Term Rewriting System R:
[X, XS, N, Y, YS, X1, X2]
active(from(X)) -> mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(minus(X, 0)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(quot(0, s(Y))) -> mark(0)
active(quot(s(X), s(Y))) -> mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) -> mark(nil)
active(zWquot(nil, XS)) -> mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) -> mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(minus(X1, X2)) -> minus(active(X1), X2)
active(minus(X1, X2)) -> minus(X1, active(X2))
active(quot(X1, X2)) -> quot(active(X1), X2)
active(quot(X1, X2)) -> quot(X1, active(X2))
active(zWquot(X1, X2)) -> zWquot(active(X1), X2)
active(zWquot(X1, X2)) -> zWquot(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))
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))
minus(mark(X1), X2) -> mark(minus(X1, X2))
minus(X1, mark(X2)) -> mark(minus(X1, X2))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
quot(mark(X1), X2) -> mark(quot(X1, X2))
quot(X1, mark(X2)) -> mark(quot(X1, X2))
quot(ok(X1), ok(X2)) -> ok(quot(X1, X2))
zWquot(mark(X1), X2) -> mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) -> mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) -> ok(zWquot(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(quot(X1, X2)) -> quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) -> zWquot(proper(X1), proper(X2))
proper(nil) -> ok(nil)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

ACTIVE(from(X)) -> CONS(X, from(s(X)))
ACTIVE(from(X)) -> FROM(s(X))
ACTIVE(from(X)) -> S(X)
ACTIVE(sel(s(N), cons(X, XS))) -> SEL(N, XS)
ACTIVE(minus(s(X), s(Y))) -> MINUS(X, Y)
ACTIVE(quot(s(X), s(Y))) -> S(quot(minus(X, Y), s(Y)))
ACTIVE(quot(s(X), s(Y))) -> QUOT(minus(X, Y), s(Y))
ACTIVE(quot(s(X), s(Y))) -> MINUS(X, Y)
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) -> CONS(quot(X, Y), zWquot(XS, YS))
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) -> QUOT(X, Y)
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) -> ZWQUOT(XS, YS)
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(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)
ACTIVE(minus(X1, X2)) -> MINUS(active(X1), X2)
ACTIVE(minus(X1, X2)) -> ACTIVE(X1)
ACTIVE(minus(X1, X2)) -> MINUS(X1, active(X2))
ACTIVE(minus(X1, X2)) -> ACTIVE(X2)
ACTIVE(quot(X1, X2)) -> QUOT(active(X1), X2)
ACTIVE(quot(X1, X2)) -> ACTIVE(X1)
ACTIVE(quot(X1, X2)) -> QUOT(X1, active(X2))
ACTIVE(quot(X1, X2)) -> ACTIVE(X2)
ACTIVE(zWquot(X1, X2)) -> ZWQUOT(active(X1), X2)
ACTIVE(zWquot(X1, X2)) -> ACTIVE(X1)
ACTIVE(zWquot(X1, X2)) -> ZWQUOT(X1, active(X2))
ACTIVE(zWquot(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)
SEL(mark(X1), X2) -> SEL(X1, X2)
SEL(X1, mark(X2)) -> SEL(X1, X2)
SEL(ok(X1), ok(X2)) -> SEL(X1, X2)
MINUS(mark(X1), X2) -> MINUS(X1, X2)
MINUS(X1, mark(X2)) -> MINUS(X1, X2)
MINUS(ok(X1), ok(X2)) -> MINUS(X1, X2)
QUOT(mark(X1), X2) -> QUOT(X1, X2)
QUOT(X1, mark(X2)) -> QUOT(X1, X2)
QUOT(ok(X1), ok(X2)) -> QUOT(X1, X2)
ZWQUOT(mark(X1), X2) -> ZWQUOT(X1, X2)
ZWQUOT(X1, mark(X2)) -> ZWQUOT(X1, X2)
ZWQUOT(ok(X1), ok(X2)) -> ZWQUOT(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(sel(X1, X2)) -> SEL(proper(X1), proper(X2))
PROPER(sel(X1, X2)) -> PROPER(X1)
PROPER(sel(X1, X2)) -> PROPER(X2)
PROPER(minus(X1, X2)) -> MINUS(proper(X1), proper(X2))
PROPER(minus(X1, X2)) -> PROPER(X1)
PROPER(minus(X1, X2)) -> PROPER(X2)
PROPER(quot(X1, X2)) -> QUOT(proper(X1), proper(X2))
PROPER(quot(X1, X2)) -> PROPER(X1)
PROPER(quot(X1, X2)) -> PROPER(X2)
PROPER(zWquot(X1, X2)) -> ZWQUOT(proper(X1), proper(X2))
PROPER(zWquot(X1, X2)) -> PROPER(X1)
PROPER(zWquot(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:

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(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(minus(X, 0)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(quot(0, s(Y))) -> mark(0)
active(quot(s(X), s(Y))) -> mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) -> mark(nil)
active(zWquot(nil, XS)) -> mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) -> mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(minus(X1, X2)) -> minus(active(X1), X2)
active(minus(X1, X2)) -> minus(X1, active(X2))
active(quot(X1, X2)) -> quot(active(X1), X2)
active(quot(X1, X2)) -> quot(X1, active(X2))
active(zWquot(X1, X2)) -> zWquot(active(X1), X2)
active(zWquot(X1, X2)) -> zWquot(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))
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))
minus(mark(X1), X2) -> mark(minus(X1, X2))
minus(X1, mark(X2)) -> mark(minus(X1, X2))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
quot(mark(X1), X2) -> mark(quot(X1, X2))
quot(X1, mark(X2)) -> mark(quot(X1, X2))
quot(ok(X1), ok(X2)) -> ok(quot(X1, X2))
zWquot(mark(X1), X2) -> mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) -> mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) -> ok(zWquot(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(quot(X1, X2)) -> quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) -> zWquot(proper(X1), proper(X2))
proper(nil) -> ok(nil)
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 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 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(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(minus(X, 0)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(quot(0, s(Y))) -> mark(0)
active(quot(s(X), s(Y))) -> mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) -> mark(nil)
active(zWquot(nil, XS)) -> mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) -> mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(minus(X1, X2)) -> minus(active(X1), X2)
active(minus(X1, X2)) -> minus(X1, active(X2))
active(quot(X1, X2)) -> quot(active(X1), X2)
active(quot(X1, X2)) -> quot(X1, active(X2))
active(zWquot(X1, X2)) -> zWquot(active(X1), X2)
active(zWquot(X1, X2)) -> zWquot(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))
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))
minus(mark(X1), X2) -> mark(minus(X1, X2))
minus(X1, mark(X2)) -> mark(minus(X1, X2))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
quot(mark(X1), X2) -> mark(quot(X1, X2))
quot(X1, mark(X2)) -> mark(quot(X1, X2))
quot(ok(X1), ok(X2)) -> ok(quot(X1, X2))
zWquot(mark(X1), X2) -> mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) -> mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) -> ok(zWquot(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(quot(X1, X2)) -> quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) -> zWquot(proper(X1), proper(X2))
proper(nil) -> ok(nil)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
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(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(minus(X, 0)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(quot(0, s(Y))) -> mark(0)
active(quot(s(X), s(Y))) -> mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) -> mark(nil)
active(zWquot(nil, XS)) -> mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) -> mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(minus(X1, X2)) -> minus(active(X1), X2)
active(minus(X1, X2)) -> minus(X1, active(X2))
active(quot(X1, X2)) -> quot(active(X1), X2)
active(quot(X1, X2)) -> quot(X1, active(X2))
active(zWquot(X1, X2)) -> zWquot(active(X1), X2)
active(zWquot(X1, X2)) -> zWquot(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))
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))
minus(mark(X1), X2) -> mark(minus(X1, X2))
minus(X1, mark(X2)) -> mark(minus(X1, X2))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
quot(mark(X1), X2) -> mark(quot(X1, X2))
quot(X1, mark(X2)) -> mark(quot(X1, X2))
quot(ok(X1), ok(X2)) -> ok(quot(X1, X2))
zWquot(mark(X1), X2) -> mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) -> mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) -> ok(zWquot(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(quot(X1, X2)) -> quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) -> zWquot(proper(X1), proper(X2))
proper(nil) -> ok(nil)
