YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/ICFP_2010/264370.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(1(3(4(x)))))
0(5(1(2(4(3(x)))))) → 0(5(2(1(4(3(x))))))
0(5(2(4(1(3(x)))))) → 0(1(5(2(4(3(x))))))
0(5(3(1(2(4(x)))))) → 0(1(5(3(2(4(x))))))
0(5(4(1(3(2(x)))))) → 0(5(4(3(1(2(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(3(1(2(0(x)))))
3(4(2(1(5(0(x)))))) → 3(4(1(2(5(0(x))))))
3(1(4(2(5(0(x)))))) → 3(4(2(5(1(0(x))))))
4(2(1(3(5(0(x)))))) → 4(2(3(5(1(0(x))))))
2(3(1(4(5(0(x)))))) → 2(1(3(4(5(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(3(1(2(0(x)))))
3(4(2(1(5(0(x)))))) → 3(4(1(2(5(0(x))))))
3(1(4(2(5(0(x)))))) → 3(4(2(5(1(0(x))))))
4(2(1(3(5(0(x)))))) → 4(2(3(5(1(0(x))))))
2(3(1(4(5(0(x)))))) → 2(1(3(4(5(0(x))))))

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52

Node 27 is start node and node 28 is final node.

Those nodes are connected through the following edges:

  • 27 to 29 labelled 4_1(0)
  • 27 to 33 labelled 3_1(0)
  • 27 to 38 labelled 3_1(0)
  • 27 to 43 labelled 4_1(0)
  • 27 to 48 labelled 2_1(0)
  • 28 to 28 labelled #_1(0)
  • 29 to 30 labelled 3_1(0)
  • 30 to 31 labelled 1_1(0)
  • 31 to 32 labelled 2_1(0)
  • 32 to 28 labelled 0_1(0)
  • 33 to 34 labelled 4_1(0)
  • 34 to 35 labelled 1_1(0)
  • 35 to 36 labelled 2_1(0)
  • 36 to 37 labelled 5_1(0)
  • 37 to 28 labelled 0_1(0)
  • 38 to 39 labelled 4_1(0)
  • 39 to 40 labelled 2_1(0)
  • 40 to 41 labelled 5_1(0)
  • 41 to 42 labelled 1_1(0)
  • 42 to 28 labelled 0_1(0)
  • 43 to 44 labelled 2_1(0)
  • 44 to 45 labelled 3_1(0)
  • 45 to 46 labelled 5_1(0)
  • 46 to 47 labelled 1_1(0)
  • 47 to 28 labelled 0_1(0)
  • 48 to 49 labelled 1_1(0)
  • 49 to 50 labelled 3_1(0)
  • 50 to 51 labelled 4_1(0)
  • 51 to 52 labelled 5_1(0)
  • 52 to 28 labelled 0_1(0)

(4) YES