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-124.srs

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

0 QTRS
↳1 DependencyPairsProof (⇔, 0 ms)
↳2 QDP
↳3 QDPOrderProof (⇔, 139 ms)
↳4 QDP
↳5 DependencyGraphProof (⇔, 0 ms)
↳6 QDP
↳7 QDPOrderProof (⇔, 15 ms)
↳8 QDP
↳9 PisEmptyProof (⇔, 0 ms)
↳10 YES

(0) Obligation:

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

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

The TRS R consists of the following rules:

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


B(c(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,-I,0A]
·x1

POL(a(x1)) =
/0A\
|-I|
\-I/
 +
/0A0A1A\
|-I0A-I|
\-I0A0A/
·x1

POL(b(x1)) =
/-I\
|0A|
\0A/
 +
/0A0A0A\
|0A1A1A|
\0A0A0A/
·x1

POL(B(x1)) = 0A +
[0A,0A,0A]
·x1

POL(c(x1)) =
/1A\
|-I|
\-I/
 +
/0A0A1A\
|-I-I-I|
\0A-I0A/
·x1

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

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

(4) Obligation:

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

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

The TRS R consists of the following rules:

a(x) → x
a(a(b(x))) → c(b(b(a(a(x)))))
b(c(x)) → a(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 2 less nodes.

(6) Obligation:

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

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

The TRS R consists of the following rules:

a(x) → x
a(a(b(x))) → c(b(b(a(a(x)))))
b(c(x)) → 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(b(x))) → A(x)
A(a(b(x))) → A(a(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( A(x1) ) = max{0, 2x1 - 1}

POL( a(x1) ) = 2x1

POL( b(x1) ) = 2x1 + 2

POL( c(x1) ) = 0


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

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

(8) Obligation:

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

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

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

(9) PisEmptyProof (EQUIVALENT transformation)

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

(10) YES