/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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, 61 ms]
(8) QDP
(9) SplitQDPProof [EQUIVALENT, 0 ms]
(10) AND
    (11) QDP
        (12) SemLabProof [SOUND, 0 ms]
        (13) QDP
        (14) DependencyGraphProof [EQUIVALENT, 0 ms]
        (15) QDP
        (16) QDPOrderProof [EQUIVALENT, 93 ms]
        (17) QDP
        (18) DependencyGraphProof [EQUIVALENT, 0 ms]
        (19) QDP
        (20) PisEmptyProof [SOUND, 0 ms]
        (21) TRUE
    (22) QDP
        (23) QDPOrderProof [EQUIVALENT, 818 ms]
        (24) QDP
        (25) PisEmptyProof [EQUIVALENT, 0 ms]
        (26) YES


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

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

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

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

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

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

Q is empty.

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

The TRS R consists of the following rules:

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

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

(9) SplitQDPProof (EQUIVALENT)
We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem

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

(10)
Complex Obligation (AND)

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

(11)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(12) SemLabProof (SOUND)
We found the following model for the rules of the TRSs R and P.
Interpretation over the domain with elements from 0 to 1.
b_{b_1}: 1
b_{a_1}: 1 + x<sub>0</sub>
A_{A_1}: 0
B_{A_1}: 0
a_{b_1}: x<sub>0</sub>
a_{a_1}: 0
By semantic labelling [SEMLAB] we obtain the following labelled QDP problem.
----------------------------------------

(13)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> B_{A_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))) -> B_{A_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1)))))))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1)))))))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1)))))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1)))))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1)))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1)))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> B_{A_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1)))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))) -> B_{A_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1)))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> B_{A_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> B_{A_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1)))))))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1)))))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1)))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> B_{A_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1)))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> B_{A_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> B_{A_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))) -> B_{A_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1)))))))))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1)))))))))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1)))))))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1)))))))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1)))))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1)))))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> B_{A_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1)))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))) -> B_{A_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1)))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> B_{A_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> B_{A_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1)))))))))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))))))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1)))))))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1)))))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> B_{A_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1)))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> B_{A_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))

The TRS R consists of the following rules:

   a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1)))))))))))))))
   a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))) -> a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1)))))))))))))))
   a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1)))))))))))))))
   a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))))))))))))
   b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> b_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1)))))))))))))))
   b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))) -> b_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1)))))))))))))))
   b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> b_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1)))))))))))))))
   b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> b_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))))))))))))

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

(14) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 46 less nodes.
----------------------------------------

(15)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1)))))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1)))))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> B_{A_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1)))))))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1)))))))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> B_{A_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))

The TRS R consists of the following rules:

   a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1)))))))))))))))
   a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))) -> a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1)))))))))))))))
   a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1)))))))))))))))
   a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))))))))))))
   b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> b_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1)))))))))))))))
   b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))) -> b_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1)))))))))))))))
   b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> b_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1)))))))))))))))
   b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> b_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))))))))))))

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

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


The following pairs can be oriented strictly and are deleted.

   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

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

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

   a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1)))))))))))))))
   a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))) -> a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1)))))))))))))))
   a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1)))))))))))))))
   a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))))))))))))
   b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> b_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1)))))))))))))))
   b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))) -> b_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1)))))))))))))))
   b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> b_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1)))))))))))))))
   b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> b_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))))))))))))


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

(17)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1)))))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1)))))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> B_{A_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1)))))))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1)))))))))
   A_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> B_{A_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))

The TRS R consists of the following rules:

   a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1)))))))))))))))
   a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))) -> a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1)))))))))))))))
   a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1)))))))))))))))
   a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))))))))))))
   b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> b_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1)))))))))))))))
   b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))) -> b_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1)))))))))))))))
   b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> b_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1)))))))))))))))
   b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> b_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))))))))))))

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

(18) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
----------------------------------------

(19)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1)))))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> B_{A_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))
   B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1)))))))))

The TRS R consists of the following rules:

   a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1)))))))))))))))
   a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))) -> a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1)))))))))))))))
   a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1)))))))))))))))
   a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))))))))))))
   b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1))))))))))) -> b_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(x1)))))))))))))))
   b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1))))))))))) -> b_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(x1)))))))))))))))
   b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1))))))))))) -> b_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(x1)))))))))))))))
   b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1))))))))))) -> b_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.1(a_{b_1}.1(x1)))))))))))))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(20) PisEmptyProof (SOUND)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
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(21)
TRUE

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(22)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(23) 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}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(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}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))))))
   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( B_{A_1}_1(x_1) ) = max{0, 2x_1 - 2}
POL( a_{a_1}_1(x_1) ) = 2x_1
POL( a_{b_1}_1(x_1) ) = x_1 + 1
POL( b_{a_1}_1(x_1) ) = x_1
POL( b_{b_1}_1(x_1) ) = 1

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

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


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(24)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(25) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
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(26)
YES