YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Trafo_06/dup17.srs

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

0 QTRS
↳1 DependencyPairsProof (⇔, 16 ms)
↳2 QDP
↳3 QDPOrderProof (⇔, 106 ms)
↳4 QDP
↳5 MRRProof (⇔, 133 ms)
↳6 QDP
↳7 DependencyGraphProof (⇔, 0 ms)
↳8 QDP
↳9 QDPOrderProof (⇔, 7 ms)
↳10 QDP
↳11 DependencyGraphProof (⇔, 0 ms)
↳12 TRUE

(0) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(3) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


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

POL(A(x1)) = x1   
POL(B(x1)) = x1   
POL(C(x1)) = 1 + x1   
POL(a(x1)) = x1   
POL(b(x1)) = x1   
POL(c(x1)) = 1 + x1   

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

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

(4) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(5) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

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

Strictly oriented rules of the TRS R:

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

Used ordering: Polynomial interpretation [POLO]:

POL(A(x1)) = 1 + x1   
POL(B(x1)) = 2 + x1   
POL(C(x1)) = 2·x1   
POL(a(x1)) = 1 + x1   
POL(b(x1)) = 2 + x1   
POL(c(x1)) = 2·x1   

(6) Obligation:

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

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

The TRS R consists of the following rules:

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

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

The TRS R consists of the following rules:

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

POL(A(x1)) = 1 + x1   
POL(B(x1)) = x1   
POL(a(x1)) = 1 + x1   
POL(b(x1)) = x1   
POL(c(x1)) = 0   

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

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

(10) Obligation:

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

B(b(a(a(x)))) → A(a(b(b(x))))

The TRS R consists of the following rules:

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

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

(11) DependencyGraphProof (EQUIVALENT transformation)

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

(12) TRUE