YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Trafo_06/un08.srs

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

0 QTRS
↳1 QTRS Reverse (⇔, 0 ms)
↳2 QTRS
↳3 RootLabelingProof (⇔, 0 ms)
↳4 QTRS
↳5 DependencyPairsProof (⇔, 20 ms)
↳6 QDP
↳7 DependencyGraphProof (⇔, 0 ms)
↳8 AND
    ↳9 QDP
        ↳10 QDPOrderProof (⇔, 37 ms)
        ↳11 QDP
        ↳12 QDPOrderProof (⇔, 136 ms)
        ↳13 QDP
        ↳14 PisEmptyProof (⇔, 0 ms)
        ↳15 YES
    ↳16 QDP
        ↳17 QDPOrderProof (⇔, 180 ms)
        ↳18 QDP
        ↳19 PisEmptyProof (⇔, 0 ms)
        ↳20 YES

(0) Obligation:

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

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

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

Q is empty.

(3) RootLabelingProof (EQUIVALENT transformation)

We used plain root labeling [ROOTLAB] with the following heuristic: LabelAll: All function symbols get labeled
As Q is empty the root labeling was sound AND complete.

(4) Obligation:

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

b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))))
b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(x)))
b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → b_{b_1}(b_{b_1}(b_{a_1}(x)))
b_{b_1}(b_{b_1}(b_{b_1}(x))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(x))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))

Q is empty.

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(6) Obligation:

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

B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))))
B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))) → B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))
B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))) → B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))
B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))) → B_{A_1}(a_{b_1}(b_{b_1}(x)))
B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))) → B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))))
B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))) → B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))
B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))) → B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))
B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))) → B_{A_1}(a_{b_1}(b_{a_1}(x)))
B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → B_{B_1}(b_{b_1}(b_{b_1}(x)))
B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → B_{B_1}(b_{b_1}(b_{a_1}(x)))
B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → B_{B_1}(b_{a_1}(x))
B_{B_1}(b_{b_1}(b_{b_1}(x))) → B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
B_{B_1}(b_{b_1}(b_{b_1}(x))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
B_{B_1}(b_{b_1}(b_{a_1}(x))) → B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
B_{B_1}(b_{b_1}(b_{a_1}(x))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))

The TRS R consists of the following rules:

b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))))
b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(x)))
b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → b_{b_1}(b_{b_1}(b_{a_1}(x)))
b_{b_1}(b_{b_1}(b_{b_1}(x))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(x))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(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 10 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → B_{B_1}(b_{b_1}(b_{b_1}(x)))
B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → B_{B_1}(b_{b_1}(b_{a_1}(x)))
B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → B_{B_1}(b_{a_1}(x))

The TRS R consists of the following rules:

b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))))
b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(x)))
b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → b_{b_1}(b_{b_1}(b_{a_1}(x)))
b_{b_1}(b_{b_1}(b_{b_1}(x))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(x))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(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_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → B_{B_1}(b_{b_1}(b_{b_1}(x)))
B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → B_{B_1}(b_{a_1}(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(B_{B_1}(x1)) = x1   
POL(a_{a_1}(x1)) = 0   
POL(a_{b_1}(x1)) = 1 + x1   
POL(b_{a_1}(x1)) = x1   
POL(b_{b_1}(x1)) = 1   

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

b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → b_{b_1}(b_{b_1}(b_{a_1}(x)))
b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(x)))
b_{b_1}(b_{b_1}(b_{b_1}(x))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(x))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))))

(11) Obligation:

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

B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → B_{B_1}(b_{b_1}(b_{a_1}(x)))

The TRS R consists of the following rules:

b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))))
b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(x)))
b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → b_{b_1}(b_{b_1}(b_{a_1}(x)))
b_{b_1}(b_{b_1}(b_{b_1}(x))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(x))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))

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

(12) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → B_{B_1}(b_{b_1}(b_{a_1}(x)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( B_{B_1}(x1) ) = 2x1 + 2

POL( b_{b_1}(x1) ) = x1 + 1

POL( b_{a_1}(x1) ) = 2x1

POL( a_{b_1}(x1) ) = 2x1 + 1

POL( a_{a_1}(x1) ) = max{0, -2}


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

b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))))
b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → b_{b_1}(b_{b_1}(b_{a_1}(x)))
b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(x)))
b_{b_1}(b_{b_1}(b_{b_1}(x))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(x))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))

(13) Obligation:

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

b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))))
b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(x)))
b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → b_{b_1}(b_{b_1}(b_{a_1}(x)))
b_{b_1}(b_{b_1}(b_{b_1}(x))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(x))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))

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

(14) PisEmptyProof (EQUIVALENT transformation)

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

(15) YES

(16) Obligation:

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

B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))) → B_{A_1}(a_{b_1}(b_{a_1}(x)))
B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))) → B_{A_1}(a_{b_1}(b_{b_1}(x)))

The TRS R consists of the following rules:

b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))))
b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(x)))
b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → b_{b_1}(b_{b_1}(b_{a_1}(x)))
b_{b_1}(b_{b_1}(b_{b_1}(x))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(x))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))

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

(17) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))) → B_{A_1}(a_{b_1}(b_{a_1}(x)))
B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))) → B_{A_1}(a_{b_1}(b_{b_1}(x)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( B_{A_1}(x1) ) = max{0, x1 - 2}

POL( a_{b_1}(x1) ) = max{0, 2x1 - 2}

POL( b_{a_1}(x1) ) = 2x1 + 2

POL( b_{b_1}(x1) ) = x1 + 2

POL( a_{a_1}(x1) ) = max{0, -2}


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

b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))))
b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → b_{b_1}(b_{b_1}(b_{a_1}(x)))
b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(x)))
b_{b_1}(b_{b_1}(b_{b_1}(x))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(x))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))

(18) Obligation:

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

b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x))))))
b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → b_{b_1}(b_{b_1}(b_{b_1}(x)))
b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → b_{b_1}(b_{b_1}(b_{a_1}(x)))
b_{b_1}(b_{b_1}(b_{b_1}(x))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{b_1}(b_{b_1}(b_{a_1}(x))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))

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

(19) PisEmptyProof (EQUIVALENT transformation)

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

(20) YES