YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) QTRS Reverse [EQUIVALENT, 0 ms]
(2) QTRS
(3) FlatCCProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) RootLabelingProof [EQUIVALENT, 0 ms]
(6) QTRS
(7) DependencyPairsProof [EQUIVALENT, 35 ms]
(8) QDP
(9) DependencyGraphProof [EQUIVALENT, 0 ms]
(10) QDP
(11) QDPOrderProof [EQUIVALENT, 477 ms]
(12) QDP
(13) DependencyGraphProof [EQUIVALENT, 0 ms]
(14) AND
    (15) QDP
        (16) UsableRulesProof [EQUIVALENT, 0 ms]
        (17) QDP
        (18) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (19) YES
    (20) QDP
        (21) UsableRulesProof [EQUIVALENT, 0 ms]
        (22) QDP
        (23) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (24) YES


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

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

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

Q is empty.

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

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

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

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

Q is empty.

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

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

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

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

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

Q is empty.

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

(5) RootLabelingProof (EQUIVALENT)
We used plain root labeling [ROOTLAB] with the following heuristic:
LabelAll: All function symbols get labeled

As Q is empty the root labeling was sound AND complete.

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

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

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

Q is empty.

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

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

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

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

The TRS R consists of the following rules:

   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))
   a_{a_1}(a_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))
   b_{a_1}(a_{a_1}(a_{c_1}(x1))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{a_1}(x1))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))
   c_{a_1}(a_{a_1}(a_{c_1}(x1))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{c_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_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1))
   a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1))
   b_{b_1}(b_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1))
   b_{b_1}(b_{c_1}(c_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1))
   b_{b_1}(b_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1))
   c_{b_1}(b_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1))
   c_{b_1}(b_{c_1}(c_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1))
   c_{b_1}(b_{c_1}(c_{c_1}(x1))) -> c_{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 1 SCC with 12 less nodes.
----------------------------------------

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

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

The TRS R consists of the following rules:

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

   B_{B_1}(b_{c_1}(c_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1))
   B_{A_1}(a_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
   C_{B_1}(b_{c_1}(c_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1))
   C_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{A_1}(x1)
   A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
   C_{B_1}(b_{c_1}(c_{b_1}(x1))) -> C_{A_1}(a_{b_1}(x1))
   C_{A_1}(a_{a_1}(a_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
   C_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1)
   A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{A_1}(a_{a_1}(x1))
   A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
   B_{A_1}(a_{a_1}(a_{a_1}(x1))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
   B_{A_1}(a_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1))
   A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{A_1}(x1)
   A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1))
   A_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{b_1}(x1))
   A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> B_{A_1}(x1)
   B_{A_1}(a_{a_1}(a_{a_1}(x1))) -> B_{A_1}(x1)
   B_{A_1}(a_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
   B_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
   B_{A_1}(a_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1))
   A_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1)
   B_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
   B_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{A_1}(x1)
   A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
   A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
   B_{A_1}(a_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
   B_{A_1}(a_{a_1}(a_{c_1}(x1))) -> B_{A_1}(a_{b_1}(b_{c_1}(x1)))
   B_{A_1}(a_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1))
   A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1))
   A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
   B_{B_1}(b_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1))
   B_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1)
   A_{A_1}(a_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
   A_{A_1}(a_{a_1}(a_{c_1}(x1))) -> B_{A_1}(a_{b_1}(b_{c_1}(x1)))
   A_{A_1}(a_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1))
   C_{A_1}(a_{a_1}(a_{a_1}(x1))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
   C_{A_1}(a_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1))
   C_{A_1}(a_{a_1}(a_{a_1}(x1))) -> B_{A_1}(x1)
   C_{A_1}(a_{a_1}(a_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
   C_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
   C_{A_1}(a_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1))
   C_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
   C_{A_1}(a_{a_1}(a_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
   C_{A_1}(a_{a_1}(a_{c_1}(x1))) -> B_{A_1}(a_{b_1}(b_{c_1}(x1)))
   C_{A_1}(a_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{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) ) = x_1
POL( B_{A_1}_1(x_1) ) = max{0, x_1 - 2}
POL( C_{A_1}_1(x_1) ) = max{0, x_1 - 2}
POL( A_{B_1}_1(x_1) ) = x_1 + 1
POL( b_{a_1}_1(x_1) ) = max{0, x_1 - 2}
POL( C_{B_1}_1(x_1) ) = x_1
POL( b_{b_1}_1(x_1) ) = max{0, x_1 - 2}
POL( b_{c_1}_1(x_1) ) = x_1
POL( c_{a_1}_1(x_1) ) = x_1 + 2
POL( c_{c_1}_1(x_1) ) = x_1 + 2
POL( a_{a_1}_1(x_1) ) = x_1 + 2
POL( a_{b_1}_1(x_1) ) = x_1 + 2
POL( c_{b_1}_1(x_1) ) = x_1 + 2
POL( a_{c_1}_1(x_1) ) = x_1 + 2
POL( B_{B_1}_1(x_1) ) = x_1

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

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


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

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

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

The TRS R consists of the following rules:

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

(14)
Complex Obligation (AND)

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

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

   C_{A_1}(a_{a_1}(x1)) -> C_{A_1}(x1)

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(16) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
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(17)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   C_{A_1}(a_{a_1}(x1)) -> C_{A_1}(x1)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(18) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*C_{A_1}(a_{a_1}(x1)) -> C_{A_1}(x1)
The graph contains the following edges 1 > 1


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(19)
YES

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(21) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
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(22)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

(23) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
The graph contains the following edges 1 > 1


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(24)
YES