Termination w.r.t. Q proof of /tmp/tmpAr2fYZ/OvConsOS_nokinds

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:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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)))
ZEROS → CONS(0, n__zeros)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
U31¹(tt, IL, M, N) → ACTIVATE(N)
ZEROS → 0¹
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:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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)))
ZEROS → CONS(0, n__zeros)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
U31¹(tt, IL, M, N) → ACTIVATE(N)
ZEROS → 0¹
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:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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))
ACTIVATE(n__length(X)) → LENGTH(X)
ISNAT(n__length(V1)) → ACTIVATE(V1)

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)
U11¹(tt, L) → ACTIVATE(L)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
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)
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNat(N))

Used ordering: Polynomial interpretation with max and min functions [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)) = x₁
POL(TAKE(x₁, x₂)) = x₁ + x₂
POL(U11(x₁, x₂)) = 1 + x₁ + x₂
POL(U11¹(x₁, x₂)) = 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) = 1
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = x₁ + x₂
POL(tt) = 0
POL(zeros) = 1

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)
0 → n__0
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
nil → n__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)))
zeros → cons(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)
zeros → n__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)
U11¹(tt, L) → ACTIVATE(L)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
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)
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))
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)
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNat(N))

The TRS R consists of the following rules:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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 4 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                  ↳ 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:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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.

ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
ISNATLIST(n__take(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)

The remaining pairs can at least be oriented weakly.

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)))
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)
U31¹(tt, IL, M, N) → ACTIVATE(M)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
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)
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(TAKE(x₁, x₂)) = x₁ + x₂
POL(U11(x₁, x₂)) = x₂
POL(U21(x₁)) = 1 + x₁
POL(U31(x₁, x₂, x₃, x₄)) = 1 + 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)) = 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)) = x₁
POL(n__nil) = 0
POL(n__s(x₁)) = x₁
POL(n__take(x₁, x₂)) = 1 + x₁ + x₂
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 1 + 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)
0 → n__0
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
nil → n__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)))
zeros → cons(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)
zeros → n__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

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                  ↳ QDP

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

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)))
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)
U31¹(tt, IL, M, N) → ACTIVATE(M)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
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)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U31¹(tt, IL, M, N) → ACTIVATE(N)

The TRS R consists of the following rules:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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 9 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                  ↳ QDP

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

ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(V) → ISNATLIST(activate(V))
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)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)

The TRS R consists of the following rules:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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.

ACTIVATE(n__isNat(X)) → ISNAT(X)

The remaining pairs can at least be oriented weakly.

ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(V) → ISNATLIST(activate(V))
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)

Used ordering: Polynomial interpretation with max and min functions [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(U11(x₁, x₂)) = x₂
POL(U21(x₁)) = 0
POL(U31(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₃ + x₄
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 1 + x₁
POL(isNatIList(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = x₁
POL(n__0) = 0
POL(n__and(x₁, x₂)) = x₂
POL(n__cons(x₁, x₂)) = x₁ + x₂
POL(n__isNat(x₁)) = 1 + x₁
POL(n__isNatIList(x₁)) = x₁
POL(n__isNatList(x₁)) = x₁
POL(n__length(x₁)) = x₁
POL(n__nil) = 0
POL(n__s(x₁)) = x₁
POL(n__take(x₁, x₂)) = 1 + x₁ + x₂
POL(n__zeros) = 1
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 1 + x₁ + x₂
POL(tt) = 0
POL(zeros) = 1

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)
0 → n__0
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
nil → n__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)))
zeros → cons(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)
zeros → n__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

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                  ↳ QDP

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

ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(V) → ISNATLIST(activate(V))
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)

The TRS R consists of the following rules:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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.

ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(V) → ISNATLIST(activate(V))

The remaining pairs can at least be oriented weakly.

ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
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)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(ISNAT(x₁)) = x₁
POL(ISNATILIST(x₁)) = 1 + x₁
POL(ISNATLIST(x₁)) = x₁
POL(U11(x₁, x₂)) = x₂
POL(U21(x₁)) = 1
POL(U31(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₄
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 1 + x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 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)) = 1 + x₁
POL(n__isNatList(x₁)) = x₁
POL(n__length(x₁)) = x₁
POL(n__nil) = 0
POL(n__s(x₁)) = x₁
POL(n__take(x₁, x₂)) = 1 + x₂
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 1 + 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)
0 → n__0
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
nil → n__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)))
zeros → cons(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)
zeros → n__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

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                  ↳ QDP

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

ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))

The TRS R consists of the following rules:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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.

ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))

The remaining pairs can at least be oriented weakly.

ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(ISNAT(x₁)) = x₁
POL(ISNATILIST(x₁)) = 1 + x₁
POL(ISNATLIST(x₁)) = 1 + x₁
POL(U11(x₁, x₂)) = 1 + x₂
POL(U21(x₁)) = 1
POL(U31(x₁, x₂, x₃, x₄)) = 1 + 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)) = 1 + x₁
POL(isNatList(x₁)) = 1 + 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)) = 1 + x₁
POL(n__isNatList(x₁)) = 1 + x₁
POL(n__length(x₁)) = 1 + x₁
POL(n__nil) = 1
POL(n__s(x₁)) = x₁
POL(n__take(x₁, x₂)) = 1 + x₁ + x₂
POL(n__zeros) = 0
POL(nil) = 1
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 1 + 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)
0 → n__0
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
nil → n__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)))
zeros → cons(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)
zeros → n__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

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                  ↳ QDP

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

ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))

The TRS R consists of the following rules:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                            ↳ QDP

                  ↳ QDP

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

ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))

The TRS R consists of the following rules:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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.

ACTIVATE(n__and(X1, X2)) → AND(X1, X2)

The remaining pairs can at least be oriented weakly.

ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
AND(tt, X) → ACTIVATE(X)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))

Used ordering: Polynomial interpretation with max and min functions [25]:

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

The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                            ↳ QDP

                  ↳ QDP

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

ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
AND(tt, X) → ACTIVATE(X)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))

The TRS R consists of the following rules:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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 ISNATLIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatList(activate(V2))) at position [0] we obtained the following new rules:

ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(y0, y1)) → AND(n__isNat(activate(y0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → AND(isNat(take(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatList(activate(y1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                            ↳ QDP

                  ↳ QDP

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

ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(y0, y1)) → AND(n__isNat(activate(y0)), n__isNatList(activate(y1)))
AND(tt, X) → ACTIVATE(X)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__take(x0, x1), y1)) → AND(isNat(take(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATLIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                            ↳ QDP

                  ↳ QDP

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

ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
AND(tt, X) → ACTIVATE(X)
ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__take(x0, x1), y1)) → AND(isNat(take(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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 ISNATILIST(n__cons(V1, V2)) → AND(isNat(activate(V1)), n__isNatIList(activate(V2))) at position [0] we obtained the following new rules:

ISNATILIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(y0, y1)) → AND(n__isNat(activate(y0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → AND(isNat(take(x0, x1)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatIList(activate(y1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                            ↳ QDP

                  ↳ QDP

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

ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATILIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatIList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
ISNATILIST(n__cons(y0, y1)) → AND(n__isNat(activate(y0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → AND(isNat(take(x0, x1)), n__isNatIList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1)))
ISNATLIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatList(activate(y1)))
ISNATILIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatIList(activate(y1)))
AND(tt, X) → ACTIVATE(X)
ISNATILIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ISNATILIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatIList(activate(y1)))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → AND(isNat(take(x0, x1)), n__isNatList(activate(y1)))
ISNATILIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
ISNATILIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatIList(activate(y1)))
ISNATLIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                            ↳ QDP

                  ↳ QDP

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

ISNATILIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatIList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → AND(isNat(take(x0, x1)), n__isNatIList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1)))
ISNATLIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatList(activate(y1)))
ISNATILIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatIList(activate(y1)))
AND(tt, X) → ACTIVATE(X)
ISNATILIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatIList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ISNATILIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatIList(activate(y1)))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → AND(isNat(take(x0, x1)), n__isNatList(activate(y1)))
ISNATILIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
ISNATILIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatIList(activate(y1)))
ISNATLIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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 instantiating [15] the rule AND(tt, X) → ACTIVATE(X) we obtained the following new rules:

AND(tt, n__isNatIList(y_5)) → ACTIVATE(n__isNatIList(y_5))
AND(tt, n__isNatList(y_4)) → ACTIVATE(n__isNatList(y_4))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                            ↳ QDP

                  ↳ QDP

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

ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATILIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatIList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → AND(isNat(take(x0, x1)), n__isNatIList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1)))
ISNATLIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ISNATILIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ISNATILIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatIList(activate(y1)))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → AND(isNat(take(x0, x1)), n__isNatList(activate(y1)))
ISNATILIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatIList(activate(y1)))
AND(tt, n__isNatIList(y_5)) → ACTIVATE(n__isNatIList(y_5))
ISNATILIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatIList(activate(y1)))
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATILIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatIList(activate(y1)))
AND(tt, n__isNatList(y_4)) → ACTIVATE(n__isNatList(y_4))
ISNATLIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                            ↳ QDPOrderProof

                                                                          ↳ QDP

                                            ↳ QDP

                  ↳ QDP

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

ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__take(x0, x1), y1)) → AND(isNat(take(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
AND(tt, n__isNatList(y_4)) → ACTIVATE(n__isNatList(y_4))
ISNATLIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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.

ISNATLIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatList(activate(y1)))

The remaining pairs can at least be oriented weakly.

ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__take(x0, x1), y1)) → AND(isNat(take(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
AND(tt, n__isNatList(y_4)) → ACTIVATE(n__isNatList(y_4))
ISNATLIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(ISNATLIST(x₁)) = x₁
POL(U11(x₁, x₂)) = x₂
POL(U21(x₁)) = 0
POL(U31(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 1 + x₁
POL(isNatIList(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = x₁
POL(n__0) = 0
POL(n__and(x₁, x₂)) = x₂
POL(n__cons(x₁, x₂)) = x₁ + x₂
POL(n__isNat(x₁)) = 1 + x₁
POL(n__isNatIList(x₁)) = x₁
POL(n__isNatList(x₁)) = x₁
POL(n__length(x₁)) = x₁
POL(n__nil) = 0
POL(n__s(x₁)) = x₁
POL(n__take(x₁, x₂)) = x₂
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 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)
0 → n__0
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
nil → n__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)))
zeros → cons(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)
zeros → n__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

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                            ↳ QDPOrderProof

                                                                          ↳ QDP

                                            ↳ QDP

                  ↳ QDP

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

ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__take(x0, x1), y1)) → AND(isNat(take(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
AND(tt, n__isNatList(y_4)) → ACTIVATE(n__isNatList(y_4))
ISNATLIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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.

ISNATLIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatList(activate(y1)))

The remaining pairs can at least be oriented weakly.

ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__take(x0, x1), y1)) → AND(isNat(take(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
AND(tt, n__isNatList(y_4)) → ACTIVATE(n__isNatList(y_4))
ISNATLIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(ISNATLIST(x₁)) = x₁
POL(U11(x₁, x₂)) = x₂
POL(U21(x₁)) = 0
POL(U31(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = x₁
POL(n__0) = 0
POL(n__and(x₁, x₂)) = x₁ + 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)) = x₁
POL(n__nil) = 0
POL(n__s(x₁)) = x₁
POL(n__take(x₁, x₂)) = x₁ + x₂
POL(n__zeros) = 1
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = x₁ + x₂
POL(tt) = 0
POL(zeros) = 1

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)
0 → n__0
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
nil → n__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)))
zeros → cons(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)
zeros → n__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

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                            ↳ QDPOrderProof

                                                                          ↳ QDP

                                            ↳ QDP

                  ↳ QDP

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

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → AND(isNat(take(x0, x1)), n__isNatList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
AND(tt, n__isNatList(y_4)) → ACTIVATE(n__isNatList(y_4))
ISNATLIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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.

ISNATLIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatList(activate(y1)))

The remaining pairs can at least be oriented weakly.

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → AND(isNat(take(x0, x1)), n__isNatList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
AND(tt, n__isNatList(y_4)) → ACTIVATE(n__isNatList(y_4))
ISNATLIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(ISNATLIST(x₁)) = x₁
POL(U11(x₁, x₂)) = 1 + x₂
POL(U21(x₁)) = x₁
POL(U31(x₁, x₂, x₃, x₄)) = x₂ + 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)) = x₂
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 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)
0 → n__0
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
nil → n__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)))
zeros → cons(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)
zeros → n__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

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                            ↳ QDPOrderProof

                                                                          ↳ QDP

                                            ↳ QDP

                  ↳ QDP

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

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__take(x0, x1), y1)) → AND(isNat(take(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
AND(tt, n__isNatList(y_4)) → ACTIVATE(n__isNatList(y_4))
ISNATLIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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.

ISNATLIST(n__cons(n__take(x0, x1), y1)) → AND(isNat(take(x0, x1)), n__isNatList(activate(y1)))

The remaining pairs can at least be oriented weakly.

