YES
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 18 ms]
(4) YES


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

   active(c) -> mark(f(g(c)))
   active(f(g(X))) -> mark(g(X))
   proper(c) -> ok(c)
   proper(f(X)) -> f(proper(X))
   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(c)
   active(f(g(x0)))
   proper(c)
   proper(f(x0))
   proper(g(x0))
   f(ok(x0))
   g(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:

   c'(active(x)) -> c'(g(f(mark(x))))
   g(f(active(X))) -> g(mark(X))
   c'(proper(x)) -> c'(ok(x))
   f(proper(X)) -> proper(f(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.

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(3) RFCMatchBoundsTRSProof (EQUIVALENT)
Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 5. This implies Q-termination of R.
The following rules were used to construct the certificate:

   c'(active(x)) -> c'(g(f(mark(x))))
   g(f(active(X))) -> g(mark(X))
   c'(proper(x)) -> c'(ok(x))
   f(proper(X)) -> proper(f(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:
32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 81, 82

Node 32 is start node and node 33 is final node.

Those nodes are connected through the following edges:

* 32 to 34 labelled c'_1(0), f_1(0), g_1(0)* 32 to 36 labelled g_1(0)* 32 to 37 labelled proper_1(0), active_1(0)* 32 to 42 labelled c'_1(1)* 32 to 45 labelled proper_1(1)* 32 to 44 labelled g_1(1)* 32 to 56 labelled proper_1(2)* 32 to 61 labelled c'_1(2)* 32 to 75 labelled c'_1(3)* 33 to 33 labelled #_1(0)* 34 to 35 labelled g_1(0)* 34 to 33 labelled ok_1(0)* 34 to 38 labelled f_1(1), g_1(1)* 34 to 39 labelled active_1(1)* 34 to 47 labelled proper_1(1)* 34 to 50 labelled g_1(2)* 34 to 58 labelled proper_1(3)* 34 to 62 labelled proper_1(2)* 35 to 36 labelled f_1(0)* 35 to 46 labelled proper_1(1)* 36 to 33 labelled mark_1(0)* 36 to 39 labelled proper_1(1)* 37 to 33 labelled f_1(0), g_1(0), top_1(0)* 37 to 40 labelled proper_1(1)* 37 to 41 labelled g_1(1)* 37 to 48 labelled proper_1(2)* 38 to 33 labelled ok_1(1)* 38 to 38 labelled f_1(1), g_1(1)* 38 to 39 labelled active_1(1)* 38 to 50 labelled g_1(2)* 38 to 58 labelled proper_1(3)* 38 to 62 labelled proper_1(2)* 39 to 33 labelled top_1(1)* 40 to 33 labelled f_1(1), g_1(1)* 40 to 40 labelled proper_1(1)* 40 to 41 labelled g_1(1)* 40 to 48 labelled proper_1(2)* 41 to 33 labelled mark_1(1)* 41 to 39 labelled proper_1(1)* 42 to 43 labelled g_1(1)* 42 to 47 labelled ok_1(1)* 42 to 59 labelled proper_1(2)* 42 to 60 labelled g_1(2)* 42 to 58 labelled ok_1(1)* 42 to 62 labelled ok_1(1)* 42 to 65 labelled f_1(2), g_1(2)* 42 to 68 labelled g_1(3)* 42 to 71 labelled proper_1(4)* 43 to 44 labelled f_1(1)* 43 to 57 labelled proper_1(2)* 44 to 39 labelled mark_1(1)* 44 to 49 labelled proper_1(2)* 45 to 39 labelled g_1(1)* 45 to 47 labelled f_1(1), g_1(1)* 45 to 58 labelled f_1(1), g_1(1)* 45 to 62 labelled f_1(1), g_1(1)* 46 to 39 labelled f_1(1)* 47 to 46 labelled g_1(1)* 48 to 39 labelled g_1(2)* 49 to 33 labelled top_1(2)* 50 to 39 labelled mark_1(2)* 50 to 49 labelled proper_1(2)* 56 to 49 labelled g_1(2)* 57 to 49 labelled f_1(2)* 58 to 49 labelled g_1(3)* 59 to 57 labelled g_1(2)* 60 to 46 labelled ok_1(2)* 60 to 63 labelled f_1(2)* 60 to 49 labelled ok_1(2)* 60 to 67 labelled active_1(3)* 61 to 59 labelled ok_1(2)* 61 to 64 labelled g_1(3)* 61 to 72 labelled g_1(4)* 61 to 71 labelled ok_1(2)* 61 to 74 labelled proper_1(5)* 62 to 58 labelled f_1(2), g_1(2)* 62 to 62 labelled f_1(2), g_1(2)* 63 to 39 labelled ok_1(2)* 63 to 49 labelled active_1(2)* 64 to 57 labelled ok_1(3)* 64 to 66 labelled f_1(3)* 64 to 67 labelled ok_1(3)* 64 to 73 labelled active_1(4)* 65 to 58 labelled ok_1(2)* 65 to 62 labelled ok_1(2)* 65 to 66 labelled g_1(3)* 65 to 69 labelled f_1(3), g_1(3)* 66 to 49 labelled ok_1(3)* 66 to 67 labelled active_1(3)* 67 to 33 labelled top_1(3)* 68 to 49 labelled mark_1(3)* 68 to 67 labelled proper_1(3)* 69 to 58 labelled ok_1(3)* 69 to 62 labelled ok_1(3)* 69 to 70 labelled g_1(4)* 69 to 69 labelled f_1(3), g_1(3)* 70 to 49 labelled ok_1(4)* 70 to 67 labelled active_1(3)* 71 to 67 labelled g_1(4)* 72 to 67 labelled mark_1(4)* 72 to 73 labelled proper_1(4)* 73 to 33 labelled top_1(4)* 74 to 73 labelled g_1(5)* 75 to 74 labelled ok_1(3)* 75 to 81 labelled g_1(4)* 81 to 73 labelled ok_1(4)* 81 to 82 labelled active_1(5)* 82 to 33 labelled top_1(5)


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(4)
YES