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-482.srs

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

0 QTRS
↳1 QTRS Reverse (⇔, 0 ms)
↳2 QTRS
↳3 DependencyPairsProof (⇔, 11 ms)
↳4 QDP
↳5 QDPOrderProof (⇔, 168 ms)
↳6 QDP
↳7 QDPOrderProof (⇔, 189 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)) → a(b(c(x)))
c(a(x)) → x
c(b(x)) → a(a(c(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:

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

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

The TRS R consists of the following rules:

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

POL(a(x1)) =
/0A\
|1A|
\-1A/
 +
/-1A-1A-1A\
|-I1A-1A|
\-I-1A-I/
·x1

POL(B(x1)) = 0A +
[-1A,-1A,1A]
·x1

POL(c(x1)) =
/0A\
|-1A|
\0A/
 +
/-1A-1A-1A\
|1A-1A1A|
\-1A-1A-1A/
·x1

POL(b(x1)) =
/0A\
|-I|
\-I/
 +
/-I-I2A\
|-I-I2A|
\-I-1A2A/
·x1

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

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

(6) Obligation:

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

A(a(x)) → B(a(x))
B(c(x)) → A(a(x))

The TRS R consists of the following rules:

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


B(c(x)) → A(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 +
[-I,-I,-1A]
·x1

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

POL(B(x1)) = -I +
[1A,-I,-1A]
·x1

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

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

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

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

(8) Obligation:

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

A(a(x)) → B(a(x))

The TRS R consists of the following rules:

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

(10) TRUE