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
Polynomial 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 pair can be strictly oriented:

ACTIVATE(nzWadr(X1, X2)) -> ZWADR(X1, X2)


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(cons(x1, x2))=  x1 + x2  
  POL(ZWADR(x1, x2))=  x1 + x2  
  POL(n__zWadr(x1, x2))=  1 + x1 + x2  
  POL(n__nil)=  0  
  POL(n__app(x1, x2))=  x1  
  POL(ACTIVATE(x1))=  x1  
  POL(APP(x1, x2))=  x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
           →DP Problem 3
Dependency Graph
       →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(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




Using the Dependency Graph the DP problem was split into 1 DP problems.


   R
DPs
       →DP Problem 1
Polo
           →DP Problem 3
DGraph
             ...
               →DP Problem 4
Polynomial Ordering
       →DP Problem 2
Remaining


Dependency Pairs:

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


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 pair can be strictly oriented:

ACTIVATE(napp(X1, X2)) -> APP(X1, X2)


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(cons(x1, x2))=  x2  
  POL(n__app(x1, x2))=  1 + x1  
  POL(ACTIVATE(x1))=  x1  
  POL(APP(x1, x2))=  x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
           →DP Problem 3
DGraph
             ...
               →DP Problem 5
Dependency Graph
       →DP Problem 2
Remaining


Dependency Pair:

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




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
Polo
       →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