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:

f(0, 1, x) → f(h(x), h(x), x)
h(0) → 0
h(g(x, y)) → y

Q is empty.


↳ QTRS
  ↳ AAECC Innermost

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

f(0, 1, x) → f(h(x), h(x), x)
h(0) → 0
h(g(x, y)) → y

Q is empty.

We have applied [15,7] to switch to innermost. The TRS R 1 is

h(0) → 0
h(g(x, y)) → y

The TRS R 2 is

f(0, 1, x) → f(h(x), h(x), x)

The signature Sigma is {f}

↳ QTRS
  ↳ AAECC Innermost
    ↳ QTRS
      ↳ DependencyPairsProof

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

f(0, 1, x) → f(h(x), h(x), x)
h(0) → 0
h(g(x, y)) → y

The set Q consists of the following terms:

f(0, 1, x0)
h(0)
h(g(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:

F(0, 1, x) → H(x)
F(0, 1, x) → F(h(x), h(x), x)

The TRS R consists of the following rules:

f(0, 1, x) → f(h(x), h(x), x)
h(0) → 0
h(g(x, y)) → y

The set Q consists of the following terms:

f(0, 1, x0)
h(0)
h(g(x0, x1))

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

↳ QTRS
  ↳ AAECC Innermost
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof

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

F(0, 1, x) → H(x)
F(0, 1, x) → F(h(x), h(x), x)

The TRS R consists of the following rules:

f(0, 1, x) → f(h(x), h(x), x)
h(0) → 0
h(g(x, y)) → y

The set Q consists of the following terms:

f(0, 1, x0)
h(0)
h(g(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
We deleted some edges using various graph approximations

↳ QTRS
  ↳ AAECC Innermost
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof

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

F(0, 1, x) → H(x)
F(0, 1, x) → F(h(x), h(x), x)

The TRS R consists of the following rules:

f(0, 1, x) → f(h(x), h(x), x)
h(0) → 0
h(g(x, y)) → y

The set Q consists of the following terms:

f(0, 1, x0)
h(0)
h(g(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 1 less node.

↳ QTRS
  ↳ AAECC Innermost
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ QDP

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

F(0, 1, x) → F(h(x), h(x), x)

The TRS R consists of the following rules:

f(0, 1, x) → f(h(x), h(x), x)
h(0) → 0
h(g(x, y)) → y

The set Q consists of the following terms:

f(0, 1, x0)
h(0)
h(g(x0, x1))

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