Termination Proof using AProVE

Term Rewriting System R:
[X, Y, X1, X2]
active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(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(from(X)) -> CONS(X, from(s(X)))
ACTIVE(from(X)) -> FROM(s(X))
ACTIVE(from(X)) -> S(X)
ACTIVE(length(cons(X, Y))) -> S(length1(Y))
ACTIVE(length(cons(X, Y))) -> LENGTH1(Y)
ACTIVE(length1(X)) -> LENGTH(X)
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)
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)
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(length(X)) -> LENGTH(proper(X))
PROPER(length(X)) -> PROPER(X)
PROPER(length1(X)) -> LENGTH1(proper(X))
PROPER(length1(X)) -> PROPER(X)
LENGTH(ok(X)) -> LENGTH(X)
LENGTH1(ok(X)) -> LENGTH1(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 eight SCCs.

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

The following remains to be proven:

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(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
FROM(ok(X)) -> FROM(X)
FROM(mark(X)) -> FROM(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
S(ok(X)) -> S(X)
S(mark(X)) -> S(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pair:
LENGTH1(ok(X)) -> LENGTH1(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pair:
LENGTH(ok(X)) -> LENGTH(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
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(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
PROPER(length1(X)) -> PROPER(X)
PROPER(length(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(from(X)) -> PROPER(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
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(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

The following remains to be proven:

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(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
FROM(ok(X)) -> FROM(X)
FROM(mark(X)) -> FROM(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
S(ok(X)) -> S(X)
S(mark(X)) -> S(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pair:
LENGTH1(ok(X)) -> LENGTH1(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pair:
LENGTH(ok(X)) -> LENGTH(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
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(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
PROPER(length1(X)) -> PROPER(X)
PROPER(length(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(from(X)) -> PROPER(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
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(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

The following remains to be proven:

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(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
FROM(ok(X)) -> FROM(X)
FROM(mark(X)) -> FROM(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
S(ok(X)) -> S(X)
S(mark(X)) -> S(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pair:
LENGTH1(ok(X)) -> LENGTH1(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pair:
LENGTH(ok(X)) -> LENGTH(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
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(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
PROPER(length1(X)) -> PROPER(X)
PROPER(length(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(from(X)) -> PROPER(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
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(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

The following remains to be proven:

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(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
FROM(ok(X)) -> FROM(X)
FROM(mark(X)) -> FROM(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
S(ok(X)) -> S(X)
S(mark(X)) -> S(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pair:
LENGTH1(ok(X)) -> LENGTH1(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pair:
LENGTH(ok(X)) -> LENGTH(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
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(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
PROPER(length1(X)) -> PROPER(X)
PROPER(length(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(from(X)) -> PROPER(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
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(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

The following remains to be proven:

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(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
FROM(ok(X)) -> FROM(X)
FROM(mark(X)) -> FROM(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
S(ok(X)) -> S(X)
S(mark(X)) -> S(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pair:
LENGTH1(ok(X)) -> LENGTH1(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pair:
LENGTH(ok(X)) -> LENGTH(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
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(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
PROPER(length1(X)) -> PROPER(X)
PROPER(length(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(from(X)) -> PROPER(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
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(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

The following remains to be proven:

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(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
FROM(ok(X)) -> FROM(X)
FROM(mark(X)) -> FROM(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
S(ok(X)) -> S(X)
S(mark(X)) -> S(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pair:
LENGTH1(ok(X)) -> LENGTH1(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pair:
LENGTH(ok(X)) -> LENGTH(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
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(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
PROPER(length1(X)) -> PROPER(X)
PROPER(length(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(from(X)) -> PROPER(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
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(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

The following remains to be proven:

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(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
FROM(ok(X)) -> FROM(X)
FROM(mark(X)) -> FROM(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
S(ok(X)) -> S(X)
S(mark(X)) -> S(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pair:
LENGTH1(ok(X)) -> LENGTH1(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pair:
LENGTH(ok(X)) -> LENGTH(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
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(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
PROPER(length1(X)) -> PROPER(X)
PROPER(length(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(from(X)) -> PROPER(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
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(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

       →DP Problem 5

         ↳Remaining Obligation(s)

       →DP Problem 6

         ↳Remaining Obligation(s)

       →DP Problem 7

         ↳Remaining Obligation(s)

       →DP Problem 8

         ↳Remaining Obligation(s)

The following remains to be proven:

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(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
FROM(ok(X)) -> FROM(X)
FROM(mark(X)) -> FROM(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
S(ok(X)) -> S(X)
S(mark(X)) -> S(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pair:
LENGTH1(ok(X)) -> LENGTH1(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pair:
LENGTH(ok(X)) -> LENGTH(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
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(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Dependency Pairs:
PROPER(length1(X)) -> PROPER(X)
PROPER(length(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(from(X)) -> PROPER(X)

Rules:

active(from(X)) -> mark(cons(X, from(s(X))))
active(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
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(length(nil)) -> mark(0)
active(length(cons(X, Y))) -> mark(s(length1(Y)))
active(length1(X)) -> mark(length(X))
active(from(X)) -> from(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
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))
proper(from(X)) -> from(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
proper(0) -> ok(0)
proper(length1(X)) -> length1(proper(X))
length(ok(X)) -> ok(length(X))
length1(ok(X)) -> ok(length1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Termination of R could not be shown.
Duration:
0:00 minutes