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:

f1(f1(x)) -> g1(f1(x))
g1(g1(x)) -> f1(x)

Q is empty.


QTRS
  ↳ DependencyPairsProof

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

f1(f1(x)) -> g1(f1(x))
g1(g1(x)) -> f1(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:

F1(f1(x)) -> G1(f1(x))
G1(g1(x)) -> F1(x)

The TRS R consists of the following rules:

f1(f1(x)) -> g1(f1(x))
g1(g1(x)) -> f1(x)

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

↳ QTRS
  ↳ DependencyPairsProof
QDP
      ↳ QDPOrderProof

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

F1(f1(x)) -> G1(f1(x))
G1(g1(x)) -> F1(x)

The TRS R consists of the following rules:

f1(f1(x)) -> g1(f1(x))
g1(g1(x)) -> f1(x)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


F1(f1(x)) -> G1(f1(x))
The remaining pairs can at least be oriented weakly.

G1(g1(x)) -> F1(x)
Used ordering: Polynomial interpretation [21]:

POL(F1(x1)) = 3 + 2·x1   
POL(G1(x1)) = 2 + x1   
POL(f1(x1)) = 1 + 2·x1   
POL(g1(x1)) = 1 + 2·x1   

The following usable rules [14] were oriented:

g1(g1(x)) -> f1(x)
f1(f1(x)) -> g1(f1(x))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof
QDP
          ↳ DependencyGraphProof

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

G1(g1(x)) -> F1(x)

The TRS R consists of the following rules:

f1(f1(x)) -> g1(f1(x))
g1(g1(x)) -> f1(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 1 less node.