YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Bouchare_06/13.srs

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

0 QTRS
↳1 DependencyPairsProof (⇔, 0 ms)
↳2 QDP
↳3 QDPOrderProof (⇔, 127 ms)
↳4 QDP
↳5 QDPOrderProof (⇔, 50 ms)
↳6 QDP
↳7 DependencyGraphProof (⇔, 0 ms)
↳8 TRUE

(0) Obligation:

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

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

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

The TRS R consists of the following rules:

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

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

(3) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


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

POL(A(x1)) = 0A +
[-I,0A,0A]
·x1

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

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

POL(B(x1)) = -I +
[0A,0A,0A]
·x1

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

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

(4) Obligation:

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

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

The TRS R consists of the following rules:

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


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

POL(A(x1)) = 0A +
[0A,0A,1A]
·x1

POL(b(x1)) =
/0A\
|0A|
\-I/
 +
/0A0A1A\
|0A0A0A|
\0A0A0A/
·x1

POL(a(x1)) =
/0A\
|0A|
\-I/
 +
/0A0A1A\
|0A0A-I|
\-I-I0A/
·x1

POL(B(x1)) = 0A +
[0A,0A,0A]
·x1

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

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

(6) Obligation:

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

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

The TRS R consists of the following rules:

a(a(b(x))) → a(x)
a(b(a(x))) → b(b(a(x)))
b(b(x)) → a(a(a(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 0 SCCs with 3 less nodes.

(8) TRUE