Termination w.r.t. Q proof of TRS/TRCSREx3_3_25_Bor03_FR.trs

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:

app(nil, YS) → YS
app(cons(X, XS), YS) → cons(X, n__app(activate(XS), YS))
from(X) → cons(X, n__from(n__s(X)))
zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
zWadr(cons(X, XS), cons(Y, YS)) → cons(app(Y, cons(X, n__nil)), n__zWadr(activate(XS), activate(YS)))
prefix(L) → cons(nil, n__zWadr(L, n__prefix(L)))
app(X1, X2) → n__app(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
nil → n__nil
zWadr(X1, X2) → n__zWadr(X1, X2)
prefix(X) → n__prefix(X)
activate(n__app(X1, X2)) → app(activate(X1), activate(X2))
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__nil) → nil
activate(n__zWadr(X1, X2)) → zWadr(activate(X1), activate(X2))
activate(n__prefix(X)) → prefix(activate(X))
activate(X) → X

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

app(nil, YS) → YS
app(cons(X, XS), YS) → cons(X, n__app(activate(XS), YS))
from(X) → cons(X, n__from(n__s(X)))
zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
zWadr(cons(X, XS), cons(Y, YS)) → cons(app(Y, cons(X, n__nil)), n__zWadr(activate(XS), activate(YS)))
prefix(L) → cons(nil, n__zWadr(L, n__prefix(L)))
app(X1, X2) → n__app(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
nil → n__nil
zWadr(X1, X2) → n__zWadr(X1, X2)
prefix(X) → n__prefix(X)
activate(n__app(X1, X2)) → app(activate(X1), activate(X2))
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__nil) → nil
activate(n__zWadr(X1, X2)) → zWadr(activate(X1), activate(X2))
activate(n__prefix(X)) → prefix(activate(X))
activate(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:

ACTIVATE(n__app(X1, X2)) → APP(activate(X1), activate(X2))
ZWADR(cons(X, XS), cons(Y, YS)) → ACTIVATE(YS)
APP(cons(X, XS), YS) → ACTIVATE(XS)
ACTIVATE(n__app(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__nil) → NIL
ACTIVATE(n__prefix(X)) → PREFIX(activate(X))
ACTIVATE(n__prefix(X)) → ACTIVATE(X)
ACTIVATE(n__app(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__from(X)) → FROM(activate(X))
ACTIVATE(n__zWadr(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__s(X)) → S(activate(X))
ACTIVATE(n__zWadr(X1, X2)) → ZWADR(activate(X1), activate(X2))
PREFIX(L) → NIL
ZWADR(cons(X, XS), cons(Y, YS)) → ACTIVATE(XS)
ACTIVATE(n__from(X)) → ACTIVATE(X)
ACTIVATE(n__zWadr(X1, X2)) → ACTIVATE(X1)
ZWADR(cons(X, XS), cons(Y, YS)) → APP(Y, cons(X, n__nil))
ACTIVATE(n__s(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

app(nil, YS) → YS
app(cons(X, XS), YS) → cons(X, n__app(activate(XS), YS))
from(X) → cons(X, n__from(n__s(X)))
zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
zWadr(cons(X, XS), cons(Y, YS)) → cons(app(Y, cons(X, n__nil)), n__zWadr(activate(XS), activate(YS)))
prefix(L) → cons(nil, n__zWadr(L, n__prefix(L)))
app(X1, X2) → n__app(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
nil → n__nil
zWadr(X1, X2) → n__zWadr(X1, X2)
prefix(X) → n__prefix(X)
activate(n__app(X1, X2)) → app(activate(X1), activate(X2))
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__nil) → nil
activate(n__zWadr(X1, X2)) → zWadr(activate(X1), activate(X2))
activate(n__prefix(X)) → prefix(activate(X))
activate(X) → X

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

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

ACTIVATE(n__app(X1, X2)) → APP(activate(X1), activate(X2))
ZWADR(cons(X, XS), cons(Y, YS)) → ACTIVATE(YS)
APP(cons(X, XS), YS) → ACTIVATE(XS)
ACTIVATE(n__app(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__nil) → NIL
ACTIVATE(n__prefix(X)) → PREFIX(activate(X))
ACTIVATE(n__prefix(X)) → ACTIVATE(X)
ACTIVATE(n__app(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__from(X)) → FROM(activate(X))
ACTIVATE(n__zWadr(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__s(X)) → S(activate(X))
ACTIVATE(n__zWadr(X1, X2)) → ZWADR(activate(X1), activate(X2))
PREFIX(L) → NIL
ZWADR(cons(X, XS), cons(Y, YS)) → ACTIVATE(XS)
ACTIVATE(n__from(X)) → ACTIVATE(X)
ACTIVATE(n__zWadr(X1, X2)) → ACTIVATE(X1)
ZWADR(cons(X, XS), cons(Y, YS)) → APP(Y, cons(X, n__nil))
ACTIVATE(n__s(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

