Termination w.r.t. Q of the given QTRS could not be shown:

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(and(tt, T)) → MARK(T)
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
ACTIVE(isNatIList(IL)) → ISNATLIST(IL)
ACTIVE(isNat(0)) → MARK(tt)
ACTIVE(isNat(s(N))) → MARK(isNat(N))
ACTIVE(isNat(s(N))) → ISNAT(N)
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
ACTIVE(isNat(length(L))) → ISNATLIST(L)
ACTIVE(isNatIList(zeros)) → MARK(tt)
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
ACTIVE(isNatIList(cons(N, IL))) → AND(isNat(N), isNatIList(IL))
ACTIVE(isNatIList(cons(N, IL))) → ISNAT(N)
ACTIVE(isNatIList(cons(N, IL))) → ISNATILIST(IL)
ACTIVE(isNatList(nil)) → MARK(tt)
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
ACTIVE(isNatList(cons(N, L))) → AND(isNat(N), isNatList(L))
ACTIVE(isNatList(cons(N, L))) → ISNAT(N)
ACTIVE(isNatList(cons(N, L))) → ISNATLIST(L)
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
ACTIVE(isNatList(take(N, IL))) → AND(isNat(N), isNatIList(IL))
ACTIVE(isNatList(take(N, IL))) → ISNAT(N)
ACTIVE(isNatList(take(N, IL))) → ISNATILIST(IL)
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(zeros) → CONS(0, zeros)
ACTIVE(take(0, IL)) → MARK(uTake1(isNatIList(IL)))
ACTIVE(take(0, IL)) → UTAKE1(isNatIList(IL))
ACTIVE(take(0, IL)) → ISNATILIST(IL)
ACTIVE(uTake1(tt)) → MARK(nil)
ACTIVE(take(s(M), cons(N, IL))) → MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
ACTIVE(take(s(M), cons(N, IL))) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNat(M), and(isNat(N), isNatIList(IL)))
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(M)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNat(N), isNatIList(IL))
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(N)
ACTIVE(take(s(M), cons(N, IL))) → ISNATILIST(IL)
ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
ACTIVE(uTake2(tt, M, N, IL)) → CONS(N, take(M, IL))
ACTIVE(uTake2(tt, M, N, IL)) → TAKE(M, IL)
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
ACTIVE(length(cons(N, L))) → ULENGTH(and(isNat(N), isNatList(L)), L)
ACTIVE(length(cons(N, L))) → AND(isNat(N), isNatList(L))
ACTIVE(length(cons(N, L))) → ISNAT(N)
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))
ACTIVE(uLength(tt, L)) → S(length(L))
ACTIVE(uLength(tt, L)) → LENGTH(L)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
MARK(and(X1, X2)) → AND(mark(X1), mark(X2))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(tt) → ACTIVE(tt)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(0) → ACTIVE(0)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → S(mark(X))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(length(X)) → LENGTH(mark(X))
MARK(length(X)) → MARK(X)
MARK(zeros) → ACTIVE(zeros)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(nil) → ACTIVE(nil)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(take(X1, X2)) → TAKE(mark(X1), mark(X2))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(uTake1(X)) → ACTIVE(uTake1(mark(X)))
MARK(uTake1(X)) → UTAKE1(mark(X))
MARK(uTake1(X)) → MARK(X)
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
MARK(uTake2(X1, X2, X3, X4)) → UTAKE2(mark(X1), X2, X3, X4)
MARK(uTake2(X1, X2, X3, X4)) → MARK(X1)
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
MARK(uLength(X1, X2)) → ULENGTH(mark(X1), X2)
MARK(uLength(X1, X2)) → MARK(X1)
AND(mark(X1), X2) → AND(X1, X2)
AND(X1, mark(X2)) → AND(X1, X2)
AND(active(X1), X2) → AND(X1, X2)
AND(X1, active(X2)) → AND(X1, X2)
ISNATILIST(mark(X)) → ISNATILIST(X)
ISNATILIST(active(X)) → ISNATILIST(X)
ISNATLIST(mark(X)) → ISNATLIST(X)
ISNATLIST(active(X)) → ISNATLIST(X)
ISNAT(mark(X)) → ISNAT(X)
ISNAT(active(X)) → ISNAT(X)
S(mark(X)) → S(X)
S(active(X)) → S(X)
LENGTH(mark(X)) → LENGTH(X)
LENGTH(active(X)) → LENGTH(X)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(X1, mark(X2)) → TAKE(X1, X2)
TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)
UTAKE1(mark(X)) → UTAKE1(X)
UTAKE1(active(X)) → UTAKE1(X)
UTAKE2(mark(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, mark(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, mark(X3), X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, X3, mark(X4)) → UTAKE2(X1, X2, X3, X4)
UTAKE2(active(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, active(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, active(X3), X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, X3, active(X4)) → UTAKE2(X1, X2, X3, X4)
ULENGTH(mark(X1), X2) → ULENGTH(X1, X2)
ULENGTH(X1, mark(X2)) → ULENGTH(X1, X2)
ULENGTH(active(X1), X2) → ULENGTH(X1, X2)
ULENGTH(X1, active(X2)) → ULENGTH(X1, X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 12 SCCs with 44 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ULENGTH(X1, mark(X2)) → ULENGTH(X1, X2)
ULENGTH(mark(X1), X2) → ULENGTH(X1, X2)
ULENGTH(active(X1), X2) → ULENGTH(X1, X2)
ULENGTH(X1, active(X2)) → ULENGTH(X1, X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ULENGTH(X1, active(X2)) → ULENGTH(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ULENGTH(x1, x2)  =  x2
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ULENGTH(X1, mark(X2)) → ULENGTH(X1, X2)
ULENGTH(mark(X1), X2) → ULENGTH(X1, X2)
ULENGTH(active(X1), X2) → ULENGTH(X1, X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ULENGTH(X1, mark(X2)) → ULENGTH(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ULENGTH(x1, x2)  =  x2
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ULENGTH(mark(X1), X2) → ULENGTH(X1, X2)
ULENGTH(active(X1), X2) → ULENGTH(X1, X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ULENGTH(active(X1), X2) → ULENGTH(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ULENGTH(x1, x2)  =  x1
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(11) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ULENGTH(mark(X1), X2) → ULENGTH(X1, X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(12) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ULENGTH(mark(X1), X2) → ULENGTH(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ULENGTH(x1, x2)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(13) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(14) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(15) TRUE

