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:

1209953, 1209954, 1209955, 1209956, 1209958, 1209957, 1209959, 1209960, 1209961, 1209962, 1209964, 1209963

Node 1209953 is start node and node 1209954 is final node.

Those nodes are connect through the following edges:

  • 1209953 to 1209954 labelled g_1(0), a(0), c_1(0), e_1(0), b(0), f_1(0), d_1(0), e_1(1), g_1(1), a(1), c_1(1), b(1), f_1(1), d_1(1), a(2), b(2), b(3)
  • 1209953 to 1209955 labelled f_1(0)
  • 1209953 to 1209957 labelled f_1(0)
  • 1209953 to 1209959 labelled f_1(1)
  • 1209953 to 1209961 labelled f_1(1)
  • 1209953 to 1209963 labelled f_1(2)
  • 1209954 to 1209954 labelled #_1(0)
  • 1209955 to 1209956 labelled c_1(0)
  • 1209956 to 1209954 labelled a(0)
  • 1209958 to 1209954 labelled b(0), a(0)
  • 1209957 to 1209958 labelled d_1(0)
  • 1209959 to 1209960 labelled c_1(1)
  • 1209960 to 1209954 labelled a(1)
  • 1209961 to 1209962 labelled d_1(1)
  • 1209962 to 1209954 labelled b(1), a(1)
  • 1209964 to 1209954 labelled b(2)
  • 1209963 to 1209964 labelled d_1(2)

(2) TRUE