Term Rewriting System R:
[YS, X, XS, X1, X2, Y, L]
app(nil, YS) -> YS
app(cons(X, XS), YS) -> cons(X, napp(activate(XS), YS))
app(X1, X2) -> napp(X1, X2)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
zWadr(nil, YS) -> nil
zWadr(XS, nil) -> nil
zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, nnil)), nzWadr(activate(XS), activate(YS)))
zWadr(X1, X2) -> nzWadr(X1, X2)
prefix(L) -> cons(nil, nzWadr(L, nprefix(L)))
prefix(X) -> nprefix(X)
s(X) -> ns(X)
nil -> nnil
activate(napp(X1, X2)) -> app(activate(X1), activate(X2))
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(nnil) -> nil
activate(nzWadr(X1, X2)) -> zWadr(activate(X1), activate(X2))
activate(nprefix(X)) -> prefix(activate(X))
activate(X) -> X

Innermost Termination of R to be shown.



   R
Removing Redundant Rules for Innermost Termination



Removing the following rules from R which left hand sides contain non normal subterms

app(nil, YS) -> YS
zWadr(nil, YS) -> nil
zWadr(XS, nil) -> nil


   R
RRRI
       →TRS2
Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(cons(X, XS), YS) -> ACTIVATE(XS)
ACTIVATE(napp(X1, X2)) -> APP(activate(X1), activate(X2))
ACTIVATE(napp(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(napp(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nfrom(X)) -> FROM(activate(X))
ACTIVATE(nfrom(X)) -> ACTIVATE(X)
ACTIVATE(ns(X)) -> S(activate(X))
ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(nnil) -> NIL
ACTIVATE(nzWadr(X1, X2)) -> ZWADR(activate(X1), activate(X2))
ACTIVATE(nzWadr(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nzWadr(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nprefix(X)) -> PREFIX(activate(X))
ACTIVATE(nprefix(X)) -> ACTIVATE(X)
ZWADR(cons(X, XS), cons(Y, YS)) -> APP(Y, cons(X, nnil))
ZWADR(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ZWADR(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
PREFIX(L) -> NIL

Furthermore, R contains one SCC.


   R
RRRI
       →TRS2
DPs
           →DP Problem 1
Narrowing Transformation


Dependency Pairs:

ZWADR(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ACTIVATE(nprefix(X)) -> ACTIVATE(X)
ACTIVATE(nzWadr(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nzWadr(X1, X2)) -> ACTIVATE(X1)
ZWADR(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ZWADR(cons(X, XS), cons(Y, YS)) -> APP(Y, cons(X, nnil))
ACTIVATE(nzWadr(X1, X2)) -> ZWADR(activate(X1), activate(X2))
ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(nfrom(X)) -> ACTIVATE(X)
ACTIVATE(napp(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(napp(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(napp(X1, X2)) -> APP(activate(X1), activate(X2))
APP(cons(X, XS), YS) -> ACTIVATE(XS)


Rules:


app(cons(X, XS), YS) -> cons(X, napp(activate(XS), YS))
app(X1, X2) -> napp(X1, X2)
activate(napp(X1, X2)) -> app(activate(X1), activate(X2))
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(nnil) -> nil
activate(nzWadr(X1, X2)) -> zWadr(activate(X1), activate(X2))
activate(nprefix(X)) -> prefix(activate(X))
activate(X) -> X
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, nnil)), nzWadr(activate(XS), activate(YS)))
zWadr(X1, X2) -> nzWadr(X1, X2)
prefix(L) -> cons(nil, nzWadr(L, nprefix(L)))
prefix(X) -> nprefix(X)
nil -> nnil
s(X) -> ns(X)





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

ACTIVATE(napp(X1, X2)) -> APP(activate(X1), activate(X2))
14 new Dependency Pairs are created:

ACTIVATE(napp(napp(X1'', X2''), X2)) -> APP(app(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(napp(nfrom(X'), X2)) -> APP(from(activate(X')), activate(X2))
ACTIVATE(napp(ns(X'), X2)) -> APP(s(activate(X')), activate(X2))
ACTIVATE(napp(nnil, X2)) -> APP(nil, activate(X2))
ACTIVATE(napp(nzWadr(X1'', X2''), X2)) -> APP(zWadr(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(napp(nprefix(X'), X2)) -> APP(prefix(activate(X')), activate(X2))
ACTIVATE(napp(X1', X2)) -> APP(X1', activate(X2))
ACTIVATE(napp(X1, napp(X1'', X2''))) -> APP(activate(X1), app(activate(X1''), activate(X2'')))
ACTIVATE(napp(X1, nfrom(X'))) -> APP(activate(X1), from(activate(X')))
ACTIVATE(napp(X1, ns(X'))) -> APP(activate(X1), s(activate(X')))
ACTIVATE(napp(X1, nnil)) -> APP(activate(X1), nil)
ACTIVATE(napp(X1, nzWadr(X1'', X2''))) -> APP(activate(X1), zWadr(activate(X1''), activate(X2'')))
ACTIVATE(napp(X1, nprefix(X'))) -> APP(activate(X1), prefix(activate(X')))
ACTIVATE(napp(X1, X2')) -> APP(activate(X1), X2')

