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:

g2(f1(x), y) -> f1(h2(x, y))
h2(x, y) -> g2(x, f1(y))

Q is empty.


↳ QTRS
  ↳ Non-Overlap Check

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

g2(f1(x), y) -> f1(h2(x, y))
h2(x, y) -> g2(x, f1(y))

Q is empty.

The TRS is non-overlapping. Hence, we can switch to innermost.

↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof

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

g2(f1(x), y) -> f1(h2(x, y))
h2(x, y) -> g2(x, f1(y))

The set Q consists of the following terms:

g2(f1(x0), x1)
h2(x0, x1)


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

H2(x, y) -> G2(x, f1(y))
G2(f1(x), y) -> H2(x, y)

The TRS R consists of the following rules:

g2(f1(x), y) -> f1(h2(x, y))
h2(x, y) -> g2(x, f1(y))

The set Q consists of the following terms:

g2(f1(x0), x1)
h2(x0, x1)

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

↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ QDPAfsSolverProof

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

H2(x, y) -> G2(x, f1(y))
G2(f1(x), y) -> H2(x, y)

The TRS R consists of the following rules:

g2(f1(x), y) -> f1(h2(x, y))
h2(x, y) -> g2(x, f1(y))

The set Q consists of the following terms:

g2(f1(x0), x1)
h2(x0, x1)

We have to consider all minimal (P,Q,R)-chains.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.

G2(f1(x), y) -> H2(x, y)
Used argument filtering: H2(x1, x2)  =  x1
G2(x1, x2)  =  x1
f1(x1)  =  f1(x1)
Used ordering: Precedence:
trivial



↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ QDPAfsSolverProof
            ↳ QDP
              ↳ DependencyGraphProof

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

H2(x, y) -> G2(x, f1(y))

The TRS R consists of the following rules:

g2(f1(x), y) -> f1(h2(x, y))
h2(x, y) -> g2(x, f1(y))

The set Q consists of the following terms:

g2(f1(x0), x1)
h2(x0, x1)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph contains 0 SCCs with 1 less node.