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
AND(tt, n__isNatList(y_4)) → ACTIVATE(n__isNatList(y_4))
ISNATLIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(ISNATLIST(x₁)) = x₁
POL(U11(x₁, x₂)) = x₂
POL(U21(x₁)) = 1 + x₁
POL(U31(x₁, x₂, x₃, x₄)) = 1 + x₂ + 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)) = 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)) = x₁
POL(n__nil) = 0
POL(n__s(x₁)) = x₁
POL(n__take(x₁, x₂)) = 1 + x₂
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 1 + 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)
0 → n__0
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
nil → n__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)))
zeros → cons(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)
zeros → n__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

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                            ↳ QDPOrderProof

                                                                          ↳ QDP

                                            ↳ QDP

                  ↳ QDP

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

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
AND(tt, n__isNatList(y_4)) → ACTIVATE(n__isNatList(y_4))
ISNATLIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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.

ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))

The remaining pairs can at least be oriented weakly.

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
AND(tt, n__isNatList(y_4)) → ACTIVATE(n__isNatList(y_4))
ISNATLIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(ISNATLIST(x₁)) = x₁
POL(U11(x₁, x₂)) = x₂
POL(U21(x₁)) = x₁
POL(U31(x₁, x₂, x₃, x₄)) = 1 + 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)) = 1 + x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 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)) = 1 + x₁
POL(n__isNatList(x₁)) = x₁
POL(n__length(x₁)) = x₁
POL(n__nil) = 0
POL(n__s(x₁)) = x₁
POL(n__take(x₁, x₂)) = 1 + x₁ + x₂
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 1 + 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)
0 → n__0
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
nil → n__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)))
zeros → cons(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)
zeros → n__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

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                            ↳ QDPOrderProof

                                                                          ↳ QDP

                                            ↳ QDP

                  ↳ QDP

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

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))
AND(tt, n__isNatList(y_4)) → ACTIVATE(n__isNatList(y_4))
ISNATLIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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.

ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))

The remaining pairs can at least be oriented weakly.

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
AND(tt, n__isNatList(y_4)) → ACTIVATE(n__isNatList(y_4))
ISNATLIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(ISNATLIST(x₁)) = x₁
POL(U11(x₁, x₂)) = x₂
POL(U21(x₁)) = 1 + x₁
POL(U31(x₁, x₂, x₃, x₄)) = 1 + 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)) = 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)) = x₁
POL(n__nil) = 1
POL(n__s(x₁)) = x₁
POL(n__take(x₁, x₂)) = 1 + x₁ + x₂
POL(n__zeros) = 0
POL(nil) = 1
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 1 + 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)
0 → n__0
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
nil → n__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)))
zeros → cons(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)
zeros → n__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

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                            ↳ QDPOrderProof

                                                                          ↳ QDP

                                            ↳ QDP

                  ↳ QDP

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

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
AND(tt, n__isNatList(y_4)) → ACTIVATE(n__isNatList(y_4))
ISNATLIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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.

ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))

The remaining pairs can at least be oriented weakly.

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
AND(tt, n__isNatList(y_4)) → ACTIVATE(n__isNatList(y_4))
ISNATLIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(ISNATLIST(x₁)) = 1 + x₁
POL(U11(x₁, x₂)) = 1 + x₂
POL(U21(x₁)) = 0
POL(U31(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 1 + x₁
POL(isNatList(x₁)) = 1 + 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)) = 1 + x₁
POL(n__isNatList(x₁)) = 1 + x₁
POL(n__length(x₁)) = 1 + x₁
POL(n__nil) = 0
POL(n__s(x₁)) = x₁
POL(n__take(x₁, x₂)) = x₂
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 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)
0 → n__0
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
nil → n__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)))
zeros → cons(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)
zeros → n__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

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                            ↳ QDPOrderProof

                                                                          ↳ QDP

                                            ↳ QDP

                  ↳ QDP

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

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
AND(tt, n__isNatList(y_4)) → ACTIVATE(n__isNatList(y_4))
ISNATLIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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.

ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))

The remaining pairs can at least be oriented weakly.

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
AND(tt, n__isNatList(y_4)) → ACTIVATE(n__isNatList(y_4))
ISNATLIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( U31(x₁, ..., x₄) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

/	0	1	\
\	0	0	/

x₄

M( n__isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( n__nil ) =

/	0	\
\	0	/

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( n__s(x₁) ) =

/	0	\
\	1	/

/	1	0	\
\	0	0	/

x₁

M( take(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	1	0	/

x₂

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( s(x₁) ) =

/	0	\
\	1	/

/	1	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( nil ) =

/	0	\
\	0	/

M( n__isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( n__length(x₁) ) =

/	0	\
\	1	/

/	1	0	\
\	0	0	/

x₁

M( n__zeros ) =

/	0	\
\	0	/

M( n__cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

M( n__isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( U11(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( n__take(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	1	0	/

x₂

M( length(x₁) ) =

/	0	\
\	1	/

/	1	0	\
\	0	0	/

x₁

M( U21(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

Tuple symbols:

M( ISNATLIST(x₁) ) =

[

]

x₁

M( AND(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( ACTIVATE(x₁) ) =

[

]

x₁

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)
0 → n__0
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
nil → n__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)))
zeros → cons(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)
zeros → n__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

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                            ↳ QDPOrderProof

                                                                          ↳ QDP

                                            ↳ QDP

                  ↳ QDP

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

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
AND(tt, n__isNatList(y_4)) → ACTIVATE(n__isNatList(y_4))
ISNATLIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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.

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))

The remaining pairs can at least be oriented weakly.

ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
AND(tt, n__isNatList(y_4)) → ACTIVATE(n__isNatList(y_4))
ISNATLIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( U31(x₁, ..., x₄) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

/	1	0	\
\	0	0	/

x₃

/	0	1	\
\	0	0	/

x₄

M( n__isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( n__nil ) =

/	0	\
\	0	/

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( n__s(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	1	0	/

x₁

M( take(x₁, x₂) ) =

/	0	\
\	1	/

/	1	0	\
\	0	1	/

x₁

/	1	0	\
\	1	0	/

x₂

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	1	0	/

x₁

M( zeros ) =

/	0	\
\	1	/

M( isNatIList(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	1	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	1	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( nil ) =

/	0	\
\	0	/

M( n__isNatIList(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	1	0	/

x₁

M( n__length(x₁) ) =

/	1	\
\	1	/

/	1	0	\
\	1	0	/

x₁

M( n__zeros ) =

/	0	\
\	1	/

M( n__cons(x₁, x₂) ) =

/	0	\
\	1	/

/	0	1	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

M( n__isNatList(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	1	0	/

x₁

M( U11(x₁, x₂) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	1	0	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	1	/

/	0	1	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( n__take(x₁, x₂) ) =

/	0	\
\	1	/

/	1	0	\
\	0	1	/

x₁

/	1	0	\
\	1	0	/

x₂

M( length(x₁) ) =

/	1	\
\	1	/

/	1	0	\
\	1	0	/

x₁

M( U21(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

Tuple symbols:

M( ISNATLIST(x₁) ) =

[

]

x₁

M( AND(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( ACTIVATE(x₁) ) =

[

]

x₁

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)
0 → n__0
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
nil → n__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)))
zeros → cons(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)
zeros → n__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

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ SemLabProof

                                                                                                                ↳ SemLabProof2

                                                                            ↳ QDPOrderProof

                                                                          ↳ QDP

                                            ↳ QDP

                  ↳ QDP

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

ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
AND(tt, n__isNatList(y_4)) → ACTIVATE(n__isNatList(y_4))
ISNATLIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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 found the following model for the rules of the TRS R. Interpretation over the domain with elements from 0 to 1.U31: 0
n__isNat: 1
activate: x₀
n__nil: 0
n__s: 0
and: x₁
take: 0
ISNATLIST: 0
tt: 1
isNatList: 1
AND: 0
zeros: 0
isNatIList: 1
s: 0
isNat: 1
nil: 0
ACTIVATE: 0
n__isNatIList: 1
n__length: 0
n__zeros: 0
n__cons: 0
n__isNatList: 1
U11: 0
0: 0
n__and: x₁
cons: 0
n__0: 0
n__take: 0
length: 0
U21: 0
By semantic labelling [33] we obtain the following labelled TRS:Q DP problem:
The TRS P consists of the following rules:

ISNATLIST.0(n__cons.0-1(n__and.1-0(x0, x1), y1)) → AND.1-1(isNat.0(and.1-0(x0, x1)), n__isNatList.1(activate.1(y1)))
ACTIVATE.1(n__isNatList.1(X)) → ISNATLIST.1(X)
ISNATLIST.0(n__cons.1-1(n__and.1-1(x0, x1), y1)) → AND.1-1(isNat.1(and.1-1(x0, x1)), n__isNatList.1(activate.1(y1)))
AND.1-1(tt., n__isNatList.0(y_4)) → ACTIVATE.1(n__isNatList.0(y_4))
ISNATLIST.0(n__cons.0-0(n__0., y1)) → AND.1-1(isNat.0(0.), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.1-1(n__and.0-1(x0, x1), y1)) → AND.1-1(isNat.1(and.0-1(x0, x1)), n__isNatList.1(activate.1(y1)))
ACTIVATE.1(n__isNatList.0(X)) → ISNATLIST.0(X)
ISNATLIST.0(n__cons.0-1(n__and.0-0(x0, x1), y1)) → AND.1-1(isNat.0(and.0-0(x0, x1)), n__isNatList.1(activate.1(y1)))
ISNATLIST.0(n__cons.0-1(x0, y1)) → AND.1-1(isNat.0(x0), n__isNatList.1(activate.1(y1)))
ISNATLIST.0(n__cons.1-0(n__and.1-1(x0, x1), y1)) → AND.1-1(isNat.1(and.1-1(x0, x1)), n__isNatList.0(activate.0(y1)))
AND.1-1(tt., n__isNatList.1(y_4)) → ACTIVATE.1(n__isNatList.1(y_4))
ISNATLIST.0(n__cons.0-0(x0, y1)) → AND.1-1(isNat.0(x0), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.1-0(x0, y1)) → AND.1-1(isNat.1(x0), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.0-0(n__and.1-0(x0, x1), y1)) → AND.1-1(isNat.0(and.1-0(x0, x1)), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.1-1(x0, y1)) → AND.1-1(isNat.1(x0), n__isNatList.1(activate.1(y1)))
ISNATLIST.0(n__cons.0-1(n__0., y1)) → AND.1-1(isNat.0(0.), n__isNatList.1(activate.1(y1)))
ISNATLIST.0(n__cons.0-0(n__and.0-0(x0, x1), y1)) → AND.1-1(isNat.0(and.0-0(x0, x1)), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.1-0(n__and.0-1(x0, x1), y1)) → AND.1-1(isNat.1(and.0-1(x0, x1)), n__isNatList.0(activate.0(y1)))

The TRS R consists of the following rules:

take.0-0(s.1(M), cons.0-0(N, IL)) → U31.1-0-1-0(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.1(M), n__isNat.0(N))), activate.0(IL), M, N)
activate.0(n__take.0-1(X1, X2)) → take.0-1(X1, X2)
U31.1-0-1-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.1-0(activate.1(M), activate.0(IL)))
and.1-0(tt., X) → activate.0(X)
activate.0(n__zeros.) → zeros.
and.0-0(X1, X2) → n__and.0-0(X1, X2)
activate.0(n__take.1-1(X1, X2)) → take.1-1(X1, X2)
U21.1(tt.) → nil.
activate.1(n__isNat.1(X)) → isNat.1(X)
activate.0(n__length.1(X)) → length.1(X)
activate.0(n__s.0(X)) → s.0(X)
length.0(X) → n__length.0(X)
isNatList.0(n__take.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.0(activate.0(V2)))
isNat.0(n__s.0(V1)) → isNat.0(activate.0(V1))
cons.1-0(X1, X2) → n__cons.1-0(X1, X2)
and.1-1(X1, X2) → n__and.1-1(X1, X2)
isNat.0(X) → n__isNat.0(X)
take.0-0(s.1(M), cons.0-1(N, IL)) → U31.1-1-1-0(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.1(M), n__isNat.0(N))), activate.1(IL), M, N)
isNatList.0(n__cons.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatList.0(activate.0(V2)))
isNatIList.0(X) → n__isNatIList.0(X)
length.0(cons.0-1(N, L)) → U11.1-1(and.1-1(isNatList.1(activate.1(L)), n__isNat.0(N)), activate.1(L))
U31.1-1-0-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.0-1(activate.0(M), activate.1(IL)))
cons.0-0(X1, X2) → n__cons.0-0(X1, X2)
activate.0(n__s.1(X)) → s.1(X)
activate.0(n__and.1-0(X1, X2)) → and.1-0(X1, X2)
s.0(X) → n__s.0(X)
activate.0(n__cons.0-0(X1, X2)) → cons.0-0(X1, X2)
U31.1-1-1-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.1-1(activate.1(M), activate.1(IL)))
cons.0-1(X1, X2) → n__cons.0-1(X1, X2)
activate.1(n__isNatList.0(X)) → isNatList.0(X)
activate.0(n__take.0-0(X1, X2)) → take.0-0(X1, X2)
and.0-1(X1, X2) → n__and.0-1(X1, X2)
isNat.0(n__length.1(V1)) → isNatList.1(activate.1(V1))
activate.1(n__isNatIList.1(X)) → isNatIList.1(X)
isNatIList.0(n__cons.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.1(activate.1(V2)))
length.0(cons.1-1(N, L)) → U11.1-1(and.1-1(isNatList.1(activate.1(L)), n__isNat.1(N)), activate.1(L))
isNatIList.1(X) → n__isNatIList.1(X)
take.0-0(X1, X2) → n__take.0-0(X1, X2)
zeros. → n__zeros.
length.0(nil.) → 0.
activate.1(n__and.0-1(X1, X2)) → and.0-1(X1, X2)
isNat.1(X) → n__isNat.1(X)
activate.0(n__take.1-0(X1, X2)) → take.1-0(X1, X2)
take.0-0(s.0(M), cons.1-1(N, IL)) → U31.1-1-0-1(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.0(M), n__isNat.1(N))), activate.1(IL), M, N)
isNatList.0(n__take.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.0(activate.0(V2)))
take.1-1(X1, X2) → n__take.1-1(X1, X2)
U31.1-0-0-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.0-0(activate.0(M), activate.0(IL)))
isNat.0(n__length.0(V1)) → isNatList.0(activate.0(V1))
isNatIList.0(n__cons.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.0(activate.0(V2)))
s.1(X) → n__s.1(X)
isNatList.0(n__cons.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatList.1(activate.1(V2)))
take.1-0(X1, X2) → n__take.1-0(X1, X2)
isNatIList.0(V) → isNatList.0(activate.0(V))
take.0-0(0., IL) → U21.1(isNatIList.0(IL))
activate.0(n__length.0(X)) → length.0(X)
activate.1(n__isNatIList.0(X)) → isNatIList.0(X)
activate.0(n__and.0-0(X1, X2)) → and.0-0(X1, X2)
isNatList.0(n__cons.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatList.0(activate.0(V2)))
isNatList.0(n__cons.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatList.1(activate.1(V2)))
take.0-0(s.1(M), cons.1-0(N, IL)) → U31.1-0-1-1(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.1(M), n__isNat.1(N))), activate.0(IL), M, N)
isNatList.0(n__take.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.1(activate.1(V2)))
U31.1-1-0-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.0-1(activate.0(M), activate.1(IL)))
isNatList.1(X) → n__isNatList.1(X)
U31.1-1-1-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.1-1(activate.1(M), activate.1(IL)))
length.1(X) → n__length.1(X)
cons.1-1(X1, X2) → n__cons.1-1(X1, X2)
take.0-1(X1, X2) → n__take.0-1(X1, X2)
and.1-0(X1, X2) → n__and.1-0(X1, X2)
activate.0(X) → X
activate.0(n__cons.0-1(X1, X2)) → cons.0-1(X1, X2)
activate.0(n__nil.) → nil.
and.1-1(tt., X) → activate.1(X)
isNatIList.1(V) → isNatList.1(activate.1(V))
take.0-1(0., IL) → U21.1(isNatIList.1(IL))
activate.1(X) → X
take.0-0(s.0(M), cons.0-0(N, IL)) → U31.1-0-0-0(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.0(M), n__isNat.0(N))), activate.0(IL), M, N)
0. → n__0.
U31.1-0-1-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.1-0(activate.1(M), activate.0(IL)))
isNat.0(n__s.1(V1)) → isNat.1(activate.1(V1))
U31.1-0-0-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.0-0(activate.0(M), activate.0(IL)))
take.0-0(s.1(M), cons.1-1(N, IL)) → U31.1-1-1-1(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.1(M), n__isNat.1(N))), activate.1(IL), M, N)
U11.1-0(tt., L) → s.0(length.0(activate.0(L)))
isNatList.0(n__nil.) → tt.
take.0-0(s.0(M), cons.0-1(N, IL)) → U31.1-1-0-0(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.0(M), n__isNat.0(N))), activate.1(IL), M, N)
isNatList.0(n__take.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.1(activate.1(V2)))
isNatIList.0(n__zeros.) → tt.
length.0(cons.1-0(N, L)) → U11.1-0(and.1-1(isNatList.0(activate.0(L)), n__isNat.1(N)), activate.0(L))
zeros. → cons.0-0(0., n__zeros.)
isNatIList.0(n__cons.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.0(activate.0(V2)))
activate.0(n__0.) → 0.
activate.1(n__isNatList.1(X)) → isNatList.1(X)
activate.0(n__cons.1-0(X1, X2)) → cons.1-0(X1, X2)
activate.1(n__and.1-1(X1, X2)) → and.1-1(X1, X2)
take.0-0(s.0(M), cons.1-0(N, IL)) → U31.1-0-0-1(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.0(M), n__isNat.1(N))), activate.0(IL), M, N)
isNat.0(n__0.) → tt.
nil. → n__nil.
U11.1-1(tt., L) → s.0(length.1(activate.1(L)))
length.0(cons.0-0(N, L)) → U11.1-0(and.1-1(isNatList.0(activate.0(L)), n__isNat.0(N)), activate.0(L))
activate.1(n__isNat.0(X)) → isNat.0(X)
activate.0(n__cons.1-1(X1, X2)) → cons.1-1(X1, X2)
isNatList.0(X) → n__isNatList.0(X)
isNatIList.0(n__cons.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.1(activate.1(V2)))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ SemLabProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                ↳ SemLabProof2

                                                                            ↳ QDPOrderProof

                                                                          ↳ QDP

                                            ↳ QDP

                  ↳ QDP

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

ISNATLIST.0(n__cons.0-1(n__and.1-0(x0, x1), y1)) → AND.1-1(isNat.0(and.1-0(x0, x1)), n__isNatList.1(activate.1(y1)))
ACTIVATE.1(n__isNatList.1(X)) → ISNATLIST.1(X)
ISNATLIST.0(n__cons.1-1(n__and.1-1(x0, x1), y1)) → AND.1-1(isNat.1(and.1-1(x0, x1)), n__isNatList.1(activate.1(y1)))
AND.1-1(tt., n__isNatList.0(y_4)) → ACTIVATE.1(n__isNatList.0(y_4))
ISNATLIST.0(n__cons.0-0(n__0., y1)) → AND.1-1(isNat.0(0.), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.1-1(n__and.0-1(x0, x1), y1)) → AND.1-1(isNat.1(and.0-1(x0, x1)), n__isNatList.1(activate.1(y1)))
ACTIVATE.1(n__isNatList.0(X)) → ISNATLIST.0(X)
ISNATLIST.0(n__cons.0-1(n__and.0-0(x0, x1), y1)) → AND.1-1(isNat.0(and.0-0(x0, x1)), n__isNatList.1(activate.1(y1)))
ISNATLIST.0(n__cons.0-1(x0, y1)) → AND.1-1(isNat.0(x0), n__isNatList.1(activate.1(y1)))
ISNATLIST.0(n__cons.1-0(n__and.1-1(x0, x1), y1)) → AND.1-1(isNat.1(and.1-1(x0, x1)), n__isNatList.0(activate.0(y1)))
AND.1-1(tt., n__isNatList.1(y_4)) → ACTIVATE.1(n__isNatList.1(y_4))
ISNATLIST.0(n__cons.0-0(x0, y1)) → AND.1-1(isNat.0(x0), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.1-0(x0, y1)) → AND.1-1(isNat.1(x0), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.0-0(n__and.1-0(x0, x1), y1)) → AND.1-1(isNat.0(and.1-0(x0, x1)), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.1-1(x0, y1)) → AND.1-1(isNat.1(x0), n__isNatList.1(activate.1(y1)))
ISNATLIST.0(n__cons.0-1(n__0., y1)) → AND.1-1(isNat.0(0.), n__isNatList.1(activate.1(y1)))
ISNATLIST.0(n__cons.0-0(n__and.0-0(x0, x1), y1)) → AND.1-1(isNat.0(and.0-0(x0, x1)), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.1-0(n__and.0-1(x0, x1), y1)) → AND.1-1(isNat.1(and.0-1(x0, x1)), n__isNatList.0(activate.0(y1)))

The TRS R consists of the following rules:

take.0-0(s.1(M), cons.0-0(N, IL)) → U31.1-0-1-0(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.1(M), n__isNat.0(N))), activate.0(IL), M, N)
activate.0(n__take.0-1(X1, X2)) → take.0-1(X1, X2)
U31.1-0-1-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.1-0(activate.1(M), activate.0(IL)))
and.1-0(tt., X) → activate.0(X)
activate.0(n__zeros.) → zeros.
and.0-0(X1, X2) → n__and.0-0(X1, X2)
activate.0(n__take.1-1(X1, X2)) → take.1-1(X1, X2)
U21.1(tt.) → nil.
activate.1(n__isNat.1(X)) → isNat.1(X)
activate.0(n__length.1(X)) → length.1(X)
activate.0(n__s.0(X)) → s.0(X)
length.0(X) → n__length.0(X)
isNatList.0(n__take.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.0(activate.0(V2)))
isNat.0(n__s.0(V1)) → isNat.0(activate.0(V1))
cons.1-0(X1, X2) → n__cons.1-0(X1, X2)
and.1-1(X1, X2) → n__and.1-1(X1, X2)
isNat.0(X) → n__isNat.0(X)
take.0-0(s.1(M), cons.0-1(N, IL)) → U31.1-1-1-0(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.1(M), n__isNat.0(N))), activate.1(IL), M, N)
isNatList.0(n__cons.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatList.0(activate.0(V2)))
isNatIList.0(X) → n__isNatIList.0(X)
length.0(cons.0-1(N, L)) → U11.1-1(and.1-1(isNatList.1(activate.1(L)), n__isNat.0(N)), activate.1(L))
U31.1-1-0-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.0-1(activate.0(M), activate.1(IL)))
cons.0-0(X1, X2) → n__cons.0-0(X1, X2)
activate.0(n__s.1(X)) → s.1(X)
activate.0(n__and.1-0(X1, X2)) → and.1-0(X1, X2)
s.0(X) → n__s.0(X)
activate.0(n__cons.0-0(X1, X2)) → cons.0-0(X1, X2)
U31.1-1-1-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.1-1(activate.1(M), activate.1(IL)))
cons.0-1(X1, X2) → n__cons.0-1(X1, X2)
activate.1(n__isNatList.0(X)) → isNatList.0(X)
activate.0(n__take.0-0(X1, X2)) → take.0-0(X1, X2)
and.0-1(X1, X2) → n__and.0-1(X1, X2)
isNat.0(n__length.1(V1)) → isNatList.1(activate.1(V1))
activate.1(n__isNatIList.1(X)) → isNatIList.1(X)
isNatIList.0(n__cons.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.1(activate.1(V2)))
length.0(cons.1-1(N, L)) → U11.1-1(and.1-1(isNatList.1(activate.1(L)), n__isNat.1(N)), activate.1(L))
isNatIList.1(X) → n__isNatIList.1(X)
take.0-0(X1, X2) → n__take.0-0(X1, X2)
zeros. → n__zeros.
length.0(nil.) → 0.
activate.1(n__and.0-1(X1, X2)) → and.0-1(X1, X2)
isNat.1(X) → n__isNat.1(X)
activate.0(n__take.1-0(X1, X2)) → take.1-0(X1, X2)
take.0-0(s.0(M), cons.1-1(N, IL)) → U31.1-1-0-1(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.0(M), n__isNat.1(N))), activate.1(IL), M, N)
isNatList.0(n__take.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.0(activate.0(V2)))
take.1-1(X1, X2) → n__take.1-1(X1, X2)
U31.1-0-0-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.0-0(activate.0(M), activate.0(IL)))
isNat.0(n__length.0(V1)) → isNatList.0(activate.0(V1))
isNatIList.0(n__cons.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.0(activate.0(V2)))
s.1(X) → n__s.1(X)
isNatList.0(n__cons.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatList.1(activate.1(V2)))
take.1-0(X1, X2) → n__take.1-0(X1, X2)
isNatIList.0(V) → isNatList.0(activate.0(V))
take.0-0(0., IL) → U21.1(isNatIList.0(IL))
activate.0(n__length.0(X)) → length.0(X)
activate.1(n__isNatIList.0(X)) → isNatIList.0(X)
activate.0(n__and.0-0(X1, X2)) → and.0-0(X1, X2)
isNatList.0(n__cons.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatList.0(activate.0(V2)))
isNatList.0(n__cons.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatList.1(activate.1(V2)))
take.0-0(s.1(M), cons.1-0(N, IL)) → U31.1-0-1-1(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.1(M), n__isNat.1(N))), activate.0(IL), M, N)
isNatList.0(n__take.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.1(activate.1(V2)))
U31.1-1-0-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.0-1(activate.0(M), activate.1(IL)))
isNatList.1(X) → n__isNatList.1(X)
U31.1-1-1-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.1-1(activate.1(M), activate.1(IL)))
length.1(X) → n__length.1(X)
cons.1-1(X1, X2) → n__cons.1-1(X1, X2)
take.0-1(X1, X2) → n__take.0-1(X1, X2)
and.1-0(X1, X2) → n__and.1-0(X1, X2)
activate.0(X) → X
activate.0(n__cons.0-1(X1, X2)) → cons.0-1(X1, X2)
activate.0(n__nil.) → nil.
and.1-1(tt., X) → activate.1(X)
isNatIList.1(V) → isNatList.1(activate.1(V))
take.0-1(0., IL) → U21.1(isNatIList.1(IL))
activate.1(X) → X
take.0-0(s.0(M), cons.0-0(N, IL)) → U31.1-0-0-0(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.0(M), n__isNat.0(N))), activate.0(IL), M, N)
0. → n__0.
U31.1-0-1-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.1-0(activate.1(M), activate.0(IL)))
isNat.0(n__s.1(V1)) → isNat.1(activate.1(V1))
U31.1-0-0-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.0-0(activate.0(M), activate.0(IL)))
take.0-0(s.1(M), cons.1-1(N, IL)) → U31.1-1-1-1(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.1(M), n__isNat.1(N))), activate.1(IL), M, N)
U11.1-0(tt., L) → s.0(length.0(activate.0(L)))
isNatList.0(n__nil.) → tt.
take.0-0(s.0(M), cons.0-1(N, IL)) → U31.1-1-0-0(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.0(M), n__isNat.0(N))), activate.1(IL), M, N)
isNatList.0(n__take.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.1(activate.1(V2)))
isNatIList.0(n__zeros.) → tt.
length.0(cons.1-0(N, L)) → U11.1-0(and.1-1(isNatList.0(activate.0(L)), n__isNat.1(N)), activate.0(L))
zeros. → cons.0-0(0., n__zeros.)
isNatIList.0(n__cons.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.0(activate.0(V2)))
activate.0(n__0.) → 0.
activate.1(n__isNatList.1(X)) → isNatList.1(X)
activate.0(n__cons.1-0(X1, X2)) → cons.1-0(X1, X2)
activate.1(n__and.1-1(X1, X2)) → and.1-1(X1, X2)
take.0-0(s.0(M), cons.1-0(N, IL)) → U31.1-0-0-1(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.0(M), n__isNat.1(N))), activate.0(IL), M, N)
isNat.0(n__0.) → tt.
nil. → n__nil.
U11.1-1(tt., L) → s.0(length.1(activate.1(L)))
length.0(cons.0-0(N, L)) → U11.1-0(and.1-1(isNatList.0(activate.0(L)), n__isNat.0(N)), activate.0(L))
activate.1(n__isNat.0(X)) → isNat.0(X)
activate.0(n__cons.1-1(X1, X2)) → cons.1-1(X1, X2)
isNatList.0(X) → n__isNatList.0(X)
isNatIList.0(n__cons.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.1(activate.1(V2)))

