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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

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 14 SCCs with 52 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

ISNAT(ok(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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

ISNAT(ok(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
ok(x1)  =  ok(x1)
active(x1)  =  x1
and(x1, x2)  =  x1
tt  =  tt
mark(x1)  =  mark
isNatIList(x1)  =  x1
isNatList(x1)  =  x1
isNat(x1)  =  x1
0  =  0
s(x1)  =  x1
length(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  x2
nil  =  nil
take(x1, x2)  =  x2
uTake1(x1)  =  x1
uTake2(x1, x2, x3, x4)  =  x1
uLength(x1, x2)  =  x2
proper(x1)  =  proper
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

proper > ok₁ > top > mark
proper > tt > nil > mark
proper > 0 > mark
proper > zeros > mark

Status:

ok₁: [1]
tt: multiset
mark: []
0: multiset
zeros: multiset
nil: multiset
proper: []
top: []

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(7) 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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(8) PisEmptyProof (EQUIVALENT transformation)

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

(9) TRUE

(10) Obligation:

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

ISNATLIST(ok(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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATLIST(ok(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
ok(x1)  =  ok(x1)
active(x1)  =  x1
and(x1, x2)  =  x1
tt  =  tt
mark(x1)  =  mark
isNatIList(x1)  =  x1
isNatList(x1)  =  x1
isNat(x1)  =  x1
0  =  0
s(x1)  =  x1
length(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  x2
nil  =  nil
take(x1, x2)  =  x2
uTake1(x1)  =  x1
uTake2(x1, x2, x3, x4)  =  x1
uLength(x1, x2)  =  x2
proper(x1)  =  proper
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

proper > ok₁ > top > mark
proper > tt > nil > mark
proper > 0 > mark
proper > zeros > mark

Status:

ok₁: [1]
tt: multiset
mark: []
0: multiset
zeros: multiset
nil: multiset
proper: []
top: []

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(12) 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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(13) PisEmptyProof (EQUIVALENT transformation)

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

(14) TRUE

(15) Obligation:

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

ISNATILIST(ok(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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(16) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATILIST(ok(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
ok(x1)  =  ok(x1)
active(x1)  =  x1
and(x1, x2)  =  x1
tt  =  tt
mark(x1)  =  mark
isNatIList(x1)  =  x1
isNatList(x1)  =  x1
isNat(x1)  =  x1
0  =  0
s(x1)  =  x1
length(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  x2
nil  =  nil
take(x1, x2)  =  x2
uTake1(x1)  =  x1
uTake2(x1, x2, x3, x4)  =  x1
uLength(x1, x2)  =  x2
proper(x1)  =  proper
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

proper > ok₁ > top > mark
proper > tt > nil > mark
proper > 0 > mark
proper > zeros > mark

Status:

ok₁: [1]
tt: multiset
mark: []
0: multiset
zeros: multiset
nil: multiset
proper: []
top: []

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(17) 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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(18) PisEmptyProof (EQUIVALENT transformation)

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

(19) TRUE

(20) Obligation:

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

ULENGTH(ok(X1), ok(X2)) → ULENGTH(X1, X2)
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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

ULENGTH(ok(X1), ok(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
ok(x1)  =  ok(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)
and(x1, x2)  =  and(x2)
tt  =  tt
isNatIList(x1)  =  x1
isNatList(x1)  =  isNatList(x1)
isNat(x1)  =  x1
0  =  0
s(x1)  =  s(x1)
length(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
take(x1, x2)  =  x2
uTake1(x1)  =  uTake1(x1)
uTake2(x1, x2, x3, x4)  =  uTake2(x1, x3, x4)
uLength(x1, x2)  =  uLength(x1, x2)
proper(x1)  =  proper(x1)
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

active₁ > and₁ > ok₁
active₁ > 0 > tt > s₁ > ok₁
active₁ > 0 > uTake1₁ > ok₁
active₁ > zeros > tt > s₁ > ok₁
active₁ > cons₂ > isNatList₁ > ok₁
active₁ > cons₂ > uTake2₃ > ok₁
active₁ > cons₂ > uLength₂ > ok₁
active₁ > nil > ok₁
proper₁ > and₁ > ok₁
proper₁ > 0 > tt > s₁ > ok₁
proper₁ > 0 > uTake1₁ > ok₁
proper₁ > zeros > tt > s₁ > ok₁
proper₁ > cons₂ > isNatList₁ > ok₁
proper₁ > cons₂ > uTake2₃ > ok₁
proper₁ > cons₂ > uLength₂ > ok₁
proper₁ > nil > ok₁

Status:

ok₁: [1]
active₁: [1]
and₁: [1]
tt: multiset
isNatList₁: [1]
0: multiset
s₁: [1]
zeros: multiset
cons₂: [1,2]
nil: multiset
uTake1₁: [1]
uTake2₃: [1,2,3]
uLength₂: [2,1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(22) 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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

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)
active(x1)  =  active(x1)
and(x1, x2)  =  and(x1, x2)
tt  =  tt
isNatIList(x1)  =  isNatIList
isNatList(x1)  =  isNatList
isNat(x1)  =  isNat
0  =  0
s(x1)  =  x1
length(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
take(x1, x2)  =  take(x1, x2)
uTake1(x1)  =  uTake1(x1)
uTake2(x1, x2, x3, x4)  =  uTake2(x1, x2, x3, x4)
uLength(x1, x2)  =  uLength(x1, x2)
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

active₁ > and₂ > mark₁
active₁ > isNatIList > tt > cons₂ > mark₁
active₁ > isNatIList > tt > cons₂ > isNatList
active₁ > isNatIList > tt > nil
active₁ > isNatIList > tt > take₂ > mark₁
active₁ > isNatIList > isNat > mark₁
active₁ > isNatIList > isNat > isNatList
active₁ > 0 > mark₁
active₁ > zeros > cons₂ > mark₁
active₁ > zeros > cons₂ > isNatList
active₁ > uTake1₁ > mark₁
active₁ > uTake1₁ > nil
active₁ > uTake2₄ > mark₁
active₁ > uLength₂ > mark₁

Status:

mark₁: [1]
active₁: [1]
and₂: [1,2]
tt: multiset
isNatIList: []
isNatList: []
isNat: multiset
0: multiset
zeros: multiset
cons₂: [2,1]
nil: multiset
take₂: [2,1]
uTake1₁: [1]
uTake2₄: [1,2,4,3]
uLength₂: [1,2]
top: []

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(24) 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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(25) PisEmptyProof (EQUIVALENT transformation)

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

(26) TRUE

(27) Obligation:

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

UTAKE2(ok(X1), ok(X2), ok(X3), ok(X4)) → UTAKE2(X1, X2, X3, X4)
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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(28) 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)  =  UTAKE2(x1, x4)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
and(x1, x2)  =  and(x1, x2)
tt  =  tt
isNatIList(x1)  =  isNatIList
isNatList(x1)  =  isNatList
isNat(x1)  =  isNat
0  =  0
s(x1)  =  x1
length(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
take(x1, x2)  =  take(x1, x2)
uTake1(x1)  =  uTake1(x1)
uTake2(x1, x2, x3, x4)  =  uTake2(x1, x2, x3, x4)
uLength(x1, x2)  =  uLength(x1, x2)
proper(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

