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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) FlatCCProof [EQUIVALENT, 0 ms]
(2) QTRS
(3) RootLabelingProof [EQUIVALENT, 1 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 50 ms]
(6) QTRS
(7) DependencyPairsProof [EQUIVALENT, 212 ms]
(8) QDP
(9) DependencyGraphProof [EQUIVALENT, 9 ms]
(10) QDP
(11) QDPOrderProof [EQUIVALENT, 438 ms]
(12) QDP
(13) DependencyGraphProof [EQUIVALENT, 0 ms]
(14) AND
    (15) QDP
        (16) QDPOrderProof [EQUIVALENT, 57 ms]
        (17) QDP
        (18) DependencyGraphProof [EQUIVALENT, 0 ms]
        (19) TRUE
    (20) QDP
        (21) QDPOrderProof [EQUIVALENT, 166 ms]
        (22) QDP
        (23) PisEmptyProof [EQUIVALENT, 0 ms]
        (24) YES


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

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

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

Q is empty.

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

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

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

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

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

Q is empty.

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

(3) 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.

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

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

   a_{a_1}(a_{a_1}(a_{a_1}(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}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{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_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))))))
   c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))))
   c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))))
   c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))))))

Q is empty.

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

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

   POL(a_{a_1}(x_1)) = x_1
   POL(a_{b_1}(x_1)) = x_1
   POL(a_{c_1}(x_1)) = 1 + x_1
   POL(b_{a_1}(x_1)) = x_1
   POL(b_{b_1}(x_1)) = x_1
   POL(b_{c_1}(x_1)) = 1 + x_1
   POL(c_{a_1}(x_1)) = x_1
   POL(c_{b_1}(x_1)) = x_1
   POL(c_{c_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:

   c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))))
   c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))))
   c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))))))




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

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

   a_{a_1}(a_{a_1}(a_{a_1}(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}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{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_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_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_{a_1}(a_{a_1}(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}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   A_{A_1}(a_{a_1}(a_{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_{b_1}(b_{a_1}(x1)))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{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_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{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_{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_{b_1}(b_{a_1}(x1)))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   C_{A_1}(a_{a_1}(a_{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_{b_1}(b_{a_1}(x1)))))
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1)))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{c_1}(x1))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1)))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{c_1}(x1))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1)))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{c_1}(x1))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{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_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(x1))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> B_{A_1}(x1)
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{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_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> B_{B_1}(b_{b_1}(x1))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> B_{B_1}(x1)
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1)))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> B_{B_1}(b_{c_1}(x1))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(x1))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> B_{A_1}(x1)
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> B_{B_1}(b_{b_1}(x1))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> B_{B_1}(x1)
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1)))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> B_{B_1}(b_{c_1}(x1))

The TRS R consists of the following rules:

   a_{a_1}(a_{a_1}(a_{a_1}(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}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{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_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_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 5 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}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   A_{A_1}(a_{a_1}(a_{a_1}(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}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   A_{A_1}(a_{a_1}(a_{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_{b_1}(b_{a_1}(x1)))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
   B_{A_1}(a_{a_1}(a_{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_{b_1}(b_{a_1}(x1)))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1)))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{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}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(x1))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> B_{A_1}(x1)
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   C_{A_1}(a_{a_1}(a_{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_{b_1}(b_{a_1}(x1)))))
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> B_{B_1}(b_{b_1}(x1))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> B_{B_1}(x1)
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1)))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1)))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{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_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(x1))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> B_{A_1}(x1)
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{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_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> B_{B_1}(b_{b_1}(x1))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> B_{B_1}(x1)
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1)))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1)))

The TRS R consists of the following rules:

   a_{a_1}(a_{a_1}(a_{a_1}(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}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{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_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_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.

