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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 19 ms]
(2) YES


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

   g(c(x1)) -> g(f(c(x1)))
   g(f(c(x1))) -> g(f(f(c(x1))))
   g(g(x1)) -> g(f(g(x1)))
   f(f(g(x1))) -> g(f(x1))

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 3. This implies Q-termination of R.
The following rules were used to construct the certificate:

   g(c(x1)) -> g(f(c(x1)))
   g(f(c(x1))) -> g(f(f(c(x1))))
   g(g(x1)) -> g(f(g(x1)))
   f(f(g(x1))) -> g(f(x1))

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, 30, 31, 38, 39

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

Those nodes are connected through the following edges:

* 1 to 3 labelled g_1(0)* 1 to 5 labelled g_1(0)* 1 to 11 labelled g_1(1)* 1 to 9 labelled g_1(1)* 1 to 14 labelled g_1(2)* 2 to 2 labelled #_1(0)* 3 to 4 labelled f_1(0)* 3 to 2 labelled f_1(0)* 3 to 8 labelled g_1(1)* 3 to 11 labelled g_1(1)* 3 to 30 labelled g_1(2)* 4 to 2 labelled c_1(0), g_1(0)* 4 to 9 labelled g_1(1)* 4 to 11 labelled g_1(1)* 4 to 14 labelled g_1(2)* 5 to 6 labelled f_1(0)* 6 to 7 labelled f_1(0)* 7 to 2 labelled c_1(0)* 8 to 2 labelled f_1(1)* 8 to 8 labelled g_1(1)* 8 to 30 labelled g_1(2)* 8 to 11 labelled g_1(1)* 9 to 10 labelled f_1(1)* 10 to 2 labelled c_1(1), g_1(1)* 10 to 9 labelled g_1(1)* 10 to 11 labelled g_1(1)* 10 to 14 labelled g_1(2)* 11 to 12 labelled f_1(1)* 12 to 13 labelled f_1(1)* 12 to 8 labelled g_1(1)* 12 to 30 labelled g_1(2), g_1(1)* 12 to 11 labelled g_1(1)* 13 to 2 labelled c_1(1)* 14 to 15 labelled f_1(2)* 15 to 16 labelled f_1(2)* 16 to 2 labelled c_1(2)* 30 to 31 labelled f_1(2)* 31 to 8 labelled g_1(2)* 31 to 30 labelled g_1(2)* 31 to 38 labelled g_1(3)* 31 to 11 labelled g_1(2), g_1(1)* 38 to 39 labelled f_1(3)* 39 to 30 labelled g_1(3)


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(2)
YES