/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) RFCMatchBoundsTRSProof [EQUIVALENT, 38 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(mark(X)))
   mark(a) -> active(a)
   mark(g(X)) -> active(g(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 4. 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(mark(X)))
   mark(a) -> active(a)
   mark(g(X)) -> active(g(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, 6, 8, 9, 11, 13, 15, 17, 18, 19, 20, 23, 25, 26, 27, 28, 39, 40, 42, 43, 44, 45, 48, 49, 50, 51

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

Those nodes are connected through the following edges:

* 1 to 6 labelled mark_1(0)* 1 to 13 labelled active_1(0)* 1 to 11 labelled active_1(0)* 1 to 2 labelled f_1(0), g_1(0), f_1(1), g_1(1)* 1 to 17 labelled active_1(1)* 1 to 25 labelled mark_1(1)* 1 to 39 labelled active_1(2)* 2 to 2 labelled #_1(0)* 6 to 8 labelled f_1(0)* 8 to 9 labelled g_1(0)* 9 to 11 labelled f_1(0)* 11 to 2 labelled a(0)* 13 to 15 labelled f_1(0)* 13 to 2 labelled g_1(0), g_1(1), f_1(1)* 13 to 19 labelled f_1(1)* 13 to 13 labelled f_1(1)* 13 to 25 labelled f_1(1)* 13 to 39 labelled f_1(1)* 13 to 42 labelled f_1(1)* 13 to 48 labelled f_1(1)* 15 to 2 labelled mark_1(0)* 15 to 19 labelled active_1(1)* 15 to 13 labelled active_1(1)* 15 to 25 labelled mark_1(1)* 15 to 39 labelled active_1(2)* 15 to 42 labelled mark_1(2)* 15 to 48 labelled active_1(3)* 17 to 18 labelled f_1(1)* 17 to 8 labelled f_1(2)* 17 to 23 labelled f_1(2)* 18 to 8 labelled mark_1(1)* 18 to 23 labelled active_1(1)* 19 to 20 labelled f_1(1)* 19 to 2 labelled a(1), f_1(2), f_1(1)* 19 to 19 labelled f_1(2)* 19 to 13 labelled f_1(2)* 19 to 25 labelled f_1(2)* 19 to 42 labelled f_1(2)* 19 to 39 labelled f_1(2)* 19 to 48 labelled f_1(2)* 20 to 2 labelled mark_1(1)* 20 to 19 labelled active_1(1)* 20 to 13 labelled active_1(1)* 20 to 25 labelled mark_1(1)* 20 to 42 labelled mark_1(2)* 20 to 39 labelled active_1(2)* 20 to 48 labelled active_1(3)* 23 to 9 labelled g_1(1)* 25 to 26 labelled f_1(1)* 26 to 27 labelled g_1(1)* 27 to 28 labelled f_1(1)* 28 to 2 labelled a(1)* 39 to 40 labelled f_1(2)* 39 to 26 labelled f_1(3)* 39 to 50 labelled f_1(3)* 40 to 26 labelled mark_1(2)* 40 to 50 labelled active_1(2)* 42 to 43 labelled f_1(2)* 43 to 44 labelled g_1(2)* 44 to 45 labelled f_1(2)* 45 to 2 labelled a(2)* 48 to 49 labelled f_1(3)* 48 to 43 labelled f_1(4)* 48 to 51 labelled f_1(4)* 49 to 43 labelled mark_1(3)* 49 to 51 labelled active_1(3)* 50 to 27 labelled g_1(2)* 51 to 44 labelled g_1(3)


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(2)
YES