YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Zantema_04/z050.srs

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

0 QTRS
↳1 Strip Symbols Proof (⇒, 0 ms)
↳2 QTRS
↳3 RFCMatchBoundsTRSProof (⇔, 0 ms)
↳4 YES

(0) Obligation:

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

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

Q is empty.

(1) Strip Symbols Proof (SOUND transformation)

We were given the following TRS:

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

By stripping symbols from the only rule of the system, we obtained the following TRS [ENDRULLIS]:

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

(2) Obligation:

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

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

Q is empty.

(3) RFCMatchBoundsTRSProof (EQUIVALENT transformation)

Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 0. This implies Q-termination of R.
The following rules were used to construct the certificate:

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

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

7, 8, 43, 44, 45, 46, 47

Node 7 is start node and node 8 is final node.

Those nodes are connected through the following edges:

  • 7 to 43 labelled a_1(0)
  • 8 to 8 labelled #_1(0)
  • 43 to 44 labelled b_1(0)
  • 44 to 45 labelled b_1(0)
  • 45 to 46 labelled a_1(0)
  • 46 to 47 labelled b_1(0)
  • 47 to 8 labelled a_1(0)

(4) YES