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-2-num-21.srs

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

0 QTRS
↳1 QTRS Reverse (⇔, 0 ms)
↳2 QTRS
↳3 DependencyPairsProof (⇔, 0 ms)
↳4 QDP
↳5 DependencyGraphProof (⇔, 0 ms)
↳6 QDP
↳7 QDPOrderProof (⇔, 50 ms)
↳8 QDP
↳9 QDPOrderProof (⇔, 37 ms)
↳10 QDP
↳11 PisEmptyProof (⇔, 0 ms)
↳12 YES

(0) Obligation:

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

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

Q is empty.

(3) DependencyPairsProof (EQUIVALENT transformation)

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

(4) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

(6) Obligation:

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

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

The TRS R consists of the following rules:

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

POL(B(x1)) = -I +
[0A,-I,0A]
·x1

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

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

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

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

(8) Obligation:

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

B(a(a(x))) → B(a(b(x)))

The TRS R consists of the following rules:

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


B(a(a(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)) = -I +
[1A,0A,0A]
·x1

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

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

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

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

(10) Obligation:

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

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

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

(11) PisEmptyProof (EQUIVALENT transformation)

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

(12) YES