(16) Obligation:

Q DP problem:
The TRS P consists of the following rules:

UTAKE2(X1, mark(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(mark(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, mark(X3), X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, X3, mark(X4)) → UTAKE2(X1, X2, X3, X4)
UTAKE2(active(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, active(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, active(X3), X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, X3, active(X4)) → UTAKE2(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

UTAKE2(X1, X2, X3, active(X4)) → UTAKE2(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
UTAKE2(x1, x2, x3, x4)  =  x4
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(18) Obligation:

Q DP problem:
The TRS P consists of the following rules:

UTAKE2(X1, mark(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(mark(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, mark(X3), X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, X3, mark(X4)) → UTAKE2(X1, X2, X3, X4)
UTAKE2(active(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, active(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, active(X3), X4) → UTAKE2(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

UTAKE2(X1, X2, X3, mark(X4)) → UTAKE2(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
UTAKE2(x1, x2, x3, x4)  =  x4
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(20) Obligation:

Q DP problem:
The TRS P consists of the following rules:

UTAKE2(X1, mark(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(mark(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, mark(X3), X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(active(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, active(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, active(X3), X4) → UTAKE2(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

UTAKE2(X1, X2, active(X3), X4) → UTAKE2(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
UTAKE2(x1, x2, x3, x4)  =  x3
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(22) Obligation:

Q DP problem:
The TRS P consists of the following rules:

UTAKE2(X1, mark(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(mark(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, X2, mark(X3), X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(active(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, active(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(23) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

UTAKE2(X1, X2, mark(X3), X4) → UTAKE2(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
UTAKE2(x1, x2, x3, x4)  =  x3
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(24) Obligation:

Q DP problem:
The TRS P consists of the following rules:

UTAKE2(X1, mark(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(mark(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(active(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(X1, active(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(25) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

UTAKE2(X1, active(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
UTAKE2(x1, x2, x3, x4)  =  x2
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(26) Obligation:

Q DP problem:
The TRS P consists of the following rules:

UTAKE2(X1, mark(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(mark(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(active(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(27) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

UTAKE2(X1, mark(X2), X3, X4) → UTAKE2(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
UTAKE2(x1, x2, x3, x4)  =  x2
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(28) Obligation:

Q DP problem:
The TRS P consists of the following rules:

UTAKE2(mark(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)
UTAKE2(active(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(29) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

UTAKE2(active(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
UTAKE2(x1, x2, x3, x4)  =  x1
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(30) Obligation:

Q DP problem:
The TRS P consists of the following rules:

UTAKE2(mark(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

UTAKE2(mark(X1), X2, X3, X4) → UTAKE2(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
UTAKE2(x1, x2, x3, x4)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(32) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(33) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(34) TRUE

(35) Obligation:

Q DP problem:
The TRS P consists of the following rules:

UTAKE1(active(X)) → UTAKE1(X)
UTAKE1(mark(X)) → UTAKE1(X)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(36) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

