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:

a(d(x)) → d(c(b(a(x))))
b(c(x)) → c(d(a(b(x))))
a(c(x)) → x
b(d(x)) → x

Q is empty.


↳ QTRS
  ↳ Overlay + Local Confluence

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

a(d(x)) → d(c(b(a(x))))
b(c(x)) → c(d(a(b(x))))
a(c(x)) → x
b(d(x)) → x

Q is empty.

The TRS is overlay and locally confluent. By [15] we can switch to innermost.

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof

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

a(d(x)) → d(c(b(a(x))))
b(c(x)) → c(d(a(b(x))))
a(c(x)) → x
b(d(x)) → x

The set Q consists of the following terms:

a(d(x0))
b(c(x0))
a(c(x0))
b(d(x0))


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

A(d(x)) → B(a(x))
B(c(x)) → A(b(x))
B(c(x)) → B(x)
A(d(x)) → A(x)

The TRS R consists of the following rules:

a(d(x)) → d(c(b(a(x))))
b(c(x)) → c(d(a(b(x))))
a(c(x)) → x
b(d(x)) → x

The set Q consists of the following terms:

a(d(x0))
b(c(x0))
a(c(x0))
b(d(x0))

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

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof

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

A(d(x)) → B(a(x))
B(c(x)) → A(b(x))
B(c(x)) → B(x)
A(d(x)) → A(x)

The TRS R consists of the following rules:

a(d(x)) → d(c(b(a(x))))
b(c(x)) → c(d(a(b(x))))
a(c(x)) → x
b(d(x)) → x

The set Q consists of the following terms:

a(d(x0))
b(c(x0))
a(c(x0))
b(d(x0))

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

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP

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

A(d(x)) → B(a(x))
B(c(x)) → A(b(x))
B(c(x)) → B(x)
A(d(x)) → A(x)

The TRS R consists of the following rules:

a(d(x)) → d(c(b(a(x))))
b(c(x)) → c(d(a(b(x))))
a(c(x)) → x
b(d(x)) → x

The set Q consists of the following terms:

a(d(x0))
b(c(x0))
a(c(x0))
b(d(x0))

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