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:

7473450, 7473451, 7473452, 7473453, 7473455, 7473454, 7473457, 7473456, 7473459, 7473458, 7473460, 7473461

Node 7473450 is start node and node 7473451 is final node.

Those nodes are connect through the following edges:

  • 7473450 to 7473451 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)
  • 7473450 to 7473452 labelled f_1(0)
  • 7473450 to 7473454 labelled f_1(0)
  • 7473450 to 7473456 labelled f_1(1)
  • 7473450 to 7473458 labelled f_1(1)
  • 7473450 to 7473460 labelled f_1(2)
  • 7473451 to 7473451 labelled #_1(0)
  • 7473452 to 7473453 labelled c_1(0)
  • 7473453 to 7473451 labelled a(0)
  • 7473455 to 7473451 labelled b(0), a(0)
  • 7473454 to 7473455 labelled d_1(0)
  • 7473457 to 7473451 labelled a(1)
  • 7473456 to 7473457 labelled c_1(1)
  • 7473459 to 7473451 labelled b(1), a(1)
  • 7473458 to 7473459 labelled d_1(1)
  • 7473460 to 7473461 labelled d_1(2)
  • 7473461 to 7473451 labelled b(2)

(2) TRUE