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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms]
(2) YES


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

   active(f(f(a))) -> mark(f(g(f(a))))
   mark(f(X)) -> active(f(X))
   mark(a) -> active(a)
   mark(g(X)) -> active(g(mark(X)))
   f(mark(X)) -> f(X)
   f(active(X)) -> f(X)
   g(mark(X)) -> g(X)
   g(active(X)) -> g(X)

The set Q consists of the following terms:

   active(f(f(a)))
   mark(f(x0))
   mark(a)
   mark(g(x0))
   f(mark(x0))
   f(active(x0))
   g(mark(x0))
   g(active(x0))


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(1) RFCMatchBoundsTRSProof (EQUIVALENT)
Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. This implies Q-termination of R.
The following rules were used to construct the certificate:

   active(f(f(a))) -> mark(f(g(f(a))))
   mark(f(X)) -> active(f(X))
   mark(a) -> active(a)
   mark(g(X)) -> active(g(mark(X)))
   f(mark(X)) -> f(X)
   f(active(X)) -> f(X)
   g(mark(X)) -> g(X)
   g(active(X)) -> g(X)

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17

Node 1 is start node and node 2 is final node.

Those nodes are connected through the following edges:

* 1 to 3 labelled mark_1(0)* 1 to 7 labelled active_1(0)* 1 to 6 labelled active_1(0)* 1 to 8 labelled active_1(0)* 1 to 2 labelled f_1(0), g_1(0), f_1(1), g_1(1)* 1 to 10 labelled active_1(1)* 1 to 13 labelled mark_1(1)* 1 to 17 labelled active_1(2)* 2 to 2 labelled #_1(0)* 3 to 4 labelled f_1(0)* 4 to 5 labelled g_1(0)* 5 to 6 labelled f_1(0)* 6 to 2 labelled a(0)* 7 to 2 labelled f_1(0), f_1(1), a(1)* 8 to 9 labelled g_1(0)* 8 to 2 labelled g_1(1)* 8 to 7 labelled g_1(1)* 8 to 11 labelled g_1(1)* 8 to 13 labelled g_1(1)* 8 to 17 labelled g_1(1)* 9 to 2 labelled mark_1(0)* 9 to 7 labelled active_1(1)* 9 to 11 labelled active_1(1)* 9 to 13 labelled mark_1(1)* 9 to 17 labelled active_1(2)* 10 to 4 labelled f_1(1)* 11 to 12 labelled g_1(1)* 11 to 2 labelled g_1(2), g_1(1)* 11 to 7 labelled g_1(2)* 11 to 11 labelled g_1(2)* 11 to 13 labelled g_1(2)* 11 to 17 labelled g_1(2)* 12 to 2 labelled mark_1(1)* 12 to 7 labelled active_1(1)* 12 to 11 labelled active_1(1)* 12 to 13 labelled mark_1(1)* 12 to 17 labelled active_1(2)* 13 to 14 labelled f_1(1)* 14 to 15 labelled g_1(1)* 15 to 16 labelled f_1(1)* 16 to 2 labelled a(1)* 17 to 14 labelled f_1(2)


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(2)
YES