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 9 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ SemLabProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                ↳ SemLabProof2

                                                                            ↳ QDPOrderProof

                                                                          ↳ QDP

                                            ↳ QDP

                  ↳ QDP

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

ISNATLIST.0(n__cons.1-0(n__and.1-1(x0, x1), y1)) → AND.1-1(isNat.1(and.1-1(x0, x1)), n__isNatList.0(activate.0(y1)))
AND.1-1(tt., n__isNatList.0(y_4)) → ACTIVATE.1(n__isNatList.0(y_4))
ISNATLIST.0(n__cons.0-0(n__0., y1)) → AND.1-1(isNat.0(0.), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.0-0(x0, y1)) → AND.1-1(isNat.0(x0), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.0-0(n__and.1-0(x0, x1), y1)) → AND.1-1(isNat.0(and.1-0(x0, x1)), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.1-0(x0, y1)) → AND.1-1(isNat.1(x0), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.0-0(n__and.0-0(x0, x1), y1)) → AND.1-1(isNat.0(and.0-0(x0, x1)), n__isNatList.0(activate.0(y1)))
ACTIVATE.1(n__isNatList.0(X)) → ISNATLIST.0(X)
ISNATLIST.0(n__cons.1-0(n__and.0-1(x0, x1), y1)) → AND.1-1(isNat.1(and.0-1(x0, x1)), n__isNatList.0(activate.0(y1)))

The TRS R consists of the following rules:

take.0-0(s.1(M), cons.0-0(N, IL)) → U31.1-0-1-0(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.1(M), n__isNat.0(N))), activate.0(IL), M, N)
activate.0(n__take.0-1(X1, X2)) → take.0-1(X1, X2)
U31.1-0-1-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.1-0(activate.1(M), activate.0(IL)))
and.1-0(tt., X) → activate.0(X)
activate.0(n__zeros.) → zeros.
and.0-0(X1, X2) → n__and.0-0(X1, X2)
activate.0(n__take.1-1(X1, X2)) → take.1-1(X1, X2)
U21.1(tt.) → nil.
activate.1(n__isNat.1(X)) → isNat.1(X)
activate.0(n__length.1(X)) → length.1(X)
activate.0(n__s.0(X)) → s.0(X)
length.0(X) → n__length.0(X)
isNatList.0(n__take.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.0(activate.0(V2)))
isNat.0(n__s.0(V1)) → isNat.0(activate.0(V1))
cons.1-0(X1, X2) → n__cons.1-0(X1, X2)
and.1-1(X1, X2) → n__and.1-1(X1, X2)
isNat.0(X) → n__isNat.0(X)
take.0-0(s.1(M), cons.0-1(N, IL)) → U31.1-1-1-0(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.1(M), n__isNat.0(N))), activate.1(IL), M, N)
isNatList.0(n__cons.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatList.0(activate.0(V2)))
isNatIList.0(X) → n__isNatIList.0(X)
length.0(cons.0-1(N, L)) → U11.1-1(and.1-1(isNatList.1(activate.1(L)), n__isNat.0(N)), activate.1(L))
U31.1-1-0-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.0-1(activate.0(M), activate.1(IL)))
cons.0-0(X1, X2) → n__cons.0-0(X1, X2)
activate.0(n__s.1(X)) → s.1(X)
activate.0(n__and.1-0(X1, X2)) → and.1-0(X1, X2)
s.0(X) → n__s.0(X)
activate.0(n__cons.0-0(X1, X2)) → cons.0-0(X1, X2)
U31.1-1-1-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.1-1(activate.1(M), activate.1(IL)))
cons.0-1(X1, X2) → n__cons.0-1(X1, X2)
activate.1(n__isNatList.0(X)) → isNatList.0(X)
activate.0(n__take.0-0(X1, X2)) → take.0-0(X1, X2)
and.0-1(X1, X2) → n__and.0-1(X1, X2)
isNat.0(n__length.1(V1)) → isNatList.1(activate.1(V1))
activate.1(n__isNatIList.1(X)) → isNatIList.1(X)
isNatIList.0(n__cons.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.1(activate.1(V2)))
length.0(cons.1-1(N, L)) → U11.1-1(and.1-1(isNatList.1(activate.1(L)), n__isNat.1(N)), activate.1(L))
isNatIList.1(X) → n__isNatIList.1(X)
take.0-0(X1, X2) → n__take.0-0(X1, X2)
zeros. → n__zeros.
length.0(nil.) → 0.
activate.1(n__and.0-1(X1, X2)) → and.0-1(X1, X2)
isNat.1(X) → n__isNat.1(X)
activate.0(n__take.1-0(X1, X2)) → take.1-0(X1, X2)
take.0-0(s.0(M), cons.1-1(N, IL)) → U31.1-1-0-1(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.0(M), n__isNat.1(N))), activate.1(IL), M, N)
isNatList.0(n__take.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.0(activate.0(V2)))
take.1-1(X1, X2) → n__take.1-1(X1, X2)
U31.1-0-0-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.0-0(activate.0(M), activate.0(IL)))
isNat.0(n__length.0(V1)) → isNatList.0(activate.0(V1))
isNatIList.0(n__cons.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.0(activate.0(V2)))
s.1(X) → n__s.1(X)
isNatList.0(n__cons.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatList.1(activate.1(V2)))
take.1-0(X1, X2) → n__take.1-0(X1, X2)
isNatIList.0(V) → isNatList.0(activate.0(V))
take.0-0(0., IL) → U21.1(isNatIList.0(IL))
activate.0(n__length.0(X)) → length.0(X)
activate.1(n__isNatIList.0(X)) → isNatIList.0(X)
activate.0(n__and.0-0(X1, X2)) → and.0-0(X1, X2)
isNatList.0(n__cons.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatList.0(activate.0(V2)))
isNatList.0(n__cons.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatList.1(activate.1(V2)))
take.0-0(s.1(M), cons.1-0(N, IL)) → U31.1-0-1-1(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.1(M), n__isNat.1(N))), activate.0(IL), M, N)
isNatList.0(n__take.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.1(activate.1(V2)))
U31.1-1-0-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.0-1(activate.0(M), activate.1(IL)))
isNatList.1(X) → n__isNatList.1(X)
U31.1-1-1-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.1-1(activate.1(M), activate.1(IL)))
length.1(X) → n__length.1(X)
cons.1-1(X1, X2) → n__cons.1-1(X1, X2)
take.0-1(X1, X2) → n__take.0-1(X1, X2)
and.1-0(X1, X2) → n__and.1-0(X1, X2)
activate.0(X) → X
activate.0(n__cons.0-1(X1, X2)) → cons.0-1(X1, X2)
activate.0(n__nil.) → nil.
and.1-1(tt., X) → activate.1(X)
isNatIList.1(V) → isNatList.1(activate.1(V))
take.0-1(0., IL) → U21.1(isNatIList.1(IL))
activate.1(X) → X
take.0-0(s.0(M), cons.0-0(N, IL)) → U31.1-0-0-0(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.0(M), n__isNat.0(N))), activate.0(IL), M, N)
0. → n__0.
U31.1-0-1-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.1-0(activate.1(M), activate.0(IL)))
isNat.0(n__s.1(V1)) → isNat.1(activate.1(V1))
U31.1-0-0-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.0-0(activate.0(M), activate.0(IL)))
take.0-0(s.1(M), cons.1-1(N, IL)) → U31.1-1-1-1(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.1(M), n__isNat.1(N))), activate.1(IL), M, N)
U11.1-0(tt., L) → s.0(length.0(activate.0(L)))
isNatList.0(n__nil.) → tt.
take.0-0(s.0(M), cons.0-1(N, IL)) → U31.1-1-0-0(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.0(M), n__isNat.0(N))), activate.1(IL), M, N)
isNatList.0(n__take.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.1(activate.1(V2)))
isNatIList.0(n__zeros.) → tt.
length.0(cons.1-0(N, L)) → U11.1-0(and.1-1(isNatList.0(activate.0(L)), n__isNat.1(N)), activate.0(L))
zeros. → cons.0-0(0., n__zeros.)
isNatIList.0(n__cons.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.0(activate.0(V2)))
activate.0(n__0.) → 0.
activate.1(n__isNatList.1(X)) → isNatList.1(X)
activate.0(n__cons.1-0(X1, X2)) → cons.1-0(X1, X2)
activate.1(n__and.1-1(X1, X2)) → and.1-1(X1, X2)
take.0-0(s.0(M), cons.1-0(N, IL)) → U31.1-0-0-1(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.0(M), n__isNat.1(N))), activate.0(IL), M, N)
isNat.0(n__0.) → tt.
nil. → n__nil.
U11.1-1(tt., L) → s.0(length.1(activate.1(L)))
length.0(cons.0-0(N, L)) → U11.1-0(and.1-1(isNatList.0(activate.0(L)), n__isNat.0(N)), activate.0(L))
activate.1(n__isNat.0(X)) → isNat.0(X)
activate.0(n__cons.1-1(X1, X2)) → cons.1-1(X1, X2)
isNatList.0(X) → n__isNatList.0(X)
isNatIList.0(n__cons.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.1(activate.1(V2)))

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.

ISNATLIST.0(n__cons.1-0(n__and.0-1(x0, x1), y1)) → AND.1-1(isNat.1(and.0-1(x0, x1)), n__isNatList.0(activate.0(y1)))

The remaining pairs can at least be oriented weakly.

