YES Termination w.r.t. Q proof of Strategy_removed_AG01_4.21.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(1) → f(g(1))
f(f(x)) → f(x)
g(0) → g(f(0))
g(g(x)) → g(x)

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:

f(1) → f(g(1))
f(f(x)) → f(x)
g(0) → g(f(0))
g(g(x)) → g(x)

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

2, 4, 6, 8, 9, 10, 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 6 labelled f_1(0)
  • 2 to 4 labelled f_1(0), g_1(0), f_1(1), g_1(1)
  • 2 to 9 labelled g_1(0)
  • 2 to 13 labelled f_1(1)
  • 2 to 15 labelled g_1(1)
  • 4 to 4 labelled #_1(0)
  • 6 to 8 labelled g_1(0)
  • 8 to 4 labelled 1(0)
  • 9 to 10 labelled f_1(0)
  • 10 to 4 labelled 0(0)
  • 13 to 14 labelled g_1(1)
  • 14 to 4 labelled 1(1)
  • 15 to 16 labelled f_1(1)
  • 16 to 4 labelled 0(1)

(2) YES