YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Secret_07_SRS/x09.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 QDPOrderProof (⇔, 628 ms)
↳6 QDP
↳7 QDPOrderProof (⇔, 240 ms)
↳8 QDP
↳9 QDPOrderProof (⇔, 399 ms)
↳10 QDP
↳11 QDPOrderProof (⇔, 176 ms)
↳12 QDP
↳13 QDPOrderProof (⇔, 318 ms)
↳14 QDP
↳15 QDPOrderProof (⇔, 1604 ms)
↳16 QDP
↳17 DependencyGraphProof (⇔, 0 ms)
↳18 TRUE

(0) Obligation:

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

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

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

The TRS R consists of the following rules:

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

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

POL(a(x1)) =
/1A\
|0A|
\0A/
 +
/0A1A0A\
|-I0A0A|
\0A0A0A/
·x1

POL(A(x1)) = 0A +
[0A,0A,0A]
·x1

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

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

(6) Obligation:

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

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

The TRS R consists of the following rules:

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

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

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

POL(A(x1)) = 0A +
[0A,-I,0A]
·x1

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

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

(8) Obligation:

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

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

The TRS R consists of the following rules:

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

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

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

POL(A(x1)) = 0A +
[0A,0A,0A]
·x1

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

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

(10) Obligation:

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

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

The TRS R consists of the following rules:

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


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

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

POL(a(x1)) =
/1A\
|-I|
\0A/
 +
/0A1A0A\
|-I0A0A|
\0A0A0A/
·x1

POL(A(x1)) = 0A +
[-I,0A,-I]
·x1

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

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

(12) Obligation:

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

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

The TRS R consists of the following rules:

b(b(a(a(a(x))))) → a(a(a(b(b(a(b(b(x))))))))
a(b(b(x))) → x
a(x) → b(b(b(x)))
a(x) → b(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(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)) = 0A +
[0A,0A,0A]
·x1

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

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

POL(A(x1)) = 1A +
[1A,1A,1A]
·x1

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

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

(14) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(15) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


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

POL(B(x1)) = 0A +
[-1A,-I,-1A]
·x1

POL(b(x1)) =
/0A\
|-1A|
\-I/
 +
/-I-I-1A\
|0A-I0A|
\-I-1A0A/
·x1

POL(a(x1)) =
/2A\
|0A|
\-1A/
 +
/-1A-1A1A\
|2A-1A0A|
\-1A0A0A/
·x1

POL(A(x1)) = 1A +
[-1A,-1A,1A]
·x1

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

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

(16) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(17) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(18) TRUE