Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

zeroscons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zerosn__zeros
take(X1, X2) → n__take(X1, X2)
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.


QTRS
  ↳ DependencyPairsProof

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

zeroscons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zerosn__zeros
take(X1, X2) → n__take(X1, X2)
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

U211(tt) → NIL
ISNATILIST(V) → ACTIVATE(V)
ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U311(tt, IL, M, N) → ACTIVATE(IL)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(V) → ISNATLIST(activate(V))
ACTIVATE(n__0) → 01
ACTIVATE(n__cons(X1, X2)) → CONS(X1, X2)
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U311(tt, IL, M, N) → ACTIVATE(M)
LENGTH(nil) → 01
ACTIVATE(n__zeros) → ZEROS
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → LENGTH(X)
TAKE(0, IL) → ISNATILIST(IL)
ACTIVATE(n__nil) → NIL
ACTIVATE(n__s(X)) → S(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
TAKE(s(M), cons(N, IL)) → U311(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
U311(tt, IL, M, N) → CONS(activate(N), n__take(activate(M), activate(IL)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
U111(tt, L) → ACTIVATE(L)
TAKE(0, IL) → U211(isNatIList(IL))
LENGTH(cons(N, L)) → ACTIVATE(L)
ISNATLIST(n__take(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
U111(tt, L) → LENGTH(activate(L))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N)))
ACTIVATE(n__isNat(X)) → ISNAT(X)
U111(tt, L) → S(length(activate(L)))
ZEROSCONS(0, n__zeros)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
U311(tt, IL, M, N) → ACTIVATE(N)
ZEROS01
ISNAT(n__length(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNat(N))

The TRS R consists of the following rules:

zeroscons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zerosn__zeros
take(X1, X2) → n__take(X1, X2)
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

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

↳ QTRS
  ↳ DependencyPairsProof
QDP
      ↳ DependencyGraphProof

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

U211(tt) → NIL
ISNATILIST(V) → ACTIVATE(V)
ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U311(tt, IL, M, N) → ACTIVATE(IL)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(V) → ISNATLIST(activate(V))
ACTIVATE(n__0) → 01
ACTIVATE(n__cons(X1, X2)) → CONS(X1, X2)
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U311(tt, IL, M, N) → ACTIVATE(M)
LENGTH(nil) → 01
ACTIVATE(n__zeros) → ZEROS
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → LENGTH(X)
TAKE(0, IL) → ISNATILIST(IL)
ACTIVATE(n__nil) → NIL
ACTIVATE(n__s(X)) → S(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
TAKE(s(M), cons(N, IL)) → U311(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
U311(tt, IL, M, N) → CONS(activate(N), n__take(activate(M), activate(IL)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
U111(tt, L) → ACTIVATE(L)
TAKE(0, IL) → U211(isNatIList(IL))
LENGTH(cons(N, L)) → ACTIVATE(L)
ISNATLIST(n__take(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
U111(tt, L) → LENGTH(activate(L))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N)))
ACTIVATE(n__isNat(X)) → ISNAT(X)
U111(tt, L) → S(length(activate(L)))
ZEROSCONS(0, n__zeros)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
U311(tt, IL, M, N) → ACTIVATE(N)
ZEROS01
ISNAT(n__length(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNat(N))

The TRS R consists of the following rules:

zeroscons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zerosn__zeros
take(X1, X2) → n__take(X1, X2)
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 12 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
QDP
          ↳ QDPOrderProof

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

ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
TAKE(s(M), cons(N, IL)) → U311(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U311(tt, IL, M, N) → ACTIVATE(IL)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(V) → ISNATLIST(activate(V))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U311(tt, IL, M, N) → ACTIVATE(M)
U111(tt, L) → ACTIVATE(L)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → LENGTH(X)
LENGTH(cons(N, L)) → ACTIVATE(L)
ISNATLIST(n__take(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
U111(tt, L) → LENGTH(activate(L))
TAKE(0, IL) → ISNATILIST(IL)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N)))
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U311(tt, IL, M, N) → ACTIVATE(N)
ISNAT(n__length(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNat(N))

The TRS R consists of the following rules:

zeroscons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zerosn__zeros
take(X1, X2) → n__take(X1, X2)
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
U111(tt, L) → ACTIVATE(L)
LENGTH(cons(N, L)) → ACTIVATE(L)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNAT(n__length(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNat(N))
The remaining pairs can at least be oriented weakly.

ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
TAKE(s(M), cons(N, IL)) → U311(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U311(tt, IL, M, N) → ACTIVATE(IL)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(V) → ISNATLIST(activate(V))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U311(tt, IL, M, N) → ACTIVATE(M)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → LENGTH(X)
ISNATLIST(n__take(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
U111(tt, L) → LENGTH(activate(L))
TAKE(0, IL) → ISNATILIST(IL)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N)))
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U311(tt, IL, M, N) → ACTIVATE(N)
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVATE(x1)) = x1   
POL(AND(x1, x2)) = x2   
POL(ISNAT(x1)) = x1   
POL(ISNATILIST(x1)) = x1   
POL(ISNATLIST(x1)) = x1   
POL(LENGTH(x1)) = 1 + x1   
POL(TAKE(x1, x2)) = x1 + x2   
POL(U11(x1, x2)) = 1 + x2   
POL(U111(x1, x2)) = 1 + x2   
POL(U21(x1)) = 0   
POL(U31(x1, x2, x3, x4)) = x2 + x3 + x4   
POL(U311(x1, x2, x3, x4)) = x2 + x3 + x4   
POL(activate(x1)) = x1   
POL(and(x1, x2)) = x2   
POL(cons(x1, x2)) = x1 + x2   
POL(isNat(x1)) = x1   
POL(isNatIList(x1)) = x1   
POL(isNatList(x1)) = x1   
POL(length(x1)) = 1 + x1   
POL(n__0) = 0   
POL(n__and(x1, x2)) = x2   
POL(n__cons(x1, x2)) = x1 + x2   
POL(n__isNat(x1)) = x1   
POL(n__isNatIList(x1)) = x1   
POL(n__isNatList(x1)) = x1   
POL(n__length(x1)) = 1 + x1   
POL(n__nil) = 0   
POL(n__s(x1)) = x1   
POL(n__take(x1, x2)) = x1 + x2   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(take(x1, x2)) = x1 + x2   
POL(tt) = 0   
POL(zeros) = 0   

The following usable rules [17] were oriented:

activate(n__zeros) → zeros
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatIList(V) → isNatList(activate(V))
activate(n__isNat(X)) → isNat(X)
isNat(n__s(V1)) → isNat(activate(V1))
and(tt, X) → activate(X)
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNat(n__length(V1)) → isNatList(activate(V1))
activate(n__isNatList(X)) → isNatList(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
length(X) → n__length(X)
0n__0
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
niln__nil
isNatIList(X) → n__isNatIList(X)
isNat(X) → n__isNat(X)
isNatList(X) → n__isNatList(X)
activate(n__nil) → nil
activate(X) → X
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
zeroscons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
isNatList(n__nil) → tt
isNatIList(n__zeros) → tt
isNat(n__0) → tt
take(X1, X2) → n__take(X1, X2)
zerosn__zeros
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
take(0, IL) → U21(isNatIList(IL))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
length(nil) → 0



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
QDP
              ↳ DependencyGraphProof

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

ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
TAKE(s(M), cons(N, IL)) → U311(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U311(tt, IL, M, N) → ACTIVATE(IL)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATILIST(V) → ISNATLIST(activate(V))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U311(tt, IL, M, N) → ACTIVATE(M)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → LENGTH(X)
ISNATLIST(n__take(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
U111(tt, L) → LENGTH(activate(L))
TAKE(0, IL) → ISNATILIST(IL)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N)))
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U311(tt, IL, M, N) → ACTIVATE(N)

The TRS R consists of the following rules:

zeroscons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zerosn__zeros
take(X1, X2) → n__take(X1, X2)
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 1 less node.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
QDP
                    ↳ Narrowing
                  ↳ QDP

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

U111(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))

The TRS R consists of the following rules:

zeroscons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zerosn__zeros
take(X1, X2) → n__take(X1, X2)
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U111(tt, L) → LENGTH(activate(L)) at position [0] we obtained the following new rules:

U111(tt, n__zeros) → LENGTH(zeros)
U111(tt, n__s(x0)) → LENGTH(s(x0))
U111(tt, n__and(x0, x1)) → LENGTH(and(x0, x1))
U111(tt, n__nil) → LENGTH(nil)
U111(tt, x0) → LENGTH(x0)
U111(tt, n__isNatIList(x0)) → LENGTH(isNatIList(x0))
U111(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U111(tt, n__isNatList(x0)) → LENGTH(isNatList(x0))
U111(tt, n__take(x0, x1)) → LENGTH(take(x0, x1))
U111(tt, n__cons(x0, x1)) → LENGTH(cons(x0, x1))
U111(tt, n__length(x0)) → LENGTH(length(x0))
U111(tt, n__0) → LENGTH(0)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ Narrowing
QDP
                        ↳ Narrowing
                  ↳ QDP