The transformation is resulting in one new DP problem:



   R
RRRI
       →TRS2
DPs
           →DP Problem 1
Nar
             ...
               →DP Problem 2
Narrowing Transformation


Dependency Pairs:

ZWADR(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ACTIVATE(napp(X1, X2')) -> APP(activate(X1), X2')
ACTIVATE(napp(X1, nprefix(X'))) -> APP(activate(X1), prefix(activate(X')))
ACTIVATE(napp(X1, nzWadr(X1'', X2''))) -> APP(activate(X1), zWadr(activate(X1''), activate(X2'')))
ACTIVATE(napp(X1, nnil)) -> APP(activate(X1), nil)
ACTIVATE(napp(X1, ns(X'))) -> APP(activate(X1), s(activate(X')))
ACTIVATE(napp(X1, nfrom(X'))) -> APP(activate(X1), from(activate(X')))
ACTIVATE(napp(X1, napp(X1'', X2''))) -> APP(activate(X1), app(activate(X1''), activate(X2'')))
ACTIVATE(napp(X1', X2)) -> APP(X1', activate(X2))
ACTIVATE(napp(nprefix(X'), X2)) -> APP(prefix(activate(X')), activate(X2))
ACTIVATE(napp(nzWadr(X1'', X2''), X2)) -> APP(zWadr(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(napp(nnil, X2)) -> APP(nil, activate(X2))
ACTIVATE(napp(ns(X'), X2)) -> APP(s(activate(X')), activate(X2))
ACTIVATE(napp(nfrom(X'), X2)) -> APP(from(activate(X')), activate(X2))
ACTIVATE(napp(napp(X1'', X2''), X2)) -> APP(app(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(nprefix(X)) -> ACTIVATE(X)
ACTIVATE(nzWadr(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nzWadr(X1, X2)) -> ACTIVATE(X1)
APP(cons(X, XS), YS) -> ACTIVATE(XS)
ZWADR(cons(X, XS), cons(Y, YS)) -> APP(Y, cons(X, nnil))
ACTIVATE(nzWadr(X1, X2)) -> ZWADR(activate(X1), activate(X2))
ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(nfrom(X)) -> ACTIVATE(X)
ACTIVATE(napp(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(napp(X1, X2)) -> ACTIVATE(X1)
ZWADR(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)


Rules:


app(cons(X, XS), YS) -> cons(X, napp(activate(XS), YS))
app(X1, X2) -> napp(X1, X2)
activate(napp(X1, X2)) -> app(activate(X1), activate(X2))
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(nnil) -> nil
activate(nzWadr(X1, X2)) -> zWadr(activate(X1), activate(X2))
activate(nprefix(X)) -> prefix(activate(X))
activate(X) -> X
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, nnil)), nzWadr(activate(XS), activate(YS)))
zWadr(X1, X2) -> nzWadr(X1, X2)
prefix(L) -> cons(nil, nzWadr(L, nprefix(L)))
prefix(X) -> nprefix(X)
nil -> nnil
s(X) -> ns(X)





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

ACTIVATE(nzWadr(X1, X2)) -> ZWADR(activate(X1), activate(X2))
14 new Dependency Pairs are created:

ACTIVATE(nzWadr(napp(X1'', X2''), X2)) -> ZWADR(app(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(nzWadr(nfrom(X'), X2)) -> ZWADR(from(activate(X')), activate(X2))
ACTIVATE(nzWadr(ns(X'), X2)) -> ZWADR(s(activate(X')), activate(X2))
ACTIVATE(nzWadr(nnil, X2)) -> ZWADR(nil, activate(X2))
ACTIVATE(nzWadr(nzWadr(X1'', X2''), X2)) -> ZWADR(zWadr(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(nzWadr(nprefix(X'), X2)) -> ZWADR(prefix(activate(X')), activate(X2))
ACTIVATE(nzWadr(X1', X2)) -> ZWADR(X1', activate(X2))
ACTIVATE(nzWadr(X1, napp(X1'', X2''))) -> ZWADR(activate(X1), app(activate(X1''), activate(X2'')))
ACTIVATE(nzWadr(X1, nfrom(X'))) -> ZWADR(activate(X1), from(activate(X')))
ACTIVATE(nzWadr(X1, ns(X'))) -> ZWADR(activate(X1), s(activate(X')))
ACTIVATE(nzWadr(X1, nnil)) -> ZWADR(activate(X1), nil)
ACTIVATE(nzWadr(X1, nzWadr(X1'', X2''))) -> ZWADR(activate(X1), zWadr(activate(X1''), activate(X2'')))
ACTIVATE(nzWadr(X1, nprefix(X'))) -> ZWADR(activate(X1), prefix(activate(X')))
ACTIVATE(nzWadr(X1, X2')) -> ZWADR(activate(X1), X2')

