YES Termination w.r.t. Q proof of Transformed_CSR_04_ExProp7_Luc06_L.ari

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

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

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

f(0) → cons(0)
f(s(0)) → f(p(s(0)))
p(s(X)) → 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) → cons(0)
f(s(0)) → f(p(s(0)))
p(s(X)) → X

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

2, 4, 5, 10, 11, 12, 14, 18, 19, 20

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

Those nodes are connected through the following edges:

  • 2 to 5 labelled cons_1(0)
  • 2 to 10 labelled f_1(0)
  • 2 to 4 labelled f_1(0), 0(0), cons_1(0), s_1(0), p_1(0), f_1(1), 0(1), cons_1(1), s_1(1), p_1(1)
  • 2 to 14 labelled cons_1(1)
  • 2 to 18 labelled f_1(1), cons_1(2)
  • 2 to 2 labelled cons_1(1)
  • 4 to 4 labelled #_1(0)
  • 5 to 4 labelled 0(0)
  • 10 to 11 labelled p_1(0)
  • 10 to 4 labelled 0(1)
  • 11 to 12 labelled s_1(0)
  • 12 to 4 labelled 0(0)
  • 14 to 4 labelled 0(1)
  • 18 to 19 labelled p_1(1)
  • 18 to 4 labelled 0(2)
  • 19 to 20 labelled s_1(1)
  • 20 to 4 labelled 0(1)

(2) YES