Termination Proof using AProVE

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.

     ↳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.

     ↳DPs

       →DP Problem 1

         ↳Argument Filtering and Ordering

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →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 pairs can be strictly oriented:

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

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

CONS(x₁, x₂) -> CONS(x₁, x₂)
mark(x₁) -> mark(x₁)
ok(x₁) -> ok(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

           →DP Problem 10

             ↳Dependency Graph

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →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.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Argument Filtering and Ordering

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →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 pairs can be strictly oriented:

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

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

APP(x₁, x₂) -> APP(x₁, x₂)
ok(x₁) -> ok(x₁)
mark(x₁) -> mark(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

           →DP Problem 11

             ↳Dependency Graph

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →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.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Argument Filtering and Ordering

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →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 pairs can be strictly oriented:

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

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

FROM(x₁) -> FROM(x₁)
mark(x₁) -> mark(x₁)
ok(x₁) -> ok(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

           →DP Problem 12

             ↳Dependency Graph

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →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.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳Argument Filtering and Ordering

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →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 pairs can be strictly oriented:

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

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

S(x₁) -> S(x₁)
ok(x₁) -> ok(x₁)
mark(x₁) -> mark(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

           →DP Problem 13

             ↳Dependency Graph

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →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.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳Argument Filtering and Ordering

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →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 pairs can be strictly oriented:

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

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

ZWADR(x₁, x₂) -> ZWADR(x₁, x₂)
mark(x₁) -> mark(x₁)
ok(x₁) -> ok(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

           →DP Problem 14

             ↳Dependency Graph

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →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.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Argument Filtering and Ordering

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →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 pairs can be strictly oriented:

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

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

PREFIX(x₁) -> PREFIX(x₁)
mark(x₁) -> mark(x₁)
ok(x₁) -> ok(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

           →DP Problem 15

             ↳Dependency Graph

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →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.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳Argument Filtering and Ordering

       →DP Problem 8

         ↳AFS

       →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 pairs can be strictly oriented:

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)

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

ACTIVE(x₁) -> ACTIVE(x₁)
s(x₁) -> s(x₁)
app(x₁, x₂) -> app(x₁, x₂)
from(x₁) -> from(x₁)
cons(x₁, x₂) -> cons(x₁, x₂)
zWadr(x₁, x₂) -> zWadr(x₁, x₂)
prefix(x₁) -> prefix(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

           →DP Problem 16

             ↳Dependency Graph

       →DP Problem 8

         ↳AFS

       →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.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳Argument Filtering and 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 pairs can be strictly oriented:

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)

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

PROPER(x₁) -> PROPER(x₁)
app(x₁, x₂) -> app(x₁, x₂)
cons(x₁, x₂) -> cons(x₁, x₂)
zWadr(x₁, x₂) -> zWadr(x₁, x₂)
prefix(x₁) -> prefix(x₁)
from(x₁) -> from(x₁)
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

           →DP Problem 17

             ↳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.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →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