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

Q is empty.

(3) 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'(half(x)) → 0'(x)
0'(s(half(x))) → 0'(x)
s(s(half(x))) → half(s(x))
0'(bits(x)) → 0'(x)
s(bits(x)) → s(half(bits(s(x))))

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

9414846, 9414847, 9414848, 9414849, 9414850, 9414851, 9414852, 9414853, 9414854, 9414855

Node 9414846 is start node and node 9414847 is final node.

Those nodes are connect through the following edges:

  • 9414846 to 9414848 labelled half_1(0)
  • 9414846 to 9414847 labelled 0'_1(0), 0'_1(1)
  • 9414846 to 9414849 labelled s_1(0)
  • 9414847 to 9414847 labelled #_1(0)
  • 9414848 to 9414847 labelled s_1(0)
  • 9414848 to 9414852 labelled half_1(1)
  • 9414848 to 9414853 labelled s_1(1)
  • 9414849 to 9414850 labelled half_1(0)
  • 9414850 to 9414851 labelled bits_1(0)
  • 9414851 to 9414847 labelled s_1(0)
  • 9414851 to 9414852 labelled half_1(1)
  • 9414851 to 9414853 labelled s_1(1)
  • 9414852 to 9414847 labelled s_1(1)
  • 9414852 to 9414852 labelled half_1(1)
  • 9414852 to 9414853 labelled s_1(1)
  • 9414853 to 9414854 labelled half_1(1)
  • 9414854 to 9414855 labelled bits_1(1)
  • 9414855 to 9414847 labelled s_1(1)
  • 9414855 to 9414852 labelled half_1(1)
  • 9414855 to 9414853 labelled s_1(1)

(4) TRUE