Termination Proof using AProVE

Term Rewriting System R:
[Z, X, Y, X1, X2]
active(fst(0, Z)) -> mark(nil)
active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(len(nil)) -> mark(0)
active(len(cons(X, Z))) -> mark(s(len(Z)))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(fst(X1, X2)) -> fst(active(X1), X2)
active(fst(X1, X2)) -> fst(X1, active(X2))
active(from(X)) -> from(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(len(X)) -> len(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
fst(mark(X1), X2) -> mark(fst(X1, X2))
fst(X1, mark(X2)) -> mark(fst(X1, X2))
fst(ok(X1), ok(X2)) -> ok(fst(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
len(mark(X)) -> mark(len(X))
len(ok(X)) -> ok(len(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(fst(X1, X2)) -> fst(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(len(X)) -> len(proper(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

ACTIVE(fst(s(X), cons(Y, Z))) -> CONS(Y, fst(X, Z))
ACTIVE(fst(s(X), cons(Y, Z))) -> FST(X, Z)
ACTIVE(from(X)) -> CONS(X, from(s(X)))
ACTIVE(from(X)) -> FROM(s(X))
ACTIVE(from(X)) -> S(X)
ACTIVE(add(s(X), Y)) -> S(add(X, Y))
ACTIVE(add(s(X), Y)) -> ADD(X, Y)
ACTIVE(len(cons(X, Z))) -> S(len(Z))
ACTIVE(len(cons(X, Z))) -> LEN(Z)
ACTIVE(cons(X1, X2)) -> CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(fst(X1, X2)) -> FST(active(X1), X2)
ACTIVE(fst(X1, X2)) -> ACTIVE(X1)
ACTIVE(fst(X1, X2)) -> FST(X1, active(X2))
ACTIVE(fst(X1, X2)) -> ACTIVE(X2)
ACTIVE(from(X)) -> FROM(active(X))
ACTIVE(from(X)) -> ACTIVE(X)
ACTIVE(add(X1, X2)) -> ADD(active(X1), X2)
ACTIVE(add(X1, X2)) -> ACTIVE(X1)
ACTIVE(add(X1, X2)) -> ADD(X1, active(X2))
ACTIVE(add(X1, X2)) -> ACTIVE(X2)
ACTIVE(len(X)) -> LEN(active(X))
ACTIVE(len(X)) -> ACTIVE(X)
CONS(mark(X1), X2) -> CONS(X1, X2)
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
FST(mark(X1), X2) -> FST(X1, X2)
FST(X1, mark(X2)) -> FST(X1, X2)
FST(ok(X1), ok(X2)) -> FST(X1, X2)
FROM(mark(X)) -> FROM(X)
FROM(ok(X)) -> FROM(X)
ADD(mark(X1), X2) -> ADD(X1, X2)
ADD(X1, mark(X2)) -> ADD(X1, X2)
ADD(ok(X1), ok(X2)) -> ADD(X1, X2)
LEN(mark(X)) -> LEN(X)
LEN(ok(X)) -> LEN(X)
PROPER(s(X)) -> S(proper(X))
PROPER(s(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(fst(X1, X2)) -> FST(proper(X1), proper(X2))
PROPER(fst(X1, X2)) -> PROPER(X1)
PROPER(fst(X1, X2)) -> PROPER(X2)
PROPER(from(X)) -> FROM(proper(X))
PROPER(from(X)) -> PROPER(X)
PROPER(add(X1, X2)) -> ADD(proper(X1), proper(X2))
PROPER(add(X1, X2)) -> PROPER(X1)
PROPER(add(X1, X2)) -> PROPER(X2)
PROPER(len(X)) -> LEN(proper(X))
PROPER(len(X)) -> PROPER(X)
S(ok(X)) -> S(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.

