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
         ↳Size-Change Principle
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳AFS


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))





We number the DPs as follows:
  1. CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
  2. CONS(mark(X1), X2) -> CONS(X1, X2)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1
2>2
{2, 1} , {2, 1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1
2>2
{2, 1} , {2, 1}
1>1
2=2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳Size-Change Principle
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳AFS


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))





We number the DPs as follows:
  1. APP(ok(X1), ok(X2)) -> APP(X1, X2)
  2. APP(X1, mark(X2)) -> APP(X1, X2)
  3. APP(mark(X1), X2) -> APP(X1, X2)
and get the following Size-Change Graph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2=2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳Size-Change Principle
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳AFS


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))





We number the DPs as follows:
  1. FROM(ok(X)) -> FROM(X)
  2. FROM(mark(X)) -> FROM(X)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳Size-Change Principle
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳AFS


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))





We number the DPs as follows:
  1. S(ok(X)) -> S(X)
  2. S(mark(X)) -> S(X)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳Size-Change Principle
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳AFS


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))





We number the DPs as follows:
  1. ZWADR(ok(X1), ok(X2)) -> ZWADR(X1, X2)
  2. ZWADR(X1, mark(X2)) -> ZWADR(X1, X2)
  3. ZWADR(mark(X1), X2) -> ZWADR(X1, X2)
and get the following Size-Change Graph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2=2
{3, 2, 1} , {3, 2, 1}
1=1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳Size-Change Principle
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳AFS


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))





We number the DPs as follows:
  1. PREFIX(ok(X)) -> PREFIX(X)
  2. PREFIX(mark(X)) -> PREFIX(X)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳Size-Change Principle
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳AFS


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))





We number the DPs as follows:
  1. ACTIVE(prefix(X)) -> ACTIVE(X)
  2. ACTIVE(zWadr(X1, X2)) -> ACTIVE(X2)
  3. ACTIVE(zWadr(X1, X2)) -> ACTIVE(X1)
  4. ACTIVE(s(X)) -> ACTIVE(X)
  5. ACTIVE(from(X)) -> ACTIVE(X)
  6. ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
  7. ACTIVE(app(X1, X2)) -> ACTIVE(X2)
  8. ACTIVE(app(X1, X2)) -> ACTIVE(X1)
and get the following Size-Change Graph(s):
{8, 7, 6, 5, 4, 3, 2, 1} , {8, 7, 6, 5, 4, 3, 2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{8, 7, 6, 5, 4, 3, 2, 1} , {8, 7, 6, 5, 4, 3, 2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
from(x1) -> from(x1)
cons(x1, x2) -> cons(x1, x2)
s(x1) -> s(x1)
app(x1, x2) -> app(x1, x2)
prefix(x1) -> prefix(x1)
zWadr(x1, x2) -> zWadr(x1, x2)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳Size-Change Principle
       →DP Problem 9
         ↳AFS


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))





We number the DPs as follows:
  1. PROPER(prefix(X)) -> PROPER(X)
  2. PROPER(zWadr(X1, X2)) -> PROPER(X2)
  3. PROPER(zWadr(X1, X2)) -> PROPER(X1)
  4. PROPER(s(X)) -> PROPER(X)
  5. PROPER(from(X)) -> PROPER(X)
  6. PROPER(cons(X1, X2)) -> PROPER(X2)
  7. PROPER(cons(X1, X2)) -> PROPER(X1)
  8. PROPER(app(X1, X2)) -> PROPER(X2)
  9. PROPER(app(X1, X2)) -> PROPER(X1)
and get the following Size-Change Graph(s):
{9, 8, 7, 6, 5, 4, 3, 2, 1} , {9, 8, 7, 6, 5, 4, 3, 2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{9, 8, 7, 6, 5, 4, 3, 2, 1} , {9, 8, 7, 6, 5, 4, 3, 2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
from(x1) -> from(x1)
cons(x1, x2) -> cons(x1, x2)
s(x1) -> s(x1)
app(x1, x2) -> app(x1, x2)
prefix(x1) -> prefix(x1)
zWadr(x1, x2) -> zWadr(x1, x2)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳Argument Filtering and Ordering


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))





The following dependency pair can be strictly oriented:

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


The following usable rules w.r.t. the AFS can be oriented:

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))
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))
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))


Used ordering: Lexicographic Path Order with Precedence:
from > mark
zWadr > app > mark
prefix > mark
prefix > nil

resulting in one new DP problem.
Used Argument Filtering System:
TOP(x1) -> x1
ok(x1) -> x1
active(x1) -> x1
prefix(x1) -> prefix(x1)
cons(x1, x2) -> x1
zWadr(x1, x2) -> zWadr(x1, x2)
app(x1, x2) -> app(x1, x2)
mark(x1) -> mark(x1)
from(x1) -> from(x1)
s(x1) -> x1
proper(x1) -> x1


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳AFS
           →DP Problem 10
             ↳Negative Polynomial Order


Dependency Pair:

TOP(ok(X)) -> TOP(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 using the given order.

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


Moreover, the following usable rules (regarding the implicit AFS) are oriented.

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))


Used ordering:
Polynomial Order with Interpretation:

POL( TOP(x1) ) = x1

POL( ok(x1) ) = x1 + 1

POL( active(x1) ) = x1

POL( app(x1, x2) ) = x2

POL( mark(x1) ) = 0

POL( cons(x1, x2) ) = x2

POL( from(x1) ) = x1

POL( s(x1) ) = x1

POL( zWadr(x1, x2) ) = x2

POL( prefix(x1) ) = x1


This results in one new DP problem. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳AFS
           →DP Problem 10
             ↳Neg POLO
             ...
               →DP Problem 11
                 ↳Dependency Graph


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.

Termination of R successfully shown.
Duration:
0:11 minutes