Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar16/TRCSRSLASHEx4_7_56_Bor03

Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

from₁(X) -> cons₂(X, n__from₁(s₁(X)))
after₂(0, XS) -> XS
after₂(s₁(N), cons₂(X, XS)) -> after₂(N, activate₁(XS))
from₁(X) -> n__from₁(X)
activate₁(n__from₁(X)) -> from₁(X)
activate₁(X) -> X

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

from₁(X) -> cons₂(X, n__from₁(s₁(X)))
after₂(0, XS) -> XS
after₂(s₁(N), cons₂(X, XS)) -> after₂(N, activate₁(XS))
from₁(X) -> n__from₁(X)
activate₁(n__from₁(X)) -> from₁(X)
activate₁(X) -> X

Q is empty.

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

AFTER₂(s₁(N), cons₂(X, XS)) -> AFTER₂(N, activate₁(XS))
AFTER₂(s₁(N), cons₂(X, XS)) -> ACTIVATE₁(XS)
ACTIVATE₁(n__from₁(X)) -> FROM₁(X)

The TRS R consists of the following rules:

from₁(X) -> cons₂(X, n__from₁(s₁(X)))
after₂(0, XS) -> XS
after₂(s₁(N), cons₂(X, XS)) -> after₂(N, activate₁(XS))
from₁(X) -> n__from₁(X)
activate₁(n__from₁(X)) -> from₁(X)
activate₁(X) -> X

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

AFTER₂(s₁(N), cons₂(X, XS)) -> AFTER₂(N, activate₁(XS))
AFTER₂(s₁(N), cons₂(X, XS)) -> ACTIVATE₁(XS)
ACTIVATE₁(n__from₁(X)) -> FROM₁(X)

The TRS R consists of the following rules:

from₁(X) -> cons₂(X, n__from₁(s₁(X)))
after₂(0, XS) -> XS
after₂(s₁(N), cons₂(X, XS)) -> after₂(N, activate₁(XS))
from₁(X) -> n__from₁(X)
activate₁(n__from₁(X)) -> from₁(X)
activate₁(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

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

AFTER₂(s₁(N), cons₂(X, XS)) -> AFTER₂(N, activate₁(XS))

The TRS R consists of the following rules:

from₁(X) -> cons₂(X, n__from₁(s₁(X)))
after₂(0, XS) -> XS
after₂(s₁(N), cons₂(X, XS)) -> after₂(N, activate₁(XS))
from₁(X) -> n__from₁(X)
activate₁(n__from₁(X)) -> from₁(X)
activate₁(X) -> X

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

The following pairs can be strictly oriented and are deleted.

AFTER₂(s₁(N), cons₂(X, XS)) -> AFTER₂(N, activate₁(XS))

The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
AFTER₂(x1, x2) = x1
s₁(x1) = s₁(x1)
cons₂(x1, x2) = x2
activate₁(x1) = activate₁(x1)
n__from₁(x1) = n__from
from₁(x1) = from

Lexicographic Path Order [19].
Precedence:

[activate₁, from] > s₁ > n_{_from}

The following usable rules [14] were oriented:

activate₁(n__from₁(X)) -> from₁(X)
activate₁(X) -> X
from₁(X) -> cons₂(X, n__from₁(s₁(X)))
from₁(X) -> n__from₁(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ PisEmptyProof

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

from₁(X) -> cons₂(X, n__from₁(s₁(X)))
after₂(0, XS) -> XS
after₂(s₁(N), cons₂(X, XS)) -> after₂(N, activate₁(XS))
from₁(X) -> n__from₁(X)
activate₁(n__from₁(X)) -> from₁(X)
activate₁(X) -> X

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