/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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:

   f(f(a)) -> c(n__f(n__g(n__f(n__a))))
   f(X) -> n__f(X)
   g(X) -> n__g(X)
   a -> n__a
   activate(n__f(X)) -> f(activate(X))
   activate(n__g(X)) -> g(activate(X))
   activate(n__a) -> a
   activate(X) -> X

Q is empty.

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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:

   f(f(a)) -> c(n__f(n__g(n__f(n__a))))
   f(X) -> n__f(X)
   g(X) -> n__g(X)
   a -> n__a
   activate(n__f(X)) -> f(activate(X))
   activate(n__g(X)) -> g(activate(X))
   activate(n__a) -> a
   activate(X) -> 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

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

Those nodes are connected through the following edges:

* 1 to 3 labelled c_1(0)* 1 to 2 labelled n__f_1(0), n__g_1(0), n__a(0), a(0), f_1(0), c_1(0), g_1(0), activate_1(0), n__a(1), n__f_1(1), n__g_1(1), a(1), f_1(1), c_1(1), g_1(1), activate_1(1), n__a(2), n__f_1(2), n__g_1(2)* 1 to 7 labelled f_1(0), g_1(0), n__f_1(1), n__g_1(1), c_1(2)* 1 to 8 labelled c_1(1)* 1 to 12 labelled f_1(1), g_1(1), n__f_1(2), n__g_1(2)* 1 to 1 labelled f_1(1), g_1(1), n__f_1(2), n__g_1(2)* 2 to 2 labelled #_1(0)* 3 to 4 labelled n__f_1(0)* 4 to 5 labelled n__g_1(0)* 5 to 6 labelled n__f_1(0)* 6 to 2 labelled n__a(0)* 7 to 2 labelled activate_1(0), a(1), f_1(1), c_1(1), n__f_1(1), n__g_1(1), n__a(1), g_1(1), activate_1(1), n__a(2), n__f_1(2), n__g_1(2)* 7 to 1 labelled f_1(1), g_1(1), n__f_1(2), n__g_1(2)* 7 to 8 labelled c_1(1)* 7 to 7 labelled c_1(2)* 8 to 9 labelled n__f_1(1)* 9 to 10 labelled n__g_1(1)* 10 to 11 labelled n__f_1(1)* 11 to 2 labelled n__a(1)* 12 to 2 labelled activate_1(1), a(1), f_1(1), c_1(1), n__f_1(1), n__g_1(1), n__a(1), g_1(1), n__a(2), n__f_1(2), n__g_1(2)* 12 to 1 labelled f_1(1), g_1(1), n__f_1(2), n__g_1(2)* 12 to 8 labelled c_1(1)* 12 to 7 labelled c_1(2)


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(2)
YES