YES
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) FlatCCProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) RootLabelingProof [EQUIVALENT, 0 ms]
(6) QTRS
(7) QTRSRRRProof [EQUIVALENT, 6 ms]
(8) QTRS
(9) QTRSRRRProof [EQUIVALENT, 2 ms]
(10) QTRS
(11) RisEmptyProof [EQUIVALENT, 0 ms]
(12) YES


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(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   b(b(b(x1))) -> a(x1)
   a(a(x1)) -> a(b(a(x1)))
   a(a(a(x1))) -> b(a(a(x1)))

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:

   b(b(b(x1))) -> a(x1)
   a(a(x1)) -> a(b(a(x1)))
   a(a(a(x1))) -> a(a(b(x1)))

Q is empty.

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(3) FlatCCProof (EQUIVALENT)
We used flat context closure [ROOTLAB]
As Q is empty the flat context closure was sound AND complete.

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(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   a(a(x1)) -> a(b(a(x1)))
   a(a(a(x1))) -> a(a(b(x1)))
   b(b(b(b(x1)))) -> b(a(x1))
   a(b(b(b(x1)))) -> a(a(x1))

Q is empty.

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(5) RootLabelingProof (EQUIVALENT)
We used plain root labeling [ROOTLAB] with the following heuristic:
LabelAll: All function symbols get labeled

As Q is empty the root labeling was sound AND complete.

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(6)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1)))
   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1)))
   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(x1))
   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(x1))
   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(x1))
   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(x1))

Q is empty.

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(7) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(a_{a_1}(x_1)) = 3 + x_1
   POL(a_{b_1}(x_1)) = x_1
   POL(b_{a_1}(x_1)) = 3 + x_1
   POL(b_{b_1}(x_1)) = 2 + x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1)))
   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1)))
   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(x1))
   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(x1))
   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(x1))
   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(x1))




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(8)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))

Q is empty.

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(9) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(a_{a_1}(x_1)) = 1 + x_1
   POL(a_{b_1}(x_1)) = x_1
   POL(b_{a_1}(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))




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(10)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(11) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(12)
YES