     ↳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(fst(0, Z)) -> mark(nil)
active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(len(nil)) -> mark(0)
active(len(cons(X, Z))) -> mark(s(len(Z)))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(fst(X1, X2)) -> fst(active(X1), X2)
active(fst(X1, X2)) -> fst(X1, active(X2))
active(from(X)) -> from(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(len(X)) -> len(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
fst(mark(X1), X2) -> mark(fst(X1, X2))
fst(X1, mark(X2)) -> mark(fst(X1, X2))
fst(ok(X1), ok(X2)) -> ok(fst(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
len(mark(X)) -> mark(len(X))
len(ok(X)) -> ok(len(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(fst(X1, X2)) -> fst(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(len(X)) -> len(proper(X))
s(ok(X)) -> ok(s(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 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)

     ↳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(fst(0, Z)) -> mark(nil)
active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(len(nil)) -> mark(0)
active(len(cons(X, Z))) -> mark(s(len(Z)))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(fst(X1, X2)) -> fst(active(X1), X2)
active(fst(X1, X2)) -> fst(X1, active(X2))
active(from(X)) -> from(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(len(X)) -> len(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
fst(mark(X1), X2) -> mark(fst(X1, X2))
fst(X1, mark(X2)) -> mark(fst(X1, X2))
fst(ok(X1), ok(X2)) -> ok(fst(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
len(mark(X)) -> mark(len(X))
len(ok(X)) -> ok(len(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(fst(X1, X2)) -> fst(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(len(X)) -> len(proper(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Argument Filtering and Ordering

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳Remaining

Dependency Pairs:

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

Rules:

active(fst(0, Z)) -> mark(nil)
active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(len(nil)) -> mark(0)
active(len(cons(X, Z))) -> mark(s(len(Z)))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(fst(X1, X2)) -> fst(active(X1), X2)
active(fst(X1, X2)) -> fst(X1, active(X2))
active(from(X)) -> from(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(len(X)) -> len(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
fst(mark(X1), X2) -> mark(fst(X1, X2))
fst(X1, mark(X2)) -> mark(fst(X1, X2))
fst(ok(X1), ok(X2)) -> ok(fst(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
len(mark(X)) -> mark(len(X))
len(ok(X)) -> ok(len(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(fst(X1, X2)) -> fst(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(len(X)) -> len(proper(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pairs can be strictly oriented:

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

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

     ↳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(fst(0, Z)) -> mark(nil)
active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(len(nil)) -> mark(0)
active(len(cons(X, Z))) -> mark(s(len(Z)))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(fst(X1, X2)) -> fst(active(X1), X2)
active(fst(X1, X2)) -> fst(X1, active(X2))
active(from(X)) -> from(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(len(X)) -> len(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
fst(mark(X1), X2) -> mark(fst(X1, X2))
fst(X1, mark(X2)) -> mark(fst(X1, X2))
fst(ok(X1), ok(X2)) -> ok(fst(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
len(mark(X)) -> mark(len(X))
len(ok(X)) -> ok(len(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(fst(X1, X2)) -> fst(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(len(X)) -> len(proper(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Argument Filtering and Ordering

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳Remaining

Dependency Pairs:

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

Rules:

active(fst(0, Z)) -> mark(nil)
active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(len(nil)) -> mark(0)
active(len(cons(X, Z))) -> mark(s(len(Z)))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(fst(X1, X2)) -> fst(active(X1), X2)
active(fst(X1, X2)) -> fst(X1, active(X2))
active(from(X)) -> from(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(len(X)) -> len(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
fst(mark(X1), X2) -> mark(fst(X1, X2))
fst(X1, mark(X2)) -> mark(fst(X1, X2))
fst(ok(X1), ok(X2)) -> ok(fst(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
len(mark(X)) -> mark(len(X))
len(ok(X)) -> ok(len(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(fst(X1, X2)) -> fst(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(len(X)) -> len(proper(X))
s(ok(X)) -> ok(s(X))
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)

