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:

1356462, 1356463, 1356464, 1356465, 1356466, 1356467, 1356468, 1356469, 1356470

Node 1356462 is start node and node 1356463 is final node.

Those nodes are connect through the following edges:

  • 1356462 to 1356464 labelled s_1(0)
  • 1356462 to 1356465 labelled f_1(0)
  • 1356462 to 1356469 labelled s_1(1)
  • 1356463 to 1356463 labelled #_1(0)
  • 1356464 to 1356463 labelled 0(0)
  • 1356465 to 1356466 labelled f_1(0)
  • 1356465 to 1356467 labelled f_1(1)
  • 1356465 to 1356469 labelled s_1(1)
  • 1356465 to 1356470 labelled s_1(2)
  • 1356466 to 1356463 labelled s_1(0)
  • 1356467 to 1356468 labelled f_1(1)
  • 1356467 to 1356467 labelled f_1(1)
  • 1356467 to 1356469 labelled s_1(1)
  • 1356467 to 1356470 labelled s_1(2)
  • 1356468 to 1356463 labelled s_1(1)
  • 1356469 to 1356463 labelled 0(1)
  • 1356470 to 1356463 labelled 0(2)

(2) TRUE