ISNATLIST.0(n__cons.1-0(n__and.1-1(x0, x1), y1)) → AND.1-1(isNat.1(and.1-1(x0, x1)), n__isNatList.0(activate.0(y1)))
AND.1-1(tt., n__isNatList.0(y_4)) → ACTIVATE.1(n__isNatList.0(y_4))
ISNATLIST.0(n__cons.0-0(n__0., y1)) → AND.1-1(isNat.0(0.), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.0-0(x0, y1)) → AND.1-1(isNat.0(x0), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.0-0(n__and.1-0(x0, x1), y1)) → AND.1-1(isNat.0(and.1-0(x0, x1)), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.1-0(x0, y1)) → AND.1-1(isNat.1(x0), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.0-0(n__and.0-0(x0, x1), y1)) → AND.1-1(isNat.0(and.0-0(x0, x1)), n__isNatList.0(activate.0(y1)))
ACTIVATE.1(n__isNatList.0(X)) → ISNATLIST.0(X)

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

activate.0(n__s.1(X)) → s.1(X)
cons.0-0(X1, X2) → n__cons.0-0(X1, X2)
U31.1-1-0-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.0-1(activate.0(M), activate.1(IL)))
length.0(cons.0-1(N, L)) → U11.1-1(and.1-1(isNatList.1(activate.1(L)), n__isNat.0(N)), activate.1(L))
isNatIList.0(X) → n__isNatIList.0(X)
take.0-0(s.1(M), cons.0-1(N, IL)) → U31.1-1-1-0(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.1(M), n__isNat.0(N))), activate.1(IL), M, N)
isNat.0(X) → n__isNat.0(X)
and.0-1(X1, X2) → n__and.0-1(X1, X2)
activate.0(n__take.0-0(X1, X2)) → take.0-0(X1, X2)
isNatList.0(n__take.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.1(activate.1(V2)))
isNatIList.0(n__cons.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.1(activate.1(V2)))
isNatList.0(n__take.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.0(activate.0(V2)))
isNatList.0(n__take.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.0(activate.0(V2)))
isNat.0(n__s.0(V1)) → isNat.0(activate.0(V1))
isNatIList.0(n__cons.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.0(activate.0(V2)))
isNat.0(n__length.0(V1)) → isNatList.0(activate.0(V1))
isNatList.0(n__take.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.1(activate.1(V2)))
activate.1(n__isNatList.0(X)) → isNatList.0(X)
and.1-1(tt., X) → activate.1(X)
isNatIList.0(n__cons.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.0(activate.0(V2)))
activate.1(n__and.1-1(X1, X2)) → and.1-1(X1, X2)
isNatList.0(n__cons.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatList.1(activate.1(V2)))
isNatIList.0(V) → isNatList.0(activate.0(V))
activate.1(n__isNatIList.0(X)) → isNatIList.0(X)
activate.1(n__isNat.0(X)) → isNat.0(X)
isNatList.0(n__cons.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatList.0(activate.0(V2)))
isNatList.0(n__cons.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatList.0(activate.0(V2)))
isNatList.0(n__cons.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatList.1(activate.1(V2)))
isNatIList.0(n__cons.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.1(activate.1(V2)))
cons.0-1(X1, X2) → n__cons.0-1(X1, X2)
U31.1-1-1-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.1-1(activate.1(M), activate.1(IL)))
activate.0(n__cons.0-0(X1, X2)) → cons.0-0(X1, X2)
s.0(X) → n__s.0(X)
and.1-0(tt., X) → activate.0(X)
activate.0(n__and.1-0(X1, X2)) → and.1-0(X1, X2)
activate.0(n__take.1-1(X1, X2)) → take.1-1(X1, X2)
U21.1(tt.) → nil.
activate.0(n__zeros.) → zeros.
and.0-0(X1, X2) → n__and.0-0(X1, X2)
U31.1-0-1-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.1-0(activate.1(M), activate.0(IL)))
take.0-0(s.1(M), cons.0-0(N, IL)) → U31.1-0-1-0(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.1(M), n__isNat.0(N))), activate.0(IL), M, N)
activate.0(n__take.0-1(X1, X2)) → take.0-1(X1, X2)
cons.1-0(X1, X2) → n__cons.1-0(X1, X2)
and.1-1(X1, X2) → n__and.1-1(X1, X2)
activate.0(n__s.0(X)) → s.0(X)
length.0(X) → n__length.0(X)
activate.1(n__isNat.1(X)) → isNat.1(X)
activate.0(n__length.1(X)) → length.1(X)
isNatList.0(X) → n__isNatList.0(X)
length.0(cons.0-0(N, L)) → U11.1-0(and.1-1(isNatList.0(activate.0(L)), n__isNat.0(N)), activate.0(L))
U11.1-1(tt., L) → s.0(length.1(activate.1(L)))
activate.0(n__cons.1-1(X1, X2)) → cons.1-1(X1, X2)
take.0-0(s.0(M), cons.0-1(N, IL)) → U31.1-1-0-0(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.0(M), n__isNat.0(N))), activate.1(IL), M, N)
isNatList.0(n__nil.) → tt.
U11.1-0(tt., L) → s.0(length.0(activate.0(L)))
take.0-0(s.1(M), cons.1-1(N, IL)) → U31.1-1-1-1(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.1(M), n__isNat.1(N))), activate.1(IL), M, N)
zeros. → cons.0-0(0., n__zeros.)
length.0(cons.1-0(N, L)) → U11.1-0(and.1-1(isNatList.0(activate.0(L)), n__isNat.1(N)), activate.0(L))
isNatIList.0(n__zeros.) → tt.
activate.0(n__cons.1-0(X1, X2)) → cons.1-0(X1, X2)
activate.1(n__isNatList.1(X)) → isNatList.1(X)
activate.0(n__0.) → 0.
nil. → n__nil.
isNat.0(n__0.) → tt.
take.0-0(s.0(M), cons.1-0(N, IL)) → U31.1-0-0-1(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.0(M), n__isNat.1(N))), activate.0(IL), M, N)
take.0-1(X1, X2) → n__take.0-1(X1, X2)
and.1-0(X1, X2) → n__and.1-0(X1, X2)
length.1(X) → n__length.1(X)
cons.1-1(X1, X2) → n__cons.1-1(X1, X2)
activate.0(n__nil.) → nil.
activate.0(X) → X
activate.0(n__cons.0-1(X1, X2)) → cons.0-1(X1, X2)
activate.1(X) → X
take.0-0(s.0(M), cons.0-0(N, IL)) → U31.1-0-0-0(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.0(M), n__isNat.0(N))), activate.0(IL), M, N)
isNatIList.1(V) → isNatList.1(activate.1(V))
take.0-1(0., IL) → U21.1(isNatIList.1(IL))
isNat.0(n__s.1(V1)) → isNat.1(activate.1(V1))
U31.1-0-0-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.0-0(activate.0(M), activate.0(IL)))
0. → n__0.
U31.1-0-1-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.1-0(activate.1(M), activate.0(IL)))
s.1(X) → n__s.1(X)
take.1-0(X1, X2) → n__take.1-0(X1, X2)
take.0-0(0., IL) → U21.1(isNatIList.0(IL))
activate.0(n__length.0(X)) → length.0(X)
activate.0(n__and.0-0(X1, X2)) → and.0-0(X1, X2)
take.0-0(s.1(M), cons.1-0(N, IL)) → U31.1-0-1-1(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.1(M), n__isNat.1(N))), activate.0(IL), M, N)
U31.1-1-0-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.0-1(activate.0(M), activate.1(IL)))
U31.1-1-1-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.1-1(activate.1(M), activate.1(IL)))
isNatList.1(X) → n__isNatList.1(X)
isNat.0(n__length.1(V1)) → isNatList.1(activate.1(V1))
activate.1(n__isNatIList.1(X)) → isNatIList.1(X)
length.0(cons.1-1(N, L)) → U11.1-1(and.1-1(isNatList.1(activate.1(L)), n__isNat.1(N)), activate.1(L))
isNatIList.1(X) → n__isNatIList.1(X)
take.0-0(X1, X2) → n__take.0-0(X1, X2)
zeros. → n__zeros.
length.0(nil.) → 0.
activate.1(n__and.0-1(X1, X2)) → and.0-1(X1, X2)
isNat.1(X) → n__isNat.1(X)
activate.0(n__take.1-0(X1, X2)) → take.1-0(X1, X2)
take.0-0(s.0(M), cons.1-1(N, IL)) → U31.1-1-0-1(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.0(M), n__isNat.1(N))), activate.1(IL), M, N)
take.1-1(X1, X2) → n__take.1-1(X1, X2)
U31.1-0-0-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.0-0(activate.0(M), activate.0(IL)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ SemLabProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                ↳ SemLabProof2

                                                                            ↳ QDPOrderProof

                                                                          ↳ QDP

                                            ↳ QDP

                  ↳ QDP

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

ISNATLIST.0(n__cons.1-0(n__and.1-1(x0, x1), y1)) → AND.1-1(isNat.1(and.1-1(x0, x1)), n__isNatList.0(activate.0(y1)))
AND.1-1(tt., n__isNatList.0(y_4)) → ACTIVATE.1(n__isNatList.0(y_4))
ISNATLIST.0(n__cons.0-0(n__0., y1)) → AND.1-1(isNat.0(0.), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.0-0(x0, y1)) → AND.1-1(isNat.0(x0), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.1-0(x0, y1)) → AND.1-1(isNat.1(x0), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.0-0(n__and.1-0(x0, x1), y1)) → AND.1-1(isNat.0(and.1-0(x0, x1)), n__isNatList.0(activate.0(y1)))
ACTIVATE.1(n__isNatList.0(X)) → ISNATLIST.0(X)
ISNATLIST.0(n__cons.0-0(n__and.0-0(x0, x1), y1)) → AND.1-1(isNat.0(and.0-0(x0, x1)), n__isNatList.0(activate.0(y1)))

The TRS R consists of the following rules:

take.0-0(s.1(M), cons.0-0(N, IL)) → U31.1-0-1-0(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.1(M), n__isNat.0(N))), activate.0(IL), M, N)
activate.0(n__take.0-1(X1, X2)) → take.0-1(X1, X2)
U31.1-0-1-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.1-0(activate.1(M), activate.0(IL)))
and.1-0(tt., X) → activate.0(X)
activate.0(n__zeros.) → zeros.
and.0-0(X1, X2) → n__and.0-0(X1, X2)
activate.0(n__take.1-1(X1, X2)) → take.1-1(X1, X2)
U21.1(tt.) → nil.
activate.1(n__isNat.1(X)) → isNat.1(X)
activate.0(n__length.1(X)) → length.1(X)
activate.0(n__s.0(X)) → s.0(X)
length.0(X) → n__length.0(X)
isNatList.0(n__take.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.0(activate.0(V2)))
isNat.0(n__s.0(V1)) → isNat.0(activate.0(V1))
cons.1-0(X1, X2) → n__cons.1-0(X1, X2)
and.1-1(X1, X2) → n__and.1-1(X1, X2)
isNat.0(X) → n__isNat.0(X)
take.0-0(s.1(M), cons.0-1(N, IL)) → U31.1-1-1-0(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.1(M), n__isNat.0(N))), activate.1(IL), M, N)
isNatList.0(n__cons.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatList.0(activate.0(V2)))
isNatIList.0(X) → n__isNatIList.0(X)
length.0(cons.0-1(N, L)) → U11.1-1(and.1-1(isNatList.1(activate.1(L)), n__isNat.0(N)), activate.1(L))
U31.1-1-0-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.0-1(activate.0(M), activate.1(IL)))
cons.0-0(X1, X2) → n__cons.0-0(X1, X2)
activate.0(n__s.1(X)) → s.1(X)
activate.0(n__and.1-0(X1, X2)) → and.1-0(X1, X2)
s.0(X) → n__s.0(X)
activate.0(n__cons.0-0(X1, X2)) → cons.0-0(X1, X2)
U31.1-1-1-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.1-1(activate.1(M), activate.1(IL)))
cons.0-1(X1, X2) → n__cons.0-1(X1, X2)
activate.1(n__isNatList.0(X)) → isNatList.0(X)
activate.0(n__take.0-0(X1, X2)) → take.0-0(X1, X2)
and.0-1(X1, X2) → n__and.0-1(X1, X2)
isNat.0(n__length.1(V1)) → isNatList.1(activate.1(V1))
activate.1(n__isNatIList.1(X)) → isNatIList.1(X)
isNatIList.0(n__cons.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.1(activate.1(V2)))
length.0(cons.1-1(N, L)) → U11.1-1(and.1-1(isNatList.1(activate.1(L)), n__isNat.1(N)), activate.1(L))
isNatIList.1(X) → n__isNatIList.1(X)
take.0-0(X1, X2) → n__take.0-0(X1, X2)
zeros. → n__zeros.
length.0(nil.) → 0.
activate.1(n__and.0-1(X1, X2)) → and.0-1(X1, X2)
isNat.1(X) → n__isNat.1(X)
activate.0(n__take.1-0(X1, X2)) → take.1-0(X1, X2)
take.0-0(s.0(M), cons.1-1(N, IL)) → U31.1-1-0-1(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.0(M), n__isNat.1(N))), activate.1(IL), M, N)
isNatList.0(n__take.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.0(activate.0(V2)))
take.1-1(X1, X2) → n__take.1-1(X1, X2)
U31.1-0-0-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.0-0(activate.0(M), activate.0(IL)))
isNat.0(n__length.0(V1)) → isNatList.0(activate.0(V1))
isNatIList.0(n__cons.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.0(activate.0(V2)))
s.1(X) → n__s.1(X)
isNatList.0(n__cons.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatList.1(activate.1(V2)))
take.1-0(X1, X2) → n__take.1-0(X1, X2)
isNatIList.0(V) → isNatList.0(activate.0(V))
take.0-0(0., IL) → U21.1(isNatIList.0(IL))
activate.0(n__length.0(X)) → length.0(X)
activate.1(n__isNatIList.0(X)) → isNatIList.0(X)
activate.0(n__and.0-0(X1, X2)) → and.0-0(X1, X2)
isNatList.0(n__cons.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatList.0(activate.0(V2)))
isNatList.0(n__cons.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatList.1(activate.1(V2)))
take.0-0(s.1(M), cons.1-0(N, IL)) → U31.1-0-1-1(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.1(M), n__isNat.1(N))), activate.0(IL), M, N)
isNatList.0(n__take.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.1(activate.1(V2)))
U31.1-1-0-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.0-1(activate.0(M), activate.1(IL)))
isNatList.1(X) → n__isNatList.1(X)
U31.1-1-1-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.1-1(activate.1(M), activate.1(IL)))
length.1(X) → n__length.1(X)
cons.1-1(X1, X2) → n__cons.1-1(X1, X2)
take.0-1(X1, X2) → n__take.0-1(X1, X2)
and.1-0(X1, X2) → n__and.1-0(X1, X2)
activate.0(X) → X
activate.0(n__cons.0-1(X1, X2)) → cons.0-1(X1, X2)
activate.0(n__nil.) → nil.
and.1-1(tt., X) → activate.1(X)
isNatIList.1(V) → isNatList.1(activate.1(V))
take.0-1(0., IL) → U21.1(isNatIList.1(IL))
activate.1(X) → X
take.0-0(s.0(M), cons.0-0(N, IL)) → U31.1-0-0-0(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.0(M), n__isNat.0(N))), activate.0(IL), M, N)
0. → n__0.
U31.1-0-1-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.1-0(activate.1(M), activate.0(IL)))
isNat.0(n__s.1(V1)) → isNat.1(activate.1(V1))
U31.1-0-0-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.0-0(activate.0(M), activate.0(IL)))
take.0-0(s.1(M), cons.1-1(N, IL)) → U31.1-1-1-1(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.1(M), n__isNat.1(N))), activate.1(IL), M, N)
U11.1-0(tt., L) → s.0(length.0(activate.0(L)))
isNatList.0(n__nil.) → tt.
take.0-0(s.0(M), cons.0-1(N, IL)) → U31.1-1-0-0(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.0(M), n__isNat.0(N))), activate.1(IL), M, N)
isNatList.0(n__take.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.1(activate.1(V2)))
isNatIList.0(n__zeros.) → tt.
length.0(cons.1-0(N, L)) → U11.1-0(and.1-1(isNatList.0(activate.0(L)), n__isNat.1(N)), activate.0(L))
zeros. → cons.0-0(0., n__zeros.)
isNatIList.0(n__cons.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.0(activate.0(V2)))
activate.0(n__0.) → 0.
activate.1(n__isNatList.1(X)) → isNatList.1(X)
activate.0(n__cons.1-0(X1, X2)) → cons.1-0(X1, X2)
activate.1(n__and.1-1(X1, X2)) → and.1-1(X1, X2)
take.0-0(s.0(M), cons.1-0(N, IL)) → U31.1-0-0-1(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.0(M), n__isNat.1(N))), activate.0(IL), M, N)
isNat.0(n__0.) → tt.
nil. → n__nil.
U11.1-1(tt., L) → s.0(length.1(activate.1(L)))
length.0(cons.0-0(N, L)) → U11.1-0(and.1-1(isNatList.0(activate.0(L)), n__isNat.0(N)), activate.0(L))
activate.1(n__isNat.0(X)) → isNat.0(X)
activate.0(n__cons.1-1(X1, X2)) → cons.1-1(X1, X2)
isNatList.0(X) → n__isNatList.0(X)
isNatIList.0(n__cons.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.1(activate.1(V2)))

