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:

f(s(x)) → s(f(f(p(s(x)))))
f(0) → 0
p(s(x)) → 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:

s(f(x)) → s(p(f(f(s(x)))))
0'(f(x)) → 0'(x)
s(p(x)) → 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:

s(f(x)) → s(p(f(f(s(x)))))
0'(f(x)) → 0'(x)
s(p(x)) → x

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

2650320, 2650321, 2650324, 2650325, 2650323, 2650322, 2650327, 2650328, 2650329, 2650326

Node 2650320 is start node and node 2650321 is final node.

Those nodes are connect through the following edges:

  • 2650320 to 2650321 labelled p_1(0), 0'_1(0), f_1(0), s_1(0), 0'_1(1), p_1(1), f_1(1), s_1(1)
  • 2650320 to 2650322 labelled s_1(0)
  • 2650320 to 2650326 labelled s_1(1)
  • 2650320 to 2650324 labelled f_1(1)
  • 2650320 to 2650328 labelled f_1(2)
  • 2650321 to 2650321 labelled #_1(0)
  • 2650324 to 2650325 labelled f_1(0)
  • 2650325 to 2650321 labelled s_1(0), p_1(1), f_1(1), s_1(1), 0'_1(1)
  • 2650325 to 2650326 labelled s_1(1)
  • 2650325 to 2650328 labelled f_1(2)
  • 2650323 to 2650324 labelled f_1(0)
  • 2650322 to 2650323 labelled p_1(0)
  • 2650327 to 2650328 labelled f_1(1)
  • 2650328 to 2650329 labelled f_1(1)
  • 2650329 to 2650321 labelled s_1(1), p_1(1), f_1(1), 0'_1(1)
  • 2650329 to 2650326 labelled s_1(1)
  • 2650329 to 2650328 labelled f_1(2)
  • 2650326 to 2650327 labelled p_1(1)

(4) TRUE