(0) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(zeros) → MARK(cons(0, zeros))
ACTIVE(zeros) → CONS(0, zeros)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
ACTIVE(U11(tt, L)) → S(length(L))
ACTIVE(U11(tt, L)) → LENGTH(L)
ACTIVE(U21(tt)) → MARK(nil)
ACTIVE(U31(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U31(tt, IL, M, N)) → CONS(N, take(M, IL))
ACTIVE(U31(tt, IL, M, N)) → TAKE(M, IL)
ACTIVE(and(tt, X)) → MARK(X)
ACTIVE(isNat(0)) → MARK(tt)
ACTIVE(isNat(length(V1))) → MARK(isNatList(V1))
ACTIVE(isNat(length(V1))) → ISNATLIST(V1)
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
ACTIVE(isNat(s(V1))) → ISNAT(V1)
ACTIVE(isNatIList(V)) → MARK(isNatList(V))
ACTIVE(isNatIList(V)) → ISNATLIST(V)
ACTIVE(isNatIList(zeros)) → MARK(tt)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(and(isNat(V1), isNatIList(V2)))
ACTIVE(isNatIList(cons(V1, V2))) → AND(isNat(V1), isNatIList(V2))
ACTIVE(isNatIList(cons(V1, V2))) → ISNAT(V1)
ACTIVE(isNatIList(cons(V1, V2))) → ISNATILIST(V2)
ACTIVE(isNatList(nil)) → MARK(tt)
ACTIVE(isNatList(cons(V1, V2))) → MARK(and(isNat(V1), isNatList(V2)))
ACTIVE(isNatList(cons(V1, V2))) → AND(isNat(V1), isNatList(V2))
ACTIVE(isNatList(cons(V1, V2))) → ISNAT(V1)
ACTIVE(isNatList(cons(V1, V2))) → ISNATLIST(V2)
ACTIVE(isNatList(take(V1, V2))) → MARK(and(isNat(V1), isNatIList(V2)))
ACTIVE(isNatList(take(V1, V2))) → AND(isNat(V1), isNatIList(V2))
ACTIVE(isNatList(take(V1, V2))) → ISNAT(V1)
ACTIVE(isNatList(take(V1, V2))) → ISNATILIST(V2)
ACTIVE(length(nil)) → MARK(0)
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
ACTIVE(length(cons(N, L))) → U111(and(isNatList(L), isNat(N)), L)
ACTIVE(length(cons(N, L))) → AND(isNatList(L), isNat(N))
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
ACTIVE(length(cons(N, L))) → ISNAT(N)
ACTIVE(take(0, IL)) → MARK(U21(isNatIList(IL)))
ACTIVE(take(0, IL)) → U211(isNatIList(IL))
ACTIVE(take(0, IL)) → ISNATILIST(IL)
ACTIVE(take(s(M), cons(N, IL))) → MARK(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
ACTIVE(take(s(M), cons(N, IL))) → U311(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNatIList(IL), and(isNat(M), isNat(N)))
ACTIVE(take(s(M), cons(N, IL))) → ISNATILIST(IL)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNat(M), isNat(N))
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(M)
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(N)
MARK(zeros) → ACTIVE(zeros)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(0) → ACTIVE(0)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U11(X1, X2)) → U111(mark(X1), X2)
MARK(U11(X1, X2)) → MARK(X1)
MARK(tt) → ACTIVE(tt)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → S(mark(X))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(length(X)) → LENGTH(mark(X))
MARK(length(X)) → MARK(X)
MARK(U21(X)) → ACTIVE(U21(mark(X)))
MARK(U21(X)) → U211(mark(X))
MARK(U21(X)) → MARK(X)
MARK(nil) → ACTIVE(nil)
MARK(U31(X1, X2, X3, X4)) → ACTIVE(U31(mark(X1), X2, X3, X4))
MARK(U31(X1, X2, X3, X4)) → U311(mark(X1), X2, X3, X4)
MARK(U31(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(take(X1, X2)) → TAKE(mark(X1), mark(X2))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
MARK(and(X1, X2)) → AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(X1, mark(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
S(mark(X)) → S(X)
S(active(X)) → S(X)
LENGTH(mark(X)) → LENGTH(X)
LENGTH(active(X)) → LENGTH(X)
U211(mark(X)) → U211(X)
U211(active(X)) → U211(X)
U311(mark(X1), X2, X3, X4) → U311(X1, X2, X3, X4)
U311(X1, mark(X2), X3, X4) → U311(X1, X2, X3, X4)
U311(X1, X2, mark(X3), X4) → U311(X1, X2, X3, X4)
U311(X1, X2, X3, mark(X4)) → U311(X1, X2, X3, X4)
U311(active(X1), X2, X3, X4) → U311(X1, X2, X3, X4)
U311(X1, active(X2), X3, X4) → U311(X1, X2, X3, X4)
U311(X1, X2, active(X3), X4) → U311(X1, X2, X3, X4)
U311(X1, X2, X3, active(X4)) → U311(X1, X2, X3, X4)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(X1, mark(X2)) → TAKE(X1, X2)
TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)
AND(mark(X1), X2) → AND(X1, X2)
AND(X1, mark(X2)) → AND(X1, X2)
AND(active(X1), X2) → AND(X1, X2)
AND(X1, active(X2)) → AND(X1, X2)
ISNAT(mark(X)) → ISNAT(X)
ISNAT(active(X)) → ISNAT(X)
ISNATLIST(mark(X)) → ISNATLIST(X)
ISNATLIST(active(X)) → ISNATLIST(X)
ISNATILIST(mark(X)) → ISNATILIST(X)
ISNATILIST(active(X)) → ISNATILIST(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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 12 SCCs with 45 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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.


