Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 RFCMatchBoundsTRSProof (⇔)
↳2 TRUE

(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:

9628472, 9628473, 9628475, 9628474, 9628476, 9628477, 9628479, 9628478, 9628480, 9628481, 9628483, 9628482

Node 9628472 is start node and node 9628473 is final node.

Those nodes are connect through the following edges:

  • 9628472 to 9628473 labelled g_1(0), a(0), c_1(0), e_1(0), b(0), f_1(0), d_1(0), g_1(1), a(1), c_1(1), b(1), f_1(1), e_1(1), d_1(1), a(2), b(2), b(3)
  • 9628472 to 9628474 labelled f_1(0)
  • 9628472 to 9628476 labelled f_1(0)
  • 9628472 to 9628478 labelled f_1(1)
  • 9628472 to 9628480 labelled f_1(1)
  • 9628472 to 9628482 labelled f_1(2)
  • 9628473 to 9628473 labelled #_1(0)
  • 9628475 to 9628473 labelled a(0)
  • 9628474 to 9628475 labelled c_1(0)
  • 9628476 to 9628477 labelled d_1(0)
  • 9628477 to 9628473 labelled b(0), a(0)
  • 9628479 to 9628473 labelled a(1)
  • 9628478 to 9628479 labelled c_1(1)
  • 9628480 to 9628481 labelled d_1(1)
  • 9628481 to 9628473 labelled b(1), a(1)
  • 9628483 to 9628473 labelled b(2)
  • 9628482 to 9628483 labelled d_1(2)

(2) TRUE