Term Rewriting System R:
[YS, X, XS, Y, L, X1, X2]
active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

ACTIVE(app(cons(X, XS), YS)) -> CONS(X, app(XS, YS))
ACTIVE(app(cons(X, XS), YS)) -> APP(XS, YS)
ACTIVE(from(X)) -> CONS(X, from(s(X)))
ACTIVE(from(X)) -> FROM(s(X))
ACTIVE(from(X)) -> S(X)
ACTIVE(zWadr(cons(X, XS), cons(Y, YS))) -> CONS(app(Y, cons(X, nil)), zWadr(XS, YS))
ACTIVE(zWadr(cons(X, XS), cons(Y, YS))) -> APP(Y, cons(X, nil))
ACTIVE(zWadr(cons(X, XS), cons(Y, YS))) -> CONS(X, nil)
ACTIVE(zWadr(cons(X, XS), cons(Y, YS))) -> ZWADR(XS, YS)
ACTIVE(prefix(L)) -> CONS(nil, zWadr(L, prefix(L)))
ACTIVE(prefix(L)) -> ZWADR(L, prefix(L))
ACTIVE(app(X1, X2)) -> APP(active(X1), X2)
ACTIVE(app(X1, X2)) -> ACTIVE(X1)
ACTIVE(app(X1, X2)) -> APP(X1, active(X2))
ACTIVE(app(X1, X2)) -> ACTIVE(X2)
ACTIVE(cons(X1, X2)) -> CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(from(X)) -> FROM(active(X))
ACTIVE(from(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> S(active(X))
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(zWadr(X1, X2)) -> ZWADR(active(X1), X2)
ACTIVE(zWadr(X1, X2)) -> ACTIVE(X1)
ACTIVE(zWadr(X1, X2)) -> ZWADR(X1, active(X2))
ACTIVE(zWadr(X1, X2)) -> ACTIVE(X2)
ACTIVE(prefix(X)) -> PREFIX(active(X))
ACTIVE(prefix(X)) -> ACTIVE(X)
APP(mark(X1), X2) -> APP(X1, X2)
APP(X1, mark(X2)) -> APP(X1, X2)
APP(ok(X1), ok(X2)) -> APP(X1, X2)
CONS(mark(X1), X2) -> CONS(X1, X2)
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
FROM(mark(X)) -> FROM(X)
FROM(ok(X)) -> FROM(X)
S(mark(X)) -> S(X)
S(ok(X)) -> S(X)
ZWADR(mark(X1), X2) -> ZWADR(X1, X2)
ZWADR(X1, mark(X2)) -> ZWADR(X1, X2)
ZWADR(ok(X1), ok(X2)) -> ZWADR(X1, X2)
PREFIX(mark(X)) -> PREFIX(X)
PREFIX(ok(X)) -> PREFIX(X)
PROPER(app(X1, X2)) -> APP(proper(X1), proper(X2))
PROPER(app(X1, X2)) -> PROPER(X1)
PROPER(app(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> CONS(proper(X1), proper(X2))
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(from(X)) -> FROM(proper(X))
PROPER(from(X)) -> PROPER(X)
PROPER(s(X)) -> S(proper(X))
PROPER(s(X)) -> PROPER(X)
PROPER(zWadr(X1, X2)) -> ZWADR(proper(X1), proper(X2))
PROPER(zWadr(X1, X2)) -> PROPER(X1)
PROPER(zWadr(X1, X2)) -> PROPER(X2)
PROPER(prefix(X)) -> PREFIX(proper(X))
PROPER(prefix(X)) -> PROPER(X)
TOP(mark(X)) -> TOP(proper(X))
TOP(mark(X)) -> PROPER(X)
TOP(ok(X)) -> TOP(active(X))
TOP(ok(X)) -> ACTIVE(X)

Furthermore, R contains nine SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pairs:

CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
CONS(mark(X1), X2) -> CONS(X1, X2)


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

CONS(ok(X1), ok(X2)) -> CONS(X1, X2)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(mark(x1))=  0  
  POL(ok(x1))=  1 + x1  
  POL(CONS(x1, x2))=  x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 10
             ↳Polynomial Ordering
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:

CONS(mark(X1), X2) -> CONS(X1, X2)


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

CONS(mark(X1), X2) -> CONS(X1, X2)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(mark(x1))=  1 + x1  
  POL(CONS(x1, x2))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 10
             ↳Polo
             ...
               →DP Problem 11
                 ↳Dependency Graph
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polynomial Ordering
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pairs:

APP(ok(X1), ok(X2)) -> APP(X1, X2)
APP(X1, mark(X2)) -> APP(X1, X2)
APP(mark(X1), X2) -> APP(X1, X2)


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

APP(ok(X1), ok(X2)) -> APP(X1, X2)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(mark(x1))=  x1  
  POL(ok(x1))=  1 + x1  
  POL(APP(x1, x2))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
           →DP Problem 12
             ↳Polynomial Ordering
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pairs:

APP(X1, mark(X2)) -> APP(X1, X2)
APP(mark(X1), X2) -> APP(X1, X2)


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

APP(mark(X1), X2) -> APP(X1, X2)


There are no usable rules using the Ce-refinement that need to be oriented.