ISNATILIST(active(X)) → ISNATILIST(X)
ISNATILIST(mark(X)) → ISNATILIST(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATILIST(x1)  =  ISNATILIST(x1)
active(x1)  =  active(x1)
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
s(x1)  =  s
length(x1)  =  length
U21(x1)  =  U21
nil  =  nil
U31(x1, x2, x3, x4)  =  U31
take(x1, x2)  =  take
and(x1, x2)  =  x2
isNat(x1)  =  isNat
isNatList(x1)  =  isNatList
isNatIList(x1)  =  isNatIList

Recursive Path Order [RPO].
Precedence:
zeros > [tt, isNat, isNatList, isNatIList] > [U11, s, length] > 0 > [ISNATILIST1, active1, mark1, cons, U21, nil, U31, take]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

(7) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(active(X)) → ISNATLIST(X)
ISNATLIST(mark(X)) → ISNATLIST(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(active(X)) → ISNATLIST(X)
ISNATLIST(mark(X)) → ISNATLIST(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATLIST(x1)  =  ISNATLIST(x1)
active(x1)  =  active(x1)
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
s(x1)  =  s
length(x1)  =  length
U21(x1)  =  U21
nil  =  nil
U31(x1, x2, x3, x4)  =  U31
take(x1, x2)  =  take
and(x1, x2)  =  x2
isNat(x1)  =  isNat
isNatList(x1)  =  isNatList
isNatIList(x1)  =  isNatIList

Recursive Path Order [RPO].
Precedence:
zeros > [tt, isNat, isNatList, isNatIList] > [U11, s, length] > 0 > [ISNATLIST1, active1, mark1, cons, U21, nil, U31, take]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

(12) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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:

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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.


ISNAT(active(X)) → ISNAT(X)
ISNAT(mark(X)) → ISNAT(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNAT(x1)  =  ISNAT(x1)
active(x1)  =  active(x1)
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
s(x1)  =  s
length(x1)  =  length
U21(x1)  =  U21
nil  =  nil
U31(x1, x2, x3, x4)  =  U31
take(x1, x2)  =  take
and(x1, x2)  =  x2
isNat(x1)  =  isNat
isNatList(x1)  =  isNatList
isNatIList(x1)  =  isNatIList

Recursive Path Order [RPO].
Precedence:
zeros > [tt, isNat, isNatList, isNatIList] > [U11, s, length] > 0 > [ISNAT1, active1, mark1, cons, U21, nil, U31, take]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

(17) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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:

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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.