UTAKE2₂ > mark₁
active₁ > isNatIList > tt > cons₂ > isNatList > and₂ > mark₁
active₁ > isNatIList > tt > cons₂ > uTake2₄ > mark₁
active₁ > isNatIList > tt > nil > mark₁
active₁ > isNatIList > isNat > isNatList > and₂ > mark₁
active₁ > 0 > mark₁
active₁ > take₂ > and₂ > mark₁
active₁ > take₂ > uTake1₁ > nil > mark₁
active₁ > take₂ > uTake2₄ > mark₁
active₁ > uLength₂ > mark₁
zeros > tt > cons₂ > isNatList > and₂ > mark₁
zeros > tt > cons₂ > uTake2₄ > mark₁
zeros > tt > nil > mark₁
zeros > 0 > mark₁
top > mark₁

Status:

UTAKE2₂: multiset
mark₁: [1]
active₁: [1]
and₂: [2,1]
tt: multiset
isNatIList: multiset
isNatList: []
isNat: []
0: multiset
zeros: multiset
cons₂: [1,2]
nil: multiset
take₂: [1,2]
uTake1₁: [1]
uTake2₄: [3,1,2,4]
uLength₂: [2,1]
top: []

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(29) Obligation:

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

UTAKE2(ok(X1), ok(X2), ok(X3), ok(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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(30) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

UTAKE2(ok(X1), ok(X2), ok(X3), ok(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)  =  UTAKE2(x1, x2, x3, x4)
ok(x1)  =  ok(x1)
active(x1)  =  x1
and(x1, x2)  =  x1
tt  =  tt
mark(x1)  =  mark
isNatIList(x1)  =  isNatIList(x1)
isNatList(x1)  =  isNatList(x1)
isNat(x1)  =  x1
0  =  0
s(x1)  =  x1
length(x1)  =  length(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
take(x1, x2)  =  x1
uTake1(x1)  =  uTake1(x1)
uTake2(x1, x2, x3, x4)  =  uTake2(x4)
uLength(x1, x2)  =  x1
proper(x1)  =  proper(x1)
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

0 > isNatIList₁ > tt > cons₂ > ok₁ > UTAKE2₄ > mark
0 > isNatIList₁ > tt > cons₂ > ok₁ > top > mark
0 > isNatIList₁ > isNatList₁ > ok₁ > UTAKE2₄ > mark
0 > isNatIList₁ > isNatList₁ > ok₁ > top > mark
0 > uTake1₁ > ok₁ > UTAKE2₄ > mark
0 > uTake1₁ > ok₁ > top > mark
proper₁ > isNatIList₁ > tt > cons₂ > ok₁ > UTAKE2₄ > mark
proper₁ > isNatIList₁ > tt > cons₂ > ok₁ > top > mark
proper₁ > isNatIList₁ > isNatList₁ > ok₁ > UTAKE2₄ > mark
proper₁ > isNatIList₁ > isNatList₁ > ok₁ > top > mark
proper₁ > length₁ > ok₁ > UTAKE2₄ > mark
proper₁ > length₁ > ok₁ > top > mark
proper₁ > zeros > cons₂ > ok₁ > UTAKE2₄ > mark
proper₁ > zeros > cons₂ > ok₁ > top > mark
proper₁ > nil > tt > cons₂ > ok₁ > UTAKE2₄ > mark
proper₁ > nil > tt > cons₂ > ok₁ > top > mark
proper₁ > uTake1₁ > ok₁ > UTAKE2₄ > mark
proper₁ > uTake1₁ > ok₁ > top > mark
proper₁ > uTake2₁ > cons₂ > ok₁ > UTAKE2₄ > mark
proper₁ > uTake2₁ > cons₂ > ok₁ > top > mark

Status:

UTAKE2₄: multiset
ok₁: [1]
tt: multiset
mark: []
isNatIList₁: [1]
isNatList₁: [1]
0: multiset
length₁: [1]
zeros: multiset
cons₂: [2,1]
nil: multiset
uTake1₁: [1]
uTake2₁: [1]
proper₁: [1]
top: multiset

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(31) 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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(32) PisEmptyProof (EQUIVALENT transformation)

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

(33) TRUE

(34) Obligation:

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

UTAKE1(ok(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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(35) 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)  =  UTAKE1(x1)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
and(x1, x2)  =  and(x1, x2)
tt  =  tt
isNatIList(x1)  =  isNatIList
isNatList(x1)  =  isNatList
isNat(x1)  =  isNat
0  =  0
s(x1)  =  x1
length(x1)  =  length(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
take(x1, x2)  =  take(x1, x2)
uTake1(x1)  =  uTake1(x1)
uTake2(x1, x2, x3, x4)  =  uTake2(x1, x2, x3, x4)
uLength(x1, x2)  =  uLength(x1, x2)
proper(x1)  =  proper(x1)
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

active₁ > isNatIList > proper₁ > tt > length₁ > and₂ > mark₁
active₁ > isNatIList > proper₁ > tt > cons₂ > mark₁
active₁ > isNatIList > proper₁ > tt > take₂ > mark₁
active₁ > isNatIList > proper₁ > isNat > mark₁
active₁ > isNatIList > proper₁ > zeros > mark₁
active₁ > isNatIList > proper₁ > nil > mark₁
active₁ > isNatIList > proper₁ > uTake1₁ > mark₁
active₁ > isNatIList > proper₁ > uTake2₄ > take₂ > mark₁
active₁ > isNatIList > proper₁ > uLength₂ > mark₁
active₁ > isNatList > proper₁ > tt > length₁ > and₂ > mark₁
active₁ > isNatList > proper₁ > tt > cons₂ > mark₁
active₁ > isNatList > proper₁ > tt > take₂ > mark₁
active₁ > isNatList > proper₁ > isNat > mark₁
active₁ > isNatList > proper₁ > zeros > mark₁
active₁ > isNatList > proper₁ > nil > mark₁
active₁ > isNatList > proper₁ > uTake1₁ > mark₁
active₁ > isNatList > proper₁ > uTake2₄ > take₂ > mark₁
active₁ > isNatList > proper₁ > uLength₂ > mark₁
active₁ > 0 > tt > length₁ > and₂ > mark₁
active₁ > 0 > tt > cons₂ > mark₁
active₁ > 0 > tt > take₂ > mark₁
active₁ > 0 > uTake1₁ > mark₁

Status:

UTAKE1₁: [1]
mark₁: multiset
active₁: [1]
and₂: [1,2]
tt: multiset
isNatIList: multiset
isNatList: []
isNat: []
0: multiset
length₁: [1]
zeros: multiset
cons₂: [1,2]
nil: multiset
take₂: [1,2]
uTake1₁: multiset
uTake2₄: [3,1,4,2]
uLength₂: [2,1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(36) Obligation:

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

UTAKE1(ok(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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(37) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

UTAKE1(ok(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
ok(x1)  =  ok(x1)
active(x1)  =  x1
and(x1, x2)  =  x1
tt  =  tt
mark(x1)  =  mark
isNatIList(x1)  =  x1
isNatList(x1)  =  x1
isNat(x1)  =  x1
0  =  0
s(x1)  =  x1
length(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  x2
nil  =  nil
take(x1, x2)  =  x2
uTake1(x1)  =  x1
uTake2(x1, x2, x3, x4)  =  x1
uLength(x1, x2)  =  x2
proper(x1)  =  proper
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

proper > ok₁ > top > mark
proper > tt > nil > mark
proper > 0 > mark
proper > zeros > mark

Status:

ok₁: [1]
tt: multiset
mark: []
0: multiset
zeros: multiset
nil: multiset
proper: []
top: []

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(38) 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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(39) PisEmptyProof (EQUIVALENT transformation)

