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(f, 0) → a(s, 0)
a(d, 0) → 0
a(d, a(s, x)) → a(s, a(s, a(d, a(p, a(s, x)))))
a(f, a(s, x)) → a(d, a(f, a(p, a(s, x))))
a(p, a(s, x)) → x

Q is empty.


↳ QTRS
  ↳ Overlay + Local Confluence

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

a(f, 0) → a(s, 0)
a(d, 0) → 0
a(d, a(s, x)) → a(s, a(s, a(d, a(p, a(s, x)))))
a(f, a(s, x)) → a(d, a(f, a(p, a(s, x))))
a(p, a(s, 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(f, 0) → a(s, 0)
a(d, 0) → 0
a(d, a(s, x)) → a(s, a(s, a(d, a(p, a(s, x)))))
a(f, a(s, x)) → a(d, a(f, a(p, a(s, x))))
a(p, a(s, x)) → x

The set Q consists of the following terms:

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

A(f, a(s, x)) → A(p, a(s, x))
A(f, a(s, x)) → A(d, a(f, a(p, a(s, x))))
A(d, a(s, x)) → A(d, a(p, a(s, x)))
A(d, a(s, x)) → A(s, a(d, a(p, a(s, x))))
A(d, a(s, x)) → A(s, a(s, a(d, a(p, a(s, x)))))
A(f, a(s, x)) → A(f, a(p, a(s, x)))
A(f, 0) → A(s, 0)
A(d, a(s, x)) → A(p, a(s, x))

The TRS R consists of the following rules:

a(f, 0) → a(s, 0)
a(d, 0) → 0
a(d, a(s, x)) → a(s, a(s, a(d, a(p, a(s, x)))))
a(f, a(s, x)) → a(d, a(f, a(p, a(s, x))))
a(p, a(s, x)) → x

The set Q consists of the following terms:

a(f, 0)
a(d, 0)
a(d, a(s, x0))
a(f, a(s, x0))
a(p, a(s, 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(f, a(s, x)) → A(p, a(s, x))
A(f, a(s, x)) → A(d, a(f, a(p, a(s, x))))
A(d, a(s, x)) → A(d, a(p, a(s, x)))
A(d, a(s, x)) → A(s, a(d, a(p, a(s, x))))
A(d, a(s, x)) → A(s, a(s, a(d, a(p, a(s, x)))))
A(f, a(s, x)) → A(f, a(p, a(s, x)))
A(f, 0) → A(s, 0)
A(d, a(s, x)) → A(p, a(s, x))

The TRS R consists of the following rules:

a(f, 0) → a(s, 0)
a(d, 0) → 0
a(d, a(s, x)) → a(s, a(s, a(d, a(p, a(s, x)))))
a(f, a(s, x)) → a(d, a(f, a(p, a(s, x))))
a(p, a(s, x)) → x

The set Q consists of the following terms:

a(f, 0)
a(d, 0)
a(d, a(s, x0))
a(f, a(s, x0))
a(p, a(s, 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
              ↳ DependencyGraphProof

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

A(f, a(s, x)) → A(p, a(s, x))
A(f, a(s, x)) → A(d, a(f, a(p, a(s, x))))
A(d, a(s, x)) → A(d, a(p, a(s, x)))
A(d, a(s, x)) → A(s, a(d, a(p, a(s, x))))
A(d, a(s, x)) → A(s, a(s, a(d, a(p, a(s, x)))))
A(f, 0) → A(s, 0)
A(f, a(s, x)) → A(f, a(p, a(s, x)))
A(d, a(s, x)) → A(p, a(s, x))

The TRS R consists of the following rules:

a(f, 0) → a(s, 0)
a(d, 0) → 0
a(d, a(s, x)) → a(s, a(s, a(d, a(p, a(s, x)))))
a(f, a(s, x)) → a(d, a(f, a(p, a(s, x))))
a(p, a(s, x)) → x

The set Q consists of the following terms:

a(f, 0)
a(d, 0)
a(d, a(s, x0))
a(f, a(s, x0))
a(p, a(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
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP

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

A(d, a(s, x)) → A(d, a(p, a(s, x)))

The TRS R consists of the following rules:

a(f, 0) → a(s, 0)
a(d, 0) → 0
a(d, a(s, x)) → a(s, a(s, a(d, a(p, a(s, x)))))
a(f, a(s, x)) → a(d, a(f, a(p, a(s, x))))
a(p, a(s, x)) → x

The set Q consists of the following terms:

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

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

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

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

A(f, a(s, x)) → A(f, a(p, a(s, x)))

The TRS R consists of the following rules:

a(f, 0) → a(s, 0)
a(d, 0) → 0
a(d, a(s, x)) → a(s, a(s, a(d, a(p, a(s, x)))))
a(f, a(s, x)) → a(d, a(f, a(p, a(s, x))))
a(p, a(s, x)) → x

The set Q consists of the following terms:

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

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