     ↳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(fst(0, Z)) -> mark(nil)
active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(len(nil)) -> mark(0)
active(len(cons(X, Z))) -> mark(s(len(Z)))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(fst(X1, X2)) -> fst(active(X1), X2)
active(fst(X1, X2)) -> fst(X1, active(X2))
active(from(X)) -> from(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(len(X)) -> len(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
fst(mark(X1), X2) -> mark(fst(X1, X2))
fst(X1, mark(X2)) -> mark(fst(X1, X2))
fst(ok(X1), ok(X2)) -> ok(fst(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
len(mark(X)) -> mark(len(X))
len(ok(X)) -> ok(len(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(fst(X1, X2)) -> fst(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(len(X)) -> len(proper(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳Argument Filtering and Ordering

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳Remaining

Dependency Pair:

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

Rules:

active(fst(0, Z)) -> mark(nil)
active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(len(nil)) -> mark(0)
active(len(cons(X, Z))) -> mark(s(len(Z)))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(fst(X1, X2)) -> fst(active(X1), X2)
active(fst(X1, X2)) -> fst(X1, active(X2))
active(from(X)) -> from(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(len(X)) -> len(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
fst(mark(X1), X2) -> mark(fst(X1, X2))
fst(X1, mark(X2)) -> mark(fst(X1, X2))
fst(ok(X1), ok(X2)) -> ok(fst(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
len(mark(X)) -> mark(len(X))
len(ok(X)) -> ok(len(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(fst(X1, X2)) -> fst(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(len(X)) -> len(proper(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pair can be strictly oriented:

S(ok(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)

     ↳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(fst(0, Z)) -> mark(nil)
active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(len(nil)) -> mark(0)
active(len(cons(X, Z))) -> mark(s(len(Z)))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(fst(X1, X2)) -> fst(active(X1), X2)
active(fst(X1, X2)) -> fst(X1, active(X2))
active(from(X)) -> from(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(len(X)) -> len(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
fst(mark(X1), X2) -> mark(fst(X1, X2))
fst(X1, mark(X2)) -> mark(fst(X1, X2))
fst(ok(X1), ok(X2)) -> ok(fst(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
len(mark(X)) -> mark(len(X))
len(ok(X)) -> ok(len(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(fst(X1, X2)) -> fst(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(len(X)) -> len(proper(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳Argument Filtering and Ordering

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳Remaining

Dependency Pairs:

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

Rules:

active(fst(0, Z)) -> mark(nil)
active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(len(nil)) -> mark(0)
active(len(cons(X, Z))) -> mark(s(len(Z)))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(fst(X1, X2)) -> fst(active(X1), X2)
active(fst(X1, X2)) -> fst(X1, active(X2))
active(from(X)) -> from(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(len(X)) -> len(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
fst(mark(X1), X2) -> mark(fst(X1, X2))
fst(X1, mark(X2)) -> mark(fst(X1, X2))
fst(ok(X1), ok(X2)) -> ok(fst(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
len(mark(X)) -> mark(len(X))
len(ok(X)) -> ok(len(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(fst(X1, X2)) -> fst(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(len(X)) -> len(proper(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pairs can be strictly oriented:

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

ADD(x₁, x₂) -> ADD(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 14

             ↳Dependency Graph

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳Remaining

Dependency Pair:

Rules:

active(fst(0, Z)) -> mark(nil)
active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(len(nil)) -> mark(0)
active(len(cons(X, Z))) -> mark(s(len(Z)))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(fst(X1, X2)) -> fst(active(X1), X2)
active(fst(X1, X2)) -> fst(X1, active(X2))
active(from(X)) -> from(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(len(X)) -> len(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
fst(mark(X1), X2) -> mark(fst(X1, X2))
fst(X1, mark(X2)) -> mark(fst(X1, X2))
fst(ok(X1), ok(X2)) -> ok(fst(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
len(mark(X)) -> mark(len(X))
len(ok(X)) -> ok(len(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(fst(X1, X2)) -> fst(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(len(X)) -> len(proper(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳Argument Filtering and Ordering

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳Remaining

Dependency Pairs:

LEN(ok(X)) -> LEN(X)
LEN(mark(X)) -> LEN(X)