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.

ISNATLIST.0(n__cons.1-0(n__and.1-1(x0, x1), y1)) → AND.1-1(isNat.1(and.1-1(x0, x1)), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.1-0(x0, y1)) → AND.1-1(isNat.1(x0), n__isNatList.0(activate.0(y1)))

The remaining pairs can at least be oriented weakly.

AND.1-1(tt., n__isNatList.0(y_4)) → ACTIVATE.1(n__isNatList.0(y_4))
ISNATLIST.0(n__cons.0-0(n__0., y1)) → AND.1-1(isNat.0(0.), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.0-0(x0, y1)) → AND.1-1(isNat.0(x0), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.0-0(n__and.1-0(x0, x1), y1)) → AND.1-1(isNat.0(and.1-0(x0, x1)), n__isNatList.0(activate.0(y1)))
ACTIVATE.1(n__isNatList.0(X)) → ISNATLIST.0(X)
ISNATLIST.0(n__cons.0-0(n__and.0-0(x0, x1), y1)) → AND.1-1(isNat.0(and.0-0(x0, x1)), n__isNatList.0(activate.0(y1)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

activate.0(n__s.1(X)) → s.1(X)
cons.0-0(X1, X2) → n__cons.0-0(X1, X2)
U31.1-1-0-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.0-1(activate.0(M), activate.1(IL)))
length.0(cons.0-1(N, L)) → U11.1-1(and.1-1(isNatList.1(activate.1(L)), n__isNat.0(N)), activate.1(L))
isNatIList.0(X) → n__isNatIList.0(X)
take.0-0(s.1(M), cons.0-1(N, IL)) → U31.1-1-1-0(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.1(M), n__isNat.0(N))), activate.1(IL), M, N)
isNat.0(X) → n__isNat.0(X)
and.0-1(X1, X2) → n__and.0-1(X1, X2)
activate.0(n__take.0-0(X1, X2)) → take.0-0(X1, X2)
isNatList.0(n__take.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.1(activate.1(V2)))
isNatIList.0(n__cons.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.1(activate.1(V2)))
isNatList.0(n__take.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.0(activate.0(V2)))
isNatList.0(n__take.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.0(activate.0(V2)))
isNat.0(n__s.0(V1)) → isNat.0(activate.0(V1))
isNatIList.0(n__cons.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.0(activate.0(V2)))
isNat.0(n__length.0(V1)) → isNatList.0(activate.0(V1))
isNatList.0(n__take.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.1(activate.1(V2)))
activate.1(n__isNatList.0(X)) → isNatList.0(X)
and.1-1(tt., X) → activate.1(X)
isNatIList.0(n__cons.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.0(activate.0(V2)))
activate.1(n__and.1-1(X1, X2)) → and.1-1(X1, X2)
isNatList.0(n__cons.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatList.1(activate.1(V2)))
isNatIList.0(V) → isNatList.0(activate.0(V))
activate.1(n__isNatIList.0(X)) → isNatIList.0(X)
activate.1(n__isNat.0(X)) → isNat.0(X)
isNatList.0(n__cons.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatList.0(activate.0(V2)))
isNatList.0(n__cons.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatList.0(activate.0(V2)))
isNatList.0(n__cons.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatList.1(activate.1(V2)))
isNatIList.0(n__cons.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.1(activate.1(V2)))
cons.0-1(X1, X2) → n__cons.0-1(X1, X2)
U31.1-1-1-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.1-1(activate.1(M), activate.1(IL)))
activate.0(n__cons.0-0(X1, X2)) → cons.0-0(X1, X2)
s.0(X) → n__s.0(X)
and.1-0(tt., X) → activate.0(X)
activate.0(n__and.1-0(X1, X2)) → and.1-0(X1, X2)
activate.0(n__take.1-1(X1, X2)) → take.1-1(X1, X2)
U21.1(tt.) → nil.
activate.0(n__zeros.) → zeros.
and.0-0(X1, X2) → n__and.0-0(X1, X2)
U31.1-0-1-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.1-0(activate.1(M), activate.0(IL)))
take.0-0(s.1(M), cons.0-0(N, IL)) → U31.1-0-1-0(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.1(M), n__isNat.0(N))), activate.0(IL), M, N)
activate.0(n__take.0-1(X1, X2)) → take.0-1(X1, X2)
cons.1-0(X1, X2) → n__cons.1-0(X1, X2)
and.1-1(X1, X2) → n__and.1-1(X1, X2)
activate.0(n__s.0(X)) → s.0(X)
length.0(X) → n__length.0(X)
activate.1(n__isNat.1(X)) → isNat.1(X)
activate.0(n__length.1(X)) → length.1(X)
isNatList.0(X) → n__isNatList.0(X)
length.0(cons.0-0(N, L)) → U11.1-0(and.1-1(isNatList.0(activate.0(L)), n__isNat.0(N)), activate.0(L))
U11.1-1(tt., L) → s.0(length.1(activate.1(L)))
activate.0(n__cons.1-1(X1, X2)) → cons.1-1(X1, X2)
take.0-0(s.0(M), cons.0-1(N, IL)) → U31.1-1-0-0(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.0(M), n__isNat.0(N))), activate.1(IL), M, N)
isNatList.0(n__nil.) → tt.
U11.1-0(tt., L) → s.0(length.0(activate.0(L)))
take.0-0(s.1(M), cons.1-1(N, IL)) → U31.1-1-1-1(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.1(M), n__isNat.1(N))), activate.1(IL), M, N)
zeros. → cons.0-0(0., n__zeros.)
length.0(cons.1-0(N, L)) → U11.1-0(and.1-1(isNatList.0(activate.0(L)), n__isNat.1(N)), activate.0(L))
isNatIList.0(n__zeros.) → tt.
activate.0(n__cons.1-0(X1, X2)) → cons.1-0(X1, X2)
activate.1(n__isNatList.1(X)) → isNatList.1(X)
activate.0(n__0.) → 0.
nil. → n__nil.
isNat.0(n__0.) → tt.
take.0-0(s.0(M), cons.1-0(N, IL)) → U31.1-0-0-1(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.0(M), n__isNat.1(N))), activate.0(IL), M, N)
take.0-1(X1, X2) → n__take.0-1(X1, X2)
and.1-0(X1, X2) → n__and.1-0(X1, X2)
length.1(X) → n__length.1(X)
cons.1-1(X1, X2) → n__cons.1-1(X1, X2)
activate.0(n__nil.) → nil.
activate.0(X) → X
activate.0(n__cons.0-1(X1, X2)) → cons.0-1(X1, X2)
activate.1(X) → X
take.0-0(s.0(M), cons.0-0(N, IL)) → U31.1-0-0-0(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.0(M), n__isNat.0(N))), activate.0(IL), M, N)
isNatIList.1(V) → isNatList.1(activate.1(V))
take.0-1(0., IL) → U21.1(isNatIList.1(IL))
isNat.0(n__s.1(V1)) → isNat.1(activate.1(V1))
U31.1-0-0-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.0-0(activate.0(M), activate.0(IL)))
0. → n__0.
U31.1-0-1-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.1-0(activate.1(M), activate.0(IL)))
s.1(X) → n__s.1(X)
take.1-0(X1, X2) → n__take.1-0(X1, X2)
take.0-0(0., IL) → U21.1(isNatIList.0(IL))
activate.0(n__length.0(X)) → length.0(X)
activate.0(n__and.0-0(X1, X2)) → and.0-0(X1, X2)
take.0-0(s.1(M), cons.1-0(N, IL)) → U31.1-0-1-1(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.1(M), n__isNat.1(N))), activate.0(IL), M, N)
U31.1-1-0-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.0-1(activate.0(M), activate.1(IL)))
U31.1-1-1-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.1-1(activate.1(M), activate.1(IL)))
isNatList.1(X) → n__isNatList.1(X)
isNat.0(n__length.1(V1)) → isNatList.1(activate.1(V1))
activate.1(n__isNatIList.1(X)) → isNatIList.1(X)
length.0(cons.1-1(N, L)) → U11.1-1(and.1-1(isNatList.1(activate.1(L)), n__isNat.1(N)), activate.1(L))
isNatIList.1(X) → n__isNatIList.1(X)
take.0-0(X1, X2) → n__take.0-0(X1, X2)
zeros. → n__zeros.
length.0(nil.) → 0.
activate.1(n__and.0-1(X1, X2)) → and.0-1(X1, X2)
isNat.1(X) → n__isNat.1(X)
activate.0(n__take.1-0(X1, X2)) → take.1-0(X1, X2)
take.0-0(s.0(M), cons.1-1(N, IL)) → U31.1-1-0-1(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.0(M), n__isNat.1(N))), activate.1(IL), M, N)
take.1-1(X1, X2) → n__take.1-1(X1, X2)
U31.1-0-0-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.0-0(activate.0(M), activate.0(IL)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ SemLabProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                ↳ SemLabProof2

                                                                            ↳ QDPOrderProof

                                                                          ↳ QDP

                                            ↳ QDP

                  ↳ QDP

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

AND.1-1(tt., n__isNatList.0(y_4)) → ACTIVATE.1(n__isNatList.0(y_4))
ISNATLIST.0(n__cons.0-0(n__0., y1)) → AND.1-1(isNat.0(0.), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.0-0(x0, y1)) → AND.1-1(isNat.0(x0), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.0-0(n__and.1-0(x0, x1), y1)) → AND.1-1(isNat.0(and.1-0(x0, x1)), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.0-0(n__and.0-0(x0, x1), y1)) → AND.1-1(isNat.0(and.0-0(x0, x1)), n__isNatList.0(activate.0(y1)))
ACTIVATE.1(n__isNatList.0(X)) → ISNATLIST.0(X)

