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(a) -> g1(h1(a))
h1(g1(x)) -> g1(h1(f1(x)))
k3(x, h1(x), a) -> h1(x)
k3(f1(x), y, x) -> f1(x)

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

f1(a) -> g1(h1(a))
h1(g1(x)) -> g1(h1(f1(x)))
k3(x, h1(x), a) -> h1(x)
k3(f1(x), y, 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(a) -> H1(a)
H1(g1(x)) -> F1(x)
H1(g1(x)) -> H1(f1(x))

The TRS R consists of the following rules:

f1(a) -> g1(h1(a))
h1(g1(x)) -> g1(h1(f1(x)))
k3(x, h1(x), a) -> h1(x)
k3(f1(x), y, x) -> f1(x)

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof

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

F1(a) -> H1(a)
H1(g1(x)) -> F1(x)
H1(g1(x)) -> H1(f1(x))

The TRS R consists of the following rules:

f1(a) -> g1(h1(a))
h1(g1(x)) -> g1(h1(f1(x)))
k3(x, h1(x), a) -> h1(x)
k3(f1(x), y, 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 1 SCC with 2 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof

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

H1(g1(x)) -> H1(f1(x))

The TRS R consists of the following rules:

f1(a) -> g1(h1(a))
h1(g1(x)) -> g1(h1(f1(x)))
k3(x, h1(x), a) -> h1(x)
k3(f1(x), y, 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.


H1(g1(x)) -> H1(f1(x))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( H1(x1) ) = x1


POL( g1(x1) ) = x1 + 1


POL( f1(x1) ) = x1


POL( a ) = 1


POL( h1(x1) ) = 0



The following usable rules [14] were oriented:

f1(a) -> g1(h1(a))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ PisEmptyProof

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

f1(a) -> g1(h1(a))
h1(g1(x)) -> g1(h1(f1(x)))
k3(x, h1(x), a) -> h1(x)
k3(f1(x), y, x) -> f1(x)

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.