/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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:

   half(0) -> 0
   half(s(s(x))) -> s(half(x))
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(s(half(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:

   0'(half(x)) -> 0'(x)
   s(s(half(x))) -> half(s(x))
   0'(s(log(x))) -> 0'(x)
   s(s(log(x))) -> half(s(log(s(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 1. This implies Q-termination of R.
The following rules were used to construct the certificate:

   0'(half(x)) -> 0'(x)
   s(s(half(x))) -> half(s(x))
   0'(s(log(x))) -> 0'(x)
   s(s(log(x))) -> half(s(log(s(x))))

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
28, 29, 30, 31, 32, 33, 34, 35, 36, 37

Node 28 is start node and node 29 is final node.

Those nodes are connected through the following edges:

* 28 to 29 labelled 0'_1(0), 0'_1(1)* 28 to 30 labelled half_1(0)* 28 to 31 labelled half_1(0)* 29 to 29 labelled #_1(0)* 30 to 29 labelled s_1(0)* 30 to 34 labelled half_1(1)* 30 to 35 labelled half_1(1)* 31 to 32 labelled s_1(0)* 32 to 33 labelled log_1(0)* 33 to 29 labelled s_1(0)* 33 to 34 labelled half_1(1)* 33 to 35 labelled half_1(1)* 34 to 29 labelled s_1(1)* 34 to 34 labelled half_1(1)* 34 to 35 labelled half_1(1)* 35 to 36 labelled s_1(1)* 36 to 37 labelled log_1(1)* 37 to 29 labelled s_1(1)* 37 to 34 labelled half_1(1)* 37 to 35 labelled half_1(1)


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(4)
YES