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

(40) TRUE

(41) 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(ok(X1), ok(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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(42) 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)
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)  =  TAKE(x1, x2)
mark(x1)  =  mark(x1)
ok(x1)  =  x1
active(x1)  =  active(x1)
and(x1, x2)  =  and(x1, x2)
tt  =  tt
isNatIList(x1)  =  isNatIList(x1)
isNatList(x1)  =  x1
isNat(x1)  =  isNat(x1)
0  =  0
s(x1)  =  x1
length(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
take(x1, x2)  =  take(x1, x2)
uTake1(x1)  =  uTake1(x1)
uTake2(x1, x2, x3, x4)  =  uTake2(x1, x2, x3, x4)
uLength(x1, x2)  =  uLength(x1, x2)
proper(x1)  =  proper(x1)
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

active₁ > and₂ > mark₁ > TAKE₂
active₁ > isNatIList₁ > isNat₁ > mark₁ > TAKE₂
active₁ > isNatIList₁ > isNat₁ > tt
active₁ > 0 > mark₁ > TAKE₂
active₁ > 0 > tt
active₁ > zeros
active₁ > take₂ > isNat₁ > mark₁ > TAKE₂
active₁ > take₂ > isNat₁ > tt
active₁ > take₂ > uTake2₄ > cons₂ > mark₁ > TAKE₂
active₁ > uTake1₁ > mark₁ > TAKE₂
active₁ > uTake1₁ > nil
active₁ > uLength₂ > mark₁ > TAKE₂
proper₁ > and₂ > mark₁ > TAKE₂
proper₁ > isNatIList₁ > isNat₁ > mark₁ > TAKE₂
proper₁ > isNatIList₁ > isNat₁ > tt
proper₁ > 0 > mark₁ > TAKE₂
proper₁ > 0 > tt
proper₁ > zeros
proper₁ > take₂ > isNat₁ > mark₁ > TAKE₂
proper₁ > take₂ > isNat₁ > tt
proper₁ > take₂ > uTake2₄ > cons₂ > mark₁ > TAKE₂
proper₁ > uTake1₁ > mark₁ > TAKE₂
proper₁ > uTake1₁ > nil
proper₁ > uLength₂ > mark₁ > TAKE₂

Status:

TAKE₂: [1,2]
mark₁: [1]
active₁: [1]
and₂: [2,1]
tt: multiset
isNatIList₁: multiset
isNat₁: multiset
0: multiset
zeros: multiset
cons₂: [1,2]
nil: multiset
take₂: [1,2]
uTake1₁: multiset
uTake2₄: [2,4,1,3]
uLength₂: [2,1]
proper₁: [1]
top: multiset

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(43) Obligation:

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

TAKE(ok(X1), ok(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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(44) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(ok(X1), ok(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)  =  TAKE(x1)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
and(x1, x2)  =  and(x1, x2)
tt  =  tt
mark(x1)  =  mark
isNatIList(x1)  =  x1
isNatList(x1)  =  isNatList(x1)
isNat(x1)  =  x1
0  =  0
s(x1)  =  x1
length(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  x1
nil  =  nil
take(x1, x2)  =  take(x1, x2)
uTake1(x1)  =  uTake1(x1)
uTake2(x1, x2, x3, x4)  =  uTake2(x1, x2, x4)
uLength(x1, x2)  =  x1
proper(x1)  =  proper(x1)
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

top > active₁ > 0 > tt > nil > ok₁
top > active₁ > 0 > tt > take₂ > ok₁
top > active₁ > 0 > mark > proper₁ > and₂ > ok₁
top > active₁ > 0 > mark > proper₁ > isNatList₁ > ok₁
top > active₁ > 0 > mark > proper₁ > zeros
top > active₁ > 0 > mark > proper₁ > uTake1₁ > ok₁
top > active₁ > 0 > mark > proper₁ > uTake2₃ > take₂ > ok₁

Status:

TAKE₁: [1]
ok₁: [1]
active₁: [1]
and₂: [1,2]
tt: multiset
mark: multiset
isNatList₁: [1]
0: multiset
zeros: multiset
nil: multiset
take₂: [1,2]
uTake1₁: [1]
uTake2₃: multiset
proper₁: [1]
top: multiset

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(45) 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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(46) PisEmptyProof (EQUIVALENT transformation)

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

(47) TRUE

(48) Obligation:

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

CONS(ok(X1), ok(X2)) → CONS(X1, X2)
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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

CONS(ok(X1), ok(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
ok(x1)  =  ok(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)
and(x1, x2)  =  and(x2)
tt  =  tt
isNatIList(x1)  =  x1
isNatList(x1)  =  isNatList(x1)
isNat(x1)  =  x1
0  =  0
s(x1)  =  s(x1)
length(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
take(x1, x2)  =  x2
uTake1(x1)  =  uTake1(x1)
uTake2(x1, x2, x3, x4)  =  uTake2(x1, x3, x4)
uLength(x1, x2)  =  uLength(x1, x2)
proper(x1)  =  proper(x1)
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

active₁ > and₁ > ok₁
active₁ > 0 > tt > s₁ > ok₁
active₁ > 0 > uTake1₁ > ok₁
active₁ > zeros > tt > s₁ > ok₁
active₁ > cons₂ > isNatList₁ > ok₁
active₁ > cons₂ > uTake2₃ > ok₁
active₁ > cons₂ > uLength₂ > ok₁
active₁ > nil > ok₁
proper₁ > and₁ > ok₁
proper₁ > 0 > tt > s₁ > ok₁
proper₁ > 0 > uTake1₁ > ok₁
proper₁ > zeros > tt > s₁ > ok₁
proper₁ > cons₂ > isNatList₁ > ok₁
proper₁ > cons₂ > uTake2₃ > ok₁
proper₁ > cons₂ > uLength₂ > ok₁
proper₁ > nil > ok₁

Status:

ok₁: [1]
active₁: [1]
and₁: [1]
tt: multiset
isNatList₁: [1]
0: multiset
s₁: [1]
zeros: multiset
cons₂: [1,2]
nil: multiset
uTake1₁: [1]
uTake2₃: [1,2,3]
uLength₂: [2,1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(50) 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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(51) 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)
active(x1)  =  active(x1)
and(x1, x2)  =  and(x1, x2)
tt  =  tt
isNatIList(x1)  =  isNatIList
isNatList(x1)  =  isNatList
isNat(x1)  =  isNat
0  =  0
s(x1)  =  x1
length(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
take(x1, x2)  =  take(x1, x2)
uTake1(x1)  =  uTake1(x1)
uTake2(x1, x2, x3, x4)  =  uTake2(x1, x2, x3, x4)
uLength(x1, x2)  =  uLength(x1, x2)
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

active₁ > and₂ > mark₁
active₁ > isNatIList > tt > cons₂ > mark₁
active₁ > isNatIList > tt > cons₂ > isNatList
active₁ > isNatIList > tt > nil
active₁ > isNatIList > tt > take₂ > mark₁
active₁ > isNatIList > isNat > mark₁
active₁ > isNatIList > isNat > isNatList
active₁ > 0 > mark₁
active₁ > zeros > cons₂ > mark₁
active₁ > zeros > cons₂ > isNatList
active₁ > uTake1₁ > mark₁
active₁ > uTake1₁ > nil
active₁ > uTake2₄ > mark₁
active₁ > uLength₂ > mark₁

