YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-561.srs

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

0 QTRS
↳1 FlatCCProof (⇔, 0 ms)
↳2 QTRS
↳3 RootLabelingProof (⇔, 0 ms)
↳4 QTRS
↳5 QTRSRRRProof (⇔, 12 ms)
↳6 QTRS
↳7 DependencyPairsProof (⇔, 0 ms)
↳8 QDP
↳9 DependencyGraphProof (⇔, 0 ms)
↳10 AND
    ↳11 QDP
        ↳12 UsableRulesProof (⇔, 0 ms)
        ↳13 QDP
        ↳14 QDPSizeChangeProof (⇔, 0 ms)
        ↳15 YES
    ↳16 QDP
        ↳17 UsableRulesProof (⇔, 0 ms)
        ↳18 QDP
        ↳19 QDPSizeChangeProof (⇔, 0 ms)
        ↳20 YES
    ↳21 QDP
        ↳22 QDPOrderProof (⇔, 59 ms)
        ↳23 QDP
        ↳24 PisEmptyProof (⇔, 0 ms)
        ↳25 YES

(0) Obligation:

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

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

Q is empty.

(1) FlatCCProof (EQUIVALENT transformation)

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

Q is empty.

(3) 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.

(4) Obligation:

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

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

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a_{a_1}(x1)) = x1   
POL(a_{b_1}(x1)) = x1   
POL(a_{c_1}(x1)) = x1   
POL(b_{a_1}(x1)) = x1   
POL(b_{b_1}(x1)) = 2 + x1   
POL(b_{c_1}(x1)) = x1   
POL(c_{a_1}(x1)) = 2 + x1   
POL(c_{b_1}(x1)) = 3 + 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}(b_{b_1}(x))) → a_{b_1}(x)
c_{a_1}(a_{b_1}(b_{b_1}(x))) → c_{b_1}(x)
c_{a_1}(a_{b_1}(b_{c_1}(x))) → c_{c_1}(x)
c_{a_1}(a_{c_1}(c_{a_1}(x))) → c_{c_1}(c_{c_1}(c_{a_1}(x)))
c_{a_1}(a_{c_1}(c_{b_1}(x))) → c_{c_1}(c_{c_1}(c_{b_1}(x)))
c_{a_1}(a_{c_1}(c_{c_1}(x))) → c_{c_1}(c_{c_1}(c_{c_1}(x)))
a_{b_1}(b_{c_1}(c_{a_1}(x))) → a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))
a_{b_1}(b_{c_1}(c_{b_1}(x))) → a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))
b_{b_1}(b_{c_1}(c_{a_1}(x))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))
b_{b_1}(b_{c_1}(c_{b_1}(x))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))
b_{b_1}(b_{c_1}(c_{c_1}(x))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))))
c_{b_1}(b_{c_1}(c_{a_1}(x))) → c_{a_1}(a_{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}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))
c_{b_1}(b_{c_1}(c_{c_1}(x))) → c_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x)))))


(6) Obligation:

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

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

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

The TRS R consists of the following rules:

a_{a_1}(a_{b_1}(b_{a_1}(x))) → a_{a_1}(x)
a_{a_1}(a_{b_1}(b_{c_1}(x))) → a_{c_1}(x)
b_{a_1}(a_{b_1}(b_{a_1}(x))) → b_{a_1}(x)
b_{a_1}(a_{b_1}(b_{b_1}(x))) → b_{b_1}(x)
b_{a_1}(a_{b_1}(b_{c_1}(x))) → b_{c_1}(x)
c_{a_1}(a_{b_1}(b_{a_1}(x))) → c_{a_1}(x)
a_{a_1}(a_{c_1}(c_{a_1}(x))) → a_{c_1}(c_{c_1}(c_{a_1}(x)))
a_{a_1}(a_{c_1}(c_{b_1}(x))) → a_{c_1}(c_{c_1}(c_{b_1}(x)))
a_{a_1}(a_{c_1}(c_{c_1}(x))) → a_{c_1}(c_{c_1}(c_{c_1}(x)))
b_{a_1}(a_{c_1}(c_{a_1}(x))) → b_{c_1}(c_{c_1}(c_{a_1}(x)))
b_{a_1}(a_{c_1}(c_{b_1}(x))) → b_{c_1}(c_{c_1}(c_{b_1}(x)))
b_{a_1}(a_{c_1}(c_{c_1}(x))) → b_{c_1}(c_{c_1}(c_{c_1}(x)))
a_{b_1}(b_{c_1}(c_{c_1}(x))) → a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{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 3 SCCs with 2 less nodes.

(10) Complex Obligation (AND)

(11) Obligation:

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

C_{A_1}(a_{b_1}(b_{a_1}(x))) → C_{A_1}(x)

The TRS R consists of the following rules:

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

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

(12) 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.

(13) Obligation:

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

C_{A_1}(a_{b_1}(b_{a_1}(x))) → C_{A_1}(x)

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

(14) 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_{b_1}(b_{a_1}(x))) → C_{A_1}(x)
    The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

A_{A_1}(a_{b_1}(b_{a_1}(x))) → A_{A_1}(x)

The TRS R consists of the following rules:

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

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

(17) 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.

(18) Obligation:

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

A_{A_1}(a_{b_1}(b_{a_1}(x))) → A_{A_1}(x)

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

(19) 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:

  • A_{A_1}(a_{b_1}(b_{a_1}(x))) → A_{A_1}(x)
    The graph contains the following edges 1 > 1

(20) YES

(21) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(22) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


A_{B_1}(b_{c_1}(c_{c_1}(x))) → A_{B_1}(b_{c_1}(x))
A_{B_1}(b_{c_1}(c_{c_1}(x))) → A_{B_1}(b_{a_1}(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_{B_1}(x1) ) = max{0, 2x1 - 1}

POL( b_{a_1}(x1) ) = x1

POL( a_{b_1}(x1) ) = x1

POL( b_{c_1}(x1) ) = x1 + 1

POL( c_{c_1}(x1) ) = 2x1 + 1

POL( a_{a_1}(x1) ) = 2x1

POL( b_{b_1}(x1) ) = 0

POL( a_{c_1}(x1) ) = 2x1 + 2

POL( c_{a_1}(x1) ) = 2

POL( c_{b_1}(x1) ) = 0


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

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

(23) Obligation:

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

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

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

(24) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(25) YES