Termination Proof using AProVE

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))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

ACTIVE(from(X)) -> CONS(X, from(s(X)))
ACTIVE(from(X)) -> FROM(s(X))
ACTIVE(from(X)) -> S(X)
ACTIVE(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.

     ↳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(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))

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 that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

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

     ↳DPs

       →DP Problem 1

         ↳AFS

           →DP Problem 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))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Argument Filtering and Ordering

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳Remaining

Dependency Pairs:

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(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))

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 that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

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

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

           →DP Problem 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))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Argument Filtering and Ordering

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳Remaining

Dependency Pairs:

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

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(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))

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 that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

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

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

           →DP Problem 13

             ↳Dependency Graph

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳Remaining

Dependency Pair:

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(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))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳Argument Filtering and Ordering

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳Remaining

Dependency Pairs:

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))

Strategy:

innermost

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 for innermost 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)

     ↳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))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳Argument Filtering and Ordering

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳Remaining

Dependency Pairs:

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))

Strategy:

innermost

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 for innermost that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

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

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 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))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Argument Filtering and Ordering

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳Remaining

Dependency Pairs:

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))

Strategy:

innermost

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 for innermost 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)

     ↳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))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳Argument Filtering and Ordering

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳Remaining

Dependency Pairs:

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))

Strategy:

innermost

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 for innermost 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)

     ↳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))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳Argument Filtering and Ordering

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳Remaining

Dependency Pairs:

ACTIVE(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))

Strategy:

innermost

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 for innermost 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)

     ↳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))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳Argument Filtering and Ordering

       →DP Problem 10

         ↳Remaining

Dependency Pairs:

PROPER(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))

Strategy:

innermost

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 for innermost 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)

     ↳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))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳AFS

       →DP Problem 10

         ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

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

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(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))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:20 minutes