YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Mixed_SRS/04.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 DependencyPairsProof (⇔, 24 ms)
↳6 QDP
↳7 DependencyGraphProof (⇔, 0 ms)
↳8 QDP
↳9 QDPOrderProof (⇔, 436 ms)
↳10 QDP
↳11 QDPOrderProof (⇔, 395 ms)
↳12 QDP
↳13 QDPOrderProof (⇔, 265 ms)
↳14 QDP
↳15 DependencyGraphProof (⇔, 0 ms)
↳16 TRUE

(0) Obligation:

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

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

Q is empty.

(5) DependencyPairsProof (EQUIVALENT transformation)

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

The TRS R consists of the following rules:

a_{a_1}(a_{a_1}(a_{a_1}(x))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))
a_{a_1}(a_{a_1}(a_{b_1}(x))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{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.

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Obligation:

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

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

The TRS R consists of the following rules:

a_{a_1}(a_{a_1}(a_{a_1}(x))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))
a_{a_1}(a_{a_1}(a_{b_1}(x))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{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.

(9) 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_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))) → A_{A_1}(a_{a_1}(a_{a_1}(x)))
A_{A_1}(a_{a_1}(a_{a_1}(x))) → B_{B_1}(b_{b_1}(b_{a_1}(x)))
B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))) → A_{A_1}(a_{a_1}(a_{a_1}(x)))
A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))) → A_{A_1}(a_{a_1}(x))
A_{A_1}(a_{a_1}(a_{b_1}(x))) → B_{B_1}(b_{b_1}(b_{b_1}(x)))
B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))) → A_{A_1}(a_{a_1}(x))
A_{A_1}(a_{a_1}(a_{b_1}(x))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))) → A_{A_1}(x)
A_{A_1}(a_{a_1}(a_{b_1}(x))) → B_{B_1}(x)
B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))) → A_{A_1}(a_{a_1}(a_{b_1}(x)))
B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))) → A_{A_1}(a_{b_1}(x))
B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))) → A_{B_1}(x)
A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))) → A_{A_1}(x)
A_{B_1}(b_{b_1}(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}(b_{a_1}(a_{b_1}(b_{b_1}(x))))) → A_{A_1}(a_{b_1}(x))
A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))) → A_{B_1}(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(A_{A_1}(x1)) = 2 + 2·x1   
POL(A_{B_1}(x1)) = 2 + 2·x1   
POL(B_{B_1}(x1)) = 2·x1   
POL(a_{a_1}(x1)) = 3 + x1   
POL(a_{b_1}(x1)) = 3 + x1   
POL(b_{a_1}(x1)) = 2 + x1   
POL(b_{b_1}(x1)) = 2 + x1   

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

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

(10) Obligation:

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

A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))
A_{A_1}(a_{a_1}(a_{a_1}(x))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))
A_{A_1}(a_{a_1}(a_{b_1}(x))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))

The TRS R consists of the following rules:

a_{a_1}(a_{a_1}(a_{a_1}(x))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))
a_{a_1}(a_{a_1}(a_{b_1}(x))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{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) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


A_{A_1}(a_{a_1}(a_{b_1}(x))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(A_{B_1}(x1)) = 0A +
[-I,0A,0A]
·x1

POL(b_{b_1}(x1)) =
/0A\
|0A|
\0A/
 +
/0A-I0A\
|-I-I0A|
\0A-I0A/
·x1

POL(b_{a_1}(x1)) =
/0A\
|0A|
\0A/
 +
/0A0A0A\
|0A-I0A|
\0A-I0A/
·x1

POL(a_{b_1}(x1)) =
/0A\
|0A|
\0A/
 +
/0A-I0A\
|1A-I0A|
\0A-I0A/
·x1

POL(A_{A_1}(x1)) = 0A +
[0A,0A,0A]
·x1

POL(a_{a_1}(x1)) =
/-I\
|0A|
\0A/
 +
/0A-I0A\
|1A0A0A|
\0A0A0A/
·x1

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

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

(12) Obligation:

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

A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))
A_{A_1}(a_{a_1}(a_{a_1}(x))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))

The TRS R consists of the following rules:

a_{a_1}(a_{a_1}(a_{a_1}(x))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))
a_{a_1}(a_{a_1}(a_{b_1}(x))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{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.


A_{A_1}(a_{a_1}(a_{a_1}(x))) → A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(A_{B_1}(x1)) = 0A +
[0A,-I,-I]
·x1

POL(b_{b_1}(x1)) =
/0A\
|0A|
\-I/
 +
/-I-I0A\
|0A0A0A|
\-I0A-I/
·x1

POL(b_{a_1}(x1)) =
/0A\
|-I|
\1A/
 +
/-I-I-I\
|-I-I-I|
\-I-I-I/
·x1

POL(a_{b_1}(x1)) =
/0A\
|-I|
\-I/
 +
/0A0A0A\
|0A0A0A|
\0A0A0A/
·x1

POL(A_{A_1}(x1)) = 1A +
[-I,-I,-I]
·x1

POL(a_{a_1}(x1)) =
/0A\
|0A|
\0A/
 +
/0A0A0A\
|0A0A0A|
\0A0A0A/
·x1

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

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

(14) Obligation:

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

A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))
A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))) → A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))

The TRS R consists of the following rules:

a_{a_1}(a_{a_1}(a_{a_1}(x))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))
a_{a_1}(a_{a_1}(a_{b_1}(x))) → a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))
a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))
b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{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 2 less nodes.

(16) TRUE