/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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(0) -> s(0)
   f(s(0)) -> s(0)
   f(s(s(x))) -> f(f(s(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(0) -> s(0)
   f(s(0)) -> s(0)
   f(s(s(x))) -> f(f(s(x)))

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 2, 3, 4, 5, 11, 12, 13, 18

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

Those nodes are connected through the following edges:

* 1 to 3 labelled s_1(0)* 1 to 4 labelled f_1(0)* 1 to 11 labelled s_1(1)* 2 to 2 labelled #_1(0)* 3 to 2 labelled 0(0)* 4 to 5 labelled f_1(0)* 4 to 11 labelled s_1(1)* 4 to 12 labelled f_1(1)* 4 to 18 labelled s_1(2)* 5 to 2 labelled s_1(0)* 11 to 2 labelled 0(1)* 12 to 13 labelled f_1(1)* 12 to 11 labelled s_1(1)* 12 to 12 labelled f_1(1)* 12 to 18 labelled s_1(2)* 13 to 2 labelled s_1(1)* 18 to 2 labelled 0(2)


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(2)
YES