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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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:

   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))))

The set Q consists of the following terms:

   half(0)
   half(s(0))
   half(s(s(x0)))
   bits(0)
   bits(s(x0))


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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:
79, 80, 81, 82, 83, 84, 85, 86, 87, 88

Node 79 is start node and node 80 is final node.

Those nodes are connected through the following edges:

* 79 to 80 labelled 0'_1(0), 0'_1(1)* 79 to 81 labelled half_1(0)* 79 to 82 labelled s_1(0)* 80 to 80 labelled #_1(0)* 81 to 80 labelled s_1(0)* 81 to 85 labelled half_1(1)* 81 to 86 labelled s_1(1)* 82 to 83 labelled half_1(0)* 83 to 84 labelled bits_1(0)* 84 to 80 labelled s_1(0)* 84 to 85 labelled half_1(1)* 84 to 86 labelled s_1(1)* 85 to 80 labelled s_1(1)* 85 to 85 labelled half_1(1)* 85 to 86 labelled s_1(1)* 86 to 87 labelled half_1(1)* 87 to 88 labelled bits_1(1)* 88 to 80 labelled s_1(1)* 88 to 85 labelled half_1(1)* 88 to 86 labelled s_1(1)


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(4)
YES