YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Bouchare_06/11.srs

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

0 QTRS
↳1 DependencyPairsProof (⇔, 0 ms)
↳2 QDP
↳3 QDPOrderProof (⇔, 116 ms)
↳4 QDP
↳5 QDPOrderProof (⇔, 71 ms)
↳6 QDP
↳7 QDPOrderProof (⇔, 51 ms)
↳8 QDP
↳9 DependencyGraphProof (⇔, 0 ms)
↳10 QDP
↳11 QDPOrderProof (⇔, 252 ms)
↳12 QDP
↳13 PisEmptyProof (⇔, 0 ms)
↳14 YES

(0) Obligation:

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

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

The TRS R consists of the following rules:

a(a(a(x))) → b(x)
b(b(x)) → a(a(x))
a(a(x)) → a(b(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(a(x))) → B(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,-I,0A]
·x1

POL(a(x1)) =
/-I\
|-I|
\0A/
 +
/1A0A0A\
|0A-I-I|
\1A0A-I/
·x1

POL(B(x1)) = -I +
[0A,0A,0A]
·x1

POL(b(x1)) =
/-I\
|0A|
\0A/
 +
/0A1A-I\
|1A0A0A|
\0A1A0A/
·x1

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

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

(4) Obligation:

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

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

The TRS R consists of the following rules:

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


B(b(x)) → A(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,0A,0A]
·x1

POL(b(x1)) =
/1A\
|1A|
\-I/
 +
/0A1A1A\
|1A-I0A|
\-I-I-I/
·x1

POL(A(x1)) = 0A +
[-I,0A,0A]
·x1

POL(a(x1)) =
/0A\
|1A|
\-I/
 +
/-I0A0A\
|0A1A1A|
\-I-I-I/
·x1

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

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

(6) Obligation:

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

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

The TRS R consists of the following rules:

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


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

POL(B(x1)) = 0A +
[-I,0A,-I]
·x1

POL(b(x1)) =
/0A\
|1A|
\0A/
 +
/0A1A0A\
|1A0A0A|
\-I0A-I/
·x1

POL(A(x1)) = -I +
[0A,-I,-I]
·x1

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

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

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

(8) Obligation:

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

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

The TRS R consists of the following rules:

a(a(a(x))) → b(x)
b(b(x)) → a(a(x))
a(a(x)) → a(b(a(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 1 SCC with 1 less node.

(10) Obligation:

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

A(a(x)) → A(b(a(x)))

The TRS R consists of the following rules:

a(a(a(x))) → b(x)
b(b(x)) → a(a(x))
a(a(x)) → a(b(a(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(x)) → A(b(a(x)))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic integers [ARCTIC,STERNAGEL_THIEMANN_RTA14]:

POL(A(x1)) = 0A +
[-1A,-1A,2A]
·x1

POL(a(x1)) =
/0A\
|-1A|
\2A/
 +
/-I-1A-1A\
|-I0A-I|
\-1A2A1A/
·x1

POL(b(x1)) =
/2A\
|-1A|
\-I/
 +
/-I2A1A\
|-I0A-I|
\1A-1A-1A/
·x1

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

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

(12) Obligation:

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

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

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

(13) PisEmptyProof (EQUIVALENT transformation)

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

(14) YES