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-283.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 (⇔, 303 ms)
↳6 QDP
↳7 DependencyGraphProof (⇔, 0 ms)
↳8 TRUE

(0) Obligation:

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

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

The TRS R consists of the following rules:

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

(4) Obligation:

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

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

The TRS R consists of the following rules:

a(x) → x
a(b(x)) → c(b(b(x)))
b(x) → a(a(c(x)))
c(c(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(x) → A(a(c(x)))
B(x) → A(c(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,-1A,-I]
·x1

POL(b(x1)) =
/0A\
|2A|
\1A/
 +
/0A0A0A\
|-1A1A1A|
\-1A1A1A/
·x1

POL(B(x1)) = 1A +
[-I,-I,-1A]
·x1

POL(a(x1)) =
/-1A\
|-1A|
\-1A/
 +
/0A0A-1A\
|-I0A-I|
\-I2A0A/
·x1

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

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

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

(6) Obligation:

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

A(b(x)) → B(b(x))

The TRS R consists of the following rules:

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

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

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) TRUE