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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) RootLabelingProof [EQUIVALENT, 0 ms]
(2) QTRS
(3) DependencyPairsProof [EQUIVALENT, 23 ms]
(4) QDP
(5) QDPOrderProof [EQUIVALENT, 506 ms]
(6) QDP
(7) DependencyGraphProof [EQUIVALENT, 0 ms]
(8) TRUE


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

   b(b(x1)) -> b(a(b(x1)))
   b(b(a(b(x1)))) -> b(a(b(a(a(b(b(x1)))))))
   b(a(b(x1))) -> b(a(a(b(x1))))
   b(a(a(b(a(b(x1)))))) -> 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_{b_1}(b_{b_1}(x1)) -> b_{a_1}(a_{b_1}(b_{b_1}(x1)))
   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(a_{b_1}(b_{a_1}(x1)))
   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(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}(b_{b_1}(x1))))
   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> b_{b_1}(b_{b_1}(x1))
   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> 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_{B_1}(b_{b_1}(x1)) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
   B_{B_1}(b_{a_1}(x1)) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{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}(b_{b_1}(x1))))
   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> B_{B_1}(b_{b_1}(x1))
   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> B_{B_1}(b_{a_1}(x1))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(5) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{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}(b_{b_1}(x1))))
   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( B_{B_1}_1(x_1) ) = 2x_1 + 2
POL( B_{A_1}_1(x_1) ) = max{0, 2x_1 - 2}
POL( a_{b_1}_1(x_1) ) = x_1 + 2
POL( b_{a_1}_1(x_1) ) = x_1
POL( b_{b_1}_1(x_1) ) = x_1 + 2
POL( a_{a_1}_1(x_1) ) = max{0, x_1 - 2}

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

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


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

   B_{B_1}(b_{b_1}(x1)) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
   B_{B_1}(b_{a_1}(x1)) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> B_{B_1}(b_{b_1}(x1))
   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> B_{B_1}(b_{a_1}(x1))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(7) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 6 less nodes.
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(8)
TRUE