YES Termination w.r.t. Q proof of Transformed_CSR_04_Ex15_Luc06_GM.ari

(0) Obligation:

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

a__f(f(a)) → a__f(g(f(a)))
mark(f(X)) → a__f(X)
mark(a) → a
mark(g(X)) → g(mark(X))
a__f(X) → f(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:

a__f(f(a)) → a__f(g(f(a)))
mark(f(X)) → a__f(X)
mark(a) → a
mark(g(X)) → g(mark(X))
a__f(X) → f(X)

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

1, 3, 5, 8, 10, 11, 13, 14, 15, 16

Node 1 is start node and node 3 is final node.

Those nodes are connected through the following edges:

  • 1 to 5 labelled a__f_1(0), f_1(1)
  • 1 to 3 labelled a__f_1(0), a(0), f_1(0), f_1(1)
  • 1 to 11 labelled g_1(0)
  • 1 to 13 labelled a__f_1(1), f_1(2)
  • 3 to 3 labelled #_1(0)
  • 5 to 8 labelled g_1(0)
  • 8 to 10 labelled f_1(0)
  • 10 to 3 labelled a(0)
  • 11 to 3 labelled mark_1(0), a__f_1(1), a(1), f_1(2)
  • 11 to 16 labelled g_1(1)
  • 11 to 13 labelled a__f_1(1), f_1(2)
  • 13 to 14 labelled g_1(1)
  • 14 to 15 labelled f_1(1)
  • 15 to 3 labelled a(1)
  • 16 to 3 labelled mark_1(1), a__f_1(1), a(1), f_1(2)
  • 16 to 16 labelled g_1(1)
  • 16 to 13 labelled a__f_1(1), f_1(2)

(2) YES