Status:

mark₁: [1]
active₁: [1]
and₂: [1,2]
tt: multiset
isNatIList: []
isNatList: []
isNat: multiset
0: multiset
zeros: multiset
cons₂: [2,1]
nil: multiset
take₂: [2,1]
uTake1₁: [1]
uTake2₄: [1,2,4,3]
uLength₂: [1,2]
top: []

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(52) 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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(53) PisEmptyProof (EQUIVALENT transformation)

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

(54) TRUE

(55) Obligation:

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

LENGTH(ok(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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

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)  =  LENGTH(x1)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
and(x1, x2)  =  and(x1, x2)
tt  =  tt
isNatIList(x1)  =  isNatIList
isNatList(x1)  =  isNatList
isNat(x1)  =  isNat
0  =  0
s(x1)  =  x1
length(x1)  =  length(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
take(x1, x2)  =  take(x1, x2)
uTake1(x1)  =  uTake1(x1)
uTake2(x1, x2, x3, x4)  =  uTake2(x1, x2, x3, x4)
uLength(x1, x2)  =  uLength(x1, x2)
proper(x1)  =  proper(x1)
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

active₁ > isNatIList > proper₁ > tt > length₁ > and₂ > mark₁
active₁ > isNatIList > proper₁ > tt > cons₂ > mark₁
active₁ > isNatIList > proper₁ > tt > take₂ > mark₁
active₁ > isNatIList > proper₁ > isNat > mark₁
active₁ > isNatIList > proper₁ > zeros > mark₁
active₁ > isNatIList > proper₁ > nil > mark₁
active₁ > isNatIList > proper₁ > uTake1₁ > mark₁
active₁ > isNatIList > proper₁ > uTake2₄ > take₂ > mark₁
active₁ > isNatIList > proper₁ > uLength₂ > mark₁
active₁ > isNatList > proper₁ > tt > length₁ > and₂ > mark₁
active₁ > isNatList > proper₁ > tt > cons₂ > mark₁
active₁ > isNatList > proper₁ > tt > take₂ > mark₁
active₁ > isNatList > proper₁ > isNat > mark₁
active₁ > isNatList > proper₁ > zeros > mark₁
active₁ > isNatList > proper₁ > nil > mark₁
active₁ > isNatList > proper₁ > uTake1₁ > mark₁
active₁ > isNatList > proper₁ > uTake2₄ > take₂ > mark₁
active₁ > isNatList > proper₁ > uLength₂ > mark₁
active₁ > 0 > tt > length₁ > and₂ > mark₁
active₁ > 0 > tt > cons₂ > mark₁
active₁ > 0 > tt > take₂ > mark₁
active₁ > 0 > uTake1₁ > mark₁

Status:

LENGTH₁: [1]
mark₁: multiset
active₁: [1]
and₂: [1,2]
tt: multiset
isNatIList: multiset
isNatList: []
isNat: []
0: multiset
length₁: [1]
zeros: multiset
cons₂: [1,2]
nil: multiset
take₂: [1,2]
uTake1₁: multiset
uTake2₄: [3,1,4,2]
uLength₂: [2,1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(57) Obligation:

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

LENGTH(ok(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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

LENGTH(ok(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
ok(x1)  =  ok(x1)
active(x1)  =  x1
and(x1, x2)  =  x1
tt  =  tt
mark(x1)  =  mark
isNatIList(x1)  =  x1
isNatList(x1)  =  x1
isNat(x1)  =  x1
0  =  0
s(x1)  =  x1
length(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  x2
nil  =  nil
take(x1, x2)  =  x2
uTake1(x1)  =  x1
uTake2(x1, x2, x3, x4)  =  x1
uLength(x1, x2)  =  x2
proper(x1)  =  proper
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

proper > ok₁ > top > mark
proper > tt > nil > mark
proper > 0 > mark
proper > zeros > mark

Status:

ok₁: [1]
tt: multiset
mark: []
0: multiset
zeros: multiset
nil: multiset
proper: []
top: []

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(59) 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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(60) PisEmptyProof (EQUIVALENT transformation)

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

(61) TRUE

(62) Obligation:

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

S(ok(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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(63) 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)  =  S(x1)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
and(x1, x2)  =  and(x1, x2)
tt  =  tt
isNatIList(x1)  =  isNatIList
isNatList(x1)  =  isNatList
isNat(x1)  =  isNat
0  =  0
s(x1)  =  x1
length(x1)  =  length(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
take(x1, x2)  =  take(x1, x2)
uTake1(x1)  =  uTake1(x1)
uTake2(x1, x2, x3, x4)  =  uTake2(x1, x2, x3, x4)
uLength(x1, x2)  =  uLength(x1, x2)
proper(x1)  =  proper(x1)
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

active₁ > isNatIList > proper₁ > tt > length₁ > and₂ > mark₁
active₁ > isNatIList > proper₁ > tt > cons₂ > mark₁
active₁ > isNatIList > proper₁ > tt > take₂ > mark₁
active₁ > isNatIList > proper₁ > isNat > mark₁
active₁ > isNatIList > proper₁ > zeros > mark₁
active₁ > isNatIList > proper₁ > nil > mark₁
active₁ > isNatIList > proper₁ > uTake1₁ > mark₁
active₁ > isNatIList > proper₁ > uTake2₄ > take₂ > mark₁
active₁ > isNatIList > proper₁ > uLength₂ > mark₁
active₁ > isNatList > proper₁ > tt > length₁ > and₂ > mark₁
active₁ > isNatList > proper₁ > tt > cons₂ > mark₁
active₁ > isNatList > proper₁ > tt > take₂ > mark₁
active₁ > isNatList > proper₁ > isNat > mark₁
active₁ > isNatList > proper₁ > zeros > mark₁
active₁ > isNatList > proper₁ > nil > mark₁
active₁ > isNatList > proper₁ > uTake1₁ > mark₁
active₁ > isNatList > proper₁ > uTake2₄ > take₂ > mark₁
active₁ > isNatList > proper₁ > uLength₂ > mark₁
active₁ > 0 > tt > length₁ > and₂ > mark₁
active₁ > 0 > tt > cons₂ > mark₁
active₁ > 0 > tt > take₂ > mark₁
active₁ > 0 > uTake1₁ > mark₁

Status:

S₁: [1]
mark₁: multiset
active₁: [1]
and₂: [1,2]
tt: multiset
isNatIList: multiset
isNatList: []
isNat: []
0: multiset
length₁: [1]
zeros: multiset
cons₂: [1,2]
nil: multiset
take₂: [1,2]
uTake1₁: multiset
uTake2₄: [3,1,4,2]
uLength₂: [2,1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(64) Obligation:

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

S(ok(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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

S(ok(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
ok(x1)  =  ok(x1)
active(x1)  =  x1
and(x1, x2)  =  x1
tt  =  tt
mark(x1)  =  mark
isNatIList(x1)  =  x1
isNatList(x1)  =  x1
isNat(x1)  =  x1
0  =  0
s(x1)  =  x1
length(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  x2
nil  =  nil
take(x1, x2)  =  x2
uTake1(x1)  =  x1
uTake2(x1, x2, x3, x4)  =  x1
uLength(x1, x2)  =  x2
proper(x1)  =  proper
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

proper > ok₁ > top > mark
proper > tt > nil > mark
proper > 0 > mark
proper > zeros > mark

Status:

ok₁: [1]
tt: multiset
mark: []
0: multiset
zeros: multiset
nil: multiset
proper: []
top: []

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(66) 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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(67) PisEmptyProof (EQUIVALENT transformation)

