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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 34 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(c(f(g(f(a)))))
   mark(f(X)) -> active(f(mark(X)))
   mark(a) -> active(a)
   mark(c(X)) -> active(c(X))
   mark(g(X)) -> active(g(mark(X)))
   f(mark(X)) -> f(X)
   f(active(X)) -> f(X)
   c(mark(X)) -> c(X)
   c(active(X)) -> c(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(c(x0))
   mark(g(x0))
   f(mark(x0))
   f(active(x0))
   c(mark(x0))
   c(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 3. This implies Q-termination of R.
The following rules were used to construct the certificate:

   active(f(f(a))) -> mark(c(f(g(f(a)))))
   mark(f(X)) -> active(f(mark(X)))
   mark(a) -> active(a)
   mark(c(X)) -> active(c(X))
   mark(g(X)) -> active(g(mark(X)))
   f(mark(X)) -> f(X)
   f(active(X)) -> f(X)
   c(mark(X)) -> c(X)
   c(active(X)) -> c(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, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28

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 8 labelled active_1(0)* 1 to 7 labelled active_1(0)* 1 to 10 labelled active_1(0)* 1 to 2 labelled f_1(0), c_1(0), g_1(0), f_1(1), c_1(1), g_1(1)* 1 to 12 labelled active_1(1)* 1 to 17 labelled mark_1(1)* 1 to 22 labelled active_1(2)* 2 to 2 labelled #_1(0)* 3 to 4 labelled c_1(0)* 4 to 5 labelled f_1(0)* 5 to 6 labelled g_1(0)* 6 to 7 labelled f_1(0)* 7 to 2 labelled a(0)* 8 to 9 labelled f_1(0)* 8 to 2 labelled c_1(0), c_1(1), f_1(1)* 8 to 13 labelled f_1(1)* 8 to 8 labelled f_1(1)* 8 to 15 labelled f_1(1)* 8 to 17 labelled f_1(1)* 8 to 22 labelled f_1(1)* 8 to 23 labelled f_1(1)* 8 to 28 labelled f_1(1)* 9 to 2 labelled mark_1(0)* 9 to 13 labelled active_1(1)* 9 to 8 labelled active_1(1)* 9 to 15 labelled active_1(1)* 9 to 17 labelled mark_1(1)* 9 to 22 labelled active_1(2)* 9 to 23 labelled mark_1(2)* 9 to 28 labelled active_1(3)* 10 to 11 labelled g_1(0)* 10 to 2 labelled g_1(1)* 10 to 13 labelled g_1(1)* 10 to 8 labelled g_1(1)* 10 to 15 labelled g_1(1)* 10 to 17 labelled g_1(1)* 10 to 22 labelled g_1(1)* 10 to 23 labelled g_1(1)* 10 to 28 labelled g_1(1)* 11 to 2 labelled mark_1(0)* 11 to 13 labelled active_1(1)* 11 to 8 labelled active_1(1)* 11 to 15 labelled active_1(1)* 11 to 17 labelled mark_1(1)* 11 to 22 labelled active_1(2)* 11 to 23 labelled mark_1(2)* 11 to 28 labelled active_1(3)* 12 to 4 labelled c_1(1)* 13 to 14 labelled f_1(1)* 13 to 2 labelled a(1), f_1(2), f_1(1)* 13 to 13 labelled f_1(2)* 13 to 8 labelled f_1(2)* 13 to 15 labelled f_1(2)* 13 to 17 labelled f_1(2)* 13 to 23 labelled f_1(2)* 13 to 22 labelled f_1(2)* 13 to 28 labelled f_1(2)* 14 to 2 labelled mark_1(1)* 14 to 13 labelled active_1(1)* 14 to 8 labelled active_1(1)* 14 to 15 labelled active_1(1)* 14 to 17 labelled mark_1(1)* 14 to 23 labelled mark_1(2)* 14 to 22 labelled active_1(2)* 14 to 28 labelled active_1(3)* 15 to 16 labelled g_1(1)* 15 to 2 labelled g_1(2), g_1(1)* 15 to 13 labelled g_1(2)* 15 to 8 labelled g_1(2)* 15 to 15 labelled g_1(2)* 15 to 17 labelled g_1(2)* 15 to 23 labelled g_1(2)* 15 to 22 labelled g_1(2)* 15 to 28 labelled g_1(2)* 16 to 2 labelled mark_1(1)* 16 to 13 labelled active_1(1)* 16 to 8 labelled active_1(1)* 16 to 15 labelled active_1(1)* 16 to 17 labelled mark_1(1)* 16 to 23 labelled mark_1(2)* 16 to 22 labelled active_1(2)* 16 to 28 labelled active_1(3)* 17 to 18 labelled c_1(1)* 18 to 19 labelled f_1(1)* 19 to 20 labelled g_1(1)* 20 to 21 labelled f_1(1)* 21 to 2 labelled a(1)* 22 to 18 labelled c_1(2)* 23 to 24 labelled c_1(2)* 24 to 25 labelled f_1(2)* 25 to 26 labelled g_1(2)* 26 to 27 labelled f_1(2)* 27 to 2 labelled a(2)* 28 to 24 labelled c_1(3)


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(2)
YES