YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/ICFP_2010/264405.srs

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

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

(0) Obligation:

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

0(1(2(3(4(x))))) → 0(2(3(1(4(x)))))
0(5(1(2(3(4(x)))))) → 0(1(2(5(3(4(x))))))
0(5(1(2(3(4(x)))))) → 0(5(2(1(3(4(x))))))
0(5(1(2(3(4(x)))))) → 5(0(2(3(1(4(x))))))
0(5(2(3(1(4(x)))))) → 0(1(5(2(3(4(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:

4(3(2(1(0(x))))) → 4(1(3(2(0(x)))))
4(3(2(1(5(0(x)))))) → 4(3(5(2(1(0(x))))))
4(3(2(1(5(0(x)))))) → 4(3(1(2(5(0(x))))))
4(3(2(1(5(0(x)))))) → 4(1(3(2(0(5(x))))))
4(1(3(2(5(0(x)))))) → 4(3(2(5(1(0(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:

4(3(2(1(0(x))))) → 4(1(3(2(0(x)))))
4(3(2(1(5(0(x)))))) → 4(3(5(2(1(0(x))))))
4(3(2(1(5(0(x)))))) → 4(3(1(2(5(0(x))))))
4(3(2(1(5(0(x)))))) → 4(1(3(2(0(5(x))))))
4(1(3(2(5(0(x)))))) → 4(3(2(5(1(0(x))))))

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21

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

Those nodes are connected through the following edges:

  • 1 to 3 labelled 4_1(0)
  • 1 to 7 labelled 4_1(0)
  • 1 to 12 labelled 4_1(0)
  • 1 to 17 labelled 4_1(0)
  • 2 to 2 labelled #_1(0), 5_1(0)
  • 3 to 4 labelled 1_1(0)
  • 4 to 5 labelled 3_1(0)
  • 5 to 6 labelled 2_1(0)
  • 6 to 2 labelled 0_1(0)
  • 7 to 8 labelled 3_1(0)
  • 8 to 9 labelled 5_1(0)
  • 9 to 10 labelled 2_1(0)
  • 10 to 11 labelled 1_1(0)
  • 11 to 2 labelled 0_1(0)
  • 12 to 13 labelled 3_1(0)
  • 13 to 14 labelled 1_1(0)
  • 14 to 15 labelled 2_1(0)
  • 15 to 16 labelled 5_1(0)
  • 16 to 2 labelled 0_1(0)
  • 17 to 18 labelled 3_1(0)
  • 18 to 19 labelled 2_1(0)
  • 19 to 20 labelled 5_1(0)
  • 20 to 21 labelled 1_1(0)
  • 21 to 2 labelled 0_1(0)

(4) YES