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

(68) TRUE

(69) 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(ok(X1), ok(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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(70) 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)
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)  =  AND(x1, x2)
mark(x1)  =  mark(x1)
ok(x1)  =  x1
active(x1)  =  active(x1)
and(x1, x2)  =  and(x1, x2)
tt  =  tt
isNatIList(x1)  =  isNatIList
isNatList(x1)  =  isNatList
isNat(x1)  =  isNat
0  =  0
s(x1)  =  x1
length(x1)  =  length(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
take(x1, x2)  =  take(x1, x2)
uTake1(x1)  =  uTake1(x1)
uTake2(x1, x2, x3, x4)  =  uTake2(x1, x2, x3, x4)
uLength(x1, x2)  =  uLength(x1, x2)
proper(x1)  =  proper(x1)
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

AND₂ > mark₁
active₁ > isNatIList > and₂ > mark₁
active₁ > isNatIList > isNatList > tt > nil > mark₁
active₁ > isNatIList > isNat > tt > nil > mark₁
active₁ > 0 > mark₁
active₁ > length₁ > and₂ > mark₁
active₁ > length₁ > isNatList > tt > nil > mark₁
active₁ > length₁ > isNat > tt > nil > mark₁
active₁ > zeros > mark₁
active₁ > take₂ > uTake1₁ > nil > mark₁
active₁ > take₂ > uTake2₄ > cons₂ > mark₁
active₁ > uLength₂ > mark₁
proper₁ > isNatIList > and₂ > mark₁
proper₁ > isNatIList > isNatList > tt > nil > mark₁
proper₁ > isNatIList > isNat > tt > nil > mark₁
proper₁ > 0 > mark₁
proper₁ > length₁ > and₂ > mark₁
proper₁ > length₁ > isNatList > tt > nil > mark₁
proper₁ > length₁ > isNat > tt > nil > mark₁
proper₁ > zeros > mark₁
proper₁ > take₂ > uTake1₁ > nil > mark₁
proper₁ > take₂ > uTake2₄ > cons₂ > mark₁
proper₁ > uLength₂ > mark₁
top > mark₁

Status:

AND₂: multiset
mark₁: [1]
active₁: [1]
and₂: [1,2]
tt: multiset
isNatIList: multiset
isNatList: multiset
isNat: multiset
0: multiset
length₁: multiset
zeros: multiset
cons₂: [2,1]
nil: multiset
take₂: [1,2]
uTake1₁: [1]
uTake2₄: [1,3,2,4]
uLength₂: [2,1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(71) Obligation:

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

AND(ok(X1), ok(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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

AND(ok(X1), ok(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)  =  AND(x1)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
and(x1, x2)  =  and(x1, x2)
tt  =  tt
mark(x1)  =  mark(x1)
isNatIList(x1)  =  x1
isNatList(x1)  =  x1
isNat(x1)  =  isNat(x1)
0  =  0
s(x1)  =  s(x1)
length(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
take(x1, x2)  =  take(x1, x2)
uTake1(x1)  =  uTake1(x1)
uTake2(x1, x2, x3, x4)  =  uTake2(x1, x2, x3, x4)
uLength(x1, x2)  =  uLength(x1, x2)
proper(x1)  =  proper(x1)
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

AND₁ > ok₁
active₁ > isNat₁ > tt > take₂ > uTake1₁ > mark₁ > ok₁
active₁ > isNat₁ > tt > take₂ > uTake2₄ > cons₂ > mark₁ > ok₁
active₁ > s₁ > and₂ > mark₁ > ok₁
active₁ > s₁ > uTake2₄ > cons₂ > mark₁ > ok₁
active₁ > zeros > 0 > mark₁ > ok₁
active₁ > zeros > cons₂ > mark₁ > ok₁
active₁ > nil > tt > take₂ > uTake1₁ > mark₁ > ok₁
active₁ > nil > tt > take₂ > uTake2₄ > cons₂ > mark₁ > ok₁
active₁ > uLength₂ > mark₁ > ok₁
proper₁ > isNat₁ > tt > take₂ > uTake1₁ > mark₁ > ok₁
proper₁ > isNat₁ > tt > take₂ > uTake2₄ > cons₂ > mark₁ > ok₁
proper₁ > s₁ > and₂ > mark₁ > ok₁
proper₁ > s₁ > uTake2₄ > cons₂ > mark₁ > ok₁
proper₁ > zeros > 0 > mark₁ > ok₁
proper₁ > zeros > cons₂ > mark₁ > ok₁
proper₁ > nil > tt > take₂ > uTake1₁ > mark₁ > ok₁
proper₁ > nil > tt > take₂ > uTake2₄ > cons₂ > mark₁ > ok₁
proper₁ > uLength₂ > mark₁ > ok₁
top > ok₁

Status:

AND₁: multiset
ok₁: [1]
active₁: [1]
and₂: [1,2]
tt: multiset
mark₁: [1]
isNat₁: multiset
0: multiset
s₁: [1]
zeros: multiset
cons₂: [1,2]
nil: multiset
take₂: [1,2]
uTake1₁: [1]
uTake2₄: [2,4,1,3]
uLength₂: [2,1]
proper₁: [1]
top: multiset

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(73) 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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(74) PisEmptyProof (EQUIVALENT transformation)

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

(75) TRUE

(76) Obligation:

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

PROPER(and(X1, X2)) → PROPER(X2)
PROPER(and(X1, X2)) → PROPER(X1)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(isNatList(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(take(X1, X2)) → PROPER(X2)
PROPER(uTake1(X)) → PROPER(X)
PROPER(uTake2(X1, X2, X3, X4)) → PROPER(X1)
PROPER(uTake2(X1, X2, X3, X4)) → PROPER(X2)
PROPER(uTake2(X1, X2, X3, X4)) → PROPER(X3)
PROPER(uTake2(X1, X2, X3, X4)) → PROPER(X4)
PROPER(uLength(X1, X2)) → PROPER(X1)
PROPER(uLength(X1, X2)) → PROPER(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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(77) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(and(X1, X2)) → PROPER(X2)
PROPER(and(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(take(X1, X2)) → PROPER(X2)
PROPER(uTake2(X1, X2, X3, X4)) → PROPER(X1)
PROPER(uTake2(X1, X2, X3, X4)) → PROPER(X2)
PROPER(uTake2(X1, X2, X3, X4)) → PROPER(X3)
PROPER(uTake2(X1, X2, X3, X4)) → PROPER(X4)
PROPER(uLength(X1, X2)) → PROPER(X1)
PROPER(uLength(X1, X2)) → PROPER(X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
and(x1, x2)  =  and(x1, x2)
isNatIList(x1)  =  x1
isNatList(x1)  =  x1
isNat(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
take(x1, x2)  =  take(x1, x2)
uTake1(x1)  =  x1
uTake2(x1, x2, x3, x4)  =  uTake2(x1, x2, x3, x4)
uLength(x1, x2)  =  uLength(x1, x2)
active(x1)  =  active(x1)
tt  =  tt
mark(x1)  =  x1
0  =  0
zeros  =  zeros
nil  =  nil
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

active₁ > and₂ > PROPER₁
active₁ > uTake2₄ > cons₂ > uLength₂ > PROPER₁
active₁ > uTake2₄ > take₂ > PROPER₁
active₁ > zeros > 0 > PROPER₁
active₁ > nil > tt > take₂ > PROPER₁
top > PROPER₁

Status:

PROPER₁: multiset
and₂: [2,1]
cons₂: multiset
take₂: multiset
uTake2₄: [3,4,1,2]
uLength₂: [1,2]
active₁: multiset
tt: multiset
0: multiset
zeros: multiset
nil: multiset
top: []