Rules:

active(fst(0, Z)) -> mark(nil)
active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(len(nil)) -> mark(0)
active(len(cons(X, Z))) -> mark(s(len(Z)))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(fst(X1, X2)) -> fst(active(X1), X2)
active(fst(X1, X2)) -> fst(X1, active(X2))
active(from(X)) -> from(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(len(X)) -> len(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
fst(mark(X1), X2) -> mark(fst(X1, X2))
fst(X1, mark(X2)) -> mark(fst(X1, X2))
fst(ok(X1), ok(X2)) -> ok(fst(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
len(mark(X)) -> mark(len(X))
len(ok(X)) -> ok(len(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(fst(X1, X2)) -> fst(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(len(X)) -> len(proper(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pairs can be strictly oriented:

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

LEN(x₁) -> LEN(x₁)
ok(x₁) -> ok(x₁)
mark(x₁) -> mark(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

           →DP Problem 15

             ↳Dependency Graph

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳Remaining

Dependency Pair:

Rules:

active(fst(0, Z)) -> mark(nil)
active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(len(nil)) -> mark(0)
active(len(cons(X, Z))) -> mark(s(len(Z)))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(fst(X1, X2)) -> fst(active(X1), X2)
active(fst(X1, X2)) -> fst(X1, active(X2))
active(from(X)) -> from(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(len(X)) -> len(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
fst(mark(X1), X2) -> mark(fst(X1, X2))
fst(X1, mark(X2)) -> mark(fst(X1, X2))
fst(ok(X1), ok(X2)) -> ok(fst(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
len(mark(X)) -> mark(len(X))
len(ok(X)) -> ok(len(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(fst(X1, X2)) -> fst(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(len(X)) -> len(proper(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳Argument Filtering and Ordering

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳Remaining

Dependency Pairs:

ACTIVE(len(X)) -> ACTIVE(X)
ACTIVE(add(X1, X2)) -> ACTIVE(X2)
ACTIVE(add(X1, X2)) -> ACTIVE(X1)
ACTIVE(from(X)) -> ACTIVE(X)
ACTIVE(fst(X1, X2)) -> ACTIVE(X2)
ACTIVE(fst(X1, X2)) -> ACTIVE(X1)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)

Rules:

active(fst(0, Z)) -> mark(nil)
active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(len(nil)) -> mark(0)
active(len(cons(X, Z))) -> mark(s(len(Z)))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(fst(X1, X2)) -> fst(active(X1), X2)
active(fst(X1, X2)) -> fst(X1, active(X2))
active(from(X)) -> from(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(len(X)) -> len(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
fst(mark(X1), X2) -> mark(fst(X1, X2))
fst(X1, mark(X2)) -> mark(fst(X1, X2))
fst(ok(X1), ok(X2)) -> ok(fst(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
len(mark(X)) -> mark(len(X))
len(ok(X)) -> ok(len(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(fst(X1, X2)) -> fst(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(len(X)) -> len(proper(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pairs can be strictly oriented:

ACTIVE(len(X)) -> ACTIVE(X)
ACTIVE(add(X1, X2)) -> ACTIVE(X2)
ACTIVE(add(X1, X2)) -> ACTIVE(X1)
ACTIVE(from(X)) -> ACTIVE(X)
ACTIVE(fst(X1, X2)) -> ACTIVE(X2)
ACTIVE(fst(X1, X2)) -> ACTIVE(X1)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)

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)
add(x₁, x₂) -> add(x₁, x₂)
fst(x₁, x₂) -> fst(x₁, x₂)
from(x₁) -> from(x₁)
len(x₁) -> len(x₁)
cons(x₁, x₂) -> cons(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 16

             ↳Dependency Graph

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳Remaining

Dependency Pair:

Rules:

active(fst(0, Z)) -> mark(nil)
active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(len(nil)) -> mark(0)
active(len(cons(X, Z))) -> mark(s(len(Z)))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(fst(X1, X2)) -> fst(active(X1), X2)
active(fst(X1, X2)) -> fst(X1, active(X2))
active(from(X)) -> from(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(len(X)) -> len(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
fst(mark(X1), X2) -> mark(fst(X1, X2))
fst(X1, mark(X2)) -> mark(fst(X1, X2))
fst(ok(X1), ok(X2)) -> ok(fst(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
len(mark(X)) -> mark(len(X))
len(ok(X)) -> ok(len(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(fst(X1, X2)) -> fst(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(len(X)) -> len(proper(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳Argument Filtering and Ordering