AND(X1, mark(X2)) → AND(X1, X2)
AND(X1, active(X2)) → AND(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AND(x1, x2)  =  AND(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
s(x1)  =  s
length(x1)  =  length
U21(x1)  =  U21
nil  =  nil
U31(x1, x2, x3, x4)  =  U31
take(x1, x2)  =  take
and(x1, x2)  =  x2
isNat(x1)  =  isNat
isNatList(x1)  =  isNatList
isNatIList(x1)  =  isNatIList

Recursive Path Order [RPO].
Precedence:
AND1 > [U21, nil]
[take, isNat] > [mark1, active1, zeros, cons, length, U31] > 0 > [tt, s, isNatList, isNatIList] > [U21, nil]
[take, isNat] > [mark1, active1, zeros, cons, length, U31] > U11 > [tt, s, isNatList, isNatIList] > [U21, nil]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

(22) Obligation:

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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.


AND(mark(X1), X2) → AND(X1, X2)
AND(active(X1), X2) → AND(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AND(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
s(x1)  =  s
length(x1)  =  length
U21(x1)  =  U21
nil  =  nil
U31(x1, x2, x3, x4)  =  U31
take(x1, x2)  =  take
and(x1, x2)  =  x2
isNat(x1)  =  isNat
isNatList(x1)  =  isNatList
isNatIList(x1)  =  isNatIList

Recursive Path Order [RPO].
Precedence:
[zeros, 0, length] > [mark1, active1, cons, U11, U31, take] > U21 > nil > tt > s
[zeros, 0, length] > [mark1, active1, cons, U11, U31, take] > isNat > [isNatList, isNatIList] > tt > s


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

(24) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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:

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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.


TAKE(X1, mark(X2)) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAKE(x1, x2)  =  TAKE(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
s(x1)  =  s
length(x1)  =  length
U21(x1)  =  U21
nil  =  nil
U31(x1, x2, x3, x4)  =  U31
take(x1, x2)  =  take
and(x1, x2)  =  x2
isNat(x1)  =  isNat
isNatList(x1)  =  isNatList
isNatIList(x1)  =  isNatIList

Recursive Path Order [RPO].
Precedence:
TAKE1 > [U21, nil]
[take, isNat] > [mark1, active1, zeros, cons, length, U31] > 0 > [tt, s, isNatList, isNatIList] > [U21, nil]
[take, isNat] > [mark1, active1, zeros, cons, length, U31] > U11 > [tt, s, isNatList, isNatIList] > [U21, nil]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

(29) Obligation:

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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.


TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(active(X1), X2) → TAKE(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAKE(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
s(x1)  =  s
length(x1)  =  length
U21(x1)  =  U21
nil  =  nil
U31(x1, x2, x3, x4)  =  U31
take(x1, x2)  =  take
and(x1, x2)  =  x2
isNat(x1)  =  isNat
isNatList(x1)  =  isNatList
isNatIList(x1)  =  isNatIList

Recursive Path Order [RPO].
Precedence:
[zeros, 0, length] > [mark1, active1, cons, U11, U31, take] > U21 > nil > tt > s
[zeros, 0, length] > [mark1, active1, cons, U11, U31, take] > isNat > [isNatList, isNatIList] > tt > s


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

(31) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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:

U311(X1, mark(X2), X3, X4) → U311(X1, X2, X3, X4)
U311(mark(X1), X2, X3, X4) → U311(X1, X2, X3, X4)
U311(X1, X2, mark(X3), X4) → U311(X1, X2, X3, X4)
U311(X1, X2, X3, mark(X4)) → U311(X1, X2, X3, X4)
U311(active(X1), X2, X3, X4) → U311(X1, X2, X3, X4)
U311(X1, active(X2), X3, X4) → U311(X1, X2, X3, X4)
U311(X1, X2, active(X3), X4) → U311(X1, X2, X3, X4)
U311(X1, X2, X3, active(X4)) → U311(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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.