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(78) Obligation:

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

PROPER(isNatIList(X)) → PROPER(X)
PROPER(isNatList(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)
PROPER(uTake1(X)) → PROPER(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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

PROPER(isNatIList(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  x1
isNatIList(x1)  =  isNatIList(x1)
isNatList(x1)  =  x1
isNat(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
uTake1(x1)  =  x1
active(x1)  =  active(x1)
and(x1, x2)  =  and(x1, x2)
tt  =  tt
mark(x1)  =  x1
0  =  0
zeros  =  zeros
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
take(x1, x2)  =  take(x1, x2)
uTake2(x1, x2, x3, x4)  =  uTake2(x1, x2, x3, x4)
uLength(x1, x2)  =  x2
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

active₁ > isNatIList₁ > and₂
active₁ > isNatIList₁ > tt > cons₂
active₁ > isNatIList₁ > tt > take₂
active₁ > zeros > 0
active₁ > zeros > cons₂
active₁ > nil
active₁ > uTake2₄ > cons₂
active₁ > uTake2₄ > take₂
proper₁ > isNatIList₁ > and₂
proper₁ > isNatIList₁ > tt > cons₂
proper₁ > isNatIList₁ > tt > take₂
proper₁ > zeros > 0
proper₁ > zeros > cons₂
proper₁ > nil
proper₁ > uTake2₄ > cons₂
proper₁ > uTake2₄ > take₂

Status:

isNatIList₁: multiset
active₁: [1]
and₂: [2,1]
tt: multiset
0: multiset
zeros: multiset
cons₂: [1,2]
nil: multiset
take₂: [2,1]
uTake2₄: [4,1,3,2]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(80) Obligation:

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

PROPER(isNatList(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)
PROPER(uTake1(X)) → PROPER(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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

PROPER(uTake1(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
isNatList(x1)  =  x1
isNat(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
uTake1(x1)  =  uTake1(x1)
active(x1)  =  active(x1)
and(x1, x2)  =  x2
tt  =  tt
mark(x1)  =  x1
isNatIList(x1)  =  isNatIList(x1)
0  =  0
zeros  =  zeros
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
take(x1, x2)  =  x2
uTake2(x1, x2, x3, x4)  =  uTake2(x1, x3, x4)
uLength(x1, x2)  =  x2
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

active₁ > uTake1₁ > PROPER₁
active₁ > tt
active₁ > isNatIList₁
active₁ > 0
active₁ > zeros
active₁ > cons₂ > uTake2₃
active₁ > nil

Status:

PROPER₁: multiset
uTake1₁: [1]
active₁: [1]
tt: multiset
isNatIList₁: [1]
0: multiset
zeros: multiset
cons₂: [1,2]
nil: multiset
uTake2₃: [3,1,2]
top: multiset

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(82) Obligation:

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

PROPER(isNatList(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(83) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(length(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
isNatList(x1)  =  x1
isNat(x1)  =  x1
s(x1)  =  x1
length(x1)  =  length(x1)
active(x1)  =  active(x1)
and(x1, x2)  =  and
tt  =  tt
mark(x1)  =  mark
isNatIList(x1)  =  x1
0  =  0
zeros  =  zeros
cons(x1, x2)  =  x1
nil  =  nil
take(x1, x2)  =  take(x1)
uTake1(x1)  =  x1
uTake2(x1, x2, x3, x4)  =  uTake2(x1, x2, x3, x4)
uLength(x1, x2)  =  uLength(x1, x2)
proper(x1)  =  proper(x1)
ok(x1)  =  ok
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

PROPER₁ > top
and > active₁ > length₁ > uLength₂ > mark > top
and > active₁ > length₁ > uLength₂ > ok > uTake2₄ > top
and > active₁ > zeros > top
and > active₁ > take₁ > mark > top
and > active₁ > take₁ > ok > uTake2₄ > top
and > proper₁ > length₁ > uLength₂ > mark > top
and > proper₁ > length₁ > uLength₂ > ok > uTake2₄ > top
and > proper₁ > zeros > top
and > proper₁ > take₁ > mark > top
and > proper₁ > take₁ > ok > uTake2₄ > top
tt > length₁ > uLength₂ > mark > top
tt > length₁ > uLength₂ > ok > uTake2₄ > top
tt > take₁ > mark > top
tt > take₁ > ok > uTake2₄ > top
0 > mark > top
0 > ok > uTake2₄ > top
nil > mark > top
nil > ok > uTake2₄ > top

Status:

PROPER₁: [1]
length₁: [1]
active₁: [1]
and: []
tt: multiset
mark: multiset
0: multiset
zeros: multiset
nil: multiset
take₁: multiset
uTake2₄: multiset
uLength₂: [2,1]
proper₁: multiset
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(84) Obligation:

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

PROPER(isNatList(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)
PROPER(s(X)) → PROPER(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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(85) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(isNatList(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  x1
isNatList(x1)  =  isNatList(x1)
isNat(x1)  =  isNat(x1)
s(x1)  =  x1
active(x1)  =  active(x1)
and(x1, x2)  =  x2
tt  =  tt
mark(x1)  =  x1
isNatIList(x1)  =  isNatIList(x1)
0  =  0
length(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  cons(x2)
nil  =  nil
take(x1, x2)  =  x2
uTake1(x1)  =  x1
uTake2(x1, x2, x3, x4)  =  x4
uLength(x1, x2)  =  uLength(x2)
proper(x1)  =  x1
ok(x1)  =  ok
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

active₁ > isNatIList₁ > isNat₁ > isNatList₁ > tt > ok
active₁ > 0 > tt > ok
active₁ > cons₁ > ok
active₁ > nil > tt > ok
active₁ > uLength₁ > ok
zeros > 0 > tt > ok
zeros > cons₁ > ok

Status:

isNatList₁: multiset
isNat₁: [1]
active₁: [1]
tt: multiset
isNatIList₁: multiset
0: multiset
zeros: multiset
cons₁: [1]
nil: multiset
uLength₁: [1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(86) Obligation:

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

PROPER(s(X)) → PROPER(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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(87) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(s(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  x1
s(x1)  =  s(x1)
active(x1)  =  x1
and(x1, x2)  =  x1
tt  =  tt
mark(x1)  =  mark
isNatIList(x1)  =  isNatIList(x1)
isNatList(x1)  =  isNatList
isNat(x1)  =  isNat
0  =  0
length(x1)  =  length
zeros  =  zeros
cons(x1, x2)  =  x1
nil  =  nil
take(x1, x2)  =  x1
uTake1(x1)  =  x1
uTake2(x1, x2, x3, x4)  =  uTake2(x1, x2, x3, x4)
uLength(x1, x2)  =  x2
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

tt > length > isNat > proper₁ > s₁ > mark
tt > length > isNat > proper₁ > isNatIList₁ > mark
tt > length > isNat > proper₁ > zeros > mark
tt > length > isNat > proper₁ > uTake2₄ > mark
isNatList > isNat > proper₁ > s₁ > mark
isNatList > isNat > proper₁ > isNatIList₁ > mark
isNatList > isNat > proper₁ > zeros > mark
isNatList > isNat > proper₁ > uTake2₄ > mark
0 > isNatIList₁ > mark
nil > mark
top > mark