   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(x1))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> B_{A_1}(x1)
   C_{A_1}(a_{a_1}(a_{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_{b_1}(b_{a_1}(x1)))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> B_{B_1}(b_{b_1}(x1))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> B_{B_1}(x1)
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1)))
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{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_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(x1))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> B_{A_1}(x1)
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{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_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> B_{B_1}(b_{b_1}(x1))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> B_{B_1}(x1)
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1)))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(A_{A_1}(x_1)) = x_1
   POL(A_{B_1}(x_1)) = x_1
   POL(B_{A_1}(x_1)) = x_1
   POL(B_{B_1}(x_1)) = x_1
   POL(C_{A_1}(x_1)) = 1 + x_1
   POL(C_{B_1}(x_1)) = 1 + x_1
   POL(a_{a_1}(x_1)) = x_1
   POL(a_{b_1}(x_1)) = x_1
   POL(a_{c_1}(x_1)) = x_1
   POL(b_{a_1}(x_1)) = x_1
   POL(b_{b_1}(x_1)) = x_1
   POL(b_{c_1}(x_1)) = x_1
   POL(c_{a_1}(x_1)) = 1 + x_1
   POL(c_{b_1}(x_1)) = 1 + x_1
   POL(c_{c_1}(x_1)) = x_1

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

   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{a_1}(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}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))


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

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

   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   A_{A_1}(a_{a_1}(a_{a_1}(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}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   A_{A_1}(a_{a_1}(a_{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_{b_1}(b_{a_1}(x1)))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
   B_{A_1}(a_{a_1}(a_{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_{b_1}(b_{a_1}(x1)))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1)))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
   B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1)))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))

The TRS R consists of the following rules:

   a_{a_1}(a_{a_1}(a_{a_1}(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}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{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_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_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 2 SCCs with 8 less nodes.
----------------------------------------

(14)
Complex Obligation (AND)

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

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

   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))

The TRS R consists of the following rules:

   a_{a_1}(a_{a_1}(a_{a_1}(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}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{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_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_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.

   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(C_{A_1}(x_1)) = x_1
   POL(C_{B_1}(x_1)) = 1 + x_1
   POL(a_{a_1}(x_1)) = 1 + x_1
   POL(a_{b_1}(x_1)) = x_1
   POL(a_{c_1}(x_1)) = 0
   POL(b_{a_1}(x_1)) = 1 + x_1
   POL(b_{b_1}(x_1)) = x_1
   POL(b_{c_1}(x_1)) = 0
   POL(c_{a_1}(x_1)) = 0
   POL(c_{b_1}(x_1)) = 0
   POL(c_{c_1}(x_1)) = 0

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

   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{a_1}(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}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))))))


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

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

   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))

The TRS R consists of the following rules:

   a_{a_1}(a_{a_1}(a_{a_1}(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}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{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_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_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 0 SCCs with 3 less nodes.
----------------------------------------

(19)
TRUE

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

(20)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A_{A_1}(a_{a_1}(a_{a_1}(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}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   A_{A_1}(a_{a_1}(a_{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_{b_1}(b_{a_1}(x1)))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
   B_{A_1}(a_{a_1}(a_{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_{b_1}(b_{a_1}(x1)))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))

The TRS R consists of the following rules:

   a_{a_1}(a_{a_1}(a_{a_1}(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}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{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_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))))))

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

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


The following pairs can be oriented strictly and are deleted.

   A_{A_1}(a_{a_1}(a_{a_1}(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}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   A_{A_1}(a_{a_1}(a_{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_{b_1}(b_{a_1}(x1)))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
   B_{A_1}(a_{a_1}(a_{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_{b_1}(b_{a_1}(x1)))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
   B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))
   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_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(B_{B_1}(x_1)) = x_1
   POL(a_{a_1}(x_1)) = 1 + x_1
   POL(a_{b_1}(x_1)) = x_1
   POL(a_{c_1}(x_1)) = 0
   POL(b_{a_1}(x_1)) = 1 + x_1
   POL(b_{b_1}(x_1)) = x_1
   POL(b_{c_1}(x_1)) = 0
   POL(c_{a_1}(x_1)) = 0
   POL(c_{b_1}(x_1)) = 1
   POL(c_{c_1}(x_1)) = 0

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

   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{a_1}(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}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))))))


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

   a_{a_1}(a_{a_1}(a_{a_1}(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}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{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_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
   c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))))
   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))))
   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))))))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(23) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
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(24)
YES