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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) RootLabelingProof [EQUIVALENT, 0 ms]
(2) QTRS
(3) DependencyPairsProof [EQUIVALENT, 0 ms]
(4) QDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) QDP
(7) QDPOrderProof [EQUIVALENT, 21 ms]
(8) QDP
(9) DependencyGraphProof [EQUIVALENT, 0 ms]
(10) QDP
(11) QDPOrderProof [EQUIVALENT, 9 ms]
(12) QDP
(13) PisEmptyProof [EQUIVALENT, 0 ms]
(14) YES


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

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

Q is empty.

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(1) 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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(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

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(3) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
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(4)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
   B_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
   B_{A_1}(a_{b_1}(b_{b_1}(x1))) -> A_{A_1}(a_{b_1}(x1))
   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{A_1}(a_{a_1}(x1))
   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1)
   A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> B_{A_1}(x1)

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   B_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{A_1}(a_{a_1}(x1))
   A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> B_{A_1}(x1)
   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1)

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(7) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{A_1}(a_{a_1}(x1))
   A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> B_{A_1}(x1)
   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(A_{A_1}(x_1)) = 1 + x_1
   POL(B_{A_1}(x_1)) = 1 + x_1
   POL(a_{a_1}(x_1)) = 1 + x_1
   POL(a_{b_1}(x_1)) = 1 + x_1
   POL(b_{a_1}(x_1)) = 1 + x_1
   POL(b_{b_1}(x_1)) = 1 + x_1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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


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(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   B_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(9) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(11) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(B_{A_1}(x_1)) = x_1
   POL(a_{a_1}(x_1)) = 0
   POL(a_{b_1}(x_1)) = x_1
   POL(b_{a_1}(x_1)) = 1
   POL(b_{b_1}(x_1)) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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


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(12)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(13) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
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(14)
YES