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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) QTRS Reverse [EQUIVALENT, 0 ms]
(2) QTRS
(3) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms]
(4) YES


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

   f(f(x)) -> f(c(f(x)))
   f(f(x)) -> f(d(f(x)))
   g(c(x)) -> x
   g(d(x)) -> x
   g(c(h(0))) -> g(d(1))
   g(c(1)) -> g(d(h(0)))
   g(h(x)) -> g(x)

Q is empty.

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(1) QTRS Reverse (EQUIVALENT)
We applied the QTRS Reverse Processor [REVERSE].
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(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f(f(x)) -> f(c(f(x)))
   f(f(x)) -> f(d(f(x)))
   c(g(x)) -> x
   d(g(x)) -> x
   0'(h(c(g(x)))) -> 1'(d(g(x)))
   1'(c(g(x))) -> 0'(h(d(g(x))))
   h(g(x)) -> g(x)

Q is empty.

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(3) 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(x)) -> f(c(f(x)))
   f(f(x)) -> f(d(f(x)))
   c(g(x)) -> x
   d(g(x)) -> x
   0'(h(c(g(x)))) -> 1'(d(g(x)))
   1'(c(g(x))) -> 0'(h(d(g(x))))
   h(g(x)) -> g(x)

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
3, 4, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40

Node 3 is start node and node 4 is final node.

Those nodes are connected through the following edges:

* 3 to 23 labelled f_1(0)* 3 to 25 labelled f_1(0)* 3 to 4 labelled f_1(0), c_1(0), d_1(0), g_1(0), 0'_1(0), h_1(0), 1'_1(0), f_1(1), c_1(1), d_1(1), g_1(1), 0'_1(1), h_1(1), 1'_1(1)* 3 to 27 labelled 1'_1(0)* 3 to 29 labelled 0'_1(0)* 3 to 32 labelled f_1(1)* 3 to 34 labelled f_1(1)* 3 to 36 labelled 1'_1(1)* 3 to 38 labelled 0'_1(1)* 4 to 4 labelled #_1(0)* 23 to 24 labelled c_1(0)* 24 to 4 labelled f_1(0)* 24 to 32 labelled f_1(1)* 24 to 34 labelled f_1(1)* 25 to 26 labelled d_1(0)* 26 to 4 labelled f_1(0)* 26 to 32 labelled f_1(1)* 26 to 34 labelled f_1(1)* 27 to 28 labelled d_1(0)* 27 to 4 labelled f_1(1), c_1(1), d_1(1), g_1(1), 0'_1(1), h_1(1), 1'_1(1)* 27 to 32 labelled f_1(1)* 27 to 34 labelled f_1(1)* 27 to 36 labelled 1'_1(1)* 27 to 38 labelled 0'_1(1)* 28 to 4 labelled g_1(0)* 29 to 30 labelled h_1(0)* 29 to 4 labelled g_1(1)* 30 to 31 labelled d_1(0)* 30 to 4 labelled f_1(1), c_1(1), d_1(1), g_1(1), 0'_1(1), h_1(1), 1'_1(1)* 30 to 32 labelled f_1(1)* 30 to 34 labelled f_1(1)* 30 to 36 labelled 1'_1(1)* 30 to 38 labelled 0'_1(1)* 31 to 4 labelled g_1(0)* 32 to 33 labelled c_1(1)* 33 to 4 labelled f_1(1)* 33 to 32 labelled f_1(1)* 33 to 34 labelled f_1(1)* 34 to 35 labelled d_1(1)* 35 to 4 labelled f_1(1)* 35 to 32 labelled f_1(1)* 35 to 34 labelled f_1(1)* 36 to 37 labelled d_1(1)* 36 to 4 labelled f_1(1), c_1(1), d_1(1), g_1(1), 0'_1(1), h_1(1), 1'_1(1)* 36 to 32 labelled f_1(1)* 36 to 34 labelled f_1(1)* 36 to 36 labelled 1'_1(1)* 36 to 38 labelled 0'_1(1)* 37 to 4 labelled g_1(1)* 38 to 39 labelled h_1(1)* 38 to 4 labelled g_1(2)* 39 to 40 labelled d_1(1)* 39 to 4 labelled f_1(1), c_1(1), d_1(1), g_1(1), 0'_1(1), h_1(1), 1'_1(1)* 39 to 32 labelled f_1(1)* 39 to 34 labelled f_1(1)* 39 to 36 labelled 1'_1(1)* 39 to 38 labelled 0'_1(1)* 40 to 4 labelled g_1(1)


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(4)
YES