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:

9135583, 9135584, 9135585, 9135586, 9135587, 9135588, 9135590, 9135591, 9135589, 9135592

Node 9135583 is start node and node 9135584 is final node.

Those nodes are connect through the following edges:

  • 9135583 to 9135584 labelled p_1(0), 0'_1(0), f_1(0), s_1(0), p_1(1), f_1(1), s_1(1), 0'_1(1)
  • 9135583 to 9135585 labelled s_1(0)
  • 9135583 to 9135589 labelled s_1(1)
  • 9135583 to 9135587 labelled f_1(1)
  • 9135583 to 9135591 labelled f_1(2)
  • 9135584 to 9135584 labelled #_1(0)
  • 9135585 to 9135586 labelled p_1(0)
  • 9135586 to 9135587 labelled f_1(0)
  • 9135587 to 9135588 labelled f_1(0)
  • 9135588 to 9135584 labelled s_1(0), p_1(1), f_1(1), s_1(1), 0'_1(1)
  • 9135588 to 9135589 labelled s_1(1)
  • 9135588 to 9135591 labelled f_1(2)
  • 9135590 to 9135591 labelled f_1(1)
  • 9135591 to 9135592 labelled f_1(1)
  • 9135589 to 9135590 labelled p_1(1)
  • 9135592 to 9135584 labelled s_1(1), p_1(1), f_1(1), 0'_1(1)
  • 9135592 to 9135589 labelled s_1(1)
  • 9135592 to 9135591 labelled f_1(2)

(4) TRUE