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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
           →DP Problem 12
             ↳Polo
             ...
               →DP Problem 13
                 ↳Polynomial Ordering
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:

APP(X1, mark(X2)) -> APP(X1, X2)


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

APP(X1, mark(X2)) -> APP(X1, X2)


There are no usable rules using the Ce-refinement that need to be oriented.

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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
           →DP Problem 12
             ↳Polo
             ...
               →DP Problem 14
                 ↳Dependency Graph
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polynomial Ordering
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pairs:

FROM(ok(X)) -> FROM(X)
FROM(mark(X)) -> FROM(X)


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

FROM(ok(X)) -> FROM(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(FROM(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(ok(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
           →DP Problem 15
             ↳Polynomial Ordering
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:

FROM(mark(X)) -> FROM(X)


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

FROM(mark(X)) -> FROM(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(FROM(x1))=  x1  
  POL(mark(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
           →DP Problem 15
             ↳Polo
             ...
               →DP Problem 16
                 ↳Dependency Graph
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polynomial Ordering
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pairs:

S(ok(X)) -> S(X)
S(mark(X)) -> S(X)


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

S(ok(X)) -> S(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(S(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(ok(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
           →DP Problem 17
             ↳Polynomial Ordering
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:

S(mark(X)) -> S(X)


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

S(mark(X)) -> S(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(S(x1))=  x1  
  POL(mark(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
           →DP Problem 17
             ↳Polo
             ...
               →DP Problem 18
                 ↳Dependency Graph
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polynomial Ordering
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pairs:

ZWADR(ok(X1), ok(X2)) -> ZWADR(X1, X2)
ZWADR(X1, mark(X2)) -> ZWADR(X1, X2)
ZWADR(mark(X1), X2) -> ZWADR(X1, X2)


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

ZWADR(ok(X1), ok(X2)) -> ZWADR(X1, X2)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ZWADR(x1, x2))=  x1  
  POL(mark(x1))=  x1  
  POL(ok(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
           →DP Problem 19
             ↳Polynomial Ordering
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pairs:

ZWADR(X1, mark(X2)) -> ZWADR(X1, X2)
ZWADR(mark(X1), X2) -> ZWADR(X1, X2)


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

ZWADR(mark(X1), X2) -> ZWADR(X1, X2)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ZWADR(x1, x2))=  x1  
  POL(mark(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
           →DP Problem 19
             ↳Polo
             ...
               →DP Problem 20
                 ↳Polynomial Ordering
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:

ZWADR(X1, mark(X2)) -> ZWADR(X1, X2)


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

ZWADR(X1, mark(X2)) -> ZWADR(X1, X2)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ZWADR(x1, x2))=  x2  
  POL(mark(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
           →DP Problem 19
             ↳Polo
             ...
               →DP Problem 21
                 ↳Dependency Graph
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polynomial Ordering
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pairs:

PREFIX(ok(X)) -> PREFIX(X)
PREFIX(mark(X)) -> PREFIX(X)


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

PREFIX(ok(X)) -> PREFIX(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(PREFIX(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(ok(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
           →DP Problem 22
             ↳Polynomial Ordering
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:

PREFIX(mark(X)) -> PREFIX(X)


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

PREFIX(mark(X)) -> PREFIX(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(PREFIX(x1))=  x1  
  POL(mark(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
           →DP Problem 22
             ↳Polo
             ...
               →DP Problem 23
                 ↳Dependency Graph
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polynomial Ordering
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pairs:

ACTIVE(prefix(X)) -> ACTIVE(X)
ACTIVE(zWadr(X1, X2)) -> ACTIVE(X2)
ACTIVE(zWadr(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(from(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(app(X1, X2)) -> ACTIVE(X2)
ACTIVE(app(X1, X2)) -> ACTIVE(X1)