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 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 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(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(minus(X, 0)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(quot(0, s(Y))) -> mark(0)
active(quot(s(X), s(Y))) -> mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) -> mark(nil)
active(zWquot(nil, XS)) -> mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) -> mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(minus(X1, X2)) -> minus(active(X1), X2)
active(minus(X1, X2)) -> minus(X1, active(X2))
active(quot(X1, X2)) -> quot(active(X1), X2)
active(quot(X1, X2)) -> quot(X1, active(X2))
active(zWquot(X1, X2)) -> zWquot(active(X1), X2)
active(zWquot(X1, X2)) -> zWquot(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))
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))
minus(mark(X1), X2) -> mark(minus(X1, X2))
minus(X1, mark(X2)) -> mark(minus(X1, X2))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
quot(mark(X1), X2) -> mark(quot(X1, X2))
quot(X1, mark(X2)) -> mark(quot(X1, X2))
quot(ok(X1), ok(X2)) -> ok(quot(X1, X2))
zWquot(mark(X1), X2) -> mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) -> mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) -> ok(zWquot(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(quot(X1, X2)) -> quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) -> zWquot(proper(X1), proper(X2))
proper(nil) -> ok(nil)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
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(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(minus(X, 0)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(quot(0, s(Y))) -> mark(0)
active(quot(s(X), s(Y))) -> mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) -> mark(nil)
active(zWquot(nil, XS)) -> mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) -> mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(minus(X1, X2)) -> minus(active(X1), X2)
active(minus(X1, X2)) -> minus(X1, active(X2))
active(quot(X1, X2)) -> quot(active(X1), X2)
active(quot(X1, X2)) -> quot(X1, active(X2))
active(zWquot(X1, X2)) -> zWquot(active(X1), X2)
active(zWquot(X1, X2)) -> zWquot(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))
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))
minus(mark(X1), X2) -> mark(minus(X1, X2))
minus(X1, mark(X2)) -> mark(minus(X1, X2))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
quot(mark(X1), X2) -> mark(quot(X1, X2))
quot(X1, mark(X2)) -> mark(quot(X1, X2))
quot(ok(X1), ok(X2)) -> ok(quot(X1, X2))
zWquot(mark(X1), X2) -> mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) -> mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) -> ok(zWquot(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(quot(X1, X2)) -> quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) -> zWquot(proper(X1), proper(X2))
proper(nil) -> ok(nil)
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 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 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(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(minus(X, 0)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(quot(0, s(Y))) -> mark(0)
active(quot(s(X), s(Y))) -> mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) -> mark(nil)
active(zWquot(nil, XS)) -> mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) -> mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(minus(X1, X2)) -> minus(active(X1), X2)
active(minus(X1, X2)) -> minus(X1, active(X2))
active(quot(X1, X2)) -> quot(active(X1), X2)
active(quot(X1, X2)) -> quot(X1, active(X2))
active(zWquot(X1, X2)) -> zWquot(active(X1), X2)
active(zWquot(X1, X2)) -> zWquot(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))
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))
minus(mark(X1), X2) -> mark(minus(X1, X2))
minus(X1, mark(X2)) -> mark(minus(X1, X2))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
quot(mark(X1), X2) -> mark(quot(X1, X2))
quot(X1, mark(X2)) -> mark(quot(X1, X2))
quot(ok(X1), ok(X2)) -> ok(quot(X1, X2))
zWquot(mark(X1), X2) -> mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) -> mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) -> ok(zWquot(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(quot(X1, X2)) -> quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) -> zWquot(proper(X1), proper(X2))
proper(nil) -> ok(nil)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
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:

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(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(minus(X, 0)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(quot(0, s(Y))) -> mark(0)
active(quot(s(X), s(Y))) -> mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) -> mark(nil)
active(zWquot(nil, XS)) -> mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) -> mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(minus(X1, X2)) -> minus(active(X1), X2)
active(minus(X1, X2)) -> minus(X1, active(X2))
active(quot(X1, X2)) -> quot(active(X1), X2)
active(quot(X1, X2)) -> quot(X1, active(X2))
active(zWquot(X1, X2)) -> zWquot(active(X1), X2)
active(zWquot(X1, X2)) -> zWquot(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))
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))
minus(mark(X1), X2) -> mark(minus(X1, X2))
minus(X1, mark(X2)) -> mark(minus(X1, X2))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
quot(mark(X1), X2) -> mark(quot(X1, X2))
quot(X1, mark(X2)) -> mark(quot(X1, X2))
quot(ok(X1), ok(X2)) -> ok(quot(X1, X2))
zWquot(mark(X1), X2) -> mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) -> mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) -> ok(zWquot(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(quot(X1, X2)) -> quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) -> zWquot(proper(X1), proper(X2))
proper(nil) -> ok(nil)
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 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 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(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(minus(X, 0)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(quot(0, s(Y))) -> mark(0)
active(quot(s(X), s(Y))) -> mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) -> mark(nil)
active(zWquot(nil, XS)) -> mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) -> mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(minus(X1, X2)) -> minus(active(X1), X2)
active(minus(X1, X2)) -> minus(X1, active(X2))
active(quot(X1, X2)) -> quot(active(X1), X2)
active(quot(X1, X2)) -> quot(X1, active(X2))
active(zWquot(X1, X2)) -> zWquot(active(X1), X2)
active(zWquot(X1, X2)) -> zWquot(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))
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))
minus(mark(X1), X2) -> mark(minus(X1, X2))
minus(X1, mark(X2)) -> mark(minus(X1, X2))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
quot(mark(X1), X2) -> mark(quot(X1, X2))
quot(X1, mark(X2)) -> mark(quot(X1, X2))
quot(ok(X1), ok(X2)) -> ok(quot(X1, X2))
zWquot(mark(X1), X2) -> mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) -> mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) -> ok(zWquot(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(quot(X1, X2)) -> quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) -> zWquot(proper(X1), proper(X2))
proper(nil) -> ok(nil)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
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:

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


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(minus(X, 0)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(quot(0, s(Y))) -> mark(0)
active(quot(s(X), s(Y))) -> mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) -> mark(nil)
active(zWquot(nil, XS)) -> mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) -> mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(minus(X1, X2)) -> minus(active(X1), X2)
active(minus(X1, X2)) -> minus(X1, active(X2))
active(quot(X1, X2)) -> quot(active(X1), X2)
active(quot(X1, X2)) -> quot(X1, active(X2))
active(zWquot(X1, X2)) -> zWquot(active(X1), X2)
active(zWquot(X1, X2)) -> zWquot(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))
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))
minus(mark(X1), X2) -> mark(minus(X1, X2))
minus(X1, mark(X2)) -> mark(minus(X1, X2))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
quot(mark(X1), X2) -> mark(quot(X1, X2))
quot(X1, mark(X2)) -> mark(quot(X1, X2))
quot(ok(X1), ok(X2)) -> ok(quot(X1, X2))
zWquot(mark(X1), X2) -> mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) -> mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) -> ok(zWquot(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(quot(X1, X2)) -> quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) -> zWquot(proper(X1), proper(X2))
proper(nil) -> ok(nil)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pairs can be strictly oriented:

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


There are no usable rules 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:
MINUS(x1, x2) -> MINUS(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(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(minus(X, 0)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(quot(0, s(Y))) -> mark(0)
active(quot(s(X), s(Y))) -> mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) -> mark(nil)
active(zWquot(nil, XS)) -> mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) -> mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(minus(X1, X2)) -> minus(active(X1), X2)
active(minus(X1, X2)) -> minus(X1, active(X2))
active(quot(X1, X2)) -> quot(active(X1), X2)
active(quot(X1, X2)) -> quot(X1, active(X2))
active(zWquot(X1, X2)) -> zWquot(active(X1), X2)
active(zWquot(X1, X2)) -> zWquot(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))
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))
minus(mark(X1), X2) -> mark(minus(X1, X2))
minus(X1, mark(X2)) -> mark(minus(X1, X2))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
quot(mark(X1), X2) -> mark(quot(X1, X2))
quot(X1, mark(X2)) -> mark(quot(X1, X2))
quot(ok(X1), ok(X2)) -> ok(quot(X1, X2))
zWquot(mark(X1), X2) -> mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) -> mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) -> ok(zWquot(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(quot(X1, X2)) -> quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) -> zWquot(proper(X1), proper(X2))
proper(nil) -> ok(nil)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
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:

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


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(minus(X, 0)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(quot(0, s(Y))) -> mark(0)
active(quot(s(X), s(Y))) -> mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) -> mark(nil)
active(zWquot(nil, XS)) -> mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) -> mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(minus(X1, X2)) -> minus(active(X1), X2)
active(minus(X1, X2)) -> minus(X1, active(X2))
active(quot(X1, X2)) -> quot(active(X1), X2)
active(quot(X1, X2)) -> quot(X1, active(X2))
active(zWquot(X1, X2)) -> zWquot(active(X1), X2)
active(zWquot(X1, X2)) -> zWquot(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))
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))
minus(mark(X1), X2) -> mark(minus(X1, X2))
minus(X1, mark(X2)) -> mark(minus(X1, X2))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
quot(mark(X1), X2) -> mark(quot(X1, X2))
quot(X1, mark(X2)) -> mark(quot(X1, X2))
quot(ok(X1), ok(X2)) -> ok(quot(X1, X2))
zWquot(mark(X1), X2) -> mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) -> mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) -> ok(zWquot(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(quot(X1, X2)) -> quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) -> zWquot(proper(X1), proper(X2))
proper(nil) -> ok(nil)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pairs can be strictly oriented:

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


There are no usable rules 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:
QUOT(x1, x2) -> QUOT(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 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(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(minus(X, 0)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(quot(0, s(Y))) -> mark(0)
active(quot(s(X), s(Y))) -> mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) -> mark(nil)
active(zWquot(nil, XS)) -> mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) -> mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(minus(X1, X2)) -> minus(active(X1), X2)
active(minus(X1, X2)) -> minus(X1, active(X2))
active(quot(X1, X2)) -> quot(active(X1), X2)
active(quot(X1, X2)) -> quot(X1, active(X2))
active(zWquot(X1, X2)) -> zWquot(active(X1), X2)
active(zWquot(X1, X2)) -> zWquot(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))
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))
minus(mark(X1), X2) -> mark(minus(X1, X2))
minus(X1, mark(X2)) -> mark(minus(X1, X2))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
quot(mark(X1), X2) -> mark(quot(X1, X2))
quot(X1, mark(X2)) -> mark(quot(X1, X2))
quot(ok(X1), ok(X2)) -> ok(quot(X1, X2))
zWquot(mark(X1), X2) -> mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) -> mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) -> ok(zWquot(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(quot(X1, X2)) -> quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) -> zWquot(proper(X1), proper(X2))
proper(nil) -> ok(nil)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
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:

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


Rules:


active(from(X)) -> mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(minus(X, 0)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(quot(0, s(Y))) -> mark(0)
active(quot(s(X), s(Y))) -> mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) -> mark(nil)
active(zWquot(nil, XS)) -> mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) -> mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(minus(X1, X2)) -> minus(active(X1), X2)
active(minus(X1, X2)) -> minus(X1, active(X2))
active(quot(X1, X2)) -> quot(active(X1), X2)
active(quot(X1, X2)) -> quot(X1, active(X2))
active(zWquot(X1, X2)) -> zWquot(active(X1), X2)
active(zWquot(X1, X2)) -> zWquot(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))
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))
minus(mark(X1), X2) -> mark(minus(X1, X2))
minus(X1, mark(X2)) -> mark(minus(X1, X2))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
quot(mark(X1), X2) -> mark(quot(X1, X2))
quot(X1, mark(X2)) -> mark(quot(X1, X2))
quot(ok(X1), ok(X2)) -> ok(quot(X1, X2))
zWquot(mark(X1), X2) -> mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) -> mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) -> ok(zWquot(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(quot(X1, X2)) -> quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) -> zWquot(proper(X1), proper(X2))
proper(nil) -> ok(nil)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pairs can be strictly oriented:

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


There are no usable rules 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:
ZWQUOT(x1, x2) -> ZWQUOT(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 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(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(minus(X, 0)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(quot(0, s(Y))) -> mark(0)
active(quot(s(X), s(Y))) -> mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) -> mark(nil)
active(zWquot(nil, XS)) -> mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) -> mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(minus(X1, X2)) -> minus(active(X1), X2)
active(minus(X1, X2)) -> minus(X1, active(X2))
active(quot(X1, X2)) -> quot(active(X1), X2)
active(quot(X1, X2)) -> quot(X1, active(X2))
active(zWquot(X1, X2)) -> zWquot(active(X1), X2)
active(zWquot(X1, X2)) -> zWquot(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))
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))
minus(mark(X1), X2) -> mark(minus(X1, X2))
minus(X1, mark(X2)) -> mark(minus(X1, X2))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
quot(mark(X1), X2) -> mark(quot(X1, X2))
quot(X1, mark(X2)) -> mark(quot(X1, X2))
quot(ok(X1), ok(X2)) -> ok(quot(X1, X2))
zWquot(mark(X1), X2) -> mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) -> mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) -> ok(zWquot(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(quot(X1, X2)) -> quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) -> zWquot(proper(X1), proper(X2))
proper(nil) -> ok(nil)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
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(zWquot(X1, X2)) -> ACTIVE(X2)
ACTIVE(zWquot(X1, X2)) -> ACTIVE(X1)
ACTIVE(quot(X1, X2)) -> ACTIVE(X2)
ACTIVE(quot(X1, X2)) -> ACTIVE(X1)
ACTIVE(minus(X1, X2)) -> ACTIVE(X2)
ACTIVE(minus(X1, X2)) -> ACTIVE(X1)
ACTIVE(sel(X1, X2)) -> ACTIVE(X2)
ACTIVE(sel(X1, X2)) -> ACTIVE(X1)
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(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(minus(X, 0)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(quot(0, s(Y))) -> mark(0)
active(quot(s(X), s(Y))) -> mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) -> mark(nil)
active(zWquot(nil, XS)) -> mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) -> mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(minus(X1, X2)) -> minus(active(X1), X2)
active(minus(X1, X2)) -> minus(X1, active(X2))
active(quot(X1, X2)) -> quot(active(X1), X2)
active(quot(X1, X2)) -> quot(X1, active(X2))
active(zWquot(X1, X2)) -> zWquot(active(X1), X2)
active(zWquot(X1, X2)) -> zWquot(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))
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))
minus(mark(X1), X2) -> mark(minus(X1, X2))
minus(X1, mark(X2)) -> mark(minus(X1, X2))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
quot(mark(X1), X2) -> mark(quot(X1, X2))
quot(X1, mark(X2)) -> mark(quot(X1, X2))
quot(ok(X1), ok(X2)) -> ok(quot(X1, X2))
zWquot(mark(X1), X2) -> mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) -> mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) -> ok(zWquot(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(quot(X1, X2)) -> quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) -> zWquot(proper(X1), proper(X2))
proper(nil) -> ok(nil)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pairs can be strictly oriented:

ACTIVE(zWquot(X1, X2)) -> ACTIVE(X2)
ACTIVE(zWquot(X1, X2)) -> ACTIVE(X1)
ACTIVE(quot(X1, X2)) -> ACTIVE(X2)
ACTIVE(quot(X1, X2)) -> ACTIVE(X1)
ACTIVE(minus(X1, X2)) -> ACTIVE(X2)
ACTIVE(minus(X1, X2)) -> ACTIVE(X1)
ACTIVE(sel(X1, X2)) -> ACTIVE(X2)
ACTIVE(sel(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(from(X)) -> ACTIVE(X)


There are no usable rules 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)
quot(x1, x2) -> quot(x1, x2)
from(x1) -> from(x1)
zWquot(x1, x2) -> zWquot(x1, x2)
sel(x1, x2) -> sel(x1, x2)
cons(x1, x2) -> cons(x1, x2)
minus(x1, x2) -> minus(x1, x2)


   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(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(minus(X, 0)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(quot(0, s(Y))) -> mark(0)
active(quot(s(X), s(Y))) -> mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) -> mark(nil)
active(zWquot(nil, XS)) -> mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) -> mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(minus(X1, X2)) -> minus(active(X1), X2)
active(minus(X1, X2)) -> minus(X1, active(X2))
active(quot(X1, X2)) -> quot(active(X1), X2)
active(quot(X1, X2)) -> quot(X1, active(X2))
active(zWquot(X1, X2)) -> zWquot(active(X1), X2)
active(zWquot(X1, X2)) -> zWquot(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))
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))
minus(mark(X1), X2) -> mark(minus(X1, X2))
minus(X1, mark(X2)) -> mark(minus(X1, X2))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
quot(mark(X1), X2) -> mark(quot(X1, X2))
quot(X1, mark(X2)) -> mark(quot(X1, X2))
quot(ok(X1), ok(X2)) -> ok(quot(X1, X2))
zWquot(mark(X1), X2) -> mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) -> mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) -> ok(zWquot(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(quot(X1, X2)) -> quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) -> zWquot(proper(X1), proper(X2))
proper(nil) -> ok(nil)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
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(zWquot(X1, X2)) -> PROPER(X2)
PROPER(zWquot(X1, X2)) -> PROPER(X1)
PROPER(quot(X1, X2)) -> PROPER(X2)
PROPER(quot(X1, X2)) -> PROPER(X1)
PROPER(minus(X1, X2)) -> PROPER(X2)
PROPER(minus(X1, X2)) -> PROPER(X1)
PROPER(sel(X1, X2)) -> PROPER(X2)
PROPER(sel(X1, X2)) -> PROPER(X1)
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(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(minus(X, 0)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(quot(0, s(Y))) -> mark(0)
active(quot(s(X), s(Y))) -> mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) -> mark(nil)
active(zWquot(nil, XS)) -> mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) -> mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(minus(X1, X2)) -> minus(active(X1), X2)
active(minus(X1, X2)) -> minus(X1, active(X2))
active(quot(X1, X2)) -> quot(active(X1), X2)
active(quot(X1, X2)) -> quot(X1, active(X2))
active(zWquot(X1, X2)) -> zWquot(active(X1), X2)
active(zWquot(X1, X2)) -> zWquot(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))
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))
minus(mark(X1), X2) -> mark(minus(X1, X2))
minus(X1, mark(X2)) -> mark(minus(X1, X2))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
quot(mark(X1), X2) -> mark(quot(X1, X2))
quot(X1, mark(X2)) -> mark(quot(X1, X2))
quot(ok(X1), ok(X2)) -> ok(quot(X1, X2))
zWquot(mark(X1), X2) -> mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) -> mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) -> ok(zWquot(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(quot(X1, X2)) -> quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) -> zWquot(proper(X1), proper(X2))
proper(nil) -> ok(nil)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following dependency pairs can be strictly oriented:

PROPER(zWquot(X1, X2)) -> PROPER(X2)
PROPER(zWquot(X1, X2)) -> PROPER(X1)
PROPER(quot(X1, X2)) -> PROPER(X2)
PROPER(quot(X1, X2)) -> PROPER(X1)
PROPER(minus(X1, X2)) -> PROPER(X2)
PROPER(minus(X1, X2)) -> PROPER(X1)
PROPER(sel(X1, X2)) -> PROPER(X2)
PROPER(sel(X1, X2)) -> PROPER(X1)
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 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)
sel(x1, x2) -> sel(x1, x2)
zWquot(x1, x2) -> zWquot(x1, x2)
cons(x1, x2) -> cons(x1, x2)
quot(x1, x2) -> quot(x1, x2)
from(x1) -> from(x1)
minus(x1, x2) -> minus(x1, x2)
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(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(minus(X, 0)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(quot(0, s(Y))) -> mark(0)
active(quot(s(X), s(Y))) -> mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) -> mark(nil)
active(zWquot(nil, XS)) -> mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) -> mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(minus(X1, X2)) -> minus(active(X1), X2)
active(minus(X1, X2)) -> minus(X1, active(X2))
active(quot(X1, X2)) -> quot(active(X1), X2)
active(quot(X1, X2)) -> quot(X1, active(X2))
active(zWquot(X1, X2)) -> zWquot(active(X1), X2)
active(zWquot(X1, X2)) -> zWquot(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))
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))
minus(mark(X1), X2) -> mark(minus(X1, X2))
minus(X1, mark(X2)) -> mark(minus(X1, X2))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
quot(mark(X1), X2) -> mark(quot(X1, X2))
quot(X1, mark(X2)) -> mark(quot(X1, X2))
quot(ok(X1), ok(X2)) -> ok(quot(X1, X2))
zWquot(mark(X1), X2) -> mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) -> mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) -> ok(zWquot(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(quot(X1, X2)) -> quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) -> zWquot(proper(X1), proper(X2))
proper(nil) -> ok(nil)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
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(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(minus(X, 0)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(quot(0, s(Y))) -> mark(0)
active(quot(s(X), s(Y))) -> mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) -> mark(nil)
active(zWquot(nil, XS)) -> mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) -> mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(minus(X1, X2)) -> minus(active(X1), X2)
active(minus(X1, X2)) -> minus(X1, active(X2))
active(quot(X1, X2)) -> quot(active(X1), X2)
active(quot(X1, X2)) -> quot(X1, active(X2))
active(zWquot(X1, X2)) -> zWquot(active(X1), X2)
active(zWquot(X1, X2)) -> zWquot(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))
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))
minus(mark(X1), X2) -> mark(minus(X1, X2))
minus(X1, mark(X2)) -> mark(minus(X1, X2))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
quot(mark(X1), X2) -> mark(quot(X1, X2))
quot(X1, mark(X2)) -> mark(quot(X1, X2))
quot(ok(X1), ok(X2)) -> ok(quot(X1, X2))
zWquot(mark(X1), X2) -> mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) -> mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) -> ok(zWquot(X1, X2))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(quot(X1, X2)) -> quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) -> zWquot(proper(X1), proper(X2))
proper(nil) -> ok(nil)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))




Termination of R could not be shown.
Duration:
0:17 minutes