app(nil, YS) → YS
app(cons(X, XS), YS) → cons(X, n__app(activate(XS), YS))
from(X) → cons(X, n__from(n__s(X)))
zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
zWadr(cons(X, XS), cons(Y, YS)) → cons(app(Y, cons(X, n__nil)), n__zWadr(activate(XS), activate(YS)))
prefix(L) → cons(nil, n__zWadr(L, n__prefix(L)))
app(X1, X2) → n__app(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
nil → n__nil
zWadr(X1, X2) → n__zWadr(X1, X2)
prefix(X) → n__prefix(X)
activate(n__app(X1, X2)) → app(activate(X1), activate(X2))
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__nil) → nil
activate(n__zWadr(X1, X2)) → zWadr(activate(X1), activate(X2))
activate(n__prefix(X)) → prefix(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We deleted some edges using various graph approximations

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

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

ACTIVATE(n__app(X1, X2)) → APP(activate(X1), activate(X2))
ZWADR(cons(X, XS), cons(Y, YS)) → ACTIVATE(YS)
APP(cons(X, XS), YS) → ACTIVATE(XS)
ACTIVATE(n__app(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__nil) → NIL
ACTIVATE(n__prefix(X)) → PREFIX(activate(X))
ACTIVATE(n__from(X)) → FROM(activate(X))
ACTIVATE(n__app(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__prefix(X)) → ACTIVATE(X)
ACTIVATE(n__zWadr(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__s(X)) → S(activate(X))
PREFIX(L) → NIL
ACTIVATE(n__zWadr(X1, X2)) → ZWADR(activate(X1), activate(X2))
ZWADR(cons(X, XS), cons(Y, YS)) → ACTIVATE(XS)
ACTIVATE(n__from(X)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ZWADR(cons(X, XS), cons(Y, YS)) → APP(Y, cons(X, n__nil))
ACTIVATE(n__zWadr(X1, X2)) → ACTIVATE(X1)

The TRS R consists of the following rules:

app(nil, YS) → YS
app(cons(X, XS), YS) → cons(X, n__app(activate(XS), YS))
from(X) → cons(X, n__from(n__s(X)))
zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
zWadr(cons(X, XS), cons(Y, YS)) → cons(app(Y, cons(X, n__nil)), n__zWadr(activate(XS), activate(YS)))
prefix(L) → cons(nil, n__zWadr(L, n__prefix(L)))
app(X1, X2) → n__app(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
nil → n__nil
zWadr(X1, X2) → n__zWadr(X1, X2)
prefix(X) → n__prefix(X)
activate(n__app(X1, X2)) → app(activate(X1), activate(X2))
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__nil) → nil
activate(n__zWadr(X1, X2)) → zWadr(activate(X1), activate(X2))
activate(n__prefix(X)) → prefix(activate(X))
activate(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 5 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

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

ACTIVATE(n__app(X1, X2)) → APP(activate(X1), activate(X2))
ZWADR(cons(X, XS), cons(Y, YS)) → ACTIVATE(YS)
APP(cons(X, XS), YS) → ACTIVATE(XS)
ACTIVATE(n__app(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__app(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__prefix(X)) → ACTIVATE(X)
ACTIVATE(n__zWadr(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__zWadr(X1, X2)) → ZWADR(activate(X1), activate(X2))
ZWADR(cons(X, XS), cons(Y, YS)) → ACTIVATE(XS)
ACTIVATE(n__from(X)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ZWADR(cons(X, XS), cons(Y, YS)) → APP(Y, cons(X, n__nil))
ACTIVATE(n__zWadr(X1, X2)) → ACTIVATE(X1)

The TRS R consists of the following rules:

app(nil, YS) → YS
app(cons(X, XS), YS) → cons(X, n__app(activate(XS), YS))
from(X) → cons(X, n__from(n__s(X)))
zWadr(nil, YS) → nil
zWadr(XS, nil) → nil
zWadr(cons(X, XS), cons(Y, YS)) → cons(app(Y, cons(X, n__nil)), n__zWadr(activate(XS), activate(YS)))
prefix(L) → cons(nil, n__zWadr(L, n__prefix(L)))
app(X1, X2) → n__app(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
nil → n__nil
zWadr(X1, X2) → n__zWadr(X1, X2)
prefix(X) → n__prefix(X)
activate(n__app(X1, X2)) → app(activate(X1), activate(X2))
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__nil) → nil
activate(n__zWadr(X1, X2)) → zWadr(activate(X1), activate(X2))
activate(n__prefix(X)) → prefix(activate(X))
activate(X) → X

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