YES Termination w.r.t. Q proof of AG01_3.35.ari

(0) Obligation:

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

g(s(x)) → f(x)
f(0) → s(0)
f(s(x)) → s(s(g(x)))
g(0) → 0

Q is empty.

(1) RFCMatchBoundsTRSProof (EQUIVALENT transformation)

Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. This implies Q-termination of R.
The following rules were used to construct the certificate:

g(s(x)) → f(x)
f(0) → s(0)
f(s(x)) → s(s(g(x)))
g(0) → 0

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

1, 4, 6, 9, 10, 12, 13, 14

Node 1 is start node and node 4 is final node.

Those nodes are connected through the following edges:

  • 1 to 4 labelled f_1(0), 0(0)
  • 1 to 6 labelled s_1(0)
  • 1 to 9 labelled s_1(0)
  • 1 to 12 labelled s_1(1)
  • 1 to 13 labelled s_1(1)
  • 4 to 4 labelled #_1(0)
  • 6 to 4 labelled 0(0)
  • 9 to 10 labelled s_1(0)
  • 10 to 4 labelled g_1(0), f_1(1), 0(1)
  • 10 to 10 labelled s_1(1)
  • 10 to 13 labelled s_1(1)
  • 12 to 4 labelled 0(1)
  • 13 to 14 labelled s_1(1)
  • 14 to 4 labelled g_1(1), f_1(1), 0(1)
  • 14 to 10 labelled s_1(1)
  • 14 to 13 labelled s_1(1)

(2) YES