       →DP Problem 9

         ↳Remaining

Dependency Pairs:

PROPER(len(X)) -> PROPER(X)
PROPER(add(X1, X2)) -> PROPER(X2)
PROPER(add(X1, X2)) -> PROPER(X1)
PROPER(from(X)) -> PROPER(X)
PROPER(fst(X1, X2)) -> PROPER(X2)
PROPER(fst(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(s(X)) -> PROPER(X)

Rules:

active(fst(0, Z)) -> mark(nil)
active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(len(nil)) -> mark(0)
active(len(cons(X, Z))) -> mark(s(len(Z)))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(fst(X1, X2)) -> fst(active(X1), X2)
active(fst(X1, X2)) -> fst(X1, active(X2))
active(from(X)) -> from(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(len(X)) -> len(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
fst(mark(X1), X2) -> mark(fst(X1, X2))
fst(X1, mark(X2)) -> mark(fst(X1, X2))
fst(ok(X1), ok(X2)) -> ok(fst(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
len(mark(X)) -> mark(len(X))
len(ok(X)) -> ok(len(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(fst(X1, X2)) -> fst(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(len(X)) -> len(proper(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

The following dependency pairs can be strictly oriented:

PROPER(len(X)) -> PROPER(X)
PROPER(add(X1, X2)) -> PROPER(X2)
PROPER(add(X1, X2)) -> PROPER(X1)
PROPER(from(X)) -> PROPER(X)
PROPER(fst(X1, X2)) -> PROPER(X2)
PROPER(fst(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(s(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)
add(x₁, x₂) -> add(x₁, x₂)
cons(x₁, x₂) -> cons(x₁, x₂)
from(x₁) -> from(x₁)
len(x₁) -> len(x₁)
fst(x₁, x₂) -> fst(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 17

             ↳Dependency Graph

       →DP Problem 9

         ↳Remaining

Dependency Pair:

Rules:

active(fst(0, Z)) -> mark(nil)
active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(len(nil)) -> mark(0)
active(len(cons(X, Z))) -> mark(s(len(Z)))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(fst(X1, X2)) -> fst(active(X1), X2)
active(fst(X1, X2)) -> fst(X1, active(X2))
active(from(X)) -> from(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(len(X)) -> len(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
fst(mark(X1), X2) -> mark(fst(X1, X2))
fst(X1, mark(X2)) -> mark(fst(X1, X2))
fst(ok(X1), ok(X2)) -> ok(fst(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
len(mark(X)) -> mark(len(X))
len(ok(X)) -> ok(len(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(fst(X1, X2)) -> fst(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(len(X)) -> len(proper(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

       →DP Problem 4

         ↳AFS

       →DP Problem 5

         ↳AFS

       →DP Problem 6

         ↳AFS

       →DP Problem 7

         ↳AFS

       →DP Problem 8

         ↳AFS

       →DP Problem 9

         ↳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(fst(0, Z)) -> mark(nil)
active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(len(nil)) -> mark(0)
active(len(cons(X, Z))) -> mark(s(len(Z)))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(fst(X1, X2)) -> fst(active(X1), X2)
active(fst(X1, X2)) -> fst(X1, active(X2))
active(from(X)) -> from(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(len(X)) -> len(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
fst(mark(X1), X2) -> mark(fst(X1, X2))
fst(X1, mark(X2)) -> mark(fst(X1, X2))
fst(ok(X1), ok(X2)) -> ok(fst(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
len(mark(X)) -> mark(len(X))
len(ok(X)) -> ok(len(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(fst(X1, X2)) -> fst(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(len(X)) -> len(proper(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Termination of R could not be shown.
Duration:
0:01 minutes