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:

7541150, 7541151, 7541152, 7541153, 7541154, 7541155, 7541156, 7541157, 7541158

Node 7541150 is start node and node 7541151 is final node.

Those nodes are connect through the following edges:

  • 7541150 to 7541152 labelled s_1(0)
  • 7541150 to 7541153 labelled f_1(0)
  • 7541150 to 7541157 labelled s_1(1)
  • 7541151 to 7541151 labelled #_1(0)
  • 7541152 to 7541151 labelled 0(0)
  • 7541153 to 7541154 labelled f_1(0)
  • 7541153 to 7541155 labelled f_1(1)
  • 7541153 to 7541157 labelled s_1(1)
  • 7541153 to 7541158 labelled s_1(2)
  • 7541154 to 7541151 labelled s_1(0)
  • 7541155 to 7541156 labelled f_1(1)
  • 7541155 to 7541155 labelled f_1(1)
  • 7541155 to 7541157 labelled s_1(1)
  • 7541155 to 7541158 labelled s_1(2)
  • 7541156 to 7541151 labelled s_1(1)
  • 7541157 to 7541151 labelled 0(1)
  • 7541158 to 7541151 labelled 0(2)

(2) TRUE