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

U111(tt, n__s(x0)) → LENGTH(s(x0))
U111(tt, n__and(x0, x1)) → LENGTH(and(x0, x1))
U111(tt, n__nil) → LENGTH(nil)
U111(tt, n__isNatIList(x0)) → LENGTH(isNatIList(x0))
U111(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U111(tt, n__take(x0, x1)) → LENGTH(take(x0, x1))
U111(tt, n__cons(x0, x1)) → LENGTH(cons(x0, x1))
U111(tt, n__length(x0)) → LENGTH(length(x0))
U111(tt, n__zeros) → LENGTH(zeros)
U111(tt, x0) → LENGTH(x0)
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U111(tt, n__isNatList(x0)) → LENGTH(isNatList(x0))
U111(tt, n__0) → LENGTH(0)

The TRS R consists of the following rules:

zeroscons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zerosn__zeros
take(X1, X2) → n__take(X1, X2)
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U111(tt, n__zeros) → LENGTH(zeros) at position [0] we obtained the following new rules:

U111(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U111(tt, n__zeros) → LENGTH(n__zeros)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
QDP
                            ↳ DependencyGraphProof
                  ↳ QDP

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

U111(tt, n__s(x0)) → LENGTH(s(x0))
U111(tt, n__and(x0, x1)) → LENGTH(and(x0, x1))
U111(tt, n__nil) → LENGTH(nil)
U111(tt, n__isNatIList(x0)) → LENGTH(isNatIList(x0))
U111(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U111(tt, n__zeros) → LENGTH(n__zeros)
U111(tt, n__cons(x0, x1)) → LENGTH(cons(x0, x1))
U111(tt, n__take(x0, x1)) → LENGTH(take(x0, x1))
U111(tt, n__length(x0)) → LENGTH(length(x0))
U111(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U111(tt, x0) → LENGTH(x0)
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U111(tt, n__isNatList(x0)) → LENGTH(isNatList(x0))
U111(tt, n__0) → LENGTH(0)

The TRS R consists of the following rules:

zeroscons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zerosn__zeros
take(X1, X2) → n__take(X1, X2)
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
QDP
                                ↳ Narrowing
                  ↳ QDP

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

U111(tt, n__s(x0)) → LENGTH(s(x0))
U111(tt, n__and(x0, x1)) → LENGTH(and(x0, x1))
U111(tt, n__nil) → LENGTH(nil)
U111(tt, n__isNatIList(x0)) → LENGTH(isNatIList(x0))
U111(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U111(tt, n__take(x0, x1)) → LENGTH(take(x0, x1))
U111(tt, n__cons(x0, x1)) → LENGTH(cons(x0, x1))
U111(tt, n__length(x0)) → LENGTH(length(x0))
U111(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U111(tt, x0) → LENGTH(x0)
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U111(tt, n__isNatList(x0)) → LENGTH(isNatList(x0))
U111(tt, n__0) → LENGTH(0)

The TRS R consists of the following rules:

zeroscons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zerosn__zeros
take(X1, X2) → n__take(X1, X2)
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U111(tt, n__s(x0)) → LENGTH(s(x0)) at position [0] we obtained the following new rules:

U111(tt, n__s(x0)) → LENGTH(n__s(x0))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
QDP
                                    ↳ DependencyGraphProof
                  ↳ QDP

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

U111(tt, n__and(x0, x1)) → LENGTH(and(x0, x1))
U111(tt, n__nil) → LENGTH(nil)
U111(tt, n__isNatIList(x0)) → LENGTH(isNatIList(x0))
U111(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U111(tt, n__cons(x0, x1)) → LENGTH(cons(x0, x1))
U111(tt, n__take(x0, x1)) → LENGTH(take(x0, x1))
U111(tt, n__length(x0)) → LENGTH(length(x0))
U111(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U111(tt, n__s(x0)) → LENGTH(n__s(x0))
U111(tt, x0) → LENGTH(x0)
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U111(tt, n__isNatList(x0)) → LENGTH(isNatList(x0))
U111(tt, n__0) → LENGTH(0)

