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

0 QTRS
↳1 QTRS Reverse (⇔)
↳2 QTRS
↳3 RFCMatchBoundsTRSProof (⇔)
↳4 TRUE

(0) Obligation:

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

half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
s(log(0)) → s(0)
log(s(x)) → s(log(half(s(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:

0'(half(x)) → 0'(x)
0'(s(half(x))) → 0'(x)
s(s(half(x))) → half(s(x))
0'(log(s(x))) → 0'(s(x))
s(log(x)) → s(half(log(s(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:

0'(half(x)) → 0'(x)
0'(s(half(x))) → 0'(x)
s(s(half(x))) → half(s(x))
0'(log(s(x))) → 0'(s(x))
s(log(x)) → s(half(log(s(x))))

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

9136377, 9136378, 9136379, 9136382, 9136380, 9136381, 9136383, 9136385, 9136386, 9136384

Node 9136377 is start node and node 9136378 is final node.

Those nodes are connect through the following edges:

  • 9136377 to 9136379 labelled half_1(0), 0'_1(0)
  • 9136377 to 9136378 labelled 0'_1(0), 0'_1(1)
  • 9136377 to 9136380 labelled s_1(0)
  • 9136377 to 9136383 labelled 0'_1(1), 0'_1(2)
  • 9136377 to 9136385 labelled 0'_1(1), 0'_1(2)
  • 9136378 to 9136378 labelled #_1(0)
  • 9136379 to 9136378 labelled s_1(0)
  • 9136379 to 9136383 labelled half_1(1)
  • 9136379 to 9136384 labelled s_1(1)
  • 9136382 to 9136378 labelled s_1(0)
  • 9136382 to 9136383 labelled half_1(1)
  • 9136382 to 9136384 labelled s_1(1)
  • 9136380 to 9136381 labelled half_1(0)
  • 9136381 to 9136382 labelled log_1(0)
  • 9136383 to 9136378 labelled s_1(1), s_1(2)
  • 9136383 to 9136383 labelled half_1(1)
  • 9136383 to 9136384 labelled s_1(1), s_1(2)
  • 9136385 to 9136386 labelled log_1(1)
  • 9136386 to 9136378 labelled s_1(1)
  • 9136386 to 9136383 labelled half_1(1)
  • 9136386 to 9136384 labelled s_1(1)
  • 9136384 to 9136385 labelled half_1(1)

(4) TRUE