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


--------------------------------------------------------------------------------


YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms]
(2) YES


----------------------------------------

(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.

----------------------------------------

(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, 6, 7, 8, 9

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 6 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 6 labelled s_1(1)* 4 to 7 labelled f_1(1)* 4 to 9 labelled s_1(2)* 5 to 2 labelled s_1(0)* 6 to 2 labelled 0(1)* 7 to 8 labelled f_1(1)* 7 to 6 labelled s_1(1)* 7 to 7 labelled f_1(1)* 7 to 9 labelled s_1(2)* 8 to 2 labelled s_1(1)* 9 to 2 labelled 0(2)


----------------------------------------

(2)
YES