YES Termination w.r.t. Q proof of AG01_3.24.ari

(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:

2, 3, 5, 8, 10, 13, 15, 16, 18

Node 2 is start node and node 3 is final node.

Those nodes are connected through the following edges:

  • 2 to 5 labelled s_1(0)
  • 2 to 8 labelled f_1(0)
  • 2 to 13 labelled s_1(1)
  • 3 to 3 labelled #_1(0)
  • 5 to 3 labelled 0(0)
  • 8 to 10 labelled f_1(0)
  • 8 to 13 labelled s_1(1)
  • 8 to 15 labelled f_1(1)
  • 8 to 18 labelled s_1(2)
  • 10 to 3 labelled s_1(0)
  • 13 to 3 labelled 0(1)
  • 15 to 16 labelled f_1(1)
  • 15 to 13 labelled s_1(1)
  • 15 to 15 labelled f_1(1)
  • 15 to 18 labelled s_1(2)
  • 16 to 3 labelled s_1(1)
  • 18 to 3 labelled 0(2)

(2) YES