Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 AND
  ↳5 QDP
  ↳6 QDP
    ↳7 QDPOrderProof (⇔)
    ↳8 QDP
    ↳9 PisEmptyProof (⇔)
    ↳10 TRUE

(0) Obligation:

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(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, prefix(L)))
app(X1, X2) → n__app(X1, X2)
from(X) → n__from(X)
nil → n__nil
zWadr(X1, X2) → n__zWadr(X1, X2)
activate(n__app(X1, X2)) → app(X1, X2)
activate(n__from(X)) → from(X)
activate(n__nil) → nil
activate(n__zWadr(X1, X2)) → zWadr(X1, X2)
activate(X) → X

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

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

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(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, prefix(L)))
app(X1, X2) → n__app(X1, X2)
from(X) → n__from(X)
nil → n__nil
zWadr(X1, X2) → n__zWadr(X1, X2)
activate(n__app(X1, X2)) → app(X1, X2)
activate(n__from(X)) → from(X)
activate(n__nil) → nil
activate(n__zWadr(X1, X2)) → zWadr(X1, X2)
activate(X) → X

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 3 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

PREFIX(L) → PREFIX(L)

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(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, prefix(L)))
app(X1, X2) → n__app(X1, X2)
from(X) → n__from(X)
nil → n__nil
zWadr(X1, X2) → n__zWadr(X1, X2)
activate(n__app(X1, X2)) → app(X1, X2)
activate(n__from(X)) → from(X)
activate(n__nil) → nil
activate(n__zWadr(X1, X2)) → zWadr(X1, X2)
activate(X) → X

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

(6) Obligation:

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

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

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(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, prefix(L)))
app(X1, X2) → n__app(X1, X2)
from(X) → n__from(X)
nil → n__nil
zWadr(X1, X2) → n__zWadr(X1, X2)
activate(n__app(X1, X2)) → app(X1, X2)
activate(n__from(X)) → from(X)
activate(n__nil) → nil
activate(n__zWadr(X1, X2)) → zWadr(X1, X2)
activate(X) → X

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

(7) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVATE(n__app(X1, X2)) → APP(X1, X2)
APP(cons(X, XS), YS) → ACTIVATE(XS)
ACTIVATE(n__zWadr(X1, X2)) → ZWADR(X1, X2)
ZWADR(cons(X, XS), cons(Y, YS)) → APP(Y, cons(X, n__nil))
ZWADR(cons(X, XS), cons(Y, YS)) → ACTIVATE(XS)
ZWADR(cons(X, XS), cons(Y, YS)) → ACTIVATE(YS)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVATE(x1)  =  ACTIVATE(x1)
n__app(x1, x2)  =  n__app(x1)
APP(x1, x2)  =  APP(x1)
cons(x1, x2)  =  cons(x1, x2)
n__zWadr(x1, x2)  =  n__zWadr(x1, x2)
ZWADR(x1, x2)  =  ZWADR(x1, x2)
n__nil  =  n__nil

Lexicographic path order with status [LPO].
Precedence:
napp1 > APP1
cons2 > ACTIVATE1 > APP1
nzWadr2 > ZWADR2 > APP1
nnil > APP1

Status:
APP1: [1]
cons2: [2,1]
napp1: [1]
nnil: []
ZWADR2: [1,2]
nzWadr2: [1,2]
ACTIVATE1: [1]

The following usable rules [FROCOS05] were oriented: none

(8) Obligation:

Q DP problem:
P is empty.
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(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, prefix(L)))
app(X1, X2) → n__app(X1, X2)
from(X) → n__from(X)
nil → n__nil
zWadr(X1, X2) → n__zWadr(X1, X2)
activate(n__app(X1, X2)) → app(X1, X2)
activate(n__from(X)) → from(X)
activate(n__nil) → nil
activate(n__zWadr(X1, X2)) → zWadr(X1, X2)
activate(X) → X

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

(9) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(10) TRUE