Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

a(d(x1)) → d(b(c(b(d(x1)))))
a(x1) → b(b(f(b(b(x1)))))
b(d(b(x1))) → a(d(x1))
d(f(x1)) → b(d(x1))

Q is empty.


↳ QTRS
  ↳ RFCMatchBoundsTRSProof

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

a(d(x1)) → d(b(c(b(d(x1)))))
a(x1) → b(b(f(b(b(x1)))))
b(d(b(x1))) → a(d(x1))
d(f(x1)) → b(d(x1))

Q is empty.

Termination of the TRS R could be shown with a Match Bound [6,7] of 4. This implies Q-termination of R.
The following rules were used to construct the certificate:

a(d(x1)) → d(b(c(b(d(x1)))))
a(x1) → b(b(f(b(b(x1)))))
b(d(b(x1))) → a(d(x1))
d(f(x1)) → b(d(x1))

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

1, 2, 6, 3, 4, 5, 10, 9, 7, 8, 14, 13, 11, 12, 18, 15, 16, 17, 22, 21, 19, 20, 24, 25, 26, 23, 27, 31, 30, 28, 29, 33, 34, 35, 32, 39, 38, 36, 37, 41, 42, 43, 40, 53, 54, 55, 52, 57, 58, 59, 56, 63, 60, 61, 62, 67, 64, 65, 66

Node 1 is start node and node 2 is final node.

Those nodes are connect through the following edges: