Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

f2(a, g1(y)) -> g1(g1(y))
f2(g1(x), a) -> f2(x, g1(a))
f2(g1(x), g1(y)) -> h3(g1(y), x, g1(y))
h3(g1(x), y, z) -> f2(y, h3(x, y, z))
h3(a, y, z) -> z

Q is empty.


QTRS
  ↳ Non-Overlap Check

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

f2(a, g1(y)) -> g1(g1(y))
f2(g1(x), a) -> f2(x, g1(a))
f2(g1(x), g1(y)) -> h3(g1(y), x, g1(y))
h3(g1(x), y, z) -> f2(y, h3(x, y, z))
h3(a, y, z) -> z

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:

f2(a, g1(y)) -> g1(g1(y))
f2(g1(x), a) -> f2(x, g1(a))
f2(g1(x), g1(y)) -> h3(g1(y), x, g1(y))
h3(g1(x), y, z) -> f2(y, h3(x, y, z))
h3(a, y, z) -> z

The set Q consists of the following terms:

f2(a, g1(x0))
f2(g1(x0), a)
f2(g1(x0), g1(x1))
h3(g1(x0), x1, x2)
h3(a, x0, x1)


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

H3(g1(x), y, z) -> H3(x, y, z)
H3(g1(x), y, z) -> F2(y, h3(x, y, z))
F2(g1(x), g1(y)) -> H3(g1(y), x, g1(y))
F2(g1(x), a) -> F2(x, g1(a))

The TRS R consists of the following rules:

f2(a, g1(y)) -> g1(g1(y))
f2(g1(x), a) -> f2(x, g1(a))
f2(g1(x), g1(y)) -> h3(g1(y), x, g1(y))
h3(g1(x), y, z) -> f2(y, h3(x, y, z))
h3(a, y, z) -> z

The set Q consists of the following terms:

f2(a, g1(x0))
f2(g1(x0), a)
f2(g1(x0), g1(x1))
h3(g1(x0), x1, x2)
h3(a, x0, x1)

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

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

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

H3(g1(x), y, z) -> H3(x, y, z)
H3(g1(x), y, z) -> F2(y, h3(x, y, z))
F2(g1(x), g1(y)) -> H3(g1(y), x, g1(y))
F2(g1(x), a) -> F2(x, g1(a))

The TRS R consists of the following rules:

f2(a, g1(y)) -> g1(g1(y))
f2(g1(x), a) -> f2(x, g1(a))
f2(g1(x), g1(y)) -> h3(g1(y), x, g1(y))
h3(g1(x), y, z) -> f2(y, h3(x, y, z))
h3(a, y, z) -> z

The set Q consists of the following terms:

f2(a, g1(x0))
f2(g1(x0), a)
f2(g1(x0), g1(x1))
h3(g1(x0), x1, x2)
h3(a, x0, x1)

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


The following pairs can be strictly oriented and are deleted.


F2(g1(x), a) -> F2(x, g1(a))
The remaining pairs can at least by weakly be oriented.

H3(g1(x), y, z) -> H3(x, y, z)
H3(g1(x), y, z) -> F2(y, h3(x, y, z))
F2(g1(x), g1(y)) -> H3(g1(y), x, g1(y))
Used ordering: Combined order from the following AFS and order.
H3(x1, x2, x3)  =  H1(x3)
g1(x1)  =  g
F2(x1, x2)  =  F1(x2)
h3(x1, x2, x3)  =  x3
a  =  a
f2(x1, x2)  =  x2

Lexicographic Path Order [19].
Precedence:
[H1, F1] > g
a > g


The following usable rules [14] were oriented:

f2(g1(x), g1(y)) -> h3(g1(y), x, g1(y))
h3(g1(x), y, z) -> f2(y, h3(x, y, z))
f2(g1(x), a) -> f2(x, g1(a))
h3(a, y, z) -> z
f2(a, g1(y)) -> g1(g1(y))



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

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

F2(g1(x), g1(y)) -> H3(g1(y), x, g1(y))
H3(g1(x), y, z) -> F2(y, h3(x, y, z))
H3(g1(x), y, z) -> H3(x, y, z)

The TRS R consists of the following rules:

f2(a, g1(y)) -> g1(g1(y))
f2(g1(x), a) -> f2(x, g1(a))
f2(g1(x), g1(y)) -> h3(g1(y), x, g1(y))
h3(g1(x), y, z) -> f2(y, h3(x, y, z))
h3(a, y, z) -> z

The set Q consists of the following terms:

f2(a, g1(x0))
f2(g1(x0), a)
f2(g1(x0), g1(x1))
h3(g1(x0), x1, x2)
h3(a, x0, x1)

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