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-499.srs

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

0 QTRS
↳1 DependencyPairsProof (⇔, 0 ms)
↳2 QDP
↳3 DependencyGraphProof (⇔, 0 ms)
↳4 QDP
↳5 QDPOrderProof (⇔, 53 ms)
↳6 QDP
↳7 QDPOrderProof (⇔, 79 ms)
↳8 QDP
↳9 DependencyGraphProof (⇔, 0 ms)
↳10 TRUE

(0) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(3) DependencyGraphProof (EQUIVALENT transformation)

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

(4) Obligation:

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

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

The TRS R consists of the following rules:

a(a(x)) → b(a(c(b(x))))
b(b(x)) → a(a(x))
c(a(x)) → 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)
A(a(x)) → A(c(b(x)))
A(a(x)) → B(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( B(x1) ) = max{0, 2x1 - 1}

POL( b(x1) ) = x1 + 1

POL( a(x1) ) = x1 + 1

POL( c(x1) ) = max{0, x1 - 1}


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(x)) → b(a(c(b(x))))
c(a(x)) → x

(6) Obligation:

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

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

The TRS R consists of the following rules:

a(a(x)) → b(a(c(b(x))))
b(b(x)) → a(a(x))
c(a(x)) → 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(c(b(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,-I]
·x1

POL(b(x1)) =
/-I\
|0A|
\0A/
 +
/0A0A-I\
|1A1A0A|
\0A1A0A/
·x1

POL(A(x1)) = -I +
[0A,-I,0A]
·x1

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

POL(c(x1)) =
/-I\
|-I|
\-I/
 +
/0A-I-I\
|0A-I-I|
\0A-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(x)) → b(a(c(b(x))))
c(a(x)) → x

(8) Obligation:

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

B(b(x)) → A(a(x))

The TRS R consists of the following rules:

a(a(x)) → b(a(c(b(x))))
b(b(x)) → a(a(x))
c(a(x)) → 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 0 SCCs with 1 less node.

(10) TRUE