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:

f2(0, y) -> y
f2(x, 0) -> x
f2(i1(x), y) -> i1(x)
f2(f2(x, y), z) -> f2(x, f2(y, z))
f2(g2(x, y), z) -> g2(f2(x, z), f2(y, z))
f2(1, g2(x, y)) -> x
f2(2, g2(x, y)) -> y

Q is empty.


QTRS
  ↳ DependencyPairsProof

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

f2(0, y) -> y
f2(x, 0) -> x
f2(i1(x), y) -> i1(x)
f2(f2(x, y), z) -> f2(x, f2(y, z))
f2(g2(x, y), z) -> g2(f2(x, z), f2(y, z))
f2(1, g2(x, y)) -> x
f2(2, g2(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:

F2(g2(x, y), z) -> F2(y, z)
F2(g2(x, y), z) -> F2(x, z)
F2(f2(x, y), z) -> F2(x, f2(y, z))
F2(f2(x, y), z) -> F2(y, z)

The TRS R consists of the following rules:

f2(0, y) -> y
f2(x, 0) -> x
f2(i1(x), y) -> i1(x)
f2(f2(x, y), z) -> f2(x, f2(y, z))
f2(g2(x, y), z) -> g2(f2(x, z), f2(y, z))
f2(1, g2(x, y)) -> x
f2(2, g2(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:

F2(g2(x, y), z) -> F2(y, z)
F2(g2(x, y), z) -> F2(x, z)
F2(f2(x, y), z) -> F2(x, f2(y, z))
F2(f2(x, y), z) -> F2(y, z)

The TRS R consists of the following rules:

f2(0, y) -> y
f2(x, 0) -> x
f2(i1(x), y) -> i1(x)
f2(f2(x, y), z) -> f2(x, f2(y, z))
f2(g2(x, y), z) -> g2(f2(x, z), f2(y, z))
f2(1, g2(x, y)) -> x
f2(2, g2(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.


F2(g2(x, y), z) -> F2(y, z)
F2(g2(x, y), z) -> F2(x, z)
F2(f2(x, y), z) -> F2(x, f2(y, z))
F2(f2(x, y), z) -> F2(y, z)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(0) = 0   
POL(1) = 0   
POL(2) = 0   
POL(F2(x1, x2)) = x1   
POL(f2(x1, x2)) = 2 + x1 + x2   
POL(g2(x1, x2)) = 1 + x1 + x2   
POL(i1(x1)) = 0   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof
QDP
          ↳ PisEmptyProof

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

f2(0, y) -> y
f2(x, 0) -> x
f2(i1(x), y) -> i1(x)
f2(f2(x, y), z) -> f2(x, f2(y, z))
f2(g2(x, y), z) -> g2(f2(x, z), f2(y, z))
f2(1, g2(x, y)) -> x
f2(2, g2(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.