YES Termination w.r.t. Q proof of Transformed_CSR_04_ExConc_Zan97_GM.ari

(0) Obligation:

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

a__f(X) → g(h(f(X)))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → g(X)
mark(h(X)) → h(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(X) → g(h(f(X)))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → g(X)
mark(h(X)) → h(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, 7, 8, 10, 11, 12, 15, 18, 19

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

Those nodes are connected through the following edges:

  • 1 to 7 labelled g_1(0)
  • 1 to 10 labelled a__f_1(0), h_1(0), f_1(1)
  • 1 to 3 labelled g_1(0), f_1(0)
  • 1 to 11 labelled g_1(1)
  • 3 to 3 labelled #_1(0)
  • 7 to 8 labelled h_1(0)
  • 8 to 3 labelled f_1(0)
  • 10 to 3 labelled mark_1(0), g_1(1)
  • 10 to 15 labelled a__f_1(1), h_1(1), f_1(2)
  • 10 to 18 labelled g_1(2)
  • 11 to 12 labelled h_1(1)
  • 12 to 10 labelled f_1(1)
  • 15 to 3 labelled mark_1(1), g_1(1)
  • 15 to 15 labelled a__f_1(1), h_1(1), f_1(2)
  • 15 to 18 labelled g_1(2)
  • 18 to 19 labelled h_1(2)
  • 19 to 15 labelled f_1(2)

(2) YES