Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Waldmann_07

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

0 QTRS

↳1 QTRS Reverse (⇔, 0 ms)

↳2 QTRS

↳3 FlatCCProof (⇔, 0 ms)

↳4 QTRS

↳5 RootLabelingProof (⇔, 0 ms)

↳6 QTRS

↳7 DependencyPairsProof (⇔, 93 ms)

↳8 QDP

↳9 DependencyGraphProof (⇔, 0 ms)

↳10 QDP

↳11 QDPOrderProof (⇔, 569 ms)

↳12 QDP

↳13 DependencyGraphProof (⇔, 0 ms)

↳14 AND

    ↳15 QDP

        ↳16 UsableRulesProof (⇔, 0 ms)

        ↳17 QDP

        ↳18 QDPSizeChangeProof (⇔, 0 ms)

        ↳19 YES

    ↳20 QDP

        ↳21 UsableRulesProof (⇔, 0 ms)

        ↳22 QDP

        ↳23 QDPSizeChangeProof (⇔, 0 ms)

        ↳24 YES

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

a(x) → x
a(a(x)) → b(a(b(c(c(b(x))))))
c(b(x)) → a(x)

Q is empty.

(1) QTRS Reverse (EQUIVALENT transformation)

We applied the QTRS Reverse Processor [REVERSE].

(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

a(x) → x
a(a(x)) → b(c(c(b(a(b(x))))))
b(c(x)) → a(x)

Q is empty.

(3) FlatCCProof (EQUIVALENT transformation)

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(x)) → a(x)
b(a(x)) → b(x)
c(a(x)) → c(x)
a(a(a(x))) → a(b(c(c(b(a(b(x)))))))
b(a(a(x))) → b(b(c(c(b(a(b(x)))))))
c(a(a(x))) → c(b(c(c(b(a(b(x)))))))
a(b(c(x))) → a(a(x))
b(b(c(x))) → b(a(x))
c(b(c(x))) → c(a(x))

Q is empty.

(5) RootLabelingProof (EQUIVALENT transformation)

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}(x)) → a_{a_1}(x)
a_{a_1}(a_{b_1}(x)) → a_{b_1}(x)
a_{a_1}(a_{c_1}(x)) → a_{c_1}(x)
b_{a_1}(a_{a_1}(x)) → b_{a_1}(x)
b_{a_1}(a_{b_1}(x)) → b_{b_1}(x)
b_{a_1}(a_{c_1}(x)) → b_{c_1}(x)
c_{a_1}(a_{a_1}(x)) → c_{a_1}(x)
c_{a_1}(a_{b_1}(x)) → c_{b_1}(x)
c_{a_1}(a_{c_1}(x)) → c_{c_1}(x)
a_{a_1}(a_{a_1}(a_{a_1}(x))) → a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))))
a_{a_1}(a_{a_1}(a_{b_1}(x))) → a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))))
a_{a_1}(a_{a_1}(a_{c_1}(x))) → a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(x))) → b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))))
b_{a_1}(a_{a_1}(a_{b_1}(x))) → b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))))
b_{a_1}(a_{a_1}(a_{c_1}(x))) → b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(x))) → c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))))
c_{a_1}(a_{a_1}(a_{b_1}(x))) → c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))))
c_{a_1}(a_{a_1}(a_{c_1}(x))) → c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{c_1}(c_{a_1}(x))) → a_{a_1}(a_{a_1}(x))
a_{b_1}(b_{c_1}(c_{b_1}(x))) → a_{a_1}(a_{b_1}(x))
a_{b_1}(b_{c_1}(c_{c_1}(x))) → a_{a_1}(a_{c_1}(x))
b_{b_1}(b_{c_1}(c_{a_1}(x))) → b_{a_1}(a_{a_1}(x))
b_{b_1}(b_{c_1}(c_{b_1}(x))) → b_{a_1}(a_{b_1}(x))
b_{b_1}(b_{c_1}(c_{c_1}(x))) → b_{a_1}(a_{c_1}(x))
c_{b_1}(b_{c_1}(c_{a_1}(x))) → c_{a_1}(a_{a_1}(x))
c_{b_1}(b_{c_1}(c_{b_1}(x))) → c_{a_1}(a_{b_1}(x))
c_{b_1}(b_{c_1}(c_{c_1}(x))) → c_{a_1}(a_{c_1}(x))

Q is empty.

