YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Bouchare_06/07.srs

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

0 QTRS
↳1 RootLabelingProof (⇔, 0 ms)
↳2 QTRS
↳3 DependencyPairsProof (⇔, 0 ms)
↳4 QDP
↳5 DependencyGraphProof (⇔, 0 ms)
↳6 QDP
↳7 QDPOrderProof (⇔, 17 ms)
↳8 QDP
↳9 DependencyGraphProof (⇔, 0 ms)
↳10 QDP
↳11 QDPOrderProof (⇔, 4 ms)
↳12 QDP
↳13 PisEmptyProof (⇔, 0 ms)
↳14 YES

(0) Obligation:

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

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

Q is empty.

(1) 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.

(2) Obligation:

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

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

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

The TRS R consists of the following rules:

b_{a_1}(a_{b_1}(x)) → b_{b_1}(b_{b_1}(x))
b_{a_1}(a_{a_1}(x)) → b_{b_1}(b_{a_1}(x))
b_{a_1}(a_{b_1}(b_{b_1}(x))) → b_{a_1}(a_{a_1}(a_{b_1}(x)))
b_{a_1}(a_{b_1}(b_{a_1}(x))) → b_{a_1}(a_{a_1}(a_{a_1}(x)))
a_{a_1}(a_{a_1}(a_{b_1}(x))) → a_{b_1}(b_{b_1}(b_{b_1}(x)))
a_{a_1}(a_{a_1}(a_{a_1}(x))) → a_{b_1}(b_{b_1}(b_{a_1}(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 1 SCC with 1 less node.

(6) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(7) 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_{a_1}(x)) → B_{A_1}(x)
B_{A_1}(a_{b_1}(b_{a_1}(x))) → A_{A_1}(a_{a_1}(x))
A_{A_1}(a_{a_1}(a_{a_1}(x))) → B_{A_1}(x)
B_{A_1}(a_{b_1}(b_{a_1}(x))) → A_{A_1}(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(A_{A_1}(x1)) = 1 + x1   
POL(B_{A_1}(x1)) = 1 + x1   
POL(a_{a_1}(x1)) = 1 + x1   
POL(a_{b_1}(x1)) = 1 + x1   
POL(b_{a_1}(x1)) = 1 + x1   
POL(b_{b_1}(x1)) = 1 + x1   

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

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

(8) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(9) DependencyGraphProof (EQUIVALENT transformation)

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

(10) Obligation:

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

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

The TRS R consists of the following rules:

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


B_{A_1}(a_{b_1}(b_{a_1}(x))) → B_{A_1}(a_{a_1}(a_{a_1}(x)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(B_{A_1}(x1)) = x1   
POL(a_{a_1}(x1)) = 0   
POL(a_{b_1}(x1)) = x1   
POL(b_{a_1}(x1)) = 1   
POL(b_{b_1}(x1)) = 0   

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

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

(12) Obligation:

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

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

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

(13) PisEmptyProof (EQUIVALENT transformation)

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

(14) YES