The TRS R consists of the following rules:

zeroscons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zerosn__zeros
take(X1, X2) → n__take(X1, X2)
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
QDP
                                        ↳ Narrowing
                  ↳ QDP

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

U111(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U111(tt, n__and(x0, x1)) → LENGTH(and(x0, x1))
U111(tt, n__nil) → LENGTH(nil)
U111(tt, n__isNatIList(x0)) → LENGTH(isNatIList(x0))
U111(tt, x0) → LENGTH(x0)
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U111(tt, n__isNatList(x0)) → LENGTH(isNatList(x0))
U111(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U111(tt, n__cons(x0, x1)) → LENGTH(cons(x0, x1))
U111(tt, n__take(x0, x1)) → LENGTH(take(x0, x1))
U111(tt, n__length(x0)) → LENGTH(length(x0))
U111(tt, n__0) → LENGTH(0)

The TRS R consists of the following rules:

zeroscons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zerosn__zeros
take(X1, X2) → n__take(X1, X2)
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U111(tt, n__nil) → LENGTH(nil) at position [0] we obtained the following new rules:

U111(tt, n__nil) → LENGTH(n__nil)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
QDP
                                            ↳ DependencyGraphProof
                  ↳ QDP

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

U111(tt, n__and(x0, x1)) → LENGTH(and(x0, x1))
U111(tt, n__isNatIList(x0)) → LENGTH(isNatIList(x0))
U111(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U111(tt, n__take(x0, x1)) → LENGTH(take(x0, x1))
U111(tt, n__cons(x0, x1)) → LENGTH(cons(x0, x1))
U111(tt, n__length(x0)) → LENGTH(length(x0))
U111(tt, n__nil) → LENGTH(n__nil)
U111(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U111(tt, x0) → LENGTH(x0)
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U111(tt, n__isNatList(x0)) → LENGTH(isNatList(x0))
U111(tt, n__0) → LENGTH(0)

The TRS R consists of the following rules:

zeroscons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zerosn__zeros
take(X1, X2) → n__take(X1, X2)
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
QDP
                                                ↳ Narrowing
                  ↳ QDP

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

U111(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U111(tt, n__and(x0, x1)) → LENGTH(and(x0, x1))
U111(tt, n__isNatIList(x0)) → LENGTH(isNatIList(x0))
U111(tt, x0) → LENGTH(x0)
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U111(tt, n__isNatList(x0)) → LENGTH(isNatList(x0))
U111(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U111(tt, n__cons(x0, x1)) → LENGTH(cons(x0, x1))
U111(tt, n__take(x0, x1)) → LENGTH(take(x0, x1))
U111(tt, n__length(x0)) → LENGTH(length(x0))
U111(tt, n__0) → LENGTH(0)

The TRS R consists of the following rules:

zeroscons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zerosn__zeros
take(X1, X2) → n__take(X1, X2)
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U111(tt, n__0) → LENGTH(0) at position [0] we obtained the following new rules:

U111(tt, n__0) → LENGTH(n__0)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
QDP
                                                    ↳ DependencyGraphProof
                  ↳ QDP

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

U111(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U111(tt, n__and(x0, x1)) → LENGTH(and(x0, x1))
U111(tt, x0) → LENGTH(x0)
U111(tt, n__isNatIList(x0)) → LENGTH(isNatIList(x0))
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U111(tt, n__0) → LENGTH(n__0)
U111(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U111(tt, n__isNatList(x0)) → LENGTH(isNatList(x0))
U111(tt, n__take(x0, x1)) → LENGTH(take(x0, x1))
U111(tt, n__cons(x0, x1)) → LENGTH(cons(x0, x1))
U111(tt, n__length(x0)) → LENGTH(length(x0))

The TRS R consists of the following rules:

zeroscons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zerosn__zeros
take(X1, X2) → n__take(X1, X2)
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
QDP
                                                        ↳ QDPOrderProof
                  ↳ QDP

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

U111(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U111(tt, n__and(x0, x1)) → LENGTH(and(x0, x1))
U111(tt, n__isNatIList(x0)) → LENGTH(isNatIList(x0))
U111(tt, x0) → LENGTH(x0)
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U111(tt, n__isNatList(x0)) → LENGTH(isNatList(x0))
U111(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U111(tt, n__cons(x0, x1)) → LENGTH(cons(x0, x1))
U111(tt, n__take(x0, x1)) → LENGTH(take(x0, x1))
U111(tt, n__length(x0)) → LENGTH(length(x0))

The TRS R consists of the following rules:

zeroscons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zerosn__zeros
take(X1, X2) → n__take(X1, X2)
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


U111(tt, n__length(x0)) → LENGTH(length(x0))
The remaining pairs can at least be oriented weakly.

U111(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U111(tt, n__and(x0, x1)) → LENGTH(and(x0, x1))
U111(tt, n__isNatIList(x0)) → LENGTH(isNatIList(x0))
U111(tt, x0) → LENGTH(x0)
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U111(tt, n__isNatList(x0)) → LENGTH(isNatList(x0))
U111(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U111(tt, n__cons(x0, x1)) → LENGTH(cons(x0, x1))
U111(tt, n__take(x0, x1)) → LENGTH(take(x0, x1))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( U31(x1, ..., x4) ) =
/0\
\0/
+
/00\
\00/
·x1+
/10\
\00/
·x2+
/00\
\00/
·x3+
/00\
\00/
·x4

M( n__isNat(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( activate(x1) ) =
/0\
\0/
+
/10\
\01/
·x1

M( n__nil ) =
/0\
\0/

M( and(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/10\
\01/
·x2

M( n__s(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( take(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/10\
\00/
·x2

M( tt ) =
/0\
\0/

M( isNatList(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( zeros ) =
/0\
\0/

M( isNatIList(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( s(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( isNat(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( nil ) =
/0\
\0/

M( n__isNatIList(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( n__length(x1) ) =
/1\
\1/
+
/00\
\00/
·x1

M( n__zeros ) =
/0\
\0/

M( n__cons(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/11\
\00/
·x2

M( n__isNatList(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U11(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( 0 ) =
/0\
\0/

M( n__and(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/10\
\01/
·x2

M( cons(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/11\
\00/
·x2

M( n__0 ) =
/0\
\0/

M( n__take(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/10\
\00/
·x2

M( length(x1) ) =
/1\
\1/
+
/00\
\00/
·x1

M( U21(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

Tuple symbols:
M( LENGTH(x1) ) = 0+
[1,0]
·x1

M( U111(x1, x2) ) = 0+
[0,0]
·x1+
[1,1]
·x2


Matrix type:
We used a basic matrix type which is not further parametrizeable.


As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:

activate(n__zeros) → zeros
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatIList(V) → isNatList(activate(V))
activate(n__isNat(X)) → isNat(X)
isNat(n__s(V1)) → isNat(activate(V1))
and(tt, X) → activate(X)
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNat(n__length(V1)) → isNatList(activate(V1))
activate(n__isNatList(X)) → isNatList(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
length(X) → n__length(X)
0n__0
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
niln__nil
isNatIList(X) → n__isNatIList(X)
isNat(X) → n__isNat(X)
isNatList(X) → n__isNatList(X)
activate(n__nil) → nil
activate(X) → X
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
zeroscons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
isNatList(n__nil) → tt
isNatIList(n__zeros) → tt
isNat(n__0) → tt
take(X1, X2) → n__take(X1, X2)
zerosn__zeros
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
take(0, IL) → U21(isNatIList(IL))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
length(nil) → 0



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
QDP
                                                            ↳ QDPOrderProof
                  ↳ QDP

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

U111(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U111(tt, n__and(x0, x1)) → LENGTH(and(x0, x1))
U111(tt, x0) → LENGTH(x0)
U111(tt, n__isNatIList(x0)) → LENGTH(isNatIList(x0))
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U111(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U111(tt, n__isNatList(x0)) → LENGTH(isNatList(x0))
U111(tt, n__take(x0, x1)) → LENGTH(take(x0, x1))
U111(tt, n__cons(x0, x1)) → LENGTH(cons(x0, x1))

The TRS R consists of the following rules:

zeroscons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zerosn__zeros
take(X1, X2) → n__take(X1, X2)
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


U111(tt, n__isNatIList(x0)) → LENGTH(isNatIList(x0))
U111(tt, n__isNatList(x0)) → LENGTH(isNatList(x0))
The remaining pairs can at least be oriented weakly.

U111(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U111(tt, n__and(x0, x1)) → LENGTH(and(x0, x1))
U111(tt, x0) → LENGTH(x0)
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U111(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U111(tt, n__take(x0, x1)) → LENGTH(take(x0, x1))
U111(tt, n__cons(x0, x1)) → LENGTH(cons(x0, x1))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( U31(x1, ..., x4) ) =
/0\
\0/
+
/00\
\00/
·x1+
/11\
\00/
·x2+
/11\
\00/
·x3+
/00\
\00/
·x4

M( n__isNat(x1) ) =
/0\
\0/
+
/01\
\01/
·x1

M( activate(x1) ) =
/0\
\0/
+
/10\
\01/
·x1

M( n__nil ) =
/0\
\0/

M( and(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/10\
\01/
·x2

M( n__s(x1) ) =
/0\
\0/
+
/00\
\11/
·x1

M( take(x1, x2) ) =
/0\
\0/
+
/01\
\00/
·x1+
/10\
\00/
·x2

M( tt ) =
/0\
\0/

M( isNatList(x1) ) =
/1\
\1/
+
/00\
\00/
·x1

M( zeros ) =
/0\
\0/

M( isNatIList(x1) ) =
/1\
\1/
+
/00\
\00/
·x1

M( s(x1) ) =
/0\
\0/
+
/00\
\11/
·x1

M( isNat(x1) ) =
/0\
\0/
+
/01\
\01/
·x1

M( nil ) =
/0\
\0/

M( n__isNatIList(x1) ) =
/1\
\1/
+
/00\
\00/
·x1

M( n__length(x1) ) =
/0\
\1/
+
/00\
\00/
·x1

M( n__zeros ) =
/0\
\0/

M( n__cons(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/11\
\00/
·x2

M( n__isNatList(x1) ) =
/1\
\1/
+
/00\
\00/
·x1

M( U11(x1, x2) ) =
/0\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( 0 ) =
/0\
\0/

M( n__and(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/10\
\01/
·x2

M( cons(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/11\
\00/
·x2

M( n__0 ) =
/0\
\0/

M( n__take(x1, x2) ) =
/0\
\0/
+
/01\
\00/
·x1+
/10\
\00/
·x2

M( length(x1) ) =
/0\
\1/
+
/00\
\00/
·x1

M( U21(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

Tuple symbols:
M( LENGTH(x1) ) = 0+
[1,0]
·x1

M( U111(x1, x2) ) = 0+
[0,0]
·x1+
[1,1]
·x2


Matrix type:
We used a basic matrix type which is not further parametrizeable.


As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:

activate(n__zeros) → zeros
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatIList(V) → isNatList(activate(V))
activate(n__isNat(X)) → isNat(X)
isNat(n__s(V1)) → isNat(activate(V1))
and(tt, X) → activate(X)
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNat(n__length(V1)) → isNatList(activate(V1))
activate(n__isNatList(X)) → isNatList(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
length(X) → n__length(X)
0n__0
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
niln__nil
isNatIList(X) → n__isNatIList(X)
isNat(X) → n__isNat(X)
isNatList(X) → n__isNatList(X)
activate(n__nil) → nil
activate(X) → X
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
zeroscons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
isNatList(n__nil) → tt
isNatIList(n__zeros) → tt
isNat(n__0) → tt
take(X1, X2) → n__take(X1, X2)
zerosn__zeros
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
take(0, IL) → U21(isNatIList(IL))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
length(nil) → 0



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
QDP
                                                                ↳ QDPOrderProof
                  ↳ QDP

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

U111(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U111(tt, n__and(x0, x1)) → LENGTH(and(x0, x1))
U111(tt, x0) → LENGTH(x0)
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U111(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U111(tt, n__cons(x0, x1)) → LENGTH(cons(x0, x1))
U111(tt, n__take(x0, x1)) → LENGTH(take(x0, x1))

The TRS R consists of the following rules:

zeroscons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zerosn__zeros
take(X1, X2) → n__take(X1, X2)
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


U111(tt, n__isNat(x0)) → LENGTH(isNat(x0))
The remaining pairs can at least be oriented weakly.

