YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/ICFP_2010/25192.srs

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

0 QTRS
↳1 RFCMatchBoundsTRSProof (⇔, 0 ms)
↳2 YES

(0) Obligation:

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

0(x) → 1(x)
4(5(4(5(x)))) → 4(4(5(5(x))))
5(5(5(5(5(5(4(4(4(4(4(4(x)))))))))))) → 2(x)

Q is empty.

(1) RFCMatchBoundsTRSProof (EQUIVALENT transformation)

Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. This implies Q-termination of R.
The following rules were used to construct the certificate:

0(x) → 1(x)
4(5(4(5(x)))) → 4(4(5(5(x))))
5(5(5(5(5(5(4(4(4(4(4(4(x)))))))))))) → 2(x)

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

1, 2, 6, 7, 10

Node 1 is start node and node 2 is final node.

Those nodes are connected through the following edges:

  • 1 to 2 labelled 1_1(0), 2_1(0)
  • 1 to 6 labelled 4_1(0)
  • 2 to 2 labelled #_1(0)
  • 6 to 7 labelled 4_1(0)
  • 7 to 10 labelled 5_1(0)
  • 7 to 2 labelled 2_1(1)
  • 10 to 2 labelled 5_1(0), 2_1(1)

(2) YES