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:

from(X) → cons(X, from(s(X)))
after(0, XS) → XS
after(s(N), cons(X, XS)) → after(N, XS)

Q is empty.


↳ QTRS
  ↳ Overlay + Local Confluence

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

from(X) → cons(X, from(s(X)))
after(0, XS) → XS
after(s(N), cons(X, XS)) → after(N, XS)

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:

from(X) → cons(X, from(s(X)))
after(0, XS) → XS
after(s(N), cons(X, XS)) → after(N, XS)

The set Q consists of the following terms:

from(x0)
after(0, x0)
after(s(x0), cons(x1, x2))


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

AFTER(s(N), cons(X, XS)) → AFTER(N, XS)
FROM(X) → FROM(s(X))

The TRS R consists of the following rules:

from(X) → cons(X, from(s(X)))
after(0, XS) → XS
after(s(N), cons(X, XS)) → after(N, XS)

The set Q consists of the following terms:

from(x0)
after(0, x0)
after(s(x0), cons(x1, x2))

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:

AFTER(s(N), cons(X, XS)) → AFTER(N, XS)
FROM(X) → FROM(s(X))

The TRS R consists of the following rules:

from(X) → cons(X, from(s(X)))
after(0, XS) → XS
after(s(N), cons(X, XS)) → after(N, XS)

The set Q consists of the following terms:

from(x0)
after(0, x0)
after(s(x0), cons(x1, x2))

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:

AFTER(s(N), cons(X, XS)) → AFTER(N, XS)
FROM(X) → FROM(s(X))

The TRS R consists of the following rules:

from(X) → cons(X, from(s(X)))
after(0, XS) → XS
after(s(N), cons(X, XS)) → after(N, XS)

The set Q consists of the following terms:

from(x0)
after(0, x0)
after(s(x0), cons(x1, x2))

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

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

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

AFTER(s(N), cons(X, XS)) → AFTER(N, XS)

The TRS R consists of the following rules:

from(X) → cons(X, from(s(X)))
after(0, XS) → XS
after(s(N), cons(X, XS)) → after(N, XS)

The set Q consists of the following terms:

from(x0)
after(0, x0)
after(s(x0), cons(x1, x2))

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


AFTER(s(N), cons(X, XS)) → AFTER(N, XS)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
AFTER(x1, x2)  =  AFTER(x2)
s(x1)  =  s
cons(x1, x2)  =  cons(x2)

Recursive Path Order [2].
Precedence:
[AFTER1, cons1]


The following usable rules [14] were oriented: none



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

Q DP problem:
P is empty.
The TRS R consists of the following rules:

from(X) → cons(X, from(s(X)))
after(0, XS) → XS
after(s(N), cons(X, XS)) → after(N, XS)

The set Q consists of the following terms:

from(x0)
after(0, x0)
after(s(x0), cons(x1, x2))

We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

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

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

FROM(X) → FROM(s(X))

The TRS R consists of the following rules:

from(X) → cons(X, from(s(X)))
after(0, XS) → XS
after(s(N), cons(X, XS)) → after(N, XS)

The set Q consists of the following terms:

from(x0)
after(0, x0)
after(s(x0), cons(x1, x2))

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