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

ACTIVE(from(X)) -> ACTIVE(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  1 + x1  
  POL(ACTIVE(x1))=  x1  
  POL(cons(x1, x2))=  x1  
  POL(s(x1))=  x1  
  POL(prefix(x1))=  x1  
  POL(app(x1, x2))=  x1 + x2  
  POL(zWadr(x1, x2))=  x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
           →DP Problem 24
             ↳Polynomial Ordering
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pairs:

ACTIVE(prefix(X)) -> ACTIVE(X)
ACTIVE(zWadr(X1, X2)) -> ACTIVE(X2)
ACTIVE(zWadr(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(app(X1, X2)) -> ACTIVE(X2)
ACTIVE(app(X1, X2)) -> ACTIVE(X1)


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

ACTIVE(cons(X1, X2)) -> ACTIVE(X1)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(cons(x1, x2))=  1 + x1  
  POL(s(x1))=  x1  
  POL(prefix(x1))=  x1  
  POL(app(x1, x2))=  x1 + x2  
  POL(zWadr(x1, x2))=  x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
           →DP Problem 24
             ↳Polo
             ...
               →DP Problem 25
                 ↳Polynomial Ordering
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pairs:

ACTIVE(prefix(X)) -> ACTIVE(X)
ACTIVE(zWadr(X1, X2)) -> ACTIVE(X2)
ACTIVE(zWadr(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(app(X1, X2)) -> ACTIVE(X2)
ACTIVE(app(X1, X2)) -> ACTIVE(X1)


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pairs can be strictly oriented:

ACTIVE(app(X1, X2)) -> ACTIVE(X2)
ACTIVE(app(X1, X2)) -> ACTIVE(X1)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(s(x1))=  x1  
  POL(prefix(x1))=  x1  
  POL(app(x1, x2))=  1 + x1 + x2  
  POL(zWadr(x1, x2))=  x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
           →DP Problem 24
             ↳Polo
             ...
               →DP Problem 26
                 ↳Polynomial Ordering
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pairs:

ACTIVE(prefix(X)) -> ACTIVE(X)
ACTIVE(zWadr(X1, X2)) -> ACTIVE(X2)
ACTIVE(zWadr(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> ACTIVE(X)


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

ACTIVE(prefix(X)) -> ACTIVE(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(s(x1))=  x1  
  POL(prefix(x1))=  1 + x1  
  POL(zWadr(x1, x2))=  x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
           →DP Problem 24
             ↳Polo
             ...
               →DP Problem 27
                 ↳Polynomial Ordering
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pairs:

ACTIVE(zWadr(X1, X2)) -> ACTIVE(X2)
ACTIVE(zWadr(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> ACTIVE(X)


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pairs can be strictly oriented:

ACTIVE(zWadr(X1, X2)) -> ACTIVE(X2)
ACTIVE(zWadr(X1, X2)) -> ACTIVE(X1)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(s(x1))=  x1  
  POL(zWadr(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
           →DP Problem 24
             ↳Polo
             ...
               →DP Problem 28
                 ↳Polynomial Ordering
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:

ACTIVE(s(X)) -> ACTIVE(X)


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

ACTIVE(s(X)) -> ACTIVE(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(s(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
           →DP Problem 24
             ↳Polo
             ...
               →DP Problem 29
                 ↳Dependency Graph
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining


Dependency Pair:


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polynomial Ordering
       →DP Problem 9
         ↳Remaining


Dependency Pairs:

PROPER(prefix(X)) -> PROPER(X)
PROPER(zWadr(X1, X2)) -> PROPER(X2)
PROPER(zWadr(X1, X2)) -> PROPER(X1)
PROPER(s(X)) -> PROPER(X)
PROPER(from(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(app(X1, X2)) -> PROPER(X2)
PROPER(app(X1, X2)) -> PROPER(X1)


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

PROPER(from(X)) -> PROPER(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  1 + x1  
  POL(PROPER(x1))=  x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(s(x1))=  x1  
  POL(prefix(x1))=  x1  
  POL(app(x1, x2))=  x1 + x2  
  POL(zWadr(x1, x2))=  x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
           →DP Problem 30
             ↳Polynomial Ordering
       →DP Problem 9
         ↳Remaining


Dependency Pairs:

PROPER(prefix(X)) -> PROPER(X)
PROPER(zWadr(X1, X2)) -> PROPER(X2)
PROPER(zWadr(X1, X2)) -> PROPER(X1)
PROPER(s(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(app(X1, X2)) -> PROPER(X2)
PROPER(app(X1, X2)) -> PROPER(X1)


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pairs can be strictly oriented:

PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(PROPER(x1))=  x1  
  POL(cons(x1, x2))=  1 + x1 + x2  
  POL(s(x1))=  x1  
  POL(prefix(x1))=  x1  
  POL(app(x1, x2))=  x1 + x2  
  POL(zWadr(x1, x2))=  x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
           →DP Problem 30
             ↳Polo
             ...
               →DP Problem 31
                 ↳Polynomial Ordering
       →DP Problem 9
         ↳Remaining


Dependency Pairs:

PROPER(prefix(X)) -> PROPER(X)
PROPER(zWadr(X1, X2)) -> PROPER(X2)
PROPER(zWadr(X1, X2)) -> PROPER(X1)
PROPER(s(X)) -> PROPER(X)
PROPER(app(X1, X2)) -> PROPER(X2)
PROPER(app(X1, X2)) -> PROPER(X1)


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pairs can be strictly oriented:

PROPER(app(X1, X2)) -> PROPER(X2)
PROPER(app(X1, X2)) -> PROPER(X1)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(PROPER(x1))=  x1  
  POL(s(x1))=  x1  
  POL(prefix(x1))=  x1  
  POL(app(x1, x2))=  1 + x1 + x2  
  POL(zWadr(x1, x2))=  x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
           →DP Problem 30
             ↳Polo
             ...
               →DP Problem 32
                 ↳Polynomial Ordering
       →DP Problem 9
         ↳Remaining


Dependency Pairs:

PROPER(prefix(X)) -> PROPER(X)
PROPER(zWadr(X1, X2)) -> PROPER(X2)
PROPER(zWadr(X1, X2)) -> PROPER(X1)
PROPER(s(X)) -> PROPER(X)


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

PROPER(prefix(X)) -> PROPER(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(PROPER(x1))=  x1  
  POL(s(x1))=  x1  
  POL(prefix(x1))=  1 + x1  
  POL(zWadr(x1, x2))=  x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
           →DP Problem 30
             ↳Polo
             ...
               →DP Problem 33
                 ↳Polynomial Ordering
       →DP Problem 9
         ↳Remaining


Dependency Pairs:

PROPER(zWadr(X1, X2)) -> PROPER(X2)
PROPER(zWadr(X1, X2)) -> PROPER(X1)
PROPER(s(X)) -> PROPER(X)


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pairs can be strictly oriented:

PROPER(zWadr(X1, X2)) -> PROPER(X2)
PROPER(zWadr(X1, X2)) -> PROPER(X1)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(PROPER(x1))=  x1  
  POL(s(x1))=  x1  
  POL(zWadr(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
           →DP Problem 30
             ↳Polo
             ...
               →DP Problem 34
                 ↳Polynomial Ordering
       →DP Problem 9
         ↳Remaining


Dependency Pair:

PROPER(s(X)) -> PROPER(X)


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pair can be strictly oriented:

PROPER(s(X)) -> PROPER(X)


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(PROPER(x1))=  x1  
  POL(s(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
           →DP Problem 30
             ↳Polo
             ...
               →DP Problem 35
                 ↳Dependency Graph
       →DP Problem 9
         ↳Remaining


Dependency Pair:


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
       →DP Problem 5
         ↳Polo
       →DP Problem 6
         ↳Polo
       →DP Problem 7
         ↳Polo
       →DP Problem 8
         ↳Polo
       →DP Problem 9
         ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))


Rules:


active(app(nil, YS)) -> mark(YS)
active(app(cons(X, XS), YS)) -> mark(cons(X, app(XS, YS)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) -> mark(nil)
active(zWadr(XS, nil)) -> mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) -> mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) -> mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) -> app(active(X1), X2)
active(app(X1, X2)) -> app(X1, active(X2))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(zWadr(X1, X2)) -> zWadr(active(X1), X2)
active(zWadr(X1, X2)) -> zWadr(X1, active(X2))
active(prefix(X)) -> prefix(active(X))
app(mark(X1), X2) -> mark(app(X1, X2))
app(X1, mark(X2)) -> mark(app(X1, X2))
app(ok(X1), ok(X2)) -> ok(app(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
zWadr(mark(X1), X2) -> mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) -> mark(zWadr(X1, X2))
zWadr(ok(X1), ok(X2)) -> ok(zWadr(X1, X2))
prefix(mark(X)) -> mark(prefix(X))
prefix(ok(X)) -> ok(prefix(X))
proper(app(X1, X2)) -> app(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(zWadr(X1, X2)) -> zWadr(proper(X1), proper(X2))
proper(prefix(X)) -> prefix(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))




Termination of R could not be shown.
Duration:
0:01 minutes