YES Termination w.r.t. Q proof of Rubio_04_gmnp.ari

(0) Obligation:

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

f(a) → f(c(a))
f(c(X)) → X
f(c(a)) → f(d(b))
f(a) → f(d(a))
f(d(X)) → X
f(c(b)) → f(d(a))
e(g(X)) → e(X)

Q is empty.

(1) RFCMatchBoundsTRSProof (EQUIVALENT transformation)

Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 3. This implies Q-termination of R.
The following rules were used to construct the certificate:

f(a) → f(c(a))
f(c(X)) → X
f(c(a)) → f(d(b))
f(a) → f(d(a))
f(d(X)) → X
f(c(b)) → f(d(a))
e(g(X)) → e(X)

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

2, 4, 5, 7, 10, 12, 13, 14, 19, 20, 21, 22

Node 2 is start node and node 4 is final node.

Those nodes are connected through the following edges:

  • 2 to 5 labelled f_1(0)
  • 2 to 4 labelled f_1(0), a(0), c_1(0), d_1(0), b(0), e_1(0), g_1(0), f_1(1), a(1), c_1(1), d_1(1), b(1), e_1(1), g_1(1), a(2), b(2), b(3)
  • 2 to 10 labelled f_1(0)
  • 2 to 13 labelled f_1(1)
  • 2 to 19 labelled f_1(1)
  • 2 to 21 labelled f_1(2)
  • 4 to 4 labelled #_1(0)
  • 5 to 7 labelled c_1(0)
  • 7 to 4 labelled a(0)
  • 10 to 12 labelled d_1(0)
  • 12 to 4 labelled b(0), a(0)
  • 13 to 14 labelled c_1(1)
  • 14 to 4 labelled a(1)
  • 19 to 20 labelled d_1(1)
  • 20 to 4 labelled b(1), a(1)
  • 21 to 22 labelled d_1(2)
  • 22 to 4 labelled b(2)

(2) YES