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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) 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(g(X)) -> mark(h(X))
   active(c) -> mark(d)
   active(h(d)) -> mark(g(c))
   mark(g(X)) -> active(g(X))
   mark(h(X)) -> active(h(X))
   mark(c) -> active(c)
   mark(d) -> active(d)
   g(mark(X)) -> g(X)
   g(active(X)) -> g(X)
   h(mark(X)) -> h(X)
   h(active(X)) -> h(X)

The set Q consists of the following terms:

   active(g(x0))
   active(c)
   active(h(d))
   mark(g(x0))
   mark(h(x0))
   mark(c)
   mark(d)
   g(mark(x0))
   g(active(x0))
   h(mark(x0))
   h(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(g(X)) -> mark(h(X))
   active(c) -> mark(d)
   active(h(d)) -> mark(g(c))
   mark(g(X)) -> active(g(X))
   mark(h(X)) -> active(h(X))
   mark(c) -> active(c)
   mark(d) -> active(d)
   g(mark(X)) -> g(X)
   g(active(X)) -> g(X)
   h(mark(X)) -> h(X)
   h(active(X)) -> h(X)

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 3, 4, 5, 6, 7, 8, 9, 10, 17, 18

Node 1 is start node and node 3 is final node.

Those nodes are connected through the following edges:

* 1 to 4 labelled mark_1(0), active_1(0), active_1(1), mark_1(1)* 1 to 5 labelled mark_1(0)* 1 to 7 labelled active_1(0)* 1 to 6 labelled active_1(0)* 1 to 3 labelled g_1(0), h_1(0), g_1(1), h_1(1)* 1 to 8 labelled active_1(1), mark_1(1)* 1 to 9 labelled active_1(2)* 1 to 10 labelled mark_1(2)* 1 to 17 labelled mark_1(3), active_1(3)* 1 to 18 labelled active_1(4)* 3 to 3 labelled #_1(0)* 4 to 3 labelled h_1(0), d(0), h_1(1)* 5 to 6 labelled g_1(0)* 6 to 3 labelled c(0), c(1)* 7 to 3 labelled g_1(0), g_1(1)* 8 to 3 labelled h_1(1), d(1)* 8 to 6 labelled g_1(1)* 9 to 3 labelled h_1(2), d(2), h_1(1)* 9 to 6 labelled g_1(2)* 10 to 6 labelled h_1(2)* 17 to 6 labelled h_1(3)* 18 to 6 labelled h_1(4)


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(2)
YES