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-3-num-535.srs

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

0 QTRS
↳1 QTRS Reverse (⇔, 0 ms)
↳2 QTRS
↳3 Overlay + Local Confluence (⇔, 0 ms)
↳4 QTRS
↳5 DependencyPairsProof (⇔, 0 ms)
↳6 QDP
↳7 QDPOrderProof (⇔, 249 ms)
↳8 QDP
↳9 QDPOrderProof (⇔, 40 ms)
↳10 QDP
↳11 DependencyGraphProof (⇔, 0 ms)
↳12 TRUE

(0) Obligation:

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

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

Q is empty.

(3) Overlay + Local Confluence (EQUIVALENT transformation)

The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.

(4) Obligation:

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

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

The set Q consists of the following terms:

b(a(a(x0)))
c(a(x0))

(5) DependencyPairsProof (EQUIVALENT transformation)

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

(6) Obligation:

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

B(a(a(x))) → C(a(c(x)))
B(a(a(x))) → C(x)
C(a(x)) → B(x)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

b(a(a(x0)))
c(a(x0))

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

POL(C(x1)) = 0A +
[0A,0A,-I]
·x1

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

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

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

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

(8) Obligation:

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

B(a(a(x))) → C(a(c(x)))
C(a(x)) → B(x)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

b(a(a(x0)))
c(a(x0))

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))) → C(a(c(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(a(x1)) =
/-I\
|1A|
\-I/
 +
/0A0A0A\
|0A0A1A|
\0A-I0A/
·x1

POL(C(x1)) = 0A +
[-I,-I,0A]
·x1

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

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

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

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

(10) Obligation:

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

C(a(x)) → B(x)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

b(a(a(x0)))
c(a(x0))

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

(11) DependencyGraphProof (EQUIVALENT transformation)

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

(12) TRUE