U311(mark(X1), X2, X3, X4) → U311(X1, X2, X3, X4)
U311(X1, X2, X3, mark(X4)) → U311(X1, X2, X3, X4)
U311(active(X1), X2, X3, X4) → U311(X1, X2, X3, X4)
U311(X1, X2, X3, active(X4)) → U311(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U311(x1, x2, x3, x4)  =  U311(x1, x4)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  x2
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
s(x1)  =  s
length(x1)  =  length
U21(x1)  =  U21
nil  =  nil
U31(x1, x2, x3, x4)  =  U31
take(x1, x2)  =  take
and(x1, x2)  =  x2
isNat(x1)  =  isNat
isNatList(x1)  =  isNatList
isNatIList(x1)  =  isNatIList

Recursive Path Order [RPO].
Precedence:
zeros > [mark1, active1] > U31^12
zeros > [mark1, active1] > 0 > [tt, U21, nil]
zeros > [mark1, active1] > [U31, take] > [tt, U21, nil]
length > [U11, s] > [isNat, isNatList, isNatIList] > [mark1, active1] > U31^12
length > [U11, s] > [isNat, isNatList, isNatIList] > [mark1, active1] > 0 > [tt, U21, nil]
length > [U11, s] > [isNat, isNatList, isNatIList] > [mark1, active1] > [U31, take] > [tt, U21, nil]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

(36) Obligation:

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

U311(X1, mark(X2), X3, X4) → U311(X1, X2, X3, X4)
U311(X1, X2, mark(X3), X4) → U311(X1, X2, X3, X4)
U311(X1, active(X2), X3, X4) → U311(X1, X2, X3, X4)
U311(X1, X2, active(X3), X4) → U311(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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.


U311(X1, X2, mark(X3), X4) → U311(X1, X2, X3, X4)
U311(X1, X2, active(X3), X4) → U311(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U311(x1, x2, x3, x4)  =  U311(x1, x3, x4)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  x2
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
s(x1)  =  s
length(x1)  =  length
U21(x1)  =  U21
nil  =  nil
U31(x1, x2, x3, x4)  =  U31
take(x1, x2)  =  take
and(x1, x2)  =  x2
isNat(x1)  =  isNat
isNatList(x1)  =  isNatList
isNatIList(x1)  =  isNatIList

Recursive Path Order [RPO].
Precedence:
[U31, take] > [U11, s, length, isNatList, isNatIList] > [0, tt, nil] > [mark1, active1] > zeros
[U31, take] > U21 > [0, tt, nil] > [mark1, active1] > zeros
isNat > [U11, s, length, isNatList, isNatIList] > [0, tt, nil] > [mark1, active1] > zeros


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

(38) Obligation:

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

U311(X1, mark(X2), X3, X4) → U311(X1, X2, X3, X4)
U311(X1, active(X2), X3, X4) → U311(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

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

(39) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U311(X1, mark(X2), X3, X4) → U311(X1, X2, X3, X4)
U311(X1, active(X2), X3, X4) → U311(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U311(x1, x2, x3, x4)  =  U311(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
s(x1)  =  s
length(x1)  =  length
U21(x1)  =  U21
nil  =  nil
U31(x1, x2, x3, x4)  =  U31
take(x1, x2)  =  take
and(x1, x2)  =  x2
isNat(x1)  =  isNat
isNatList(x1)  =  isNatList
isNatIList(x1)  =  isNatIList

Recursive Path Order [RPO].
Precedence:
[U31^11, mark1, active1] > [tt, isNat, isNatList, isNatIList] > length > [zeros, cons, U11, s, U31, take] > [U21, nil]
[U31^11, mark1, active1] > [tt, isNat, isNatList, isNatIList] > length > 0 > [U21, nil]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

(40) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

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

(41) PisEmptyProof (EQUIVALENT transformation)

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

(42) TRUE

(43) Obligation:

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

U211(active(X)) → U211(X)
U211(mark(X)) → U211(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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.


