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

0 QTRS
↳1 RFCMatchBoundsTRSProof (⇔)
↳2 TRUE

(0) Obligation:

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

f(0) → s(0)
f(s(0)) → s(0)
f(s(s(x))) → f(f(s(x)))

Q is empty.

(1) 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:

f(0) → s(0)
f(s(0)) → s(0)
f(s(s(x))) → f(f(s(x)))

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

9695266, 9695267, 9695268, 9695270, 9695269, 9695271, 9695272, 9695273, 9695274

Node 9695266 is start node and node 9695267 is final node.

Those nodes are connect through the following edges:

  • 9695266 to 9695268 labelled s_1(0)
  • 9695266 to 9695269 labelled f_1(0)
  • 9695266 to 9695273 labelled s_1(1)
  • 9695267 to 9695267 labelled #_1(0)
  • 9695268 to 9695267 labelled 0(0)
  • 9695270 to 9695267 labelled s_1(0)
  • 9695269 to 9695270 labelled f_1(0)
  • 9695269 to 9695271 labelled f_1(1)
  • 9695269 to 9695273 labelled s_1(1)
  • 9695269 to 9695274 labelled s_1(2)
  • 9695271 to 9695272 labelled f_1(1)
  • 9695271 to 9695271 labelled f_1(1)
  • 9695271 to 9695273 labelled s_1(1)
  • 9695271 to 9695274 labelled s_1(2)
  • 9695272 to 9695267 labelled s_1(1)
  • 9695273 to 9695267 labelled 0(1)
  • 9695274 to 9695267 labelled 0(2)

(2) TRUE