Status:

s₁: [1]
tt: multiset
mark: multiset
isNatIList₁: multiset
isNatList: []
isNat: []
0: multiset
length: []
zeros: multiset
nil: multiset
uTake2₄: [4,2,1,3]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(88) 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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(89) PisEmptyProof (EQUIVALENT transformation)

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

(90) TRUE

(91) Obligation:

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

ACTIVE(and(X1, X2)) → ACTIVE(X2)
ACTIVE(and(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X2)
ACTIVE(uTake1(X)) → ACTIVE(X)
ACTIVE(uTake2(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(uLength(X1, X2)) → ACTIVE(X1)

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(92) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(and(X1, X2)) → ACTIVE(X2)
ACTIVE(and(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
and(x1, x2)  =  and(x1, x2)
s(x1)  =  x1
length(x1)  =  x1
cons(x1, x2)  =  x1
take(x1, x2)  =  take(x1, x2)
uTake1(x1)  =  x1
uTake2(x1, x2, x3, x4)  =  x1
uLength(x1, x2)  =  x1
active(x1)  =  x1
tt  =  tt
mark(x1)  =  mark
isNatIList(x1)  =  isNatIList
isNatList(x1)  =  isNatList(x1)
isNat(x1)  =  isNat(x1)
0  =  0
zeros  =  zeros
nil  =  nil
proper(x1)  =  proper(x1)
ok(x1)  =  ok
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

nil > tt > take₂ > ok > mark
proper₁ > take₂ > ok > mark
proper₁ > isNatIList > isNatList₁ > and₂ > ok > mark
proper₁ > isNat₁ > isNatList₁ > and₂ > ok > mark
proper₁ > zeros > 0 > ok > mark
top > mark

Status:

and₂: [1,2]
take₂: [2,1]
tt: multiset
mark: []
isNatIList: multiset
isNatList₁: multiset
isNat₁: multiset
0: multiset
zeros: multiset
nil: multiset
proper₁: [1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(93) Obligation:

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

ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(uTake1(X)) → ACTIVE(X)
ACTIVE(uTake2(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(uLength(X1, X2)) → ACTIVE(X1)

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(94) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(cons(X1, X2)) → ACTIVE(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
s(x1)  =  x1
length(x1)  =  x1
cons(x1, x2)  =  cons(x1)
uTake1(x1)  =  x1
uTake2(x1, x2, x3, x4)  =  x1
uLength(x1, x2)  =  x1
active(x1)  =  active(x1)
and(x1, x2)  =  and(x2)
tt  =  tt
mark(x1)  =  mark
isNatIList(x1)  =  x1
isNatList(x1)  =  x1
isNat(x1)  =  isNat
0  =  0
zeros  =  zeros
nil  =  nil
take(x1, x2)  =  take(x1, x2)
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

ACTIVE₁ > mark
active₁ > and₁ > mark
active₁ > zeros > cons₁ > mark
active₁ > zeros > 0 > tt > mark
active₁ > nil > tt > mark
active₁ > take₂ > isNat > tt > mark
proper₁ > and₁ > mark
proper₁ > zeros > cons₁ > mark
proper₁ > zeros > 0 > tt > mark
proper₁ > nil > tt > mark
proper₁ > take₂ > isNat > tt > mark
top > mark

Status:

ACTIVE₁: [1]
cons₁: multiset
active₁: [1]
and₁: multiset
tt: multiset
mark: multiset
isNat: multiset
0: multiset
zeros: multiset
nil: multiset
take₂: multiset
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(95) Obligation:

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

ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(uTake1(X)) → ACTIVE(X)
ACTIVE(uTake2(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(uLength(X1, X2)) → ACTIVE(X1)

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(96) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(uLength(X1, X2)) → ACTIVE(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
uTake1(x1)  =  x1
uTake2(x1, x2, x3, x4)  =  x1
uLength(x1, x2)  =  uLength(x1, x2)
active(x1)  =  active(x1)
and(x1, x2)  =  and(x1, x2)
tt  =  tt
mark(x1)  =  x1
isNatIList(x1)  =  isNatIList
isNatList(x1)  =  isNatList
isNat(x1)  =  isNat
0  =  0
zeros  =  zeros
cons(x1, x2)  =  x2
nil  =  nil
take(x1, x2)  =  take
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

active₁ > and₂ > uLength₂
active₁ > 0 > isNatIList > isNat > isNatList > uLength₂
active₁ > nil > tt > take > isNat > isNatList > uLength₂
zeros > tt > take > isNat > isNatList > uLength₂
zeros > 0 > isNatIList > isNat > isNatList > uLength₂
proper₁ > and₂ > uLength₂
proper₁ > 0 > isNatIList > isNat > isNatList > uLength₂
proper₁ > take > isNat > isNatList > uLength₂
top > uLength₂

Status:

uLength₂: [1,2]
active₁: [1]
and₂: [1,2]
tt: multiset
isNatIList: multiset
isNatList: []
isNat: multiset
0: multiset
zeros: multiset
nil: multiset
take: multiset
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(97) Obligation:

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

ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(uTake1(X)) → ACTIVE(X)
ACTIVE(uTake2(X1, X2, X3, X4)) → ACTIVE(X1)

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(98) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(uTake2(X1, X2, X3, X4)) → ACTIVE(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
s(x1)  =  x1
length(x1)  =  x1
uTake1(x1)  =  x1
uTake2(x1, x2, x3, x4)  =  uTake2(x1, x2, x3, x4)
active(x1)  =  active(x1)
and(x1, x2)  =  and
tt  =  tt
mark(x1)  =  mark
isNatIList(x1)  =  x1
isNatList(x1)  =  isNatList(x1)
isNat(x1)  =  isNat
0  =  0
zeros  =  zeros
cons(x1, x2)  =  x2
nil  =  nil
take(x1, x2)  =  take(x1, x2)
uLength(x1, x2)  =  x1
proper(x1)  =  proper(x1)
ok(x1)  =  ok
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

ACTIVE₁ > mark
proper₁ > and > active₁ > uTake2₄ > take₂ > ok > mark
proper₁ > and > active₁ > isNatList₁ > tt > mark
proper₁ > and > active₁ > isNatList₁ > ok > mark
proper₁ > and > active₁ > isNat > tt > mark
proper₁ > and > active₁ > isNat > ok > mark
proper₁ > and > active₁ > zeros > 0 > tt > mark
proper₁ > and > active₁ > zeros > 0 > ok > mark
proper₁ > and > active₁ > nil > mark
top > mark

Status:

ACTIVE₁: [1]
uTake2₄: multiset
active₁: multiset
and: []
tt: multiset
mark: []
isNatList₁: multiset
isNat: multiset
0: multiset
zeros: multiset
nil: multiset
take₂: [1,2]
proper₁: [1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(99) Obligation:

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

ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(uTake1(X)) → ACTIVE(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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