UTAKE1(mark(X)) → UTAKE1(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
UTAKE1(x1)  =  x1
active(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(37) Obligation:

Q DP problem:
The TRS P consists of the following rules:

UTAKE1(active(X)) → UTAKE1(X)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(38) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

UTAKE1(active(X)) → UTAKE1(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
UTAKE1(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(39) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(40) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(41) TRUE

(42) Obligation:

Q DP problem:
The TRS P consists of the following rules:

TAKE(X1, mark(X2)) → TAKE(X1, X2)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(43) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(X1, active(X2)) → TAKE(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAKE(x1, x2)  =  x2
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(44) Obligation:

Q DP problem:
The TRS P consists of the following rules:

TAKE(X1, mark(X2)) → TAKE(X1, X2)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(active(X1), X2) → TAKE(X1, X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(45) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(X1, mark(X2)) → TAKE(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAKE(x1, x2)  =  x2
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(46) Obligation:

Q DP problem:
The TRS P consists of the following rules:

TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(active(X1), X2) → TAKE(X1, X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(47) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(active(X1), X2) → TAKE(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAKE(x1, x2)  =  x1
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(48) Obligation:

Q DP problem:
The TRS P consists of the following rules:

TAKE(mark(X1), X2) → TAKE(X1, X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(49) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(mark(X1), X2) → TAKE(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAKE(x1, x2)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(50) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(51) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(52) TRUE

(53) Obligation:

Q DP problem:
The TRS P consists of the following rules:

CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(54) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(X1, active(X2)) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  x2
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(55) Obligation:

Q DP problem:
The TRS P consists of the following rules:

CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(56) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(X1, mark(X2)) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  x2
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(57) Obligation:

Q DP problem:
The TRS P consists of the following rules:

CONS(mark(X1), X2) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(58) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(active(X1), X2) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  x1
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(59) Obligation:

Q DP problem:
The TRS P consists of the following rules:

CONS(mark(X1), X2) → CONS(X1, X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(60) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(mark(X1), X2) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(61) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(62) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(63) TRUE

(64) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LENGTH(active(X)) → LENGTH(X)
LENGTH(mark(X)) → LENGTH(X)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(65) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

LENGTH(mark(X)) → LENGTH(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
LENGTH(x1)  =  x1
active(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(66) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LENGTH(active(X)) → LENGTH(X)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(67) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

LENGTH(active(X)) → LENGTH(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
LENGTH(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(68) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(69) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(70) TRUE

(71) Obligation:

Q DP problem:
The TRS P consists of the following rules:

S(active(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(72) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(mark(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1)  =  x1
active(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(73) Obligation:

Q DP problem:
The TRS P consists of the following rules:

S(active(X)) → S(X)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(74) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(active(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(75) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(76) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(77) TRUE

(78) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNAT(active(X)) → ISNAT(X)
ISNAT(mark(X)) → ISNAT(X)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(79) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNAT(mark(X)) → ISNAT(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNAT(x1)  =  x1
active(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(80) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNAT(active(X)) → ISNAT(X)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(81) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNAT(active(X)) → ISNAT(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNAT(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(82) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(83) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(84) TRUE

(85) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNATLIST(active(X)) → ISNATLIST(X)
ISNATLIST(mark(X)) → ISNATLIST(X)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(86) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATLIST(mark(X)) → ISNATLIST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATLIST(x1)  =  x1
active(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(87) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNATLIST(active(X)) → ISNATLIST(X)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(88) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATLIST(active(X)) → ISNATLIST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATLIST(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(89) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(90) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(91) TRUE

(92) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNATILIST(active(X)) → ISNATILIST(X)
ISNATILIST(mark(X)) → ISNATILIST(X)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(93) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATILIST(mark(X)) → ISNATILIST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATILIST(x1)  =  x1
active(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(94) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNATILIST(active(X)) → ISNATILIST(X)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(95) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATILIST(active(X)) → ISNATILIST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATILIST(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(96) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(97) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(98) TRUE

(99) Obligation:

Q DP problem:
The TRS P consists of the following rules:

AND(X1, mark(X2)) → AND(X1, X2)
AND(mark(X1), X2) → AND(X1, X2)
AND(active(X1), X2) → AND(X1, X2)
AND(X1, active(X2)) → AND(X1, X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(100) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AND(X1, active(X2)) → AND(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AND(x1, x2)  =  x2
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(101) Obligation:

Q DP problem:
The TRS P consists of the following rules:

AND(X1, mark(X2)) → AND(X1, X2)
AND(mark(X1), X2) → AND(X1, X2)
AND(active(X1), X2) → AND(X1, X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(102) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AND(X1, mark(X2)) → AND(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AND(x1, x2)  =  x2
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(103) Obligation:

Q DP problem:
The TRS P consists of the following rules:

AND(mark(X1), X2) → AND(X1, X2)
AND(active(X1), X2) → AND(X1, X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(104) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AND(active(X1), X2) → AND(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AND(x1, x2)  =  x1
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(105) Obligation:

Q DP problem:
The TRS P consists of the following rules:

AND(mark(X1), X2) → AND(X1, X2)

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(106) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AND(mark(X1), X2) → AND(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AND(x1, x2)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(107) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(108) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(109) TRUE

(110) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(and(tt, T)) → MARK(T)
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(s(N))) → MARK(isNat(N))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(length(X)) → MARK(X)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(take(0, IL)) → MARK(uTake1(isNatIList(IL)))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(uTake1(X)) → ACTIVE(uTake1(mark(X)))
ACTIVE(take(s(M), cons(N, IL))) → MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
MARK(uTake1(X)) → MARK(X)
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
MARK(uTake2(X1, X2, X3, X4)) → MARK(X1)
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
MARK(uLength(X1, X2)) → MARK(X1)
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(111) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(uTake1(X)) → ACTIVE(uTake1(mark(X)))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK
ACTIVE(x1)  =  x1
and(x1, x2)  =  and
isNatIList(x1)  =  isNatIList
isNatList(x1)  =  isNatList
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
zeros  =  zeros
cons(x1, x2)  =  cons
take(x1, x2)  =  take
uTake1(x1)  =  uTake1
uTake2(x1, x2, x3, x4)  =  uTake2
uLength(x1, x2)  =  uLength

Lexicographic Path Order [LPO].
Precedence:

[MARK, and, isNatIList, isNatList, isNat, length, zeros, take, uTake2, uLength] > s
[MARK, and, isNatIList, isNatList, isNat, length, zeros, take, uTake2, uLength] > cons
[MARK, and, isNatIList, isNatList, isNat, length, zeros, take, uTake2, uLength] > uTake1

The following usable rules [FROCOS05] were oriented:

take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uTake1(active(X)) → uTake1(X)
uTake1(mark(X)) → uTake1(X)

(112) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(and(X1, X2)) → ACTIVE(and(mark(X1), mark(X2)))
ACTIVE(and(tt, T)) → MARK(T)
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X2)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNatIList(IL)) → MARK(isNatList(IL))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(s(N))) → MARK(isNat(N))
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNat(length(L))) → MARK(isNatList(L))
ACTIVE(isNatIList(cons(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(cons(N, L))) → MARK(and(isNat(N), isNatList(L)))
MARK(length(X)) → MARK(X)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(isNatList(take(N, IL))) → MARK(and(isNat(N), isNatIList(IL)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(take(0, IL)) → MARK(uTake1(isNatIList(IL)))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
ACTIVE(take(s(M), cons(N, IL))) → MARK(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
MARK(uTake1(X)) → MARK(X)
MARK(uTake2(X1, X2, X3, X4)) → ACTIVE(uTake2(mark(X1), X2, X3, X4))
ACTIVE(uTake2(tt, M, N, IL)) → MARK(cons(N, take(M, IL)))
MARK(uTake2(X1, X2, X3, X4)) → MARK(X1)
MARK(uLength(X1, X2)) → ACTIVE(uLength(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(uLength(and(isNat(N), isNatList(L)), L))
MARK(uLength(X1, X2)) → MARK(X1)
ACTIVE(uLength(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
mark(and(X1, X2)) → active(and(mark(X1), mark(X2)))
mark(tt) → active(tt)
mark(isNatIList(X)) → active(isNatIList(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNat(X)) → active(isNat(X))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(nil) → active(nil)
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(uTake1(X)) → active(uTake1(mark(X)))
mark(uTake2(X1, X2, X3, X4)) → active(uTake2(mark(X1), X2, X3, X4))
mark(uLength(X1, X2)) → active(uLength(mark(X1), X2))
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
uTake1(mark(X)) → uTake1(X)
uTake1(active(X)) → uTake1(X)
uTake2(mark(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, mark(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, mark(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, mark(X4)) → uTake2(X1, X2, X3, X4)
uTake2(active(X1), X2, X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, active(X2), X3, X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, active(X3), X4) → uTake2(X1, X2, X3, X4)
uTake2(X1, X2, X3, active(X4)) → uTake2(X1, X2, X3, X4)
uLength(mark(X1), X2) → uLength(X1, X2)
uLength(X1, mark(X2)) → uLength(X1, X2)
uLength(active(X1), X2) → uLength(X1, X2)
uLength(X1, active(X2)) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.