YES Termination w.r.t. Q proof of HirokawaMiddeldorp_04_t004.ari

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

0 QTRS
↳1 QTRS Reverse (⇔, 0 ms)
↳2 QTRS
↳3 RFCMatchBoundsTRSProof (⇔, 0 ms)
↳4 YES

(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:

2, 4, 9, 10, 11, 12, 13, 14, 15, 16

Node 2 is start node and node 4 is final node.

Those nodes are connected through the following edges:

  • 2 to 9 labelled s_1(0)
  • 2 to 4 labelled 0'_1(0), s_1(0), f_1(0), p_1(0), 0'_1(1), s_1(1), f_1(1), p_1(1)
  • 2 to 13 labelled s_1(1)
  • 2 to 11 labelled f_1(1)
  • 2 to 15 labelled f_1(2)
  • 4 to 4 labelled #_1(0)
  • 9 to 10 labelled p_1(0)
  • 10 to 11 labelled f_1(0)
  • 11 to 12 labelled f_1(0)
  • 12 to 4 labelled s_1(0), s_1(1), f_1(1), p_1(1), 0'_1(1)
  • 12 to 13 labelled s_1(1)
  • 12 to 15 labelled f_1(2)
  • 13 to 14 labelled p_1(1)
  • 14 to 15 labelled f_1(1)
  • 15 to 16 labelled f_1(1)
  • 16 to 4 labelled s_1(1), f_1(1), p_1(1), 0'_1(1)
  • 16 to 13 labelled s_1(1)
  • 16 to 15 labelled f_1(2)

(4) YES