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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Node 68 is start node and node 69 is final node.

Those nodes are connected through the following edges:

* 68 to 69 labelled 0'_1(0), 0'_1(1)* 68 to 70 labelled half_1(0)* 68 to 71 labelled s_1(0)* 69 to 69 labelled #_1(0)* 70 to 69 labelled s_1(0)* 70 to 74 labelled half_1(1)* 70 to 75 labelled s_1(1)* 71 to 72 labelled half_1(0)* 72 to 73 labelled bits_1(0)* 73 to 69 labelled s_1(0)* 73 to 74 labelled half_1(1)* 73 to 75 labelled s_1(1)* 74 to 69 labelled s_1(1)* 74 to 74 labelled half_1(1)* 74 to 75 labelled s_1(1)* 75 to 76 labelled half_1(1)* 76 to 77 labelled bits_1(1)* 77 to 69 labelled s_1(1)* 77 to 74 labelled half_1(1)* 77 to 75 labelled s_1(1)


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(4)
YES