U111(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U111(tt, n__and(x0, x1)) → LENGTH(and(x0, x1))
U111(tt, x0) → LENGTH(x0)
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U111(tt, n__cons(x0, x1)) → LENGTH(cons(x0, x1))
U111(tt, n__take(x0, x1)) → LENGTH(take(x0, x1))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( U31(x1, ..., x4) ) =
/0\
\0/
+
/00\
\00/
·x1+
/11\
\00/
·x2+
/00\
\00/
·x3+
/00\
\00/
·x4

M( n__isNat(x1) ) =
/0\
\1/
+
/00\
\00/
·x1

M( activate(x1) ) =
/0\
\0/
+
/10\
\01/
·x1

M( n__nil ) =
/0\
\0/

M( and(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/10\
\01/
·x2

M( n__s(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( take(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/10\
\01/
·x2

M( tt ) =
/0\
\0/

M( isNatList(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( zeros ) =
/0\
\0/

M( isNatIList(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( s(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( isNat(x1) ) =
/0\
\1/
+
/00\
\00/
·x1

M( nil ) =
/0\
\0/

M( n__isNatIList(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( n__length(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( n__zeros ) =
/0\
\0/

M( n__cons(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/11\
\00/
·x2

M( n__isNatList(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U11(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( 0 ) =
/0\
\0/

M( n__and(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/10\
\01/
·x2

M( cons(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/11\
\00/
·x2

M( n__0 ) =
/0\
\0/

M( n__take(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/10\
\01/
·x2

M( length(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U21(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

Tuple symbols:
M( LENGTH(x1) ) = 0+
[1,0]
·x1

M( U111(x1, x2) ) = 0+
[0,0]
·x1+
[1,1]
·x2


Matrix type:
We used a basic matrix type which is not further parametrizeable.


As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:

activate(n__zeros) → zeros
and(X1, X2) → n__and(X1, X2)
activate(n__0) → 0
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatIList(V) → isNatList(activate(V))
activate(n__isNat(X)) → isNat(X)
isNat(n__s(V1)) → isNat(activate(V1))
and(tt, X) → activate(X)
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__and(X1, X2)) → and(X1, X2)
isNat(n__length(V1)) → isNatList(activate(V1))
activate(n__isNatList(X)) → isNatList(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
length(X) → n__length(X)
0n__0
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
niln__nil
isNatIList(X) → n__isNatIList(X)
isNat(X) → n__isNat(X)
isNatList(X) → n__isNatList(X)
activate(n__nil) → nil
activate(X) → X
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
zeroscons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
isNatList(n__nil) → tt
isNatIList(n__zeros) → tt
isNat(n__0) → tt
take(X1, X2) → n__take(X1, X2)
zerosn__zeros
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
take(0, IL) → U21(isNatIList(IL))
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
length(nil) → 0



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
QDP
                  ↳ QDP

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

U111(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U111(tt, n__and(x0, x1)) → LENGTH(and(x0, x1))
U111(tt, x0) → LENGTH(x0)
LENGTH(cons(N, L)) → U111(and(isNatList(activate(L)), n__isNat(N)), activate(L))
U111(tt, n__take(x0, x1)) → LENGTH(take(x0, x1))
U111(tt, n__cons(x0, x1)) → LENGTH(cons(x0, x1))

The TRS R consists of the following rules:

zeroscons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zerosn__zeros
take(X1, X2) → n__take(X1, X2)
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
QDP

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

ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
TAKE(s(M), cons(N, IL)) → U311(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U311(tt, IL, M, N) → ACTIVATE(IL)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATILIST(V) → ISNATLIST(activate(V))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
U311(tt, IL, M, N) → ACTIVATE(M)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__take(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
TAKE(0, IL) → ISNATILIST(IL)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N)))
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U311(tt, IL, M, N) → ACTIVATE(N)

The TRS R consists of the following rules:

zeroscons(0, n__zeros)
U11(tt, L) → s(length(activate(L)))
U21(tt) → nil
U31(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → isNatList(activate(V1))
isNat(n__s(V1)) → isNat(activate(V1))
isNatIList(V) → isNatList(activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → and(isNat(activate(V1)), n__isNatList(activate(V2)))
isNatList(n__take(V1, V2)) → and(isNat(activate(V1)), n__isNatIList(activate(V2)))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(activate(L)), n__isNat(N)), activate(L))
take(0, IL) → U21(isNatIList(IL))
take(s(M), cons(N, IL)) → U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N)
zerosn__zeros
take(X1, X2) → n__take(X1, X2)
0n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
niln__nil
isNatList(X) → n__isNatList(X)
isNat(X) → n__isNat(X)
and(X1, X2) → n__and(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIList(X)) → isNatIList(X)
activate(n__nil) → nil
activate(n__isNatList(X)) → isNatList(X)
activate(n__isNat(X)) → isNat(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(X) → X

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