YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Zantema_06/16.srs

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

0 QTRS
↳1 FlatCCProof (⇔, 0 ms)
↳2 QTRS
↳3 RootLabelingProof (⇔, 0 ms)
↳4 QTRS
↳5 QTRSRRRProof (⇔, 46 ms)
↳6 QTRS
↳7 DependencyPairsProof (⇔, 2 ms)
↳8 QDP
↳9 DependencyGraphProof (⇔, 6 ms)
↳10 AND
    ↳11 QDP
        ↳12 UsableRulesProof (⇔, 0 ms)
        ↳13 QDP
        ↳14 QDPSizeChangeProof (⇔, 0 ms)
        ↳15 YES
    ↳16 QDP
        ↳17 QDPOrderProof (⇔, 17 ms)
        ↳18 QDP
        ↳19 PisEmptyProof (⇔, 0 ms)
        ↳20 YES

(0) Obligation:

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

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

Q is empty.

(1) FlatCCProof (EQUIVALENT transformation)

We used flat context closure [ROOTLAB] As Q is empty the flat context closure was sound AND complete.

(2) Obligation:

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

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

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

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a_{a_1}(x1)) = 1 + x1   
POL(a_{b_1}(x1)) = x1   
POL(b_{a_1}(x1)) = x1   
POL(b_{b_1}(x1)) = x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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


(6) Obligation:

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

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

Q is empty.

(7) DependencyPairsProof (EQUIVALENT transformation)

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

(8) Obligation:

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

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

The TRS R consists of the following rules:

a_{a_1}(a_{a_1}(a_{a_1}(x))) → a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x)))))
a_{a_1}(a_{a_1}(a_{b_1}(x))) → a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x)))))
b_{a_1}(a_{b_1}(x)) → b_{b_1}(b_{b_1}(b_{b_1}(x)))
a_{b_1}(b_{a_1}(a_{b_1}(x))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))
b_{b_1}(b_{a_1}(a_{b_1}(x))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x)))))
b_{a_1}(a_{b_1}(x)) → b_{b_1}(b_{b_1}(x))
b_{a_1}(a_{b_1}(b_{a_1}(x))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x))))
b_{a_1}(a_{b_1}(b_{b_1}(x))) → b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x))))
b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x)))) → b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))))
b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x)))) → b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{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_{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_{b_1}(b_{b_1}(b_{b_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 2 SCCs with 38 less nodes.

(10) Complex Obligation (AND)

(11) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(12) 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.

(13) Obligation:

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

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

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • B_{B_1}(b_{a_1}(a_{b_1}(x))) → B_{B_1}(x)
    The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

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

The TRS R consists of the following rules:

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


A_{A_1}(a_{a_1}(a_{b_1}(x))) → A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x)))))
A_{A_1}(a_{a_1}(a_{a_1}(x))) → A_{A_1}(a_{b_1}(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(A_{A_1}(x1)) = x1   
POL(a_{a_1}(x1)) = 1 + x1   
POL(a_{b_1}(x1)) = 1   
POL(b_{a_1}(x1)) = 0   
POL(b_{b_1}(x1)) = 0   

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

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

(18) Obligation:

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

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