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:

f3(f3(x, y, z), u, f3(x, y, v)) -> f3(x, y, f3(z, u, v))
f3(x, y, y) -> y
f3(x, y, g1(y)) -> x
f3(x, x, y) -> x
f3(g1(x), x, y) -> y

Q is empty.


QTRS
  ↳ DependencyPairsProof

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

f3(f3(x, y, z), u, f3(x, y, v)) -> f3(x, y, f3(z, u, v))
f3(x, y, y) -> y
f3(x, y, g1(y)) -> x
f3(x, x, y) -> x
f3(g1(x), x, y) -> y

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:

F3(f3(x, y, z), u, f3(x, y, v)) -> F3(x, y, f3(z, u, v))
F3(f3(x, y, z), u, f3(x, y, v)) -> F3(z, u, v)

The TRS R consists of the following rules:

f3(f3(x, y, z), u, f3(x, y, v)) -> f3(x, y, f3(z, u, v))
f3(x, y, y) -> y
f3(x, y, g1(y)) -> x
f3(x, x, y) -> x
f3(g1(x), x, y) -> y

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:

F3(f3(x, y, z), u, f3(x, y, v)) -> F3(x, y, f3(z, u, v))
F3(f3(x, y, z), u, f3(x, y, v)) -> F3(z, u, v)

The TRS R consists of the following rules:

f3(f3(x, y, z), u, f3(x, y, v)) -> f3(x, y, f3(z, u, v))
f3(x, y, y) -> y
f3(x, y, g1(y)) -> x
f3(x, x, y) -> x
f3(g1(x), x, y) -> y

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.


F3(f3(x, y, z), u, f3(x, y, v)) -> F3(x, y, f3(z, u, v))
F3(f3(x, y, z), u, f3(x, y, v)) -> F3(z, u, v)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(F3(x1, x2, x3)) = x1 + 2·x2 + x3   
POL(f3(x1, x2, x3)) = 1 + x1 + 2·x2 + x3   
POL(g1(x1)) = 0   

The following usable rules [14] were oriented:

f3(g1(x), x, y) -> y
f3(x, y, g1(y)) -> x
f3(x, y, y) -> y
f3(x, x, y) -> x
f3(f3(x, y, z), u, f3(x, y, v)) -> f3(x, y, f3(z, u, v))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof
QDP
          ↳ PisEmptyProof

Q DP problem:
P is empty.
The TRS R consists of the following rules:

f3(f3(x, y, z), u, f3(x, y, v)) -> f3(x, y, f3(z, u, v))
f3(x, y, y) -> y
f3(x, y, g1(y)) -> x
f3(x, x, y) -> x
f3(g1(x), x, y) -> y

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.