YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Zantema_06/12.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 (⇔, 30 ms)
↳6 QDP
↳7 DependencyGraphProof (⇔, 0 ms)
↳8 QDP
↳9 QDPOrderProof (⇔, 19 ms)
↳10 QDP
↳11 QDPOrderProof (⇔, 14 ms)
↳12 QDP
↳13 QDPOrderProof (⇔, 332 ms)
↳14 QDP
↳15 DependencyGraphProof (⇔, 0 ms)
↳16 TRUE

(0) Obligation:

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

a(l(x)) → l(a(x))
a(c(x)) → c(a(x))
c(a(r(x))) → r(a(x))
l(r(a(a(x)))) → a(a(l(c(c(c(r(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(l(x)) → L(a(x))
A(l(x)) → A(x)
A(c(x)) → C(a(x))
A(c(x)) → A(x)
C(a(r(x))) → A(x)
L(r(a(a(x)))) → A(a(l(c(c(c(r(x)))))))
L(r(a(a(x)))) → A(l(c(c(c(r(x))))))
L(r(a(a(x)))) → L(c(c(c(r(x)))))
L(r(a(a(x)))) → C(c(c(r(x))))
L(r(a(a(x)))) → C(c(r(x)))
L(r(a(a(x)))) → C(r(x))

The TRS R consists of the following rules:

a(l(x)) → l(a(x))
a(c(x)) → c(a(x))
c(a(r(x))) → r(a(x))
l(r(a(a(x)))) → a(a(l(c(c(c(r(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 4 less nodes.

(4) Obligation:

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

L(r(a(a(x)))) → A(a(l(c(c(c(r(x)))))))
A(l(x)) → L(a(x))
L(r(a(a(x)))) → A(l(c(c(c(r(x))))))
A(l(x)) → A(x)
A(c(x)) → C(a(x))
C(a(r(x))) → A(x)
A(c(x)) → A(x)

The TRS R consists of the following rules:

a(l(x)) → l(a(x))
a(c(x)) → c(a(x))
c(a(r(x))) → r(a(x))
l(r(a(a(x)))) → a(a(l(c(c(c(r(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(c(x)) → C(a(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(A(x1)) = 1 + x1   
POL(C(x1)) = x1   
POL(L(x1)) = 1 + x1   
POL(a(x1)) = x1   
POL(c(x1)) = x1   
POL(l(x1)) = x1   
POL(r(x1)) = 1 + x1   

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

l(r(a(a(x)))) → a(a(l(c(c(c(r(x)))))))
a(l(x)) → l(a(x))
a(c(x)) → c(a(x))
c(a(r(x))) → r(a(x))

(6) Obligation:

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

L(r(a(a(x)))) → A(a(l(c(c(c(r(x)))))))
A(l(x)) → L(a(x))
L(r(a(a(x)))) → A(l(c(c(c(r(x))))))
A(l(x)) → A(x)
C(a(r(x))) → A(x)
A(c(x)) → A(x)

The TRS R consists of the following rules:

a(l(x)) → l(a(x))
a(c(x)) → c(a(x))
c(a(r(x))) → r(a(x))
l(r(a(a(x)))) → a(a(l(c(c(c(r(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 1 SCC with 1 less node.

(8) Obligation:

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

A(l(x)) → L(a(x))
L(r(a(a(x)))) → A(a(l(c(c(c(r(x)))))))
A(l(x)) → A(x)
A(c(x)) → A(x)
L(r(a(a(x)))) → A(l(c(c(c(r(x))))))

The TRS R consists of the following rules:

a(l(x)) → l(a(x))
a(c(x)) → c(a(x))
c(a(r(x))) → r(a(x))
l(r(a(a(x)))) → a(a(l(c(c(c(r(x)))))))

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

(9) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


A(l(x)) → A(x)
L(r(a(a(x)))) → A(l(c(c(c(r(x))))))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(A(x1)) = 1 + x1   
POL(L(x1)) = 1 + x1   
POL(a(x1)) = 1 + x1   
POL(c(x1)) = x1   
POL(l(x1)) = 1 + x1   
POL(r(x1)) = 1 + x1   

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

l(r(a(a(x)))) → a(a(l(c(c(c(r(x)))))))
a(l(x)) → l(a(x))
a(c(x)) → c(a(x))
c(a(r(x))) → r(a(x))

(10) Obligation:

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

A(l(x)) → L(a(x))
L(r(a(a(x)))) → A(a(l(c(c(c(r(x)))))))
A(c(x)) → A(x)

The TRS R consists of the following rules:

a(l(x)) → l(a(x))
a(c(x)) → c(a(x))
c(a(r(x))) → r(a(x))
l(r(a(a(x)))) → a(a(l(c(c(c(r(x)))))))

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

(11) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


A(c(x)) → A(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(A(x1)) = x1   
POL(L(x1)) = 0   
POL(a(x1)) = x1   
POL(c(x1)) = 1 + x1   
POL(l(x1)) = 0   
POL(r(x1)) = 0   

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

l(r(a(a(x)))) → a(a(l(c(c(c(r(x)))))))
a(l(x)) → l(a(x))
a(c(x)) → c(a(x))
c(a(r(x))) → r(a(x))

(12) Obligation:

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

A(l(x)) → L(a(x))
L(r(a(a(x)))) → A(a(l(c(c(c(r(x)))))))

The TRS R consists of the following rules:

a(l(x)) → l(a(x))
a(c(x)) → c(a(x))
c(a(r(x))) → r(a(x))
l(r(a(a(x)))) → a(a(l(c(c(c(r(x)))))))

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

(13) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


L(r(a(a(x)))) → A(a(l(c(c(c(r(x)))))))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO,RATPOLO]:

POL(A(x1)) = x1   
POL(L(x1)) = [1/2]x1   
POL(a(x1)) = [4]x1   
POL(c(x1)) = [1/4]x1   
POL(l(x1)) = [2]x1   
POL(r(x1)) = [1/4]   
The value of delta used in the strict ordering is 3/32.
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

l(r(a(a(x)))) → a(a(l(c(c(c(r(x)))))))
a(l(x)) → l(a(x))
a(c(x)) → c(a(x))
c(a(r(x))) → r(a(x))

(14) Obligation:

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

A(l(x)) → L(a(x))

The TRS R consists of the following rules:

a(l(x)) → l(a(x))
a(c(x)) → c(a(x))
c(a(r(x))) → r(a(x))
l(r(a(a(x)))) → a(a(l(c(c(c(r(x)))))))

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

(15) DependencyGraphProof (EQUIVALENT transformation)

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

(16) TRUE