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:

app2(nil, YS) -> YS
app2(cons2(X, XS), YS) -> cons2(X, n__app2(activate1(XS), YS))
from1(X) -> cons2(X, n__from1(n__s1(X)))
zWadr2(nil, YS) -> nil
zWadr2(XS, nil) -> nil
zWadr2(cons2(X, XS), cons2(Y, YS)) -> cons2(app2(Y, cons2(X, n__nil)), n__zWadr2(activate1(XS), activate1(YS)))
prefix1(L) -> cons2(nil, n__zWadr2(L, n__prefix1(L)))
app2(X1, X2) -> n__app2(X1, X2)
from1(X) -> n__from1(X)
s1(X) -> n__s1(X)
nil -> n__nil
zWadr2(X1, X2) -> n__zWadr2(X1, X2)
prefix1(X) -> n__prefix1(X)
activate1(n__app2(X1, X2)) -> app2(activate1(X1), activate1(X2))
activate1(n__from1(X)) -> from1(activate1(X))
activate1(n__s1(X)) -> s1(activate1(X))
activate1(n__nil) -> nil
activate1(n__zWadr2(X1, X2)) -> zWadr2(activate1(X1), activate1(X2))
activate1(n__prefix1(X)) -> prefix1(activate1(X))
activate1(X) -> X

Q is empty.


QTRS
  ↳ DependencyPairsProof

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

app2(nil, YS) -> YS
app2(cons2(X, XS), YS) -> cons2(X, n__app2(activate1(XS), YS))
from1(X) -> cons2(X, n__from1(n__s1(X)))
zWadr2(nil, YS) -> nil
zWadr2(XS, nil) -> nil
zWadr2(cons2(X, XS), cons2(Y, YS)) -> cons2(app2(Y, cons2(X, n__nil)), n__zWadr2(activate1(XS), activate1(YS)))
prefix1(L) -> cons2(nil, n__zWadr2(L, n__prefix1(L)))
app2(X1, X2) -> n__app2(X1, X2)
from1(X) -> n__from1(X)
s1(X) -> n__s1(X)
nil -> n__nil
zWadr2(X1, X2) -> n__zWadr2(X1, X2)
prefix1(X) -> n__prefix1(X)
activate1(n__app2(X1, X2)) -> app2(activate1(X1), activate1(X2))
activate1(n__from1(X)) -> from1(activate1(X))
activate1(n__s1(X)) -> s1(activate1(X))
activate1(n__nil) -> nil
activate1(n__zWadr2(X1, X2)) -> zWadr2(activate1(X1), activate1(X2))
activate1(n__prefix1(X)) -> prefix1(activate1(X))
activate1(X) -> X

Q is empty.

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

ACTIVATE1(n__app2(X1, X2)) -> APP2(activate1(X1), activate1(X2))
ZWADR2(cons2(X, XS), cons2(Y, YS)) -> ACTIVATE1(YS)
APP2(cons2(X, XS), YS) -> ACTIVATE1(XS)
ACTIVATE1(n__app2(X1, X2)) -> ACTIVATE1(X2)
ACTIVATE1(n__nil) -> NIL
ACTIVATE1(n__prefix1(X)) -> PREFIX1(activate1(X))
ACTIVATE1(n__prefix1(X)) -> ACTIVATE1(X)
ACTIVATE1(n__app2(X1, X2)) -> ACTIVATE1(X1)
ACTIVATE1(n__from1(X)) -> FROM1(activate1(X))
ACTIVATE1(n__zWadr2(X1, X2)) -> ACTIVATE1(X2)
ACTIVATE1(n__s1(X)) -> S1(activate1(X))
ACTIVATE1(n__zWadr2(X1, X2)) -> ZWADR2(activate1(X1), activate1(X2))
PREFIX1(L) -> NIL
ZWADR2(cons2(X, XS), cons2(Y, YS)) -> ACTIVATE1(XS)
ACTIVATE1(n__from1(X)) -> ACTIVATE1(X)
ACTIVATE1(n__zWadr2(X1, X2)) -> ACTIVATE1(X1)
ZWADR2(cons2(X, XS), cons2(Y, YS)) -> APP2(Y, cons2(X, n__nil))
ACTIVATE1(n__s1(X)) -> ACTIVATE1(X)

The TRS R consists of the following rules:

app2(nil, YS) -> YS
app2(cons2(X, XS), YS) -> cons2(X, n__app2(activate1(XS), YS))
from1(X) -> cons2(X, n__from1(n__s1(X)))
zWadr2(nil, YS) -> nil
zWadr2(XS, nil) -> nil
zWadr2(cons2(X, XS), cons2(Y, YS)) -> cons2(app2(Y, cons2(X, n__nil)), n__zWadr2(activate1(XS), activate1(YS)))
prefix1(L) -> cons2(nil, n__zWadr2(L, n__prefix1(L)))
app2(X1, X2) -> n__app2(X1, X2)
from1(X) -> n__from1(X)
s1(X) -> n__s1(X)
nil -> n__nil
zWadr2(X1, X2) -> n__zWadr2(X1, X2)
prefix1(X) -> n__prefix1(X)
activate1(n__app2(X1, X2)) -> app2(activate1(X1), activate1(X2))
activate1(n__from1(X)) -> from1(activate1(X))
activate1(n__s1(X)) -> s1(activate1(X))
activate1(n__nil) -> nil
activate1(n__zWadr2(X1, X2)) -> zWadr2(activate1(X1), activate1(X2))
activate1(n__prefix1(X)) -> prefix1(activate1(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:

ACTIVATE1(n__app2(X1, X2)) -> APP2(activate1(X1), activate1(X2))
ZWADR2(cons2(X, XS), cons2(Y, YS)) -> ACTIVATE1(YS)
APP2(cons2(X, XS), YS) -> ACTIVATE1(XS)
ACTIVATE1(n__app2(X1, X2)) -> ACTIVATE1(X2)
ACTIVATE1(n__nil) -> NIL
ACTIVATE1(n__prefix1(X)) -> PREFIX1(activate1(X))
ACTIVATE1(n__prefix1(X)) -> ACTIVATE1(X)
ACTIVATE1(n__app2(X1, X2)) -> ACTIVATE1(X1)
ACTIVATE1(n__from1(X)) -> FROM1(activate1(X))
ACTIVATE1(n__zWadr2(X1, X2)) -> ACTIVATE1(X2)
ACTIVATE1(n__s1(X)) -> S1(activate1(X))
ACTIVATE1(n__zWadr2(X1, X2)) -> ZWADR2(activate1(X1), activate1(X2))
PREFIX1(L) -> NIL
ZWADR2(cons2(X, XS), cons2(Y, YS)) -> ACTIVATE1(XS)
ACTIVATE1(n__from1(X)) -> ACTIVATE1(X)
ACTIVATE1(n__zWadr2(X1, X2)) -> ACTIVATE1(X1)
ZWADR2(cons2(X, XS), cons2(Y, YS)) -> APP2(Y, cons2(X, n__nil))
ACTIVATE1(n__s1(X)) -> ACTIVATE1(X)

The TRS R consists of the following rules:

app2(nil, YS) -> YS
app2(cons2(X, XS), YS) -> cons2(X, n__app2(activate1(XS), YS))
from1(X) -> cons2(X, n__from1(n__s1(X)))
zWadr2(nil, YS) -> nil
zWadr2(XS, nil) -> nil
zWadr2(cons2(X, XS), cons2(Y, YS)) -> cons2(app2(Y, cons2(X, n__nil)), n__zWadr2(activate1(XS), activate1(YS)))
prefix1(L) -> cons2(nil, n__zWadr2(L, n__prefix1(L)))
app2(X1, X2) -> n__app2(X1, X2)
from1(X) -> n__from1(X)
s1(X) -> n__s1(X)
nil -> n__nil
zWadr2(X1, X2) -> n__zWadr2(X1, X2)
prefix1(X) -> n__prefix1(X)
activate1(n__app2(X1, X2)) -> app2(activate1(X1), activate1(X2))
activate1(n__from1(X)) -> from1(activate1(X))
activate1(n__s1(X)) -> s1(activate1(X))
activate1(n__nil) -> nil
activate1(n__zWadr2(X1, X2)) -> zWadr2(activate1(X1), activate1(X2))
activate1(n__prefix1(X)) -> prefix1(activate1(X))
activate1(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph contains 1 SCC with 5 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
QDP

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

ACTIVATE1(n__app2(X1, X2)) -> APP2(activate1(X1), activate1(X2))
ZWADR2(cons2(X, XS), cons2(Y, YS)) -> ACTIVATE1(YS)
APP2(cons2(X, XS), YS) -> ACTIVATE1(XS)
ACTIVATE1(n__app2(X1, X2)) -> ACTIVATE1(X2)
ACTIVATE1(n__app2(X1, X2)) -> ACTIVATE1(X1)
ACTIVATE1(n__prefix1(X)) -> ACTIVATE1(X)
ACTIVATE1(n__zWadr2(X1, X2)) -> ACTIVATE1(X2)
ACTIVATE1(n__zWadr2(X1, X2)) -> ZWADR2(activate1(X1), activate1(X2))
ZWADR2(cons2(X, XS), cons2(Y, YS)) -> ACTIVATE1(XS)
ACTIVATE1(n__from1(X)) -> ACTIVATE1(X)
ACTIVATE1(n__s1(X)) -> ACTIVATE1(X)
ZWADR2(cons2(X, XS), cons2(Y, YS)) -> APP2(Y, cons2(X, n__nil))
ACTIVATE1(n__zWadr2(X1, X2)) -> ACTIVATE1(X1)

The TRS R consists of the following rules:

app2(nil, YS) -> YS
app2(cons2(X, XS), YS) -> cons2(X, n__app2(activate1(XS), YS))
from1(X) -> cons2(X, n__from1(n__s1(X)))
zWadr2(nil, YS) -> nil
zWadr2(XS, nil) -> nil
zWadr2(cons2(X, XS), cons2(Y, YS)) -> cons2(app2(Y, cons2(X, n__nil)), n__zWadr2(activate1(XS), activate1(YS)))
prefix1(L) -> cons2(nil, n__zWadr2(L, n__prefix1(L)))
app2(X1, X2) -> n__app2(X1, X2)
from1(X) -> n__from1(X)
s1(X) -> n__s1(X)
nil -> n__nil
zWadr2(X1, X2) -> n__zWadr2(X1, X2)
prefix1(X) -> n__prefix1(X)
activate1(n__app2(X1, X2)) -> app2(activate1(X1), activate1(X2))
activate1(n__from1(X)) -> from1(activate1(X))
activate1(n__s1(X)) -> s1(activate1(X))
activate1(n__nil) -> nil
activate1(n__zWadr2(X1, X2)) -> zWadr2(activate1(X1), activate1(X2))
activate1(n__prefix1(X)) -> prefix1(activate1(X))
activate1(X) -> X

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