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-82.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 (⇔, 179 ms)
↳6 QDP
↳7 DependencyGraphProof (⇔, 0 ms)
↳8 AND
    ↳9 QDP
        ↳10 QDPOrderProof (⇔, 73 ms)
        ↳11 QDP
        ↳12 PisEmptyProof (⇔, 0 ms)
        ↳13 YES
    ↳14 QDP
        ↳15 UsableRulesProof (⇔, 0 ms)
        ↳16 QDP
        ↳17 QDPSizeChangeProof (⇔, 0 ms)
        ↳18 YES

(0) Obligation:

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

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

The TRS R consists of the following rules:

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

The TRS R consists of the following rules:

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


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(A(x1)) = 0A +
[0A,0A,0A]
·x1

POL(a(x1)) =
/1A\
|0A|
\-I/
 +
/0A0A0A\
|0A1A1A|
\-I1A0A/
·x1

POL(C(x1)) = 0A +
[-I,0A,0A]
·x1

POL(b(x1)) =
/1A\
|-I|
\0A/
 +
/0A-I0A\
|-I0A0A|
\0A0A1A/
·x1

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

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

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

(6) Obligation:

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

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

The TRS R consists of the following rules:

a(x) → x
a(a(x)) → b(a(b(c(x))))
b(x) → x
c(b(x)) → a(c(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 2 SCCs with 1 less node.

(8) Complex Obligation (AND)

(9) Obligation:

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

A(a(x)) → A(b(c(x)))

The TRS R consists of the following rules:

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

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

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.


A(a(x)) → A(b(c(x)))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(A(x1)) = -I +
[0A,-I,-I]
·x1

POL(a(x1)) =
/1A\
|0A|
\0A/
 +
/1A1A0A\
|1A0A0A|
\0A-I0A/
·x1

POL(b(x1)) =
/0A\
|1A|
\-I/
 +
/0A0A-I\
|0A1A0A|
\-I-I0A/
·x1

POL(c(x1)) =
/0A\
|0A|
\0A/
 +
/-I0A-I\
|-I0A-I|
\0A0A0A/
·x1

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

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

(11) Obligation:

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

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

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

(12) PisEmptyProof (EQUIVALENT transformation)

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

(13) YES

(14) Obligation:

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

C(b(x)) → C(x)

The TRS R consists of the following rules:

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

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

(15) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(16) Obligation:

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

C(b(x)) → C(x)

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

(17) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • C(b(x)) → C(x)
    The graph contains the following edges 1 > 1

(18) YES