YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Trafo_06/un09.srs

(0) Obligation:

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

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

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) Obligation:

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

B(b(b(a(b(x))))) → B(a(b(b(a(b(x))))))
B(a(b(b(x)))) → B(b(a(b(a(b(x))))))
B(a(b(b(x)))) → B(a(b(a(b(x)))))
B(a(b(b(x)))) → B(a(b(x)))
B(a(b(a(a(b(b(x))))))) → B(a(a(b(a(a(b(b(b(a(b(x)))))))))))
B(a(b(a(a(b(b(x))))))) → B(a(a(b(b(b(a(b(x))))))))
B(a(b(a(a(b(b(x))))))) → B(b(b(a(b(x)))))
B(a(b(a(a(b(b(x))))))) → B(b(a(b(x))))
B(a(b(a(a(b(b(x))))))) → B(a(b(x)))

The TRS R consists of the following rules:

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

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

(3) DependencyGraphProof (EQUIVALENT transformation)

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

(4) Obligation:

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

B(a(b(b(x)))) → B(b(a(b(a(b(x))))))
B(b(b(a(b(x))))) → B(a(b(b(a(b(x))))))
B(a(b(b(x)))) → B(a(b(a(b(x)))))
B(a(b(b(x)))) → B(a(b(x)))
B(a(b(a(a(b(b(x))))))) → B(b(b(a(b(x)))))
B(a(b(a(a(b(b(x))))))) → B(b(a(b(x))))
B(a(b(a(a(b(b(x))))))) → B(a(b(x)))

The TRS R consists of the following rules:

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

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

(5) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


B(a(b(a(a(b(b(x))))))) → B(b(b(a(b(x)))))
B(a(b(a(a(b(b(x))))))) → B(b(a(b(x))))
B(a(b(a(a(b(b(x))))))) → B(a(b(x)))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(B(x1)) = 0A +
[0A,0A,0A]
·x1

POL(a(x1)) =
/1A\
|0A|
\0A/
+
/0A-I0A\
|0A0A0A|
\1A0A0A/
·x1

POL(b(x1)) =
/0A\
|0A|
\0A/
+
/-I-I-I\
|0A0A0A|
\0A0A-I/
·x1

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

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

(6) Obligation:

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

B(a(b(b(x)))) → B(b(a(b(a(b(x))))))
B(b(b(a(b(x))))) → B(a(b(b(a(b(x))))))
B(a(b(b(x)))) → B(a(b(a(b(x)))))
B(a(b(b(x)))) → B(a(b(x)))

The TRS R consists of the following rules:

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

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

(7) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


B(b(b(a(b(x))))) → B(a(b(b(a(b(x))))))
B(a(b(b(x)))) → B(a(b(a(b(x)))))
B(a(b(b(x)))) → B(a(b(x)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( B(x1) ) = max{0, 2x1 - 2}

POL( b(x1) ) = x1 + 2

POL( a(x1) ) = max{0, x1 - 2}


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

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

(8) Obligation:

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

B(a(b(b(x)))) → B(b(a(b(a(b(x))))))

The TRS R consists of the following rules:

b(b(b(a(b(x))))) → b(a(b(b(a(b(x))))))
b(a(b(b(x)))) → b(b(a(b(a(b(x))))))
b(a(b(a(a(b(b(x))))))) → b(a(a(b(a(a(b(b(b(a(b(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 0 SCCs with 1 less node.

(10) TRUE