Term Rewriting System R:
[X, L, X1, X2]
active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(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))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(X))
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(incr(cons(X, L))) -> CONS(s(X), incr(L))
ACTIVE(incr(cons(X, L))) -> S(X)
ACTIVE(incr(cons(X, L))) -> INCR(L)
ACTIVE(zeros) -> CONS(0, zeros)
ACTIVE(incr(X)) -> INCR(active(X))
ACTIVE(incr(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(tail(X)) -> TAIL(active(X))
ACTIVE(tail(X)) -> ACTIVE(X)
INCR(mark(X)) -> INCR(X)
INCR(ok(X)) -> INCR(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)
TAIL(mark(X)) -> TAIL(X)
TAIL(ok(X)) -> TAIL(X)
PROPER(incr(X)) -> INCR(proper(X))
PROPER(incr(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(tail(X)) -> TAIL(proper(X))
PROPER(tail(X)) -> PROPER(X)
TOP(mark(X)) -> TOP(proper(X))
TOP(mark(X)) -> PROPER(X)
TOP(ok(X)) -> TOP(active(X))
TOP(ok(X)) -> ACTIVE(X)

Furthermore, R contains nine SCCs.

`   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`
`         ↳Remaining`

Dependency Pairs:

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

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(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))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pairs can be strictly oriented:

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

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
CONS(x1, x2) -> CONS(x1, x2)
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 10`
`             ↳Dependency Graph`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳AFS`
`       →DP Problem 5`
`         ↳AFS`
`       →DP Problem 6`
`         ↳AFS`
`       →DP Problem 7`
`         ↳AFS`
`       →DP Problem 8`
`         ↳AFS`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pair:

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(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))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(X))
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`
`         ↳Remaining`

Dependency Pairs:

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

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(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))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pairs can be strictly oriented:

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

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
S(x1) -> S(x1)
ok(x1) -> ok(x1)
mark(x1) -> mark(x1)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`           →DP Problem 11`
`             ↳Dependency Graph`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳AFS`
`       →DP Problem 5`
`         ↳AFS`
`       →DP Problem 6`
`         ↳AFS`
`       →DP Problem 7`
`         ↳AFS`
`       →DP Problem 8`
`         ↳AFS`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pair:

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(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))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(X))
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`
`         ↳Remaining`

Dependency Pairs:

INCR(ok(X)) -> INCR(X)
INCR(mark(X)) -> INCR(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(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))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pairs can be strictly oriented:

INCR(ok(X)) -> INCR(X)
INCR(mark(X)) -> INCR(X)

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
INCR(x1) -> INCR(x1)
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`           →DP Problem 12`
`             ↳Dependency Graph`
`       →DP Problem 4`
`         ↳AFS`
`       →DP Problem 5`
`         ↳AFS`
`       →DP Problem 6`
`         ↳AFS`
`       →DP Problem 7`
`         ↳AFS`
`       →DP Problem 8`
`         ↳AFS`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pair:

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(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))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(X))
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`
`         ↳Remaining`

Dependency Pairs:

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(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))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pairs can be strictly oriented:

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
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 13`
`             ↳Dependency Graph`
`       →DP Problem 5`
`         ↳AFS`
`       →DP Problem 6`
`         ↳AFS`
`       →DP Problem 7`
`         ↳AFS`
`       →DP Problem 8`
`         ↳AFS`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pair:

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(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))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(X))
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`
`         ↳Remaining`

Dependency Pairs:

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(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))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pairs can be strictly oriented:

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
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 14`
`             ↳Dependency Graph`
`       →DP Problem 6`
`         ↳AFS`
`       →DP Problem 7`
`         ↳AFS`
`       →DP Problem 8`
`         ↳AFS`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pair:

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(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))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(X))
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`
`         ↳Remaining`

Dependency Pairs:

TAIL(ok(X)) -> TAIL(X)
TAIL(mark(X)) -> TAIL(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(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))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pairs can be strictly oriented:

TAIL(ok(X)) -> TAIL(X)
TAIL(mark(X)) -> TAIL(X)

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
TAIL(x1) -> TAIL(x1)
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 15`
`             ↳Dependency Graph`
`       →DP Problem 7`
`         ↳AFS`
`       →DP Problem 8`
`         ↳AFS`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pair:

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(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))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(X))
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`
`         ↳Remaining`

Dependency Pairs:

ACTIVE(tail(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(incr(X)) -> ACTIVE(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(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))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pairs can be strictly oriented:

ACTIVE(tail(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(incr(X)) -> ACTIVE(X)

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
ACTIVE(x1) -> ACTIVE(x1)
s(x1) -> s(x1)
incr(x1) -> incr(x1)
tail(x1) -> tail(x1)
cons(x1, x2) -> cons(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 16`
`             ↳Dependency Graph`
`       →DP Problem 8`
`         ↳AFS`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pair:

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(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))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(X))
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`
`         ↳Remaining`

Dependency Pairs:

PROPER(tail(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(incr(X)) -> PROPER(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(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))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pairs can be strictly oriented:

PROPER(tail(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(incr(X)) -> PROPER(X)

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
PROPER(x1) -> PROPER(x1)
cons(x1, x2) -> cons(x1, x2)
incr(x1) -> incr(x1)
tail(x1) -> tail(x1)
s(x1) -> s(x1)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳AFS`
`       →DP Problem 5`
`         ↳AFS`
`       →DP Problem 6`
`         ↳AFS`
`       →DP Problem 7`
`         ↳AFS`
`       →DP Problem 8`
`         ↳AFS`
`           →DP Problem 17`
`             ↳Dependency Graph`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pair:

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(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))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(X))
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`
`         ↳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(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(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))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)