/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) FlatCCProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) RootLabelingProof [EQUIVALENT, 0 ms]
(6) QTRS
(7) DependencyPairsProof [EQUIVALENT, 32 ms]
(8) QDP
(9) DependencyGraphProof [EQUIVALENT, 0 ms]
(10) QDP
(11) QDPOrderProof [EQUIVALENT, 58 ms]
(12) QDP
(13) DependencyGraphProof [EQUIVALENT, 0 ms]
(14) TRUE


----------------------------------------

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

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

Q is empty.

----------------------------------------

(1) QTRS Reverse (EQUIVALENT)
We applied the QTRS Reverse Processor [REVERSE].
----------------------------------------

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

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

Q is empty.

----------------------------------------

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

----------------------------------------

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

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

Q is empty.

----------------------------------------

(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.

----------------------------------------

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

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

Q is empty.

----------------------------------------

(7) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
----------------------------------------

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

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

The TRS R consists of the following rules:

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

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

(9) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 10 less nodes.
----------------------------------------

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

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

The TRS R consists of the following rules:

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

   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_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_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
   A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1)
   A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1))
   A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
   A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1))
   A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1)
   A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_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(A_{B_1}(x_1)) = 1 + x_1
   POL(B_{A_1}(x_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:

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


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

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

The TRS R consists of the following rules:

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

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

(13) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
----------------------------------------

(14)
TRUE