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 QTRSRRRProof (⇔, 33 ms)

↳8 QTRS

↳9 DependencyPairsProof (⇔, 0 ms)

↳10 QDP

↳11 DependencyGraphProof (⇔, 0 ms)

↳12 QDP

↳13 QDPOrderProof (⇔, 303 ms)

↳14 QDP

↳15 DependencyGraphProof (⇔, 0 ms)

↳16 TRUE

(0) Obligation:

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

a(x) → b(b(c(x)))
a(b(x)) → x
a(c(b(x))) → a(a(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) → c(b(b(x)))
b(a(x)) → x
b(c(a(x))) → a(a(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(c(b(b(x))))
c(a(x)) → c(c(b(b(x))))
b(a(x)) → b(c(b(b(x))))
a(b(a(x))) → a(x)
c(b(a(x))) → c(x)
b(b(a(x))) → b(x)
a(b(c(a(x)))) → a(a(a(a(x))))
c(b(c(a(x)))) → c(a(a(a(x))))
b(b(c(a(x)))) → b(a(a(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_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x))))
a_{a_1}(a_{c_1}(x)) → a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x))))
a_{a_1}(a_{b_1}(x)) → a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x))))
c_{a_1}(a_{a_1}(x)) → c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x))))
c_{a_1}(a_{c_1}(x)) → c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x))))
c_{a_1}(a_{b_1}(x)) → c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x))))
b_{a_1}(a_{a_1}(x)) → b_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x))))
b_{a_1}(a_{c_1}(x)) → b_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x))))
b_{a_1}(a_{b_1}(x)) → b_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x))))
a_{b_1}(b_{a_1}(a_{a_1}(x))) → a_{a_1}(x)
a_{b_1}(b_{a_1}(a_{c_1}(x))) → a_{c_1}(x)
a_{b_1}(b_{a_1}(a_{b_1}(x))) → a_{b_1}(x)
c_{b_1}(b_{a_1}(a_{a_1}(x))) → c_{a_1}(x)
c_{b_1}(b_{a_1}(a_{c_1}(x))) → c_{c_1}(x)
c_{b_1}(b_{a_1}(a_{b_1}(x))) → c_{b_1}(x)
b_{b_1}(b_{a_1}(a_{a_1}(x))) → b_{a_1}(x)
b_{b_1}(b_{a_1}(a_{c_1}(x))) → b_{c_1}(x)
b_{b_1}(b_{a_1}(a_{b_1}(x))) → b_{b_1}(x)
a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))
a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x)))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x))))
a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x)))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))
c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))) → c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))
c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x)))) → c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x))))
c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x)))) → c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))
b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x)))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))
b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x)))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x))))
b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x)))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a_{a_1}(x₁)) = x₁
POL(a_{b_1}(x₁)) = 1 + x₁
POL(a_{c_1}(x₁)) = x₁
POL(b_{a_1}(x₁)) = x₁
POL(b_{b_1}(x₁)) = x₁
POL(b_{c_1}(x₁)) = x₁
POL(c_{a_1}(x₁)) = x₁
POL(c_{b_1}(x₁)) = x₁
POL(c_{c_1}(x₁)) = x₁

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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

(8) Obligation:

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

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

Q is empty.

(9) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(10) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(11) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(12) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

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

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

POL( B_{B_1}(x₁) ) = 2x₁ + 2

POL( C_{B_1}(x₁) ) = max{0, 2x₁ - 2}

POL( A_{A_1}(x₁) ) = 2x₁ + 2

POL( a_{c_1}(x₁) ) = x₁ + 2

POL( c_{b_1}(x₁) ) = x₁

POL( B_{A_1}(x₁) ) = 2x₁ + 2

POL( C_{A_1}(x₁) ) = 2x₁ + 2

POL( b_{a_1}(x₁) ) = x₁ + 1

POL( a_{a_1}(x₁) ) = x₁ + 1

POL( b_{c_1}(x₁) ) = x₁ + 2

POL( b_{b_1}(x₁) ) = max{0, x₁ - 1}

POL( c_{a_1}(x₁) ) = x₁ + 2

POL( a_{b_1}(x₁) ) = 0

POL( c_{c_1}(x₁) ) = max{0, -2}

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

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

(14) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(15) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 10 less nodes.

(0) Obligation:

(1) QTRS Reverse (EQUIVALENT transformation)

(2) Obligation:

(3) FlatCCProof (EQUIVALENT transformation)

(4) Obligation:

(5) RootLabelingProof (EQUIVALENT transformation)

(6) Obligation:

(7) QTRSRRRProof (EQUIVALENT transformation)

(8) Obligation:

(9) DependencyPairsProof (EQUIVALENT transformation)

(10) Obligation:

(11) DependencyGraphProof (EQUIVALENT transformation)

(12) Obligation:

(13) QDPOrderProof (EQUIVALENT transformation)

(14) Obligation:

(15) DependencyGraphProof (EQUIVALENT transformation)

(16) TRUE