YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Trafo_06/dup05.srs

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

0 QTRS
↳1 QTRS Reverse (⇔, 0 ms)
↳2 QTRS
↳3 DependencyPairsProof (⇔, 30 ms)
↳4 QDP
↳5 DependencyGraphProof (⇔, 0 ms)
↳6 AND
    ↳7 QDP
        ↳8 UsableRulesProof (⇔, 0 ms)
        ↳9 QDP
        ↳10 QDPOrderProof (⇔, 61 ms)
        ↳11 QDP
        ↳12 DependencyGraphProof (⇔, 0 ms)
        ↳13 AND
            ↳14 QDP
                ↳15 QDPOrderProof (⇔, 1 ms)
                ↳16 QDP
                ↳17 PisEmptyProof (⇔, 0 ms)
                ↳18 YES
            ↳19 QDP
                ↳20 QDPOrderProof (⇔, 14 ms)
                ↳21 QDP
                ↳22 PisEmptyProof (⇔, 0 ms)
                ↳23 YES
    ↳24 QDP
        ↳25 QDPOrderProof (⇔, 41 ms)
        ↳26 QDP
        ↳27 PisEmptyProof (⇔, 0 ms)
        ↳28 YES

(0) Obligation:

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

a(a(s(s(x)))) → s(s(a(a(x))))
b(b(a(a(b(b(s(s(x)))))))) → a(a(b(b(s(s(a(a(x))))))))
b(b(a(a(b(b(b(b(x)))))))) → a(a(b(b(a(a(b(b(x))))))))
a(a(b(b(a(a(a(a(x)))))))) → b(b(a(a(b(b(a(a(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:

s(s(a(a(x)))) → a(a(s(s(x))))
s(s(b(b(a(a(b(b(x)))))))) → a(a(s(s(b(b(a(a(x))))))))
b(b(b(b(a(a(b(b(x)))))))) → b(b(a(a(b(b(a(a(x))))))))
a(a(a(a(b(b(a(a(x)))))))) → a(a(b(b(a(a(b(b(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:

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

The TRS R consists of the following rules:

s(s(a(a(x)))) → a(a(s(s(x))))
s(s(b(b(a(a(b(b(x)))))))) → a(a(s(s(b(b(a(a(x))))))))
b(b(b(b(a(a(b(b(x)))))))) → b(b(a(a(b(b(a(a(x))))))))
a(a(a(a(b(b(a(a(x)))))))) → a(a(b(b(a(a(b(b(x))))))))

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 9 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

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

The TRS R consists of the following rules:

s(s(a(a(x)))) → a(a(s(s(x))))
s(s(b(b(a(a(b(b(x)))))))) → a(a(s(s(b(b(a(a(x))))))))
b(b(b(b(a(a(b(b(x)))))))) → b(b(a(a(b(b(a(a(x))))))))
a(a(a(a(b(b(a(a(x)))))))) → a(a(b(b(a(a(b(b(x))))))))

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

(8) 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.

(9) Obligation:

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

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

The TRS R consists of the following rules:

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


B(b(b(b(a(a(b(b(x)))))))) → B(a(a(b(b(a(a(x)))))))
B(b(b(b(a(a(b(b(x)))))))) → A(a(b(b(a(a(x))))))
A(a(a(a(b(b(a(a(x)))))))) → A(b(b(a(a(b(b(x)))))))
A(a(a(a(b(b(a(a(x)))))))) → B(b(a(a(b(b(x))))))
B(b(b(b(a(a(b(b(x)))))))) → A(b(b(a(a(x)))))
A(a(a(a(b(b(a(a(x)))))))) → B(a(a(b(b(x)))))
B(b(b(b(a(a(b(b(x)))))))) → B(b(a(a(x))))
B(b(b(b(a(a(b(b(x)))))))) → B(a(a(x)))
B(b(b(b(a(a(b(b(x)))))))) → A(a(x))
A(a(a(a(b(b(a(a(x)))))))) → A(a(b(b(x))))
A(a(a(a(b(b(a(a(x)))))))) → A(b(b(x)))
A(a(a(a(b(b(a(a(x)))))))) → B(b(x))
B(b(b(b(a(a(b(b(x)))))))) → A(x)
A(a(a(a(b(b(a(a(x)))))))) → 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)) = 1 + x1   

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

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

(11) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(12) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.

(13) Complex Obligation (AND)

(14) Obligation:

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

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

The TRS R consists of the following rules:

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

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

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

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

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

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

(16) Obligation:

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

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

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

(17) PisEmptyProof (EQUIVALENT transformation)

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

(18) YES

(19) Obligation:

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

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

The TRS R consists of the following rules:

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

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

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

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

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

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

(21) Obligation:

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

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

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

(22) PisEmptyProof (EQUIVALENT transformation)

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

(23) YES

(24) Obligation:

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

S(s(a(a(x)))) → S(x)
S(s(a(a(x)))) → S(s(x))
S(s(b(b(a(a(b(b(x)))))))) → S(s(b(b(a(a(x))))))

The TRS R consists of the following rules:

s(s(a(a(x)))) → a(a(s(s(x))))
s(s(b(b(a(a(b(b(x)))))))) → a(a(s(s(b(b(a(a(x))))))))
b(b(b(b(a(a(b(b(x)))))))) → b(b(a(a(b(b(a(a(x))))))))
a(a(a(a(b(b(a(a(x)))))))) → a(a(b(b(a(a(b(b(x))))))))

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

(25) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


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

POL(S(x1)) = x1   
POL(a(x1)) = 1 + x1   
POL(b(x1)) = 1 + x1   
POL(s(x1)) = x1   

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

s(s(a(a(x)))) → a(a(s(s(x))))
s(s(b(b(a(a(b(b(x)))))))) → a(a(s(s(b(b(a(a(x))))))))
a(a(a(a(b(b(a(a(x)))))))) → a(a(b(b(a(a(b(b(x))))))))
b(b(b(b(a(a(b(b(x)))))))) → b(b(a(a(b(b(a(a(x))))))))

(26) Obligation:

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

s(s(a(a(x)))) → a(a(s(s(x))))
s(s(b(b(a(a(b(b(x)))))))) → a(a(s(s(b(b(a(a(x))))))))
b(b(b(b(a(a(b(b(x)))))))) → b(b(a(a(b(b(a(a(x))))))))
a(a(a(a(b(b(a(a(x)))))))) → a(a(b(b(a(a(b(b(x))))))))

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

(27) PisEmptyProof (EQUIVALENT transformation)

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

(28) YES