(7) DependencyPairsProof (EQUIVALENT transformation)

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}(x)) → B_{A_1}(x)
B_{A_1}(a_{b_1}(x)) → B_{B_1}(x)
C_{A_1}(a_{a_1}(x)) → C_{A_1}(x)
C_{A_1}(a_{b_1}(x)) → C_{B_1}(x)
A_{A_1}(a_{a_1}(a_{a_1}(x))) → A_{B_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))))
A_{A_1}(a_{a_1}(a_{a_1}(x))) → C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))
A_{A_1}(a_{a_1}(a_{a_1}(x))) → B_{A_1}(a_{b_1}(b_{a_1}(x)))
A_{A_1}(a_{a_1}(a_{a_1}(x))) → A_{B_1}(b_{a_1}(x))
A_{A_1}(a_{a_1}(a_{a_1}(x))) → B_{A_1}(x)
A_{A_1}(a_{a_1}(a_{b_1}(x))) → A_{B_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))))
A_{A_1}(a_{a_1}(a_{b_1}(x))) → C_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))
A_{A_1}(a_{a_1}(a_{b_1}(x))) → B_{A_1}(a_{b_1}(b_{b_1}(x)))
A_{A_1}(a_{a_1}(a_{b_1}(x))) → A_{B_1}(b_{b_1}(x))
A_{A_1}(a_{a_1}(a_{b_1}(x))) → B_{B_1}(x)
A_{A_1}(a_{a_1}(a_{c_1}(x))) → A_{B_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))))))
A_{A_1}(a_{a_1}(a_{c_1}(x))) → C_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x))))
A_{A_1}(a_{a_1}(a_{c_1}(x))) → B_{A_1}(a_{b_1}(b_{c_1}(x)))
A_{A_1}(a_{a_1}(a_{c_1}(x))) → A_{B_1}(b_{c_1}(x))
B_{A_1}(a_{a_1}(a_{a_1}(x))) → B_{B_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))))
B_{A_1}(a_{a_1}(a_{a_1}(x))) → C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))
B_{A_1}(a_{a_1}(a_{a_1}(x))) → B_{A_1}(a_{b_1}(b_{a_1}(x)))
B_{A_1}(a_{a_1}(a_{a_1}(x))) → A_{B_1}(b_{a_1}(x))
B_{A_1}(a_{a_1}(a_{a_1}(x))) → B_{A_1}(x)
B_{A_1}(a_{a_1}(a_{b_1}(x))) → B_{B_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))))
B_{A_1}(a_{a_1}(a_{b_1}(x))) → C_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))
B_{A_1}(a_{a_1}(a_{b_1}(x))) → B_{A_1}(a_{b_1}(b_{b_1}(x)))
B_{A_1}(a_{a_1}(a_{b_1}(x))) → A_{B_1}(b_{b_1}(x))
B_{A_1}(a_{a_1}(a_{b_1}(x))) → B_{B_1}(x)
B_{A_1}(a_{a_1}(a_{c_1}(x))) → B_{B_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))))))
B_{A_1}(a_{a_1}(a_{c_1}(x))) → C_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x))))
B_{A_1}(a_{a_1}(a_{c_1}(x))) → B_{A_1}(a_{b_1}(b_{c_1}(x)))
B_{A_1}(a_{a_1}(a_{c_1}(x))) → A_{B_1}(b_{c_1}(x))
C_{A_1}(a_{a_1}(a_{a_1}(x))) → C_{B_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))))
C_{A_1}(a_{a_1}(a_{a_1}(x))) → C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))
C_{A_1}(a_{a_1}(a_{a_1}(x))) → B_{A_1}(a_{b_1}(b_{a_1}(x)))
C_{A_1}(a_{a_1}(a_{a_1}(x))) → A_{B_1}(b_{a_1}(x))
C_{A_1}(a_{a_1}(a_{a_1}(x))) → B_{A_1}(x)
C_{A_1}(a_{a_1}(a_{b_1}(x))) → C_{B_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))))
C_{A_1}(a_{a_1}(a_{b_1}(x))) → C_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))
C_{A_1}(a_{a_1}(a_{b_1}(x))) → B_{A_1}(a_{b_1}(b_{b_1}(x)))
C_{A_1}(a_{a_1}(a_{b_1}(x))) → A_{B_1}(b_{b_1}(x))
C_{A_1}(a_{a_1}(a_{b_1}(x))) → B_{B_1}(x)
C_{A_1}(a_{a_1}(a_{c_1}(x))) → C_{B_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))))))
C_{A_1}(a_{a_1}(a_{c_1}(x))) → C_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x))))
C_{A_1}(a_{a_1}(a_{c_1}(x))) → B_{A_1}(a_{b_1}(b_{c_1}(x)))
C_{A_1}(a_{a_1}(a_{c_1}(x))) → A_{B_1}(b_{c_1}(x))
A_{B_1}(b_{c_1}(c_{a_1}(x))) → A_{A_1}(a_{a_1}(x))
A_{B_1}(b_{c_1}(c_{a_1}(x))) → A_{A_1}(x)
A_{B_1}(b_{c_1}(c_{b_1}(x))) → A_{A_1}(a_{b_1}(x))
A_{B_1}(b_{c_1}(c_{b_1}(x))) → A_{B_1}(x)
A_{B_1}(b_{c_1}(c_{c_1}(x))) → A_{A_1}(a_{c_1}(x))
B_{B_1}(b_{c_1}(c_{a_1}(x))) → B_{A_1}(a_{a_1}(x))
B_{B_1}(b_{c_1}(c_{a_1}(x))) → A_{A_1}(x)
B_{B_1}(b_{c_1}(c_{b_1}(x))) → B_{A_1}(a_{b_1}(x))
B_{B_1}(b_{c_1}(c_{b_1}(x))) → A_{B_1}(x)
B_{B_1}(b_{c_1}(c_{c_1}(x))) → B_{A_1}(a_{c_1}(x))
C_{B_1}(b_{c_1}(c_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
C_{B_1}(b_{c_1}(c_{a_1}(x))) → A_{A_1}(x)
C_{B_1}(b_{c_1}(c_{b_1}(x))) → C_{A_1}(a_{b_1}(x))
C_{B_1}(b_{c_1}(c_{b_1}(x))) → A_{B_1}(x)
C_{B_1}(b_{c_1}(c_{c_1}(x))) → C_{A_1}(a_{c_1}(x))