U211(active(X)) → U211(X)
U211(mark(X)) → U211(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U211(x1)  =  U211(x1)
active(x1)  =  active(x1)
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
s(x1)  =  s
length(x1)  =  length
U21(x1)  =  U21
nil  =  nil
U31(x1, x2, x3, x4)  =  U31
take(x1, x2)  =  take
and(x1, x2)  =  x2
isNat(x1)  =  isNat
isNatList(x1)  =  isNatList
isNatIList(x1)  =  isNatIList

Recursive Path Order [RPO].
Precedence:
zeros > [tt, isNat, isNatList, isNatIList] > [U11, s, length] > 0 > [U21^11, active1, mark1, cons, U21, nil, U31, take]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

(45) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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:

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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.


LENGTH(active(X)) → LENGTH(X)
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)
active(x1)  =  active(x1)
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
s(x1)  =  s
length(x1)  =  length
U21(x1)  =  U21
nil  =  nil
U31(x1, x2, x3, x4)  =  U31
take(x1, x2)  =  take
and(x1, x2)  =  x2
isNat(x1)  =  isNat
isNatList(x1)  =  isNatList
isNatIList(x1)  =  isNatIList

Recursive Path Order [RPO].
Precedence:
zeros > [tt, isNat, isNatList, isNatIList] > [U11, s, length] > 0 > [LENGTH1, active1, mark1, cons, U21, nil, U31, take]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

(50) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

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

(51) PisEmptyProof (EQUIVALENT transformation)

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

(52) TRUE

(53) Obligation:

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

S(active(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

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

(54) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


S(active(X)) → S(X)
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)
active(x1)  =  active(x1)
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
s(x1)  =  s
length(x1)  =  length
U21(x1)  =  U21
nil  =  nil
U31(x1, x2, x3, x4)  =  U31
take(x1, x2)  =  take
and(x1, x2)  =  x2
isNat(x1)  =  isNat
isNatList(x1)  =  isNatList
isNatIList(x1)  =  isNatIList

Recursive Path Order [RPO].
Precedence:
zeros > [tt, isNat, isNatList, isNatIList] > [U11, s, length] > 0 > [S1, active1, mark1, cons, U21, nil, U31, take]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

(55) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

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

(56) PisEmptyProof (EQUIVALENT transformation)

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

(57) TRUE

(58) Obligation:

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

U111(X1, mark(X2)) → U111(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

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

(59) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U111(X1, mark(X2)) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U111(x1, x2)  =  U111(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
s(x1)  =  s
length(x1)  =  length
U21(x1)  =  U21
nil  =  nil
U31(x1, x2, x3, x4)  =  U31
take(x1, x2)  =  take
and(x1, x2)  =  x2
isNat(x1)  =  isNat
isNatList(x1)  =  isNatList
isNatIList(x1)  =  isNatIList

Recursive Path Order [RPO].
Precedence:
U11^11 > [U21, nil]
[take, isNat] > [mark1, active1, zeros, cons, length, U31] > 0 > [tt, s, isNatList, isNatIList] > [U21, nil]
[take, isNat] > [mark1, active1, zeros, cons, length, U31] > U11 > [tt, s, isNatList, isNatIList] > [U21, nil]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

(60) Obligation:

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

U111(mark(X1), X2) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

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

(61) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U111(mark(X1), X2) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U111(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
s(x1)  =  s
length(x1)  =  length
U21(x1)  =  U21
nil  =  nil
U31(x1, x2, x3, x4)  =  U31
take(x1, x2)  =  take
and(x1, x2)  =  x2
isNat(x1)  =  isNat
isNatList(x1)  =  isNatList
isNatIList(x1)  =  isNatIList

Recursive Path Order [RPO].
Precedence:
[zeros, 0, length] > [mark1, active1, cons, U11, U31, take] > U21 > nil > tt > s
[zeros, 0, length] > [mark1, active1, cons, U11, U31, take] > isNat > [isNatList, isNatIList] > tt > s


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

(62) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

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

(63) PisEmptyProof (EQUIVALENT transformation)

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

(64) TRUE

(65) Obligation:

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

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

(66) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  CONS(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
s(x1)  =  s
length(x1)  =  length
U21(x1)  =  U21
nil  =  nil
U31(x1, x2, x3, x4)  =  U31
take(x1, x2)  =  take
and(x1, x2)  =  x2
isNat(x1)  =  isNat
isNatList(x1)  =  isNatList
isNatIList(x1)  =  isNatIList

Recursive Path Order [RPO].
Precedence:
CONS1 > [U21, nil]
[take, isNat] > [mark1, active1, zeros, cons, length, U31] > 0 > [tt, s, isNatList, isNatIList] > [U21, nil]
[take, isNat] > [mark1, active1, zeros, cons, length, U31] > U11 > [tt, s, isNatList, isNatIList] > [U21, nil]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

(67) Obligation:

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

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

(68) 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)
CONS(active(X1), X2) → CONS(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
s(x1)  =  s
length(x1)  =  length
U21(x1)  =  U21
nil  =  nil
U31(x1, x2, x3, x4)  =  U31
take(x1, x2)  =  take
and(x1, x2)  =  x2
isNat(x1)  =  isNat
isNatList(x1)  =  isNatList
isNatIList(x1)  =  isNatIList

Recursive Path Order [RPO].
Precedence:
[zeros, 0, length] > [mark1, active1, cons, U11, U31, take] > U21 > nil > tt > s
[zeros, 0, length] > [mark1, active1, cons, U11, U31, take] > isNat > [isNatList, isNatIList] > tt > s


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

(69) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

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

(70) PisEmptyProof (EQUIVALENT transformation)

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

(71) TRUE

(72) Obligation:

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

MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(U31(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
MARK(U11(X1, X2)) → MARK(X1)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(and(tt, X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNat(length(V1))) → MARK(isNatList(V1))
MARK(length(X)) → MARK(X)
MARK(U21(X)) → ACTIVE(U21(mark(X)))
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
MARK(U21(X)) → MARK(X)
MARK(U31(X1, X2, X3, X4)) → ACTIVE(U31(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(V)) → MARK(isNatList(V))
MARK(U31(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(and(isNat(V1), isNatIList(V2)))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNatList(cons(V1, V2))) → MARK(and(isNat(V1), isNatList(V2)))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNatList(take(V1, V2))) → MARK(and(isNat(V1), isNatIList(V2)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(take(0, IL)) → MARK(U21(isNatIList(IL)))
ACTIVE(take(s(M), cons(N, IL))) → MARK(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

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

(73) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(s(X)) → ACTIVE(s(mark(X)))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK
cons(x1, x2)  =  cons
ACTIVE(x1)  =  x1
mark(x1)  =  mark
U11(x1, x2)  =  U11
tt  =  tt
s(x1)  =  s
length(x1)  =  length
zeros  =  zeros
0  =  0
U31(x1, x2, x3, x4)  =  U31
take(x1, x2)  =  take
and(x1, x2)  =  and
isNat(x1)  =  isNat
isNatList(x1)  =  isNatList
U21(x1)  =  U21
isNatIList(x1)  =  isNatIList
active(x1)  =  active
nil  =  nil

Recursive Path Order [RPO].
Precedence:
[mark, active] > [MARK, cons, U11, length, zeros, 0, U31, take, and, isNat, isNatList, U21, isNatIList, nil] > tt > s


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

(74) Obligation:

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

MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(U31(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
MARK(U11(X1, X2)) → MARK(X1)
ACTIVE(and(tt, X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNat(length(V1))) → MARK(isNatList(V1))
MARK(length(X)) → MARK(X)
MARK(U21(X)) → ACTIVE(U21(mark(X)))
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
MARK(U21(X)) → MARK(X)
MARK(U31(X1, X2, X3, X4)) → ACTIVE(U31(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(V)) → MARK(isNatList(V))
MARK(U31(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(and(isNat(V1), isNatIList(V2)))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNatList(cons(V1, V2))) → MARK(and(isNat(V1), isNatList(V2)))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNatList(take(V1, V2))) → MARK(and(isNat(V1), isNatIList(V2)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(take(0, IL)) → MARK(U21(isNatIList(IL)))
ACTIVE(take(s(M), cons(N, IL))) → MARK(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

