Termination Proof using AProVE

Term Rewriting System R:
[X, XS, N, X1, X2]
active(from(X)) -> mark(cons(X, from(s(X))))
active(head(cons(X, XS))) -> mark(X)
active(2nd(cons(X, XS))) -> mark(head(XS))
active(take(0, XS)) -> mark(nil)
active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(2nd(X)) -> 2nd(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(2nd(X)) -> 2nd(proper(X))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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(from(X)) -> CONS(X, from(s(X)))
ACTIVE(from(X)) -> FROM(s(X))
ACTIVE(from(X)) -> S(X)
ACTIVE(2nd(cons(X, XS))) -> HEAD(XS)
ACTIVE(take(s(N), cons(X, XS))) -> CONS(X, take(N, XS))
ACTIVE(take(s(N), cons(X, XS))) -> TAKE(N, XS)
ACTIVE(sel(s(N), cons(X, XS))) -> SEL(N, XS)
ACTIVE(from(X)) -> FROM(active(X))
ACTIVE(from(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> S(active(X))
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(head(X)) -> HEAD(active(X))
ACTIVE(head(X)) -> ACTIVE(X)
ACTIVE(2nd(X)) -> 2ND(active(X))
ACTIVE(2nd(X)) -> ACTIVE(X)
ACTIVE(take(X1, X2)) -> TAKE(active(X1), X2)
ACTIVE(take(X1, X2)) -> ACTIVE(X1)
ACTIVE(take(X1, X2)) -> TAKE(X1, active(X2))
ACTIVE(take(X1, X2)) -> ACTIVE(X2)
ACTIVE(sel(X1, X2)) -> SEL(active(X1), X2)
ACTIVE(sel(X1, X2)) -> ACTIVE(X1)
ACTIVE(sel(X1, X2)) -> SEL(X1, active(X2))
ACTIVE(sel(X1, X2)) -> ACTIVE(X2)
FROM(mark(X)) -> FROM(X)
FROM(ok(X)) -> FROM(X)
CONS(mark(X1), X2) -> CONS(X1, X2)
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
S(mark(X)) -> S(X)
S(ok(X)) -> S(X)
HEAD(mark(X)) -> HEAD(X)
HEAD(ok(X)) -> HEAD(X)
2ND(mark(X)) -> 2ND(X)
2ND(ok(X)) -> 2ND(X)
TAKE(mark(X1), X2) -> TAKE(X1, X2)
TAKE(X1, mark(X2)) -> TAKE(X1, X2)
TAKE(ok(X1), ok(X2)) -> TAKE(X1, X2)
SEL(mark(X1), X2) -> SEL(X1, X2)
SEL(X1, mark(X2)) -> SEL(X1, X2)
SEL(ok(X1), ok(X2)) -> SEL(X1, X2)
PROPER(from(X)) -> FROM(proper(X))
PROPER(from(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> CONS(proper(X1), proper(X2))
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(s(X)) -> S(proper(X))
PROPER(s(X)) -> PROPER(X)
PROPER(head(X)) -> HEAD(proper(X))
PROPER(head(X)) -> PROPER(X)
PROPER(2nd(X)) -> 2ND(proper(X))
PROPER(2nd(X)) -> PROPER(X)
PROPER(take(X1, X2)) -> TAKE(proper(X1), proper(X2))
PROPER(take(X1, X2)) -> PROPER(X1)
PROPER(take(X1, X2)) -> PROPER(X2)
PROPER(sel(X1, X2)) -> SEL(proper(X1), proper(X2))
PROPER(sel(X1, X2)) -> PROPER(X1)
PROPER(sel(X1, X2)) -> PROPER(X2)
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 10 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

         ↳AFS

       →DP Problem 10

         ↳Remaining

Dependency Pairs:

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

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(head(cons(X, XS))) -> mark(X)
active(2nd(cons(X, XS))) -> mark(head(XS))
active(take(0, XS)) -> mark(nil)
active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(2nd(X)) -> 2nd(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(2nd(X)) -> 2nd(proper(X))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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 11

             ↳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

         ↳AFS

       →DP Problem 10

         ↳Remaining

Dependency Pair:

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(head(cons(X, XS))) -> mark(X)
active(2nd(cons(X, XS))) -> mark(head(XS))
active(take(0, XS)) -> mark(nil)
active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(2nd(X)) -> 2nd(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(2nd(X)) -> 2nd(proper(X))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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

         ↳AFS

       →DP Problem 10

         ↳Remaining

Dependency Pairs:

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

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(head(cons(X, XS))) -> mark(X)
active(2nd(cons(X, XS))) -> mark(head(XS))
active(take(0, XS)) -> mark(nil)
active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(2nd(X)) -> 2nd(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(2nd(X)) -> 2nd(proper(X))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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 12