The TRS R consists of the following rules:

a_{a_1}(a_{a_1}(x)) → a_{a_1}(x)
a_{a_1}(a_{b_1}(x)) → a_{b_1}(x)
a_{a_1}(a_{c_1}(x)) → a_{c_1}(x)
b_{a_1}(a_{a_1}(x)) → b_{a_1}(x)
b_{a_1}(a_{b_1}(x)) → b_{b_1}(x)
b_{a_1}(a_{c_1}(x)) → b_{c_1}(x)
c_{a_1}(a_{a_1}(x)) → c_{a_1}(x)
c_{a_1}(a_{b_1}(x)) → c_{b_1}(x)
c_{a_1}(a_{c_1}(x)) → c_{c_1}(x)
a_{a_1}(a_{a_1}(a_{a_1}(x))) → a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))))
a_{a_1}(a_{a_1}(a_{b_1}(x))) → a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))))
a_{a_1}(a_{a_1}(a_{c_1}(x))) → a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(x))) → b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))))
b_{a_1}(a_{a_1}(a_{b_1}(x))) → b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))))
b_{a_1}(a_{a_1}(a_{c_1}(x))) → b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(x))) → c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))))
c_{a_1}(a_{a_1}(a_{b_1}(x))) → c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))))
c_{a_1}(a_{a_1}(a_{c_1}(x))) → c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{c_1}(c_{a_1}(x))) → a_{a_1}(a_{a_1}(x))
a_{b_1}(b_{c_1}(c_{b_1}(x))) → a_{a_1}(a_{b_1}(x))
a_{b_1}(b_{c_1}(c_{c_1}(x))) → a_{a_1}(a_{c_1}(x))
b_{b_1}(b_{c_1}(c_{a_1}(x))) → b_{a_1}(a_{a_1}(x))
b_{b_1}(b_{c_1}(c_{b_1}(x))) → b_{a_1}(a_{b_1}(x))
b_{b_1}(b_{c_1}(c_{c_1}(x))) → b_{a_1}(a_{c_1}(x))
c_{b_1}(b_{c_1}(c_{a_1}(x))) → c_{a_1}(a_{a_1}(x))
c_{b_1}(b_{c_1}(c_{b_1}(x))) → c_{a_1}(a_{b_1}(x))
c_{b_1}(b_{c_1}(c_{c_1}(x))) → c_{a_1}(a_{c_1}(x))

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

(9) DependencyGraphProof (EQUIVALENT transformation)

