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:

a1(d1(x)) -> d1(c1(b1(a1(x))))
b1(c1(x)) -> c1(d1(a1(b1(x))))
a1(c1(x)) -> x
b1(d1(x)) -> x

Q is empty.


QTRS
  ↳ Non-Overlap Check

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

a1(d1(x)) -> d1(c1(b1(a1(x))))
b1(c1(x)) -> c1(d1(a1(b1(x))))
a1(c1(x)) -> x
b1(d1(x)) -> x

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:

a1(d1(x)) -> d1(c1(b1(a1(x))))
b1(c1(x)) -> c1(d1(a1(b1(x))))
a1(c1(x)) -> x
b1(d1(x)) -> x

The set Q consists of the following terms:

a1(d1(x0))
b1(c1(x0))
a1(c1(x0))
b1(d1(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:

A1(d1(x)) -> B1(a1(x))
B1(c1(x)) -> A1(b1(x))
B1(c1(x)) -> B1(x)
A1(d1(x)) -> A1(x)

The TRS R consists of the following rules:

a1(d1(x)) -> d1(c1(b1(a1(x))))
b1(c1(x)) -> c1(d1(a1(b1(x))))
a1(c1(x)) -> x
b1(d1(x)) -> x

The set Q consists of the following terms:

a1(d1(x0))
b1(c1(x0))
a1(c1(x0))
b1(d1(x0))

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

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

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

A1(d1(x)) -> B1(a1(x))
B1(c1(x)) -> A1(b1(x))
B1(c1(x)) -> B1(x)
A1(d1(x)) -> A1(x)

The TRS R consists of the following rules:

a1(d1(x)) -> d1(c1(b1(a1(x))))
b1(c1(x)) -> c1(d1(a1(b1(x))))
a1(c1(x)) -> x
b1(d1(x)) -> x

The set Q consists of the following terms:

a1(d1(x0))
b1(c1(x0))
a1(c1(x0))
b1(d1(x0))

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