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(a(b(x1))) → b(c(b(a(a(a(x1))))))
a(c(x1)) → x1

Q is empty.


↳ QTRS
  ↳ Overlay + Local Confluence
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

a(a(b(x1))) → b(c(b(a(a(a(x1))))))
a(c(x1)) → x1

Q is empty.

The TRS is overlay and locally confluent. By [19] we can switch to innermost.

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

a(a(b(x1))) → b(c(b(a(a(a(x1))))))
a(c(x1)) → x1

The set Q consists of the following terms:

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


Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

A(a(b(x1))) → A(x1)
A(a(b(x1))) → A(a(x1))
A(a(b(x1))) → A(a(a(x1)))

The TRS R consists of the following rules:

a(a(b(x1))) → b(c(b(a(a(a(x1))))))
a(c(x1)) → x1

The set Q consists of the following terms:

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

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

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ MNOCProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

A(a(b(x1))) → A(x1)
A(a(b(x1))) → A(a(x1))
A(a(b(x1))) → A(a(a(x1)))

The TRS R consists of the following rules:

a(a(b(x1))) → b(c(b(a(a(a(x1))))))
a(c(x1)) → x1

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
We use the modular non-overlap check [17] to decrease Q to the empty set.

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ MNOCProof
            ↳ QDP
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

A(a(b(x1))) → A(x1)
A(a(b(x1))) → A(a(x1))
A(a(b(x1))) → A(a(a(x1)))

The TRS R consists of the following rules:

a(a(b(x1))) → b(c(b(a(a(a(x1))))))
a(c(x1)) → x1

Q is empty.
We have to consider all (P,Q,R)-chains.
We have reversed the following QTRS:
The set of rules R is

a(a(b(x1))) → b(c(b(a(a(a(x1))))))
a(c(x1)) → x1

The set Q is empty.
We have obtained the following QTRS:

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

The set Q is empty.

↳ QTRS
  ↳ Overlay + Local Confluence
  ↳ QTRS Reverse
    ↳ QTRS
  ↳ QTRS Reverse

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

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

Q is empty.

We have reversed the following QTRS:
The set of rules R is

a(a(b(x1))) → b(c(b(a(a(a(x1))))))
a(c(x1)) → x1

The set Q is empty.
We have obtained the following QTRS:

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

The set Q is empty.

↳ QTRS
  ↳ Overlay + Local Confluence
  ↳ QTRS Reverse
  ↳ QTRS Reverse
    ↳ QTRS

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

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

Q is empty.