YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Zantema_04/z034.srs

(0) Obligation:

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

b(a(a(b(a(x))))) → a(a(b(b(a(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:

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

Q is empty.

(3) RFCMatchBoundsTRSProof (EQUIVALENT transformation)

Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. This implies Q-termination of R.
The following rules were used to construct the certificate:

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

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58

Node 33 is start node and node 34 is final node.

Those nodes are connected through the following edges:

  • 33 to 35 labelled b_1(0)
  • 34 to 34 labelled #_1(0)
  • 35 to 36 labelled a_1(0)
  • 36 to 37 labelled a_1(0)
  • 37 to 38 labelled b_1(0)
  • 38 to 39 labelled b_1(0)
  • 39 to 40 labelled a_1(0)
  • 39 to 47 labelled b_1(1)
  • 40 to 34 labelled a_1(0)
  • 40 to 41 labelled b_1(1)
  • 41 to 42 labelled a_1(1)
  • 42 to 43 labelled a_1(1)
  • 43 to 44 labelled b_1(1)
  • 44 to 45 labelled b_1(1)
  • 45 to 46 labelled a_1(1)
  • 45 to 53 labelled b_1(2)
  • 46 to 34 labelled a_1(1)
  • 46 to 41 labelled b_1(1)
  • 47 to 48 labelled a_1(1)
  • 48 to 49 labelled a_1(1)
  • 49 to 50 labelled b_1(1)
  • 50 to 51 labelled b_1(1)
  • 51 to 52 labelled a_1(1)
  • 51 to 53 labelled b_1(2)
  • 52 to 44 labelled a_1(1)
  • 52 to 41 labelled b_1(1)
  • 53 to 54 labelled a_1(2)
  • 54 to 55 labelled a_1(2)
  • 55 to 56 labelled b_1(2)
  • 56 to 57 labelled b_1(2)
  • 57 to 58 labelled a_1(2)
  • 57 to 53 labelled b_1(2)
  • 58 to 44 labelled a_1(2)
  • 58 to 41 labelled b_1(1)

(4) YES