YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Zantema_04/z024.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 QDPSizeChangeProof (⇔, 0 ms)
↳8 YES

(0) Obligation:

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

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

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

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

The TRS R consists of the following rules:

a_{b_1}(b_{a_1}(a_{a_1}(x))) → a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x))))))
a_{b_1}(b_{a_1}(a_{b_1}(x))) → a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x))))))
b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x)))) → b_{a_1}(a_{b_1}(b_{a_1}(x)))
b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x)))) → b_{a_1}(a_{b_1}(b_{b_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 5 less nodes.

(6) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(7) 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_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x)))) → B_{A_1}(a_{b_1}(b_{a_1}(x)))
    The graph contains the following edges 1 > 1

(8) YES