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}(x)) → B_{B_1}(x)
B_{B_1}(b_{c_1}(c_{a_1}(x))) → B_{A_1}(a_{a_1}(x))
B_{A_1}(a_{a_1}(x)) → B_{A_1}(x)
B_{A_1}(a_{a_1}(a_{a_1}(x))) → C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))
C_{B_1}(b_{c_1}(c_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
C_{A_1}(a_{a_1}(x)) → C_{A_1}(x)
C_{A_1}(a_{b_1}(x)) → C_{B_1}(x)
C_{B_1}(b_{c_1}(c_{a_1}(x))) → A_{A_1}(x)
A_{A_1}(a_{a_1}(a_{a_1}(x))) → C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))
C_{B_1}(b_{c_1}(c_{b_1}(x))) → C_{A_1}(a_{b_1}(x))
C_{A_1}(a_{a_1}(a_{a_1}(x))) → C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))
C_{B_1}(b_{c_1}(c_{b_1}(x))) → A_{B_1}(x)
A_{B_1}(b_{c_1}(c_{a_1}(x))) → A_{A_1}(a_{a_1}(x))
A_{A_1}(a_{a_1}(a_{a_1}(x))) → B_{A_1}(a_{b_1}(b_{a_1}(x)))
B_{A_1}(a_{a_1}(a_{a_1}(x))) → B_{A_1}(a_{b_1}(b_{a_1}(x)))
B_{A_1}(a_{a_1}(a_{a_1}(x))) → A_{B_1}(b_{a_1}(x))
A_{B_1}(b_{c_1}(c_{a_1}(x))) → A_{A_1}(x)
A_{A_1}(a_{a_1}(a_{a_1}(x))) → A_{B_1}(b_{a_1}(x))
A_{B_1}(b_{c_1}(c_{b_1}(x))) → A_{A_1}(a_{b_1}(x))
A_{A_1}(a_{a_1}(a_{a_1}(x))) → B_{A_1}(x)
B_{A_1}(a_{a_1}(a_{a_1}(x))) → B_{A_1}(x)
B_{A_1}(a_{a_1}(a_{b_1}(x))) → C_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))
B_{A_1}(a_{a_1}(a_{b_1}(x))) → B_{A_1}(a_{b_1}(b_{b_1}(x)))
B_{A_1}(a_{a_1}(a_{b_1}(x))) → A_{B_1}(b_{b_1}(x))
A_{B_1}(b_{c_1}(c_{b_1}(x))) → A_{B_1}(x)
B_{A_1}(a_{a_1}(a_{b_1}(x))) → B_{B_1}(x)
B_{B_1}(b_{c_1}(c_{a_1}(x))) → A_{A_1}(x)
A_{A_1}(a_{a_1}(a_{b_1}(x))) → C_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))
A_{A_1}(a_{a_1}(a_{b_1}(x))) → B_{A_1}(a_{b_1}(b_{b_1}(x)))
B_{A_1}(a_{a_1}(a_{c_1}(x))) → C_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x))))
B_{A_1}(a_{a_1}(a_{c_1}(x))) → B_{A_1}(a_{b_1}(b_{c_1}(x)))
B_{A_1}(a_{a_1}(a_{c_1}(x))) → A_{B_1}(b_{c_1}(x))
A_{A_1}(a_{a_1}(a_{b_1}(x))) → A_{B_1}(b_{b_1}(x))
A_{A_1}(a_{a_1}(a_{b_1}(x))) → B_{B_1}(x)
B_{B_1}(b_{c_1}(c_{b_1}(x))) → B_{A_1}(a_{b_1}(x))
B_{B_1}(b_{c_1}(c_{b_1}(x))) → A_{B_1}(x)
A_{A_1}(a_{a_1}(a_{c_1}(x))) → C_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x))))
A_{A_1}(a_{a_1}(a_{c_1}(x))) → B_{A_1}(a_{b_1}(b_{c_1}(x)))
A_{A_1}(a_{a_1}(a_{c_1}(x))) → A_{B_1}(b_{c_1}(x))
C_{A_1}(a_{a_1}(a_{a_1}(x))) → B_{A_1}(a_{b_1}(b_{a_1}(x)))
C_{A_1}(a_{a_1}(a_{a_1}(x))) → A_{B_1}(b_{a_1}(x))
C_{A_1}(a_{a_1}(a_{a_1}(x))) → B_{A_1}(x)
C_{A_1}(a_{a_1}(a_{b_1}(x))) → C_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))
C_{A_1}(a_{a_1}(a_{b_1}(x))) → B_{A_1}(a_{b_1}(b_{b_1}(x)))
C_{A_1}(a_{a_1}(a_{b_1}(x))) → A_{B_1}(b_{b_1}(x))
C_{A_1}(a_{a_1}(a_{b_1}(x))) → B_{B_1}(x)
C_{A_1}(a_{a_1}(a_{c_1}(x))) → C_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x))))
C_{A_1}(a_{a_1}(a_{c_1}(x))) → B_{A_1}(a_{b_1}(b_{c_1}(x)))
C_{A_1}(a_{a_1}(a_{c_1}(x))) → A_{B_1}(b_{c_1}(x))

The TRS R consists of the following rules:

a_{a_1}(a_{a_1}(x)) → a_{a_1}(x)
a_{a_1}(a_{b_1}(x)) → a_{b_1}(x)
a_{a_1}(a_{c_1}(x)) → a_{c_1}(x)
b_{a_1}(a_{a_1}(x)) → b_{a_1}(x)
b_{a_1}(a_{b_1}(x)) → b_{b_1}(x)
b_{a_1}(a_{c_1}(x)) → b_{c_1}(x)
c_{a_1}(a_{a_1}(x)) → c_{a_1}(x)
c_{a_1}(a_{b_1}(x)) → c_{b_1}(x)
c_{a_1}(a_{c_1}(x)) → c_{c_1}(x)
a_{a_1}(a_{a_1}(a_{a_1}(x))) → a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))))
a_{a_1}(a_{a_1}(a_{b_1}(x))) → a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))))
a_{a_1}(a_{a_1}(a_{c_1}(x))) → a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(x))) → b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))))
b_{a_1}(a_{a_1}(a_{b_1}(x))) → b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))))
b_{a_1}(a_{a_1}(a_{c_1}(x))) → b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(x))) → c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))))
c_{a_1}(a_{a_1}(a_{b_1}(x))) → c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))))
c_{a_1}(a_{a_1}(a_{c_1}(x))) → c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{c_1}(c_{a_1}(x))) → a_{a_1}(a_{a_1}(x))
a_{b_1}(b_{c_1}(c_{b_1}(x))) → a_{a_1}(a_{b_1}(x))
a_{b_1}(b_{c_1}(c_{c_1}(x))) → a_{a_1}(a_{c_1}(x))
b_{b_1}(b_{c_1}(c_{a_1}(x))) → b_{a_1}(a_{a_1}(x))
b_{b_1}(b_{c_1}(c_{b_1}(x))) → b_{a_1}(a_{b_1}(x))
b_{b_1}(b_{c_1}(c_{c_1}(x))) → b_{a_1}(a_{c_1}(x))
c_{b_1}(b_{c_1}(c_{a_1}(x))) → c_{a_1}(a_{a_1}(x))
c_{b_1}(b_{c_1}(c_{b_1}(x))) → c_{a_1}(a_{b_1}(x))
c_{b_1}(b_{c_1}(c_{c_1}(x))) → c_{a_1}(a_{c_1}(x))

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

(11) QDPOrderProof (EQUIVALENT transformation)

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}(x))) → B_{A_1}(a_{a_1}(x))
B_{A_1}(a_{a_1}(a_{a_1}(x))) → C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))
C_{B_1}(b_{c_1}(c_{a_1}(x))) → C_{A_1}(a_{a_1}(x))
C_{B_1}(b_{c_1}(c_{a_1}(x))) → A_{A_1}(x)
A_{A_1}(a_{a_1}(a_{a_1}(x))) → C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))
C_{B_1}(b_{c_1}(c_{b_1}(x))) → C_{A_1}(a_{b_1}(x))
C_{A_1}(a_{a_1}(a_{a_1}(x))) → C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))
C_{B_1}(b_{c_1}(c_{b_1}(x))) → A_{B_1}(x)
A_{B_1}(b_{c_1}(c_{a_1}(x))) → A_{A_1}(a_{a_1}(x))
A_{A_1}(a_{a_1}(a_{a_1}(x))) → B_{A_1}(a_{b_1}(b_{a_1}(x)))
B_{A_1}(a_{a_1}(a_{a_1}(x))) → B_{A_1}(a_{b_1}(b_{a_1}(x)))
B_{A_1}(a_{a_1}(a_{a_1}(x))) → A_{B_1}(b_{a_1}(x))
A_{B_1}(b_{c_1}(c_{a_1}(x))) → A_{A_1}(x)
A_{A_1}(a_{a_1}(a_{a_1}(x))) → A_{B_1}(b_{a_1}(x))
A_{B_1}(b_{c_1}(c_{b_1}(x))) → A_{A_1}(a_{b_1}(x))
A_{A_1}(a_{a_1}(a_{a_1}(x))) → B_{A_1}(x)
B_{A_1}(a_{a_1}(a_{a_1}(x))) → B_{A_1}(x)
B_{A_1}(a_{a_1}(a_{b_1}(x))) → C_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))
B_{A_1}(a_{a_1}(a_{b_1}(x))) → B_{A_1}(a_{b_1}(b_{b_1}(x)))
B_{A_1}(a_{a_1}(a_{b_1}(x))) → A_{B_1}(b_{b_1}(x))
A_{B_1}(b_{c_1}(c_{b_1}(x))) → A_{B_1}(x)
B_{A_1}(a_{a_1}(a_{b_1}(x))) → B_{B_1}(x)
B_{B_1}(b_{c_1}(c_{a_1}(x))) → A_{A_1}(x)
A_{A_1}(a_{a_1}(a_{b_1}(x))) → C_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))
A_{A_1}(a_{a_1}(a_{b_1}(x))) → B_{A_1}(a_{b_1}(b_{b_1}(x)))
B_{A_1}(a_{a_1}(a_{c_1}(x))) → C_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x))))
B_{A_1}(a_{a_1}(a_{c_1}(x))) → B_{A_1}(a_{b_1}(b_{c_1}(x)))
B_{A_1}(a_{a_1}(a_{c_1}(x))) → A_{B_1}(b_{c_1}(x))
A_{A_1}(a_{a_1}(a_{b_1}(x))) → A_{B_1}(b_{b_1}(x))
A_{A_1}(a_{a_1}(a_{b_1}(x))) → B_{B_1}(x)
B_{B_1}(b_{c_1}(c_{b_1}(x))) → B_{A_1}(a_{b_1}(x))
B_{B_1}(b_{c_1}(c_{b_1}(x))) → A_{B_1}(x)
A_{A_1}(a_{a_1}(a_{c_1}(x))) → C_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x))))
A_{A_1}(a_{a_1}(a_{c_1}(x))) → B_{A_1}(a_{b_1}(b_{c_1}(x)))
A_{A_1}(a_{a_1}(a_{c_1}(x))) → A_{B_1}(b_{c_1}(x))
C_{A_1}(a_{a_1}(a_{a_1}(x))) → B_{A_1}(a_{b_1}(b_{a_1}(x)))
C_{A_1}(a_{a_1}(a_{a_1}(x))) → A_{B_1}(b_{a_1}(x))
C_{A_1}(a_{a_1}(a_{a_1}(x))) → B_{A_1}(x)
C_{A_1}(a_{a_1}(a_{b_1}(x))) → C_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))
C_{A_1}(a_{a_1}(a_{b_1}(x))) → B_{A_1}(a_{b_1}(b_{b_1}(x)))
C_{A_1}(a_{a_1}(a_{b_1}(x))) → A_{B_1}(b_{b_1}(x))
C_{A_1}(a_{a_1}(a_{b_1}(x))) → B_{B_1}(x)
C_{A_1}(a_{a_1}(a_{c_1}(x))) → C_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x))))
C_{A_1}(a_{a_1}(a_{c_1}(x))) → B_{A_1}(a_{b_1}(b_{c_1}(x)))
C_{A_1}(a_{a_1}(a_{c_1}(x))) → A_{B_1}(b_{c_1}(x))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( A_{A_1}(x₁) ) = x₁

