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))

Innermost Termination of R to be shown. 



   R
     ↳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. 


   R
     ↳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:

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))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost w.r.t. to the AFS 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)


   R
     ↳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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳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:

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))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost w.r.t. to the AFS 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)


   R
     ↳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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳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:

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


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost w.r.t. to the AFS 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)


   R
     ↳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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳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:

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


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost w.r.t. to the AFS 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)


   R
     ↳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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳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:

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


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost w.r.t. to the AFS 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)


   R
     ↳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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳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:

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))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost w.r.t. to the AFS 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)


   R
     ↳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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳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))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost w.r.t. to the AFS 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)


   R
     ↳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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳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))


Strategy:

innermost




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 for innermost w.r.t. to the AFS 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)


   R
     ↳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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳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))


Strategy:

innermost




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 for innermost w.r.t. to the AFS 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)


   R
     ↳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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳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))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:07 minutes