The TRS R consists of the following rules:

take.0-0(s.1(M), cons.0-0(N, IL)) → U31.1-0-1-0(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.1(M), n__isNat.0(N))), activate.0(IL), M, N)
activate.0(n__take.0-1(X1, X2)) → take.0-1(X1, X2)
U31.1-0-1-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.1-0(activate.1(M), activate.0(IL)))
and.1-0(tt., X) → activate.0(X)
activate.0(n__zeros.) → zeros.
and.0-0(X1, X2) → n__and.0-0(X1, X2)
activate.0(n__take.1-1(X1, X2)) → take.1-1(X1, X2)
U21.1(tt.) → nil.
activate.1(n__isNat.1(X)) → isNat.1(X)
activate.0(n__length.1(X)) → length.1(X)
activate.0(n__s.0(X)) → s.0(X)
length.0(X) → n__length.0(X)
isNatList.0(n__take.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.0(activate.0(V2)))
isNat.0(n__s.0(V1)) → isNat.0(activate.0(V1))
cons.1-0(X1, X2) → n__cons.1-0(X1, X2)
and.1-1(X1, X2) → n__and.1-1(X1, X2)
isNat.0(X) → n__isNat.0(X)
take.0-0(s.1(M), cons.0-1(N, IL)) → U31.1-1-1-0(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.1(M), n__isNat.0(N))), activate.1(IL), M, N)
isNatList.0(n__cons.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatList.0(activate.0(V2)))
isNatIList.0(X) → n__isNatIList.0(X)
length.0(cons.0-1(N, L)) → U11.1-1(and.1-1(isNatList.1(activate.1(L)), n__isNat.0(N)), activate.1(L))
U31.1-1-0-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.0-1(activate.0(M), activate.1(IL)))
cons.0-0(X1, X2) → n__cons.0-0(X1, X2)
activate.0(n__s.1(X)) → s.1(X)
activate.0(n__and.1-0(X1, X2)) → and.1-0(X1, X2)
s.0(X) → n__s.0(X)
activate.0(n__cons.0-0(X1, X2)) → cons.0-0(X1, X2)
U31.1-1-1-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.1-1(activate.1(M), activate.1(IL)))
cons.0-1(X1, X2) → n__cons.0-1(X1, X2)
activate.1(n__isNatList.0(X)) → isNatList.0(X)
activate.0(n__take.0-0(X1, X2)) → take.0-0(X1, X2)
and.0-1(X1, X2) → n__and.0-1(X1, X2)
isNat.0(n__length.1(V1)) → isNatList.1(activate.1(V1))
activate.1(n__isNatIList.1(X)) → isNatIList.1(X)
isNatIList.0(n__cons.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.1(activate.1(V2)))
length.0(cons.1-1(N, L)) → U11.1-1(and.1-1(isNatList.1(activate.1(L)), n__isNat.1(N)), activate.1(L))
isNatIList.1(X) → n__isNatIList.1(X)
take.0-0(X1, X2) → n__take.0-0(X1, X2)
zeros. → n__zeros.
length.0(nil.) → 0.
activate.1(n__and.0-1(X1, X2)) → and.0-1(X1, X2)
isNat.1(X) → n__isNat.1(X)
activate.0(n__take.1-0(X1, X2)) → take.1-0(X1, X2)
take.0-0(s.0(M), cons.1-1(N, IL)) → U31.1-1-0-1(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.0(M), n__isNat.1(N))), activate.1(IL), M, N)
isNatList.0(n__take.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.0(activate.0(V2)))
take.1-1(X1, X2) → n__take.1-1(X1, X2)
U31.1-0-0-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.0-0(activate.0(M), activate.0(IL)))
isNat.0(n__length.0(V1)) → isNatList.0(activate.0(V1))
isNatIList.0(n__cons.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.0(activate.0(V2)))
s.1(X) → n__s.1(X)
isNatList.0(n__cons.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatList.1(activate.1(V2)))
take.1-0(X1, X2) → n__take.1-0(X1, X2)
isNatIList.0(V) → isNatList.0(activate.0(V))
take.0-0(0., IL) → U21.1(isNatIList.0(IL))
activate.0(n__length.0(X)) → length.0(X)
activate.1(n__isNatIList.0(X)) → isNatIList.0(X)
activate.0(n__and.0-0(X1, X2)) → and.0-0(X1, X2)
isNatList.0(n__cons.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatList.0(activate.0(V2)))
isNatList.0(n__cons.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatList.1(activate.1(V2)))
take.0-0(s.1(M), cons.1-0(N, IL)) → U31.1-0-1-1(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.1(M), n__isNat.1(N))), activate.0(IL), M, N)
isNatList.0(n__take.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.1(activate.1(V2)))
U31.1-1-0-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.0-1(activate.0(M), activate.1(IL)))
isNatList.1(X) → n__isNatList.1(X)
U31.1-1-1-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.1-1(activate.1(M), activate.1(IL)))
length.1(X) → n__length.1(X)
cons.1-1(X1, X2) → n__cons.1-1(X1, X2)
take.0-1(X1, X2) → n__take.0-1(X1, X2)
and.1-0(X1, X2) → n__and.1-0(X1, X2)
activate.0(X) → X
activate.0(n__cons.0-1(X1, X2)) → cons.0-1(X1, X2)
activate.0(n__nil.) → nil.
and.1-1(tt., X) → activate.1(X)
isNatIList.1(V) → isNatList.1(activate.1(V))
take.0-1(0., IL) → U21.1(isNatIList.1(IL))
activate.1(X) → X
take.0-0(s.0(M), cons.0-0(N, IL)) → U31.1-0-0-0(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.0(M), n__isNat.0(N))), activate.0(IL), M, N)
0. → n__0.
U31.1-0-1-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.1-0(activate.1(M), activate.0(IL)))
isNat.0(n__s.1(V1)) → isNat.1(activate.1(V1))
U31.1-0-0-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.0-0(activate.0(M), activate.0(IL)))
take.0-0(s.1(M), cons.1-1(N, IL)) → U31.1-1-1-1(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.1(M), n__isNat.1(N))), activate.1(IL), M, N)
U11.1-0(tt., L) → s.0(length.0(activate.0(L)))
isNatList.0(n__nil.) → tt.
take.0-0(s.0(M), cons.0-1(N, IL)) → U31.1-1-0-0(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.0(M), n__isNat.0(N))), activate.1(IL), M, N)
isNatList.0(n__take.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.1(activate.1(V2)))
isNatIList.0(n__zeros.) → tt.
length.0(cons.1-0(N, L)) → U11.1-0(and.1-1(isNatList.0(activate.0(L)), n__isNat.1(N)), activate.0(L))
zeros. → cons.0-0(0., n__zeros.)
isNatIList.0(n__cons.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.0(activate.0(V2)))
activate.0(n__0.) → 0.
activate.1(n__isNatList.1(X)) → isNatList.1(X)
activate.0(n__cons.1-0(X1, X2)) → cons.1-0(X1, X2)
activate.1(n__and.1-1(X1, X2)) → and.1-1(X1, X2)
take.0-0(s.0(M), cons.1-0(N, IL)) → U31.1-0-0-1(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.0(M), n__isNat.1(N))), activate.0(IL), M, N)
isNat.0(n__0.) → tt.
nil. → n__nil.
U11.1-1(tt., L) → s.0(length.1(activate.1(L)))
length.0(cons.0-0(N, L)) → U11.1-0(and.1-1(isNatList.0(activate.0(L)), n__isNat.0(N)), activate.0(L))
activate.1(n__isNat.0(X)) → isNat.0(X)
activate.0(n__cons.1-1(X1, X2)) → cons.1-1(X1, X2)
isNatList.0(X) → n__isNatList.0(X)
isNatIList.0(n__cons.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.1(activate.1(V2)))

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.

ISNATLIST.0(n__cons.0-0(n__and.1-0(x0, x1), y1)) → AND.1-1(isNat.0(and.1-0(x0, x1)), n__isNatList.0(activate.0(y1)))

The remaining pairs can at least be oriented weakly.

AND.1-1(tt., n__isNatList.0(y_4)) → ACTIVATE.1(n__isNatList.0(y_4))
ISNATLIST.0(n__cons.0-0(n__0., y1)) → AND.1-1(isNat.0(0.), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.0-0(x0, y1)) → AND.1-1(isNat.0(x0), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.0-0(n__and.0-0(x0, x1), y1)) → AND.1-1(isNat.0(and.0-0(x0, x1)), n__isNatList.0(activate.0(y1)))
ACTIVATE.1(n__isNatList.0(X)) → ISNATLIST.0(X)

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

activate.0(n__s.1(X)) → s.1(X)
cons.0-0(X1, X2) → n__cons.0-0(X1, X2)
U31.1-1-0-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.0-1(activate.0(M), activate.1(IL)))
length.0(cons.0-1(N, L)) → U11.1-1(and.1-1(isNatList.1(activate.1(L)), n__isNat.0(N)), activate.1(L))
isNatIList.0(X) → n__isNatIList.0(X)
take.0-0(s.1(M), cons.0-1(N, IL)) → U31.1-1-1-0(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.1(M), n__isNat.0(N))), activate.1(IL), M, N)
isNat.0(X) → n__isNat.0(X)
and.0-1(X1, X2) → n__and.0-1(X1, X2)
activate.0(n__take.0-0(X1, X2)) → take.0-0(X1, X2)
isNatList.0(n__take.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.1(activate.1(V2)))
isNatIList.0(n__cons.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.1(activate.1(V2)))
isNatList.0(n__take.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.0(activate.0(V2)))
isNatList.0(n__take.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.0(activate.0(V2)))
isNat.0(n__s.0(V1)) → isNat.0(activate.0(V1))
isNatIList.0(n__cons.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.0(activate.0(V2)))
isNat.0(n__length.0(V1)) → isNatList.0(activate.0(V1))
isNatList.0(n__take.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.1(activate.1(V2)))
activate.1(n__isNatList.0(X)) → isNatList.0(X)
and.1-1(tt., X) → activate.1(X)
isNatIList.0(n__cons.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.0(activate.0(V2)))
activate.1(n__and.1-1(X1, X2)) → and.1-1(X1, X2)
isNatList.0(n__cons.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatList.1(activate.1(V2)))
isNatIList.0(V) → isNatList.0(activate.0(V))
activate.1(n__isNatIList.0(X)) → isNatIList.0(X)
activate.1(n__isNat.0(X)) → isNat.0(X)
isNatList.0(n__cons.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatList.0(activate.0(V2)))
isNatList.0(n__cons.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatList.0(activate.0(V2)))
isNatList.0(n__cons.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatList.1(activate.1(V2)))
isNatIList.0(n__cons.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.1(activate.1(V2)))
cons.0-1(X1, X2) → n__cons.0-1(X1, X2)
U31.1-1-1-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.1-1(activate.1(M), activate.1(IL)))
activate.0(n__cons.0-0(X1, X2)) → cons.0-0(X1, X2)
s.0(X) → n__s.0(X)
and.1-0(tt., X) → activate.0(X)
activate.0(n__and.1-0(X1, X2)) → and.1-0(X1, X2)
activate.0(n__take.1-1(X1, X2)) → take.1-1(X1, X2)
U21.1(tt.) → nil.
activate.0(n__zeros.) → zeros.
and.0-0(X1, X2) → n__and.0-0(X1, X2)
U31.1-0-1-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.1-0(activate.1(M), activate.0(IL)))
take.0-0(s.1(M), cons.0-0(N, IL)) → U31.1-0-1-0(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.1(M), n__isNat.0(N))), activate.0(IL), M, N)
activate.0(n__take.0-1(X1, X2)) → take.0-1(X1, X2)
cons.1-0(X1, X2) → n__cons.1-0(X1, X2)
and.1-1(X1, X2) → n__and.1-1(X1, X2)
activate.0(n__s.0(X)) → s.0(X)
length.0(X) → n__length.0(X)
activate.1(n__isNat.1(X)) → isNat.1(X)
activate.0(n__length.1(X)) → length.1(X)
isNatList.0(X) → n__isNatList.0(X)
length.0(cons.0-0(N, L)) → U11.1-0(and.1-1(isNatList.0(activate.0(L)), n__isNat.0(N)), activate.0(L))
U11.1-1(tt., L) → s.0(length.1(activate.1(L)))
activate.0(n__cons.1-1(X1, X2)) → cons.1-1(X1, X2)
take.0-0(s.0(M), cons.0-1(N, IL)) → U31.1-1-0-0(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.0(M), n__isNat.0(N))), activate.1(IL), M, N)
isNatList.0(n__nil.) → tt.
U11.1-0(tt., L) → s.0(length.0(activate.0(L)))
take.0-0(s.1(M), cons.1-1(N, IL)) → U31.1-1-1-1(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.1(M), n__isNat.1(N))), activate.1(IL), M, N)
zeros. → cons.0-0(0., n__zeros.)
length.0(cons.1-0(N, L)) → U11.1-0(and.1-1(isNatList.0(activate.0(L)), n__isNat.1(N)), activate.0(L))
isNatIList.0(n__zeros.) → tt.
activate.0(n__cons.1-0(X1, X2)) → cons.1-0(X1, X2)
activate.1(n__isNatList.1(X)) → isNatList.1(X)
activate.0(n__0.) → 0.
nil. → n__nil.
isNat.0(n__0.) → tt.
take.0-0(s.0(M), cons.1-0(N, IL)) → U31.1-0-0-1(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.0(M), n__isNat.1(N))), activate.0(IL), M, N)
take.0-1(X1, X2) → n__take.0-1(X1, X2)
and.1-0(X1, X2) → n__and.1-0(X1, X2)
length.1(X) → n__length.1(X)
cons.1-1(X1, X2) → n__cons.1-1(X1, X2)
activate.0(n__nil.) → nil.
activate.0(X) → X
activate.0(n__cons.0-1(X1, X2)) → cons.0-1(X1, X2)
activate.1(X) → X
take.0-0(s.0(M), cons.0-0(N, IL)) → U31.1-0-0-0(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.0(M), n__isNat.0(N))), activate.0(IL), M, N)
isNatIList.1(V) → isNatList.1(activate.1(V))
take.0-1(0., IL) → U21.1(isNatIList.1(IL))
isNat.0(n__s.1(V1)) → isNat.1(activate.1(V1))
U31.1-0-0-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.0-0(activate.0(M), activate.0(IL)))
0. → n__0.
U31.1-0-1-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.1-0(activate.1(M), activate.0(IL)))
s.1(X) → n__s.1(X)
take.1-0(X1, X2) → n__take.1-0(X1, X2)
take.0-0(0., IL) → U21.1(isNatIList.0(IL))
activate.0(n__length.0(X)) → length.0(X)
activate.0(n__and.0-0(X1, X2)) → and.0-0(X1, X2)
take.0-0(s.1(M), cons.1-0(N, IL)) → U31.1-0-1-1(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.1(M), n__isNat.1(N))), activate.0(IL), M, N)
U31.1-1-0-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.0-1(activate.0(M), activate.1(IL)))
U31.1-1-1-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.1-1(activate.1(M), activate.1(IL)))
isNatList.1(X) → n__isNatList.1(X)
isNat.0(n__length.1(V1)) → isNatList.1(activate.1(V1))
activate.1(n__isNatIList.1(X)) → isNatIList.1(X)
length.0(cons.1-1(N, L)) → U11.1-1(and.1-1(isNatList.1(activate.1(L)), n__isNat.1(N)), activate.1(L))
isNatIList.1(X) → n__isNatIList.1(X)
take.0-0(X1, X2) → n__take.0-0(X1, X2)
zeros. → n__zeros.
length.0(nil.) → 0.
activate.1(n__and.0-1(X1, X2)) → and.0-1(X1, X2)
isNat.1(X) → n__isNat.1(X)
activate.0(n__take.1-0(X1, X2)) → take.1-0(X1, X2)
take.0-0(s.0(M), cons.1-1(N, IL)) → U31.1-1-0-1(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.0(M), n__isNat.1(N))), activate.1(IL), M, N)
take.1-1(X1, X2) → n__take.1-1(X1, X2)
U31.1-0-0-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.0-0(activate.0(M), activate.0(IL)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ SemLabProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                ↳ SemLabProof2

