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


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YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 0 ms]
(4) QTRS
(5) DependencyPairsProof [EQUIVALENT, 88 ms]
(6) QDP
(7) QDPOrderProof [EQUIVALENT, 369 ms]
(8) QDP
(9) DependencyGraphProof [EQUIVALENT, 0 ms]
(10) AND
    (11) QDP
        (12) QDPOrderProof [EQUIVALENT, 217 ms]
        (13) QDP
        (14) PisEmptyProof [EQUIVALENT, 0 ms]
        (15) YES
    (16) QDP
        (17) QDPOrderProof [EQUIVALENT, 48 ms]
        (18) QDP
        (19) PisEmptyProof [EQUIVALENT, 0 ms]
        (20) YES


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

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

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

Q is empty.

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

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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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


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

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

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

The TRS R consists of the following rules:

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

(10)
Complex Obligation (AND)

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

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

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

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.

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

POL( A_{A_1}_1(x_1) ) = 2x_1
POL( B_{A_1}_1(x_1) ) = max{0, 2x_1 - 2}
POL( A_{B_1}_1(x_1) ) = 2x_1 + 1
POL( b_{a_1}_1(x_1) ) = max{0, x_1 - 2}
POL( B_{B_1}_1(x_1) ) = 2x_1 + 1
POL( b_{c_1}_1(x_1) ) = max{0, 2x_1 - 2}
POL( c_{a_1}_1(x_1) ) = max{0, -2}
POL( a_{b_1}_1(x_1) ) = x_1 + 2
POL( b_{b_1}_1(x_1) ) = x_1 + 2
POL( a_{a_1}_1(x_1) ) = x_1 + 2
POL( a_{c_1}_1(x_1) ) = 2
POL( c_{b_1}_1(x_1) ) = max{0, -2}
POL( c_{c_1}_1(x_1) ) = max{0, -2}

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

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


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

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

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

(14) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------

(15)
YES

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

(16)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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

   POL(C_{A_1}(x_1)) = 1 + x_1
   POL(C_{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)) = 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:

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


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

(18)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

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

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