The transformation is resulting in one new DP problem:



   R
RRRI
       →TRS2
DPs
           →DP Problem 1
Nar
             ...
               →DP Problem 3
Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

ACTIVATE(nzWadr(X1, X2')) -> ZWADR(activate(X1), X2')
ACTIVATE(nzWadr(X1, nprefix(X'))) -> ZWADR(activate(X1), prefix(activate(X')))
ACTIVATE(nzWadr(X1, nzWadr(X1'', X2''))) -> ZWADR(activate(X1), zWadr(activate(X1''), activate(X2'')))
ACTIVATE(nzWadr(X1, nnil)) -> ZWADR(activate(X1), nil)
ACTIVATE(nzWadr(X1, ns(X'))) -> ZWADR(activate(X1), s(activate(X')))
ACTIVATE(nzWadr(X1, nfrom(X'))) -> ZWADR(activate(X1), from(activate(X')))
ACTIVATE(nzWadr(X1, napp(X1'', X2''))) -> ZWADR(activate(X1), app(activate(X1''), activate(X2'')))
ACTIVATE(nzWadr(X1', X2)) -> ZWADR(X1', activate(X2))
ACTIVATE(nzWadr(nprefix(X'), X2)) -> ZWADR(prefix(activate(X')), activate(X2))
ACTIVATE(nzWadr(nzWadr(X1'', X2''), X2)) -> ZWADR(zWadr(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(nzWadr(nnil, X2)) -> ZWADR(nil, activate(X2))
ACTIVATE(nzWadr(ns(X'), X2)) -> ZWADR(s(activate(X')), activate(X2))
ACTIVATE(nzWadr(nfrom(X'), X2)) -> ZWADR(from(activate(X')), activate(X2))
ZWADR(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ZWADR(cons(X, XS), cons(Y, YS)) -> APP(Y, cons(X, nnil))
ACTIVATE(nzWadr(napp(X1'', X2''), X2)) -> ZWADR(app(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(napp(X1, X2')) -> APP(activate(X1), X2')
ACTIVATE(napp(X1, nprefix(X'))) -> APP(activate(X1), prefix(activate(X')))
ACTIVATE(napp(X1, nzWadr(X1'', X2''))) -> APP(activate(X1), zWadr(activate(X1''), activate(X2'')))
ACTIVATE(napp(X1, nnil)) -> APP(activate(X1), nil)
ACTIVATE(napp(X1, ns(X'))) -> APP(activate(X1), s(activate(X')))
ACTIVATE(napp(X1, nfrom(X'))) -> APP(activate(X1), from(activate(X')))
ACTIVATE(napp(X1, napp(X1'', X2''))) -> APP(activate(X1), app(activate(X1''), activate(X2'')))
ACTIVATE(napp(X1', X2)) -> APP(X1', activate(X2))
ACTIVATE(napp(nprefix(X'), X2)) -> APP(prefix(activate(X')), activate(X2))
ACTIVATE(napp(nzWadr(X1'', X2''), X2)) -> APP(zWadr(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(napp(nnil, X2)) -> APP(nil, activate(X2))
ACTIVATE(napp(ns(X'), X2)) -> APP(s(activate(X')), activate(X2))
ACTIVATE(napp(nfrom(X'), X2)) -> APP(from(activate(X')), activate(X2))
APP(cons(X, XS), YS) -> ACTIVATE(XS)
ACTIVATE(napp(napp(X1'', X2''), X2)) -> APP(app(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(nprefix(X)) -> ACTIVATE(X)
ACTIVATE(nzWadr(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nzWadr(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(nfrom(X)) -> ACTIVATE(X)
ACTIVATE(napp(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(napp(X1, X2)) -> ACTIVATE(X1)
ZWADR(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)


Rules:


app(cons(X, XS), YS) -> cons(X, napp(activate(XS), YS))
app(X1, X2) -> napp(X1, X2)
activate(napp(X1, X2)) -> app(activate(X1), activate(X2))
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(nnil) -> nil
activate(nzWadr(X1, X2)) -> zWadr(activate(X1), activate(X2))
activate(nprefix(X)) -> prefix(activate(X))
activate(X) -> X
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, nnil)), nzWadr(activate(XS), activate(YS)))
zWadr(X1, X2) -> nzWadr(X1, X2)
prefix(L) -> cons(nil, nzWadr(L, nprefix(L)))
prefix(X) -> nprefix(X)
nil -> nnil
s(X) -> ns(X)




The Proof could not be continued due to a Timeout.
Innermost Termination of R could not be shown.
Duration:
1:00 minutes