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:

a2(f, 0) -> a2(s, 0)
a2(d, 0) -> 0
a2(d, a2(s, x)) -> a2(s, a2(s, a2(d, a2(p, a2(s, x)))))
a2(f, a2(s, x)) -> a2(d, a2(f, a2(p, a2(s, x))))
a2(p, a2(s, x)) -> x

Q is empty.


QTRS
  ↳ Non-Overlap Check

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

a2(f, 0) -> a2(s, 0)
a2(d, 0) -> 0
a2(d, a2(s, x)) -> a2(s, a2(s, a2(d, a2(p, a2(s, x)))))
a2(f, a2(s, x)) -> a2(d, a2(f, a2(p, a2(s, x))))
a2(p, a2(s, 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:

a2(f, 0) -> a2(s, 0)
a2(d, 0) -> 0
a2(d, a2(s, x)) -> a2(s, a2(s, a2(d, a2(p, a2(s, x)))))
a2(f, a2(s, x)) -> a2(d, a2(f, a2(p, a2(s, x))))
a2(p, a2(s, x)) -> x

The set Q consists of the following terms:

a2(f, 0)
a2(d, 0)
a2(d, a2(s, x0))
a2(f, a2(s, x0))
a2(p, a2(s, 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:

A2(f, a2(s, x)) -> A2(p, a2(s, x))
A2(f, a2(s, x)) -> A2(d, a2(f, a2(p, a2(s, x))))
A2(d, a2(s, x)) -> A2(d, a2(p, a2(s, x)))
A2(d, a2(s, x)) -> A2(s, a2(d, a2(p, a2(s, x))))
A2(d, a2(s, x)) -> A2(s, a2(s, a2(d, a2(p, a2(s, x)))))
A2(f, a2(s, x)) -> A2(f, a2(p, a2(s, x)))
A2(f, 0) -> A2(s, 0)
A2(d, a2(s, x)) -> A2(p, a2(s, x))

The TRS R consists of the following rules:

a2(f, 0) -> a2(s, 0)
a2(d, 0) -> 0
a2(d, a2(s, x)) -> a2(s, a2(s, a2(d, a2(p, a2(s, x)))))
a2(f, a2(s, x)) -> a2(d, a2(f, a2(p, a2(s, x))))
a2(p, a2(s, x)) -> x

The set Q consists of the following terms:

a2(f, 0)
a2(d, 0)
a2(d, a2(s, x0))
a2(f, a2(s, x0))
a2(p, a2(s, x0))

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

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

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

A2(f, a2(s, x)) -> A2(p, a2(s, x))
A2(f, a2(s, x)) -> A2(d, a2(f, a2(p, a2(s, x))))
A2(d, a2(s, x)) -> A2(d, a2(p, a2(s, x)))
A2(d, a2(s, x)) -> A2(s, a2(d, a2(p, a2(s, x))))
A2(d, a2(s, x)) -> A2(s, a2(s, a2(d, a2(p, a2(s, x)))))
A2(f, a2(s, x)) -> A2(f, a2(p, a2(s, x)))
A2(f, 0) -> A2(s, 0)
A2(d, a2(s, x)) -> A2(p, a2(s, x))

The TRS R consists of the following rules:

a2(f, 0) -> a2(s, 0)
a2(d, 0) -> 0
a2(d, a2(s, x)) -> a2(s, a2(s, a2(d, a2(p, a2(s, x)))))
a2(f, a2(s, x)) -> a2(d, a2(f, a2(p, a2(s, x))))
a2(p, a2(s, x)) -> x

The set Q consists of the following terms:

a2(f, 0)
a2(d, 0)
a2(d, a2(s, x0))
a2(f, a2(s, x0))
a2(p, a2(s, x0))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 2 SCCs with 6 less nodes.

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

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

A2(d, a2(s, x)) -> A2(d, a2(p, a2(s, x)))

The TRS R consists of the following rules:

a2(f, 0) -> a2(s, 0)
a2(d, 0) -> 0
a2(d, a2(s, x)) -> a2(s, a2(s, a2(d, a2(p, a2(s, x)))))
a2(f, a2(s, x)) -> a2(d, a2(f, a2(p, a2(s, x))))
a2(p, a2(s, x)) -> x

The set Q consists of the following terms:

a2(f, 0)
a2(d, 0)
a2(d, a2(s, x0))
a2(f, a2(s, x0))
a2(p, a2(s, x0))

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

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

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

A2(f, a2(s, x)) -> A2(f, a2(p, a2(s, x)))

The TRS R consists of the following rules:

a2(f, 0) -> a2(s, 0)
a2(d, 0) -> 0
a2(d, a2(s, x)) -> a2(s, a2(s, a2(d, a2(p, a2(s, x)))))
a2(f, a2(s, x)) -> a2(d, a2(f, a2(p, a2(s, x))))
a2(p, a2(s, x)) -> x

The set Q consists of the following terms:

a2(f, 0)
a2(d, 0)
a2(d, a2(s, x0))
a2(f, a2(s, x0))
a2(p, a2(s, x0))

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