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:

f(f(X)) → f(a(b(f(X))))
f(a(g(X))) → b(X)
b(X) → a(X)

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

f(f(X)) → f(a(b(f(X))))
f(a(g(X))) → b(X)
b(X) → a(X)

Q is empty.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

F(f(X)) → B(f(X))
F(a(g(X))) → B(X)
F(f(X)) → F(a(b(f(X))))

The TRS R consists of the following rules:

f(f(X)) → f(a(b(f(X))))
f(a(g(X))) → b(X)
b(X) → a(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof

Q DP problem:
The TRS P consists of the following rules:

F(f(X)) → B(f(X))
F(a(g(X))) → B(X)
F(f(X)) → F(a(b(f(X))))

The TRS R consists of the following rules:

f(f(X)) → f(a(b(f(X))))
f(a(g(X))) → b(X)
b(X) → a(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We deleted some edges using various graph approximations

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

F(f(X)) → B(f(X))
F(a(g(X))) → B(X)
F(f(X)) → F(a(b(f(X))))

The TRS R consists of the following rules:

f(f(X)) → f(a(b(f(X))))
f(a(g(X))) → b(X)
b(X) → a(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 3 less nodes.