75 labelled f_1(1)* 46 to 83 labelled proper_1(2)* 47 to 47 labelled #_1(0)* 48 to 49 labelled g_1(0)* 48 to 47 labelled ok_1(0)* 48 to 53 labelled f_1(1), c_1(1), g_1(1), d_1(1), h_1(1)* 48 to 54 labelled active_1(1)* 48 to 68 labelled active_1(2)* 48 to 69 labelled d_1(2)* 48 to 70 labelled d_1(2)* 48 to 79 labelled f_1(2)* 48 to 84 labelled proper_1(1)* 48 to 87 labelled proper_1(3)* 48 to 91 labelled proper_1(2)* 49 to 50 labelled f_1(0)* 49 to 67 labelled proper_1(1)* 50 to 51 labelled c_1(0)* 50 to 66 labelled proper_1(1)* 51 to 47 labelled mark_1(0)* 51 to 55 labelled f_1(1), h_1(1)* 51 to 54 labelled proper_1(1)* 51 to 68 labelled proper_1(2)* 52 to 47 labelled f_1(0), h_1(0), c_1(0), g_1(0), d_1(0), top_1(0)* 52 to 56 labelled f_1(1)* 52 to 60 labelled active_1(1), proper_1(1)* 52 to 58 labelled d_1(1)* 52 to 55 labelled d_1(1)* 52 to 72 labelled proper_1(2)* 53 to 47 labelled ok_1(1)* 53 to 53 labelled f_1(1), c_1(1), g_1(1), d_1(1), h_1(1)* 53 to 54 labelled active_1(1)* 53 to 68 labelled active_1(2)* 53 to 69 labelled d_1(2)* 53 to 70 labelled d_1(2)* 53 to 79 labelled f_1(2)* 53 to 87 labelled proper_1(3)* 53 to 91 labelled proper_1(2)* 54 to 47 labelled top_1(1)* 55 to 47 labelled mark_1(1)* 55 to 55 labelled f_1(1), h_1(1)* 55 to 54 labelled proper_1(1)* 55 to 68 labelled proper_1(2)* 56 to 57 labelled g_1(1)* 56 to 88 labelled proper_1(2)* 57 to 58 labelled f_1(1)* 57 to 85 labelled proper_1(2)* 58 to 59 labelled c_1(1)* 58 to 73 labelled proper_1(2)* 59 to 47 labelled mark_1(1)* 59 to 55 labelled f_1(1), h_1(1)* 59 to 54 labelled proper_1(1)* 59 to 68 labelled proper_1(2)* 60 to 47 labelled f_1(1), h_1(1), c_1(1), g_1(1), d_1(1)* 60 to 56 labelled f_1(1)* 60 to 60 labelled active_1(1), proper_1(1)* 60 to 58 labelled d_1(1)* 60 to 55 labelled d_1(1)* 60 to 72 labelled proper_1(2)* 61 to 54 labelled f_1(1), h_1(1)* 61 to 66 labelled d_1(1)* 61 to 68 labelled f_1(1), h_1(1), d_1(1)* 61 to 84 labelled f_1(1), c_1(1), g_1(1), d_1(1), h_1(1)* 61 to 87 labelled f_1(1), c_1(1), g_1(1), d_1(1), h_1(1)* 61 to 91 labelled f_1(1), c_1(1), g_1(1), d_1(1), h_1(1)* 62 to 54 labelled mark_1(1)* 62 to 74 labelled proper_1(2)* 62 to 68 labelled mark_1(1)* 62 to 69 labelled f_1(2), h_1(2)* 62 to 92 labelled proper_1(3)* 63 to 64 labelled c_1(1)* 63 to 86 labelled proper_1(2)* 64 to 54 labelled mark_1(1)* 64 to 74 labelled proper_1(2)* 64 to 68 labelled mark_1(1)* 64 to 69 labelled f_1(2), h_1(2)* 64 to 92 labelled proper_1(3)* 65 to 54 labelled d_1(1)* 66 to 54 