             ↳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

         ↳AFS

       →DP Problem 10

         ↳Remaining

Dependency Pair:

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(head(cons(X, XS))) -> mark(X)
active(2nd(cons(X, XS))) -> mark(head(XS))
active(take(0, XS)) -> mark(nil)
active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(2nd(X)) -> 2nd(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(2nd(X)) -> 2nd(proper(X))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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

         ↳AFS

       →DP Problem 10

         ↳Remaining

Dependency Pairs:

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

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(head(cons(X, XS))) -> mark(X)
active(2nd(cons(X, XS))) -> mark(head(XS))
active(take(0, XS)) -> mark(nil)
active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(2nd(X)) -> 2nd(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(2nd(X)) -> 2nd(proper(X))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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 13

             ↳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

         ↳AFS

       →DP Problem 10

         ↳Remaining

Dependency Pair:

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(head(cons(X, XS))) -> mark(X)
active(2nd(cons(X, XS))) -> mark(head(XS))
active(take(0, XS)) -> mark(nil)
active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(2nd(X)) -> 2nd(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(2nd(X)) -> 2nd(proper(X))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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

         ↳AFS

       →DP Problem 10

         ↳Remaining

Dependency Pairs:

HEAD(ok(X)) -> HEAD(X)
HEAD(mark(X)) -> HEAD(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(head(cons(X, XS))) -> mark(X)
active(2nd(cons(X, XS))) -> mark(head(XS))
active(take(0, XS)) -> mark(nil)
active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(2nd(X)) -> 2nd(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(2nd(X)) -> 2nd(proper(X))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pairs can be strictly oriented:

HEAD(ok(X)) -> HEAD(X)
HEAD(mark(X)) -> HEAD(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:

HEAD(x₁) -> HEAD(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 14

             ↳Dependency Graph

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳Remaining

Dependency Pair:

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(head(cons(X, XS))) -> mark(X)
active(2nd(cons(X, XS))) -> mark(head(XS))
active(take(0, XS)) -> mark(nil)
active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(2nd(X)) -> 2nd(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(2nd(X)) -> 2nd(proper(X))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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

         ↳AFS

       →DP Problem 10

         ↳Remaining

Dependency Pairs:

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

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(head(cons(X, XS))) -> mark(X)
active(2nd(cons(X, XS))) -> mark(head(XS))
active(take(0, XS)) -> mark(nil)
active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(2nd(X)) -> 2nd(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(2nd(X)) -> 2nd(proper(X))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pairs can be strictly oriented:

TAKE(ok(X1), ok(X2)) -> TAKE(X1, X2)
TAKE(X1, mark(X2)) -> TAKE(X1, X2)
TAKE(mark(X1), X2) -> TAKE(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:

TAKE(x₁, x₂) -> TAKE(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 15

             ↳Dependency Graph

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳Remaining

Dependency Pair:

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(head(cons(X, XS))) -> mark(X)
active(2nd(cons(X, XS))) -> mark(head(XS))
active(take(0, XS)) -> mark(nil)
active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(2nd(X)) -> 2nd(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(2nd(X)) -> 2nd(proper(X))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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

         ↳AFS

       →DP Problem 10

         ↳Remaining

Dependency Pairs:

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

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(head(cons(X, XS))) -> mark(X)
active(2nd(cons(X, XS))) -> mark(head(XS))
active(take(0, XS)) -> mark(nil)
active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(2nd(X)) -> 2nd(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(2nd(X)) -> 2nd(proper(X))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pairs can be strictly oriented:

SEL(ok(X1), ok(X2)) -> SEL(X1, X2)
SEL(X1, mark(X2)) -> SEL(X1, X2)
SEL(mark(X1), X2) -> SEL(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:

SEL(x₁, x₂) -> SEL(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 6

         ↳AFS

           →DP Problem 16

             ↳Dependency Graph

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳Remaining

Dependency Pair:

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(head(cons(X, XS))) -> mark(X)
active(2nd(cons(X, XS))) -> mark(head(XS))
active(take(0, XS)) -> mark(nil)
active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(2nd(X)) -> 2nd(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(2nd(X)) -> 2nd(proper(X))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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

         ↳AFS

       →DP Problem 10

         ↳Remaining

Dependency Pairs:

2ND(ok(X)) -> 2ND(X)
2ND(mark(X)) -> 2ND(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(head(cons(X, XS))) -> mark(X)
active(2nd(cons(X, XS))) -> mark(head(XS))
active(take(0, XS)) -> mark(nil)
active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(2nd(X)) -> 2nd(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(2nd(X)) -> 2nd(proper(X))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pairs can be strictly oriented:

2ND(ok(X)) -> 2ND(X)
2ND(mark(X)) -> 2ND(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:

2ND(x₁) -> 2ND(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 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

           →DP Problem 17

             ↳Dependency Graph

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳Remaining

Dependency Pair:

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(head(cons(X, XS))) -> mark(X)
active(2nd(cons(X, XS))) -> mark(head(XS))
active(take(0, XS)) -> mark(nil)
active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(2nd(X)) -> 2nd(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(2nd(X)) -> 2nd(proper(X))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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

         ↳AFS

       →DP Problem 10

         ↳Remaining

Dependency Pairs:

ACTIVE(sel(X1, X2)) -> ACTIVE(X2)
ACTIVE(sel(X1, X2)) -> ACTIVE(X1)
ACTIVE(take(X1, X2)) -> ACTIVE(X2)
ACTIVE(take(X1, X2)) -> ACTIVE(X1)
ACTIVE(2nd(X)) -> ACTIVE(X)
ACTIVE(head(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(from(X)) -> ACTIVE(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(head(cons(X, XS))) -> mark(X)
active(2nd(cons(X, XS))) -> mark(head(XS))
active(take(0, XS)) -> mark(nil)
active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(2nd(X)) -> 2nd(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(2nd(X)) -> 2nd(proper(X))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pairs can be strictly oriented:

ACTIVE(sel(X1, X2)) -> ACTIVE(X2)
ACTIVE(sel(X1, X2)) -> ACTIVE(X1)
ACTIVE(take(X1, X2)) -> ACTIVE(X2)
ACTIVE(take(X1, X2)) -> ACTIVE(X1)
ACTIVE(2nd(X)) -> ACTIVE(X)
ACTIVE(head(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(from(X)) -> ACTIVE(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:

ACTIVE(x₁) -> ACTIVE(x₁)
s(x₁) -> s(x₁)
take(x₁, x₂) -> take(x₁, x₂)
from(x₁) -> from(x₁)
sel(x₁, x₂) -> sel(x₁, x₂)
cons(x₁, x₂) -> cons(x₁, x₂)
2nd(x₁) -> 2nd(x₁)
head(x₁) -> head(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 18

             ↳Dependency Graph

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳Remaining

Dependency Pair:

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(head(cons(X, XS))) -> mark(X)
active(2nd(cons(X, XS))) -> mark(head(XS))
active(take(0, XS)) -> mark(nil)
active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(2nd(X)) -> 2nd(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(2nd(X)) -> 2nd(proper(X))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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

         ↳Argument Filtering and Ordering

       →DP Problem 10

         ↳Remaining

Dependency Pairs:

PROPER(sel(X1, X2)) -> PROPER(X2)
PROPER(sel(X1, X2)) -> PROPER(X1)
PROPER(take(X1, X2)) -> PROPER(X2)
PROPER(take(X1, X2)) -> PROPER(X1)
PROPER(2nd(X)) -> PROPER(X)
PROPER(head(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(from(X)) -> PROPER(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(head(cons(X, XS))) -> mark(X)
active(2nd(cons(X, XS))) -> mark(head(XS))
active(take(0, XS)) -> mark(nil)
active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(2nd(X)) -> 2nd(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(2nd(X)) -> 2nd(proper(X))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pairs can be strictly oriented:

PROPER(sel(X1, X2)) -> PROPER(X2)
PROPER(sel(X1, X2)) -> PROPER(X1)
PROPER(take(X1, X2)) -> PROPER(X2)
PROPER(take(X1, X2)) -> PROPER(X1)
PROPER(2nd(X)) -> PROPER(X)
PROPER(head(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(from(X)) -> PROPER(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:

PROPER(x₁) -> PROPER(x₁)
head(x₁) -> head(x₁)
take(x₁, x₂) -> take(x₁, x₂)
sel(x₁, x₂) -> sel(x₁, x₂)
cons(x₁, x₂) -> cons(x₁, x₂)
from(x₁) -> from(x₁)
2nd(x₁) -> 2nd(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 9

         ↳AFS

           →DP Problem 19

             ↳Dependency Graph

       →DP Problem 10

         ↳Remaining

Dependency Pair:

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(head(cons(X, XS))) -> mark(X)
active(2nd(cons(X, XS))) -> mark(head(XS))
active(take(0, XS)) -> mark(nil)
active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(2nd(X)) -> 2nd(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(2nd(X)) -> 2nd(proper(X))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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

         ↳AFS

       →DP Problem 10

         ↳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(from(X)) -> mark(cons(X, from(s(X))))
active(head(cons(X, XS))) -> mark(X)
active(2nd(cons(X, XS))) -> mark(head(XS))
active(take(0, XS)) -> mark(nil)
active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(2nd(X)) -> 2nd(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(2nd(X)) -> 2nd(proper(X))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Termination of R could not be shown.
Duration:
0:05 minutes