ACTIVE(uTake1(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
s(x1)  =  x1
length(x1)  =  x1
uTake1(x1)  =  uTake1(x1)
active(x1)  =  active(x1)
and(x1, x2)  =  and(x1, x2)
tt  =  tt
mark(x1)  =  x1
isNatIList(x1)  =  isNatIList
isNatList(x1)  =  isNatList
isNat(x1)  =  isNat
0  =  0
zeros  =  zeros
cons(x1, x2)  =  cons(x2)
nil  =  nil
take(x1, x2)  =  x2
uTake2(x1, x2, x3, x4)  =  uTake2(x4)
uLength(x1, x2)  =  uLength(x1, x2)
proper(x1)  =  proper(x1)
ok(x1)  =  ok
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

proper₁ > isNatList > isNatIList > tt
proper₁ > isNatList > isNatIList > ok > uTake1₁
proper₁ > isNatList > isNatIList > ok > and₂
proper₁ > isNat > tt
proper₁ > isNat > ok > uTake1₁
proper₁ > isNat > ok > and₂
proper₁ > 0 > isNatIList > tt
proper₁ > 0 > isNatIList > ok > uTake1₁
proper₁ > 0 > isNatIList > ok > and₂
proper₁ > zeros > tt
proper₁ > zeros > ok > uTake1₁
proper₁ > zeros > ok > and₂
proper₁ > cons₁ > ok > uTake1₁
proper₁ > cons₁ > ok > and₂
proper₁ > nil
proper₁ > uTake2₁ > ok > uTake1₁
proper₁ > uTake2₁ > ok > and₂
proper₁ > uLength₂ > ok > uTake1₁
proper₁ > uLength₂ > ok > and₂
top > active₁ > isNatList > isNatIList > tt
top > active₁ > isNatList > isNatIList > ok > uTake1₁
top > active₁ > isNatList > isNatIList > ok > and₂
top > active₁ > isNat > tt
top > active₁ > isNat > ok > uTake1₁
top > active₁ > isNat > ok > and₂
top > active₁ > 0 > isNatIList > tt
top > active₁ > 0 > isNatIList > ok > uTake1₁
top > active₁ > 0 > isNatIList > ok > and₂
top > active₁ > zeros > tt
top > active₁ > zeros > ok > uTake1₁
top > active₁ > zeros > ok > and₂
top > active₁ > cons₁ > ok > uTake1₁
top > active₁ > cons₁ > ok > and₂
top > active₁ > nil
top > active₁ > uTake2₁ > ok > uTake1₁
top > active₁ > uTake2₁ > ok > and₂
top > active₁ > uLength₂ > ok > uTake1₁
top > active₁ > uLength₂ > ok > and₂

Status:

ACTIVE₁: [1]
uTake1₁: multiset
active₁: [1]
and₂: [2,1]
tt: multiset
isNatIList: []
isNatList: []
isNat: []
0: multiset
zeros: multiset
cons₁: [1]
nil: multiset
uTake2₁: multiset
uLength₂: multiset
proper₁: [1]
ok: []
top: multiset

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(101) Obligation:

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

ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

ACTIVE(length(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
s(x1)  =  x1
length(x1)  =  length(x1)
active(x1)  =  active(x1)
and(x1, x2)  =  x1
tt  =  tt
mark(x1)  =  mark
isNatIList(x1)  =  isNatIList(x1)
isNatList(x1)  =  isNatList
isNat(x1)  =  x1
0  =  0
zeros  =  zeros
cons(x1, x2)  =  x1
nil  =  nil
take(x1, x2)  =  x2
uTake1(x1)  =  x1
uTake2(x1, x2, x3, x4)  =  uTake2
uLength(x1, x2)  =  uLength(x1, x2)
proper(x1)  =  proper(x1)
ok(x1)  =  ok
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

active₁ > isNatList > isNatIList₁ > mark
active₁ > isNatList > ok > length₁ > ACTIVE₁ > mark
active₁ > isNatList > ok > top > mark
active₁ > 0 > isNatIList₁ > mark
active₁ > 0 > ok > length₁ > ACTIVE₁ > mark
active₁ > 0 > ok > top > mark
active₁ > zeros > mark
active₁ > uLength₂ > ok > length₁ > ACTIVE₁ > mark
active₁ > uLength₂ > ok > top > mark
tt > ok > length₁ > ACTIVE₁ > mark
tt > ok > top > mark
nil > ok > length₁ > ACTIVE₁ > mark
nil > ok > top > mark
uTake2 > ok > length₁ > ACTIVE₁ > mark
uTake2 > ok > top > mark
proper₁ > isNatList > isNatIList₁ > mark
proper₁ > isNatList > ok > length₁ > ACTIVE₁ > mark
proper₁ > isNatList > ok > top > mark
proper₁ > 0 > isNatIList₁ > mark
proper₁ > 0 > ok > length₁ > ACTIVE₁ > mark
proper₁ > 0 > ok > top > mark
proper₁ > zeros > mark
proper₁ > uLength₂ > ok > length₁ > ACTIVE₁ > mark
proper₁ > uLength₂ > ok > top > mark

Status:

ACTIVE₁: multiset
length₁: [1]
active₁: [1]
tt: multiset
mark: multiset
isNatIList₁: multiset
isNatList: multiset
0: multiset
zeros: multiset
nil: multiset
uTake2: []
uLength₂: multiset
proper₁: [1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(103) Obligation:

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

ACTIVE(s(X)) → ACTIVE(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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

ACTIVE(s(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
s(x1)  =  s(x1)
active(x1)  =  x1
and(x1, x2)  =  x1
tt  =  tt
mark(x1)  =  mark
isNatIList(x1)  =  isNatIList(x1)
isNatList(x1)  =  isNatList
isNat(x1)  =  isNat
0  =  0
length(x1)  =  length
zeros  =  zeros
cons(x1, x2)  =  x1
nil  =  nil
take(x1, x2)  =  x1
uTake1(x1)  =  x1
uTake2(x1, x2, x3, x4)  =  uTake2(x1, x2, x3, x4)
uLength(x1, x2)  =  x2
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Precedence:

tt > length > isNat > proper₁ > s₁ > mark
tt > length > isNat > proper₁ > isNatIList₁ > mark
tt > length > isNat > proper₁ > zeros > mark
tt > length > isNat > proper₁ > uTake2₄ > mark
isNatList > isNat > proper₁ > s₁ > mark
isNatList > isNat > proper₁ > isNatIList₁ > mark
isNatList > isNat > proper₁ > zeros > mark
isNatList > isNat > proper₁ > uTake2₄ > mark
0 > isNatIList₁ > mark
nil > mark
top > mark

Status:

s₁: [1]
tt: multiset
mark: multiset
isNatIList₁: multiset
isNatList: []
isNat: []
0: multiset
length: []
zeros: multiset
nil: multiset
uTake2₄: [4,2,1,3]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(105) 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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(106) PisEmptyProof (EQUIVALENT transformation)

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

(107) TRUE

(108) Obligation:

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

TOP(ok(X)) → TOP(active(X))
TOP(mark(X)) → TOP(proper(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)))
active(and(X1, X2)) → and(active(X1), X2)
active(and(X1, X2)) → and(X1, active(X2))
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(uTake1(X)) → uTake1(active(X))
active(uTake2(X1, X2, X3, X4)) → uTake2(active(X1), X2, X3, X4)
active(uLength(X1, X2)) → uLength(active(X1), X2)
and(mark(X1), X2) → mark(and(X1, X2))
and(X1, mark(X2)) → mark(and(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
uTake1(mark(X)) → mark(uTake1(X))
uTake2(mark(X1), X2, X3, X4) → mark(uTake2(X1, X2, X3, X4))
uLength(mark(X1), X2) → mark(uLength(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNatIList(X)) → isNatIList(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(uTake1(X)) → uTake1(proper(X))
proper(uTake2(X1, X2, X3, X4)) → uTake2(proper(X1), proper(X2), proper(X3), proper(X4))
proper(uLength(X1, X2)) → uLength(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
isNatList(ok(X)) → ok(isNatList(X))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
uTake1(ok(X)) → ok(uTake1(X))
uTake2(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(uTake2(X1, X2, X3, X4))
uLength(ok(X1), ok(X2)) → ok(uLength(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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