labelled c_1(1)* 66 to 68 labelled c_1(1)* 67 to 66 labelled f_1(1)* 68 to 54 labelled f_1(2), h_1(2)* 68 to 68 labelled f_1(2), h_1(2)* 69 to 54 labelled mark_1(2)* 69 to 68 labelled mark_1(2)* 69 to 74 labelled proper_1(2)* 69 to 89 labelled f_1(3), h_1(3)* 69 to 92 labelled proper_1(3)* 69 to 97 labelled proper_1(4)* 70 to 71 labelled c_1(2)* 70 to 90 labelled proper_1(3)* 71 to 54 labelled mark_1(2)* 71 to 68 labelled mark_1(2)* 71 to 74 labelled proper_1(2)* 71 to 89 labelled f_1(3), h_1(3)* 71 to 92 labelled proper_1(3)* 71 to 97 labelled proper_1(4)* 72 to 54 labelled d_1(2)* 72 to 68 labelled d_1(2)* 72 to 73 labelled d_1(2)* 72 to 88 labelled f_1(2)* 73 to 54 labelled c_1(2)* 73 to 68 labelled c_1(2)* 74 to 47 labelled top_1(2)* 75 to 76 labelled g_1(1)* 75 to 95 labelled proper_1(2)* 76 to 77 labelled f_1(1)* 76 to 93 labelled proper_1(2)* 77 to 78 labelled c_1(1)* 77 to 86 labelled proper_1(2)* 78 to 54 labelled mark_1(1)* 78 to 74 labelled proper_1(2)* 78 to 68 labelled mark_1(1)* 78 to 69 labelled f_1(2), h_1(2)* 78 to 92 labelled proper_1(3)* 79 to 80 labelled g_1(2)* 79 to 96 labelled proper_1(3)* 80 to 81 labelled f_1(2)* 80 to 94 labelled proper_1(3)* 81 to 82 labelled c_1(2)* 81 to 90 labelled proper_1(3)* 82 to 54 labelled mark_1(2)* 82 to 74 labelled proper_1(2)* 82 to 68 labelled mark_1(2)* 82 to 89 labelled f_1(3), h_1(3)* 82 to 92 labelled proper_1(3)* 82 to 97 labelled proper_1(4)* 83 to 74 labelled d_1(2)* 83 to 86 labelled d_1(2)* 83 to 92 labelled d_1(2)* 83 to 95 labelled f_1(2)* 84 to 67 labelled g_1(1)* 85 to 73 labelled f_1(2)* 86 to 74 labelled c_1(2)* 86 to 92 labelled c_1(2)* 87 to 74 labelled d_1(3)* 87 to 90 labelled d_1(3)* 87 to 92 labelled d_1(3)* 87 to 96 labelled f_1(3)* 87 to 97 labelled d_1(3)* 88 to 85 labelled g_1(2)* 89 to 54 labelled mark_1(3)* 89 to 68 labelled mark_1(3)* 89 to 89 labelled f_1(3), h_1(3)* 89 to 74 labelled proper_1(2)* 89 to 92 labelled proper_1(3)* 89 to 97 labelled proper_1(4)* 90 to 74 labelled c_1(3)* 90 to 92 labelled c_1(3)* 90 to 97 labelled c_1(3)* 91 to 87 labelled f_1(2), c_1(2), g_1(2), d_1(2), h_1(2)* 91 to 91 labelled f_1(2), c_1(2), g_1(2), d_1(2), h_1(2)* 92 to 74 labelled f_1(3), h_1(3)* 92 to 92 labelled f_1(3), h_1(3)* 92 to 97 labelled f_1(3), h_1(3)* 93 to 86 labelled f_1(2)* 94 to 90 labelled f_1(3)* 95 to 93 labelled g_1(2)* 96 to 94 labelled g_1(3)* 97 to 92 labelled f_1(4), h_1(4)* 97 to 97 labelled f_1(4), h_1(4)


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(4)
YES