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:

p(f(f(x))) → q(f(g(x)))
p(g(g(x))) → q(g(f(x)))
q(f(f(x))) → p(f(g(x)))
q(g(g(x))) → p(g(f(x)))

Q is empty.


↳ QTRS
  ↳ AAECC Innermost

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

p(f(f(x))) → q(f(g(x)))
p(g(g(x))) → q(g(f(x)))
q(f(f(x))) → p(f(g(x)))
q(g(g(x))) → p(g(f(x)))

Q is empty.

We have applied [15,7] to switch to innermost. The TRS R 1 is none

The TRS R 2 is

p(f(f(x))) → q(f(g(x)))
p(g(g(x))) → q(g(f(x)))
q(f(f(x))) → p(f(g(x)))
q(g(g(x))) → p(g(f(x)))

The signature Sigma is {q, p}

↳ QTRS
  ↳ AAECC Innermost
    ↳ QTRS
      ↳ DependencyPairsProof

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

p(f(f(x))) → q(f(g(x)))
p(g(g(x))) → q(g(f(x)))
q(f(f(x))) → p(f(g(x)))
q(g(g(x))) → p(g(f(x)))

The set Q consists of the following terms:

p(f(f(x0)))
p(g(g(x0)))
q(f(f(x0)))
q(g(g(x0)))


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

Q(g(g(x))) → P(g(f(x)))
P(f(f(x))) → Q(f(g(x)))
P(g(g(x))) → Q(g(f(x)))
Q(f(f(x))) → P(f(g(x)))

The TRS R consists of the following rules:

p(f(f(x))) → q(f(g(x)))
p(g(g(x))) → q(g(f(x)))
q(f(f(x))) → p(f(g(x)))
q(g(g(x))) → p(g(f(x)))

The set Q consists of the following terms:

p(f(f(x0)))
p(g(g(x0)))
q(f(f(x0)))
q(g(g(x0)))

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

↳ QTRS
  ↳ AAECC Innermost
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof

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

Q(g(g(x))) → P(g(f(x)))
P(f(f(x))) → Q(f(g(x)))
P(g(g(x))) → Q(g(f(x)))
Q(f(f(x))) → P(f(g(x)))

The TRS R consists of the following rules:

p(f(f(x))) → q(f(g(x)))
p(g(g(x))) → q(g(f(x)))
q(f(f(x))) → p(f(g(x)))
q(g(g(x))) → p(g(f(x)))

The set Q consists of the following terms:

p(f(f(x0)))
p(g(g(x0)))
q(f(f(x0)))
q(g(g(x0)))

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

↳ QTRS
  ↳ AAECC Innermost
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof

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

Q(g(g(x))) → P(g(f(x)))
P(f(f(x))) → Q(f(g(x)))
P(g(g(x))) → Q(g(f(x)))
Q(f(f(x))) → P(f(g(x)))

The TRS R consists of the following rules:

p(f(f(x))) → q(f(g(x)))
p(g(g(x))) → q(g(f(x)))
q(f(f(x))) → p(f(g(x)))
q(g(g(x))) → p(g(f(x)))

The set Q consists of the following terms:

p(f(f(x0)))
p(g(g(x0)))
q(f(f(x0)))
q(g(g(x0)))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 4 less nodes.