POL( B_{A_1}(x₁) ) = max{0, x₁ - 2}

POL( C_{A_1}(x₁) ) = max{0, x₁ - 2}

POL( A_{B_1}(x₁) ) = x₁ + 1

POL( b_{a_1}(x₁) ) = max{0, x₁ - 2}

POL( C_{B_1}(x₁) ) = x₁

POL( b_{b_1}(x₁) ) = max{0, x₁ - 2}

POL( b_{c_1}(x₁) ) = x₁

POL( c_{a_1}(x₁) ) = x₁ + 2

POL( c_{c_1}(x₁) ) = x₁ + 2

POL( a_{a_1}(x₁) ) = x₁ + 2

POL( a_{b_1}(x₁) ) = x₁ + 2

POL( c_{b_1}(x₁) ) = x₁ + 2

POL( a_{c_1}(x₁) ) = x₁ + 2

POL( B_{B_1}(x₁) ) = x₁

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

a_{a_1}(a_{b_1}(x)) → a_{b_1}(x)
a_{b_1}(b_{c_1}(c_{a_1}(x))) → a_{a_1}(a_{a_1}(x))
a_{a_1}(a_{a_1}(x)) → a_{a_1}(x)
a_{b_1}(b_{c_1}(c_{b_1}(x))) → a_{a_1}(a_{b_1}(x))
a_{a_1}(a_{c_1}(x)) → a_{c_1}(x)
a_{a_1}(a_{a_1}(a_{a_1}(x))) → a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))))
a_{a_1}(a_{a_1}(a_{b_1}(x))) → a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))))
a_{a_1}(a_{a_1}(a_{c_1}(x))) → a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))))))
b_{a_1}(a_{b_1}(x)) → b_{b_1}(x)
b_{b_1}(b_{c_1}(c_{a_1}(x))) → b_{a_1}(a_{a_1}(x))
b_{a_1}(a_{a_1}(x)) → b_{a_1}(x)
b_{b_1}(b_{c_1}(c_{b_1}(x))) → b_{a_1}(a_{b_1}(x))
b_{a_1}(a_{c_1}(x)) → b_{c_1}(x)
b_{a_1}(a_{a_1}(a_{a_1}(x))) → b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))))
b_{a_1}(a_{a_1}(a_{b_1}(x))) → b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))))
b_{a_1}(a_{a_1}(a_{c_1}(x))) → b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{c_1}(c_{c_1}(x))) → a_{a_1}(a_{c_1}(x))
b_{b_1}(b_{c_1}(c_{c_1}(x))) → b_{a_1}(a_{c_1}(x))
c_{a_1}(a_{b_1}(x)) → c_{b_1}(x)
c_{b_1}(b_{c_1}(c_{a_1}(x))) → c_{a_1}(a_{a_1}(x))
c_{a_1}(a_{a_1}(x)) → c_{a_1}(x)
c_{b_1}(b_{c_1}(c_{b_1}(x))) → c_{a_1}(a_{b_1}(x))
c_{a_1}(a_{a_1}(a_{a_1}(x))) → c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))))
c_{a_1}(a_{a_1}(a_{b_1}(x))) → c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))))
c_{a_1}(a_{a_1}(a_{c_1}(x))) → c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))))))
c_{b_1}(b_{c_1}(c_{c_1}(x))) → c_{a_1}(a_{c_1}(x))
c_{a_1}(a_{c_1}(x)) → c_{c_1}(x)

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

B_{A_1}(a_{b_1}(x)) → B_{B_1}(x)
B_{A_1}(a_{a_1}(x)) → B_{A_1}(x)
C_{A_1}(a_{a_1}(x)) → C_{A_1}(x)
C_{A_1}(a_{b_1}(x)) → C_{B_1}(x)

The TRS R consists of the following rules:

a_{a_1}(a_{a_1}(x)) → a_{a_1}(x)
a_{a_1}(a_{b_1}(x)) → a_{b_1}(x)
a_{a_1}(a_{c_1}(x)) → a_{c_1}(x)
b_{a_1}(a_{a_1}(x)) → b_{a_1}(x)
b_{a_1}(a_{b_1}(x)) → b_{b_1}(x)
b_{a_1}(a_{c_1}(x)) → b_{c_1}(x)
c_{a_1}(a_{a_1}(x)) → c_{a_1}(x)
c_{a_1}(a_{b_1}(x)) → c_{b_1}(x)
c_{a_1}(a_{c_1}(x)) → c_{c_1}(x)
a_{a_1}(a_{a_1}(a_{a_1}(x))) → a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))))
a_{a_1}(a_{a_1}(a_{b_1}(x))) → a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))))
a_{a_1}(a_{a_1}(a_{c_1}(x))) → a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(x))) → b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))))
b_{a_1}(a_{a_1}(a_{b_1}(x))) → b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))))
b_{a_1}(a_{a_1}(a_{c_1}(x))) → b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(x))) → c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))))
c_{a_1}(a_{a_1}(a_{b_1}(x))) → c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))))
c_{a_1}(a_{a_1}(a_{c_1}(x))) → c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{c_1}(c_{a_1}(x))) → a_{a_1}(a_{a_1}(x))
a_{b_1}(b_{c_1}(c_{b_1}(x))) → a_{a_1}(a_{b_1}(x))
a_{b_1}(b_{c_1}(c_{c_1}(x))) → a_{a_1}(a_{c_1}(x))
b_{b_1}(b_{c_1}(c_{a_1}(x))) → b_{a_1}(a_{a_1}(x))
b_{b_1}(b_{c_1}(c_{b_1}(x))) → b_{a_1}(a_{b_1}(x))
b_{b_1}(b_{c_1}(c_{c_1}(x))) → b_{a_1}(a_{c_1}(x))
c_{b_1}(b_{c_1}(c_{a_1}(x))) → c_{a_1}(a_{a_1}(x))
c_{b_1}(b_{c_1}(c_{b_1}(x))) → c_{a_1}(a_{b_1}(x))
c_{b_1}(b_{c_1}(c_{c_1}(x))) → c_{a_1}(a_{c_1}(x))

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

(13) DependencyGraphProof (EQUIVALENT transformation)

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}(x)) → C_{A_1}(x)

The TRS R consists of the following rules:

a_{a_1}(a_{a_1}(x)) → a_{a_1}(x)
a_{a_1}(a_{b_1}(x)) → a_{b_1}(x)
a_{a_1}(a_{c_1}(x)) → a_{c_1}(x)
b_{a_1}(a_{a_1}(x)) → b_{a_1}(x)
b_{a_1}(a_{b_1}(x)) → b_{b_1}(x)
b_{a_1}(a_{c_1}(x)) → b_{c_1}(x)
c_{a_1}(a_{a_1}(x)) → c_{a_1}(x)
c_{a_1}(a_{b_1}(x)) → c_{b_1}(x)
c_{a_1}(a_{c_1}(x)) → c_{c_1}(x)
a_{a_1}(a_{a_1}(a_{a_1}(x))) → a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))))
a_{a_1}(a_{a_1}(a_{b_1}(x))) → a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))))
a_{a_1}(a_{a_1}(a_{c_1}(x))) → a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(x))) → b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))))
b_{a_1}(a_{a_1}(a_{b_1}(x))) → b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))))
b_{a_1}(a_{a_1}(a_{c_1}(x))) → b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(x))) → c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))))
c_{a_1}(a_{a_1}(a_{b_1}(x))) → c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))))
c_{a_1}(a_{a_1}(a_{c_1}(x))) → c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{c_1}(c_{a_1}(x))) → a_{a_1}(a_{a_1}(x))
a_{b_1}(b_{c_1}(c_{b_1}(x))) → a_{a_1}(a_{b_1}(x))
a_{b_1}(b_{c_1}(c_{c_1}(x))) → a_{a_1}(a_{c_1}(x))
b_{b_1}(b_{c_1}(c_{a_1}(x))) → b_{a_1}(a_{a_1}(x))
b_{b_1}(b_{c_1}(c_{b_1}(x))) → b_{a_1}(a_{b_1}(x))
b_{b_1}(b_{c_1}(c_{c_1}(x))) → b_{a_1}(a_{c_1}(x))
c_{b_1}(b_{c_1}(c_{a_1}(x))) → c_{a_1}(a_{a_1}(x))
c_{b_1}(b_{c_1}(c_{b_1}(x))) → c_{a_1}(a_{b_1}(x))
c_{b_1}(b_{c_1}(c_{c_1}(x))) → c_{a_1}(a_{c_1}(x))

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

(16) UsableRulesProof (EQUIVALENT transformation)

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.

