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:

f(x, y) → x
g(a) → h(a, b, a)
i(x) → f(x, x)
h(x, x, y) → g(x)

Q is empty.


↳ QTRS
  ↳ AAECC Innermost

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

f(x, y) → x
g(a) → h(a, b, a)
i(x) → f(x, x)
h(x, x, y) → g(x)

Q is empty.

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

f(x, y) → x
i(x) → f(x, x)

The TRS R 2 is

g(a) → h(a, b, a)
h(x, x, y) → g(x)

The signature Sigma is {h, g}

↳ QTRS
  ↳ AAECC Innermost
    ↳ QTRS
      ↳ DependencyPairsProof

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

f(x, y) → x
g(a) → h(a, b, a)
i(x) → f(x, x)
h(x, x, y) → g(x)

The set Q consists of the following terms:

f(x0, x1)
g(a)
i(x0)
h(x0, 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:

G(a) → H(a, b, a)
H(x, x, y) → G(x)
I(x) → F(x, x)

The TRS R consists of the following rules:

f(x, y) → x
g(a) → h(a, b, a)
i(x) → f(x, x)
h(x, x, y) → g(x)

The set Q consists of the following terms:

f(x0, x1)
g(a)
i(x0)
h(x0, 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:

G(a) → H(a, b, a)
H(x, x, y) → G(x)
I(x) → F(x, x)

The TRS R consists of the following rules:

f(x, y) → x
g(a) → h(a, b, a)
i(x) → f(x, x)
h(x, x, y) → g(x)

The set Q consists of the following terms:

f(x0, x1)
g(a)
i(x0)
h(x0, 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:

G(a) → H(a, b, a)
H(x, x, y) → G(x)
I(x) → F(x, x)

The TRS R consists of the following rules:

f(x, y) → x
g(a) → h(a, b, a)
i(x) → f(x, x)
h(x, x, y) → g(x)

The set Q consists of the following terms:

f(x0, x1)
g(a)
i(x0)
h(x0, x0, x1)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 3 less nodes.