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(s(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, prefix(L)))
nil -> nnil
activate(napp(X1, X2)) -> app(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nnil) -> nil
activate(nzWadr(X1, X2)) -> zWadr(X1, X2)
activate(X) -> X

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains two SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering
       →DP Problem 2
         ↳Remaining


Dependency Pairs:

ZWADR(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
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(X1, X2)
ACTIVATE(napp(X1, X2)) -> APP(X1, X2)
APP(cons(X, XS), YS) -> ACTIVATE(XS)


Rules:


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(s(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, prefix(L)))
nil -> nnil
activate(napp(X1, X2)) -> app(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nnil) -> nil
activate(nzWadr(X1, X2)) -> zWadr(X1, X2)
activate(X) -> X


Strategy:

innermost




The following dependency pairs can be strictly oriented:

ZWADR(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
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(X1, X2)
ACTIVATE(napp(X1, X2)) -> APP(X1, X2)
APP(cons(X, XS), YS) -> ACTIVATE(XS)


There are no usable rules for innermost that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{cons, ZWADR, nzWadr, nnil} > {APP, ACTIVATE}

resulting in one new DP problem.
Used Argument Filtering System:
ACTIVATE(x1) -> ACTIVATE(x1)
APP(x1, x2) -> APP(x1, x2)
napp(x1, x2) -> napp(x1, x2)
cons(x1, x2) -> cons(x1, x2)
nzWadr(x1, x2) -> nzWadr(x1, x2)
ZWADR(x1, x2) -> ZWADR(x1, x2)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 3
             ↳Dependency Graph
       →DP Problem 2
         ↳Remaining


Dependency Pair:


Rules:


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(s(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, prefix(L)))
nil -> nnil
activate(napp(X1, X2)) -> app(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nnil) -> nil
activate(nzWadr(X1, X2)) -> zWadr(X1, X2)
activate(X) -> X


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

PREFIX(L) -> PREFIX(L)


Rules:


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(s(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, prefix(L)))
nil -> nnil
activate(napp(X1, X2)) -> app(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nnil) -> nil
activate(nzWadr(X1, X2)) -> zWadr(X1, X2)
activate(X) -> X


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:00 minutes