(17) Obligation:

Q DP problem:
The TRS P consists of the following rules:

C_{A_1}(a_{a_1}(x)) → C_{A_1}(x)

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

(18) QDPSizeChangeProof (EQUIVALENT transformation)

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}(x)) → C_{A_1}(x)
The graph contains the following edges 1 > 1

(19) YES

(20) Obligation:

Q DP problem:
The TRS P consists of the following rules:

B_{A_1}(a_{a_1}(x)) → B_{A_1}(x)

The TRS R consists of the following rules:

a_{a_1}(a_{a_1}(x)) → a_{a_1}(x)
a_{a_1}(a_{b_1}(x)) → a_{b_1}(x)
a_{a_1}(a_{c_1}(x)) → a_{c_1}(x)
b_{a_1}(a_{a_1}(x)) → b_{a_1}(x)
b_{a_1}(a_{b_1}(x)) → b_{b_1}(x)
b_{a_1}(a_{c_1}(x)) → b_{c_1}(x)
c_{a_1}(a_{a_1}(x)) → c_{a_1}(x)
c_{a_1}(a_{b_1}(x)) → c_{b_1}(x)
c_{a_1}(a_{c_1}(x)) → c_{c_1}(x)
a_{a_1}(a_{a_1}(a_{a_1}(x))) → a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))))
a_{a_1}(a_{a_1}(a_{b_1}(x))) → a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))))
a_{a_1}(a_{a_1}(a_{c_1}(x))) → a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))))))
b_{a_1}(a_{a_1}(a_{a_1}(x))) → b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))))
b_{a_1}(a_{a_1}(a_{b_1}(x))) → b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))))
b_{a_1}(a_{a_1}(a_{c_1}(x))) → b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))))))
c_{a_1}(a_{a_1}(a_{a_1}(x))) → c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))))
c_{a_1}(a_{a_1}(a_{b_1}(x))) → c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))))
c_{a_1}(a_{a_1}(a_{c_1}(x))) → c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))))))
a_{b_1}(b_{c_1}(c_{a_1}(x))) → a_{a_1}(a_{a_1}(x))
a_{b_1}(b_{c_1}(c_{b_1}(x))) → a_{a_1}(a_{b_1}(x))
a_{b_1}(b_{c_1}(c_{c_1}(x))) → a_{a_1}(a_{c_1}(x))
b_{b_1}(b_{c_1}(c_{a_1}(x))) → b_{a_1}(a_{a_1}(x))
b_{b_1}(b_{c_1}(c_{b_1}(x))) → b_{a_1}(a_{b_1}(x))
b_{b_1}(b_{c_1}(c_{c_1}(x))) → b_{a_1}(a_{c_1}(x))
c_{b_1}(b_{c_1}(c_{a_1}(x))) → c_{a_1}(a_{a_1}(x))
c_{b_1}(b_{c_1}(c_{b_1}(x))) → c_{a_1}(a_{b_1}(x))
c_{b_1}(b_{c_1}(c_{c_1}(x))) → c_{a_1}(a_{c_1}(x))

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

(21) UsableRulesProof (EQUIVALENT transformation)

(22) Obligation:

Q DP problem:
The TRS P consists of the following rules:

B_{A_1}(a_{a_1}(x)) → B_{A_1}(x)

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

(23) QDPSizeChangeProof (EQUIVALENT transformation)

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}(x)) → B_{A_1}(x)
The graph contains the following edges 1 > 1

(0) Obligation:

(1) QTRS Reverse (EQUIVALENT transformation)

(2) Obligation:

(3) FlatCCProof (EQUIVALENT transformation)

(4) Obligation:

(5) RootLabelingProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyPairsProof (EQUIVALENT transformation)

(8) Obligation:

(9) DependencyGraphProof (EQUIVALENT transformation)

(10) Obligation:

(11) QDPOrderProof (EQUIVALENT transformation)

(12) Obligation:

(13) DependencyGraphProof (EQUIVALENT transformation)

(14) Complex Obligation (AND)

(15) Obligation:

(16) UsableRulesProof (EQUIVALENT transformation)

(17) Obligation:

(18) QDPSizeChangeProof (EQUIVALENT transformation)

(19) YES

(20) Obligation:

(21) UsableRulesProof (EQUIVALENT transformation)

(22) Obligation:

(23) QDPSizeChangeProof (EQUIVALENT transformation)

(24) YES