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

(75) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK
cons(x1, x2)  =  cons
ACTIVE(x1)  =  x1
mark(x1)  =  mark
U11(x1, x2)  =  U11
tt  =  tt
s(x1)  =  s
length(x1)  =  length
zeros  =  zeros
0  =  0
U31(x1, x2, x3, x4)  =  U31
take(x1, x2)  =  take
and(x1, x2)  =  and
isNat(x1)  =  isNat
isNatList(x1)  =  isNatList
U21(x1)  =  U21
isNatIList(x1)  =  isNatIList
active(x1)  =  x1
nil  =  nil

Recursive Path Order [RPO].
Precedence:
[MARK, mark, U11, tt, s, length, zeros, 0, U31, take, and, isNat, isNatList, U21, isNatIList, nil] > cons


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

(76) Obligation:

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

ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(U31(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
MARK(U11(X1, X2)) → MARK(X1)
ACTIVE(and(tt, X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNat(length(V1))) → MARK(isNatList(V1))
MARK(length(X)) → MARK(X)
MARK(U21(X)) → ACTIVE(U21(mark(X)))
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
MARK(U21(X)) → MARK(X)
MARK(U31(X1, X2, X3, X4)) → ACTIVE(U31(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(V)) → MARK(isNatList(V))
MARK(U31(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(and(isNat(V1), isNatIList(V2)))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNatList(cons(V1, V2))) → MARK(and(isNat(V1), isNatList(V2)))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNatList(take(V1, V2))) → MARK(and(isNat(V1), isNatIList(V2)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(take(0, IL)) → MARK(U21(isNatIList(IL)))
ACTIVE(take(s(M), cons(N, IL))) → MARK(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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.


MARK(U21(X)) → ACTIVE(U21(mark(X)))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
U11(x1, x2)  =  U11
tt  =  tt
MARK(x1)  =  MARK
s(x1)  =  s
length(x1)  =  length
cons(x1, x2)  =  cons
zeros  =  zeros
0  =  0
mark(x1)  =  mark
U31(x1, x2, x3, x4)  =  U31
take(x1, x2)  =  take
and(x1, x2)  =  and
isNat(x1)  =  isNat
isNatList(x1)  =  isNatList
U21(x1)  =  U21
isNatIList(x1)  =  isNatIList
active(x1)  =  active
nil  =  nil

Recursive Path Order [RPO].
Precedence:
cons > [U11, tt, MARK, length, zeros, 0, U31, take, and, isNat, isNatList, isNatIList, nil] > [mark, U21, active] > s


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

(78) Obligation:

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

ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(U31(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
MARK(U11(X1, X2)) → MARK(X1)
ACTIVE(and(tt, X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNat(length(V1))) → MARK(isNatList(V1))
MARK(length(X)) → MARK(X)
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
MARK(U21(X)) → MARK(X)
MARK(U31(X1, X2, X3, X4)) → ACTIVE(U31(mark(X1), X2, X3, X4))
ACTIVE(isNatIList(V)) → MARK(isNatList(V))
MARK(U31(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(and(isNat(V1), isNatIList(V2)))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNatList(cons(V1, V2))) → MARK(and(isNat(V1), isNatList(V2)))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(isNatList(take(V1, V2))) → MARK(and(isNat(V1), isNatIList(V2)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(take(0, IL)) → MARK(U21(isNatIList(IL)))
ACTIVE(take(s(M), cons(N, IL))) → MARK(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U21(X)) → active(U21(mark(X)))
mark(nil) → active(nil)
mark(U31(X1, X2, X3, X4)) → active(U31(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, mark(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, mark(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, mark(X4)) → U31(X1, X2, X3, X4)
U31(active(X1), X2, X3, X4) → U31(X1, X2, X3, X4)
U31(X1, active(X2), X3, X4) → U31(X1, X2, X3, X4)
U31(X1, X2, active(X3), X4) → U31(X1, X2, X3, X4)
U31(X1, X2, X3, active(X4)) → U31(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

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