                                                                            ↳ QDPOrderProof

                                                                          ↳ QDP

                                            ↳ QDP

                  ↳ QDP

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

AND.1-1(tt., n__isNatList.0(y_4)) → ACTIVATE.1(n__isNatList.0(y_4))
ISNATLIST.0(n__cons.0-0(n__0., y1)) → AND.1-1(isNat.0(0.), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.0-0(x0, y1)) → AND.1-1(isNat.0(x0), n__isNatList.0(activate.0(y1)))
ACTIVATE.1(n__isNatList.0(X)) → ISNATLIST.0(X)
ISNATLIST.0(n__cons.0-0(n__and.0-0(x0, x1), y1)) → AND.1-1(isNat.0(and.0-0(x0, x1)), n__isNatList.0(activate.0(y1)))

The TRS R consists of the following rules:

take.0-0(s.1(M), cons.0-0(N, IL)) → U31.1-0-1-0(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.1(M), n__isNat.0(N))), activate.0(IL), M, N)
activate.0(n__take.0-1(X1, X2)) → take.0-1(X1, X2)
U31.1-0-1-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.1-0(activate.1(M), activate.0(IL)))
and.1-0(tt., X) → activate.0(X)
activate.0(n__zeros.) → zeros.
and.0-0(X1, X2) → n__and.0-0(X1, X2)
activate.0(n__take.1-1(X1, X2)) → take.1-1(X1, X2)
U21.1(tt.) → nil.
activate.1(n__isNat.1(X)) → isNat.1(X)
activate.0(n__length.1(X)) → length.1(X)
activate.0(n__s.0(X)) → s.0(X)
length.0(X) → n__length.0(X)
isNatList.0(n__take.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.0(activate.0(V2)))
isNat.0(n__s.0(V1)) → isNat.0(activate.0(V1))
cons.1-0(X1, X2) → n__cons.1-0(X1, X2)
and.1-1(X1, X2) → n__and.1-1(X1, X2)
isNat.0(X) → n__isNat.0(X)
take.0-0(s.1(M), cons.0-1(N, IL)) → U31.1-1-1-0(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.1(M), n__isNat.0(N))), activate.1(IL), M, N)
isNatList.0(n__cons.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatList.0(activate.0(V2)))
isNatIList.0(X) → n__isNatIList.0(X)
length.0(cons.0-1(N, L)) → U11.1-1(and.1-1(isNatList.1(activate.1(L)), n__isNat.0(N)), activate.1(L))
U31.1-1-0-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.0-1(activate.0(M), activate.1(IL)))
cons.0-0(X1, X2) → n__cons.0-0(X1, X2)
activate.0(n__s.1(X)) → s.1(X)
activate.0(n__and.1-0(X1, X2)) → and.1-0(X1, X2)
s.0(X) → n__s.0(X)
activate.0(n__cons.0-0(X1, X2)) → cons.0-0(X1, X2)
U31.1-1-1-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.1-1(activate.1(M), activate.1(IL)))
cons.0-1(X1, X2) → n__cons.0-1(X1, X2)
activate.1(n__isNatList.0(X)) → isNatList.0(X)
activate.0(n__take.0-0(X1, X2)) → take.0-0(X1, X2)
and.0-1(X1, X2) → n__and.0-1(X1, X2)
isNat.0(n__length.1(V1)) → isNatList.1(activate.1(V1))
activate.1(n__isNatIList.1(X)) → isNatIList.1(X)
isNatIList.0(n__cons.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.1(activate.1(V2)))
length.0(cons.1-1(N, L)) → U11.1-1(and.1-1(isNatList.1(activate.1(L)), n__isNat.1(N)), activate.1(L))
isNatIList.1(X) → n__isNatIList.1(X)
take.0-0(X1, X2) → n__take.0-0(X1, X2)
zeros. → n__zeros.
length.0(nil.) → 0.
activate.1(n__and.0-1(X1, X2)) → and.0-1(X1, X2)
isNat.1(X) → n__isNat.1(X)
activate.0(n__take.1-0(X1, X2)) → take.1-0(X1, X2)
take.0-0(s.0(M), cons.1-1(N, IL)) → U31.1-1-0-1(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.0(M), n__isNat.1(N))), activate.1(IL), M, N)
isNatList.0(n__take.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.0(activate.0(V2)))
take.1-1(X1, X2) → n__take.1-1(X1, X2)
U31.1-0-0-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.0-0(activate.0(M), activate.0(IL)))
isNat.0(n__length.0(V1)) → isNatList.0(activate.0(V1))
isNatIList.0(n__cons.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.0(activate.0(V2)))
s.1(X) → n__s.1(X)
isNatList.0(n__cons.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatList.1(activate.1(V2)))
take.1-0(X1, X2) → n__take.1-0(X1, X2)
isNatIList.0(V) → isNatList.0(activate.0(V))
take.0-0(0., IL) → U21.1(isNatIList.0(IL))
activate.0(n__length.0(X)) → length.0(X)
activate.1(n__isNatIList.0(X)) → isNatIList.0(X)
activate.0(n__and.0-0(X1, X2)) → and.0-0(X1, X2)
isNatList.0(n__cons.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatList.0(activate.0(V2)))
isNatList.0(n__cons.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatList.1(activate.1(V2)))
take.0-0(s.1(M), cons.1-0(N, IL)) → U31.1-0-1-1(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.1(M), n__isNat.1(N))), activate.0(IL), M, N)
isNatList.0(n__take.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.1(activate.1(V2)))
U31.1-1-0-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.0-1(activate.0(M), activate.1(IL)))
isNatList.1(X) → n__isNatList.1(X)
U31.1-1-1-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.1-1(activate.1(M), activate.1(IL)))
length.1(X) → n__length.1(X)
cons.1-1(X1, X2) → n__cons.1-1(X1, X2)
take.0-1(X1, X2) → n__take.0-1(X1, X2)
and.1-0(X1, X2) → n__and.1-0(X1, X2)
activate.0(X) → X
activate.0(n__cons.0-1(X1, X2)) → cons.0-1(X1, X2)
activate.0(n__nil.) → nil.
and.1-1(tt., X) → activate.1(X)
isNatIList.1(V) → isNatList.1(activate.1(V))
take.0-1(0., IL) → U21.1(isNatIList.1(IL))
activate.1(X) → X
take.0-0(s.0(M), cons.0-0(N, IL)) → U31.1-0-0-0(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.0(M), n__isNat.0(N))), activate.0(IL), M, N)
0. → n__0.
U31.1-0-1-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.1-0(activate.1(M), activate.0(IL)))
isNat.0(n__s.1(V1)) → isNat.1(activate.1(V1))
U31.1-0-0-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.0-0(activate.0(M), activate.0(IL)))
take.0-0(s.1(M), cons.1-1(N, IL)) → U31.1-1-1-1(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.1(M), n__isNat.1(N))), activate.1(IL), M, N)
U11.1-0(tt., L) → s.0(length.0(activate.0(L)))
isNatList.0(n__nil.) → tt.
take.0-0(s.0(M), cons.0-1(N, IL)) → U31.1-1-0-0(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.0(M), n__isNat.0(N))), activate.1(IL), M, N)
isNatList.0(n__take.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.1(activate.1(V2)))
isNatIList.0(n__zeros.) → tt.
length.0(cons.1-0(N, L)) → U11.1-0(and.1-1(isNatList.0(activate.0(L)), n__isNat.1(N)), activate.0(L))
zeros. → cons.0-0(0., n__zeros.)
isNatIList.0(n__cons.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.0(activate.0(V2)))
activate.0(n__0.) → 0.
activate.1(n__isNatList.1(X)) → isNatList.1(X)
activate.0(n__cons.1-0(X1, X2)) → cons.1-0(X1, X2)
activate.1(n__and.1-1(X1, X2)) → and.1-1(X1, X2)
take.0-0(s.0(M), cons.1-0(N, IL)) → U31.1-0-0-1(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.0(M), n__isNat.1(N))), activate.0(IL), M, N)
isNat.0(n__0.) → tt.
nil. → n__nil.
U11.1-1(tt., L) → s.0(length.1(activate.1(L)))
length.0(cons.0-0(N, L)) → U11.1-0(and.1-1(isNatList.0(activate.0(L)), n__isNat.0(N)), activate.0(L))
activate.1(n__isNat.0(X)) → isNat.0(X)
activate.0(n__cons.1-1(X1, X2)) → cons.1-1(X1, X2)
isNatList.0(X) → n__isNatList.0(X)
isNatIList.0(n__cons.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.1(activate.1(V2)))

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.

ISNATLIST.0(n__cons.0-0(n__and.0-0(x0, x1), y1)) → AND.1-1(isNat.0(and.0-0(x0, x1)), n__isNatList.0(activate.0(y1)))

The remaining pairs can at least be oriented weakly.

AND.1-1(tt., n__isNatList.0(y_4)) → ACTIVATE.1(n__isNatList.0(y_4))
ISNATLIST.0(n__cons.0-0(n__0., y1)) → AND.1-1(isNat.0(0.), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.0-0(x0, y1)) → AND.1-1(isNat.0(x0), n__isNatList.0(activate.0(y1)))
ACTIVATE.1(n__isNatList.0(X)) → ISNATLIST.0(X)

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

activate.0(n__s.1(X)) → s.1(X)
cons.0-0(X1, X2) → n__cons.0-0(X1, X2)
U31.1-1-0-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.0-1(activate.0(M), activate.1(IL)))
length.0(cons.0-1(N, L)) → U11.1-1(and.1-1(isNatList.1(activate.1(L)), n__isNat.0(N)), activate.1(L))
isNatIList.0(X) → n__isNatIList.0(X)
take.0-0(s.1(M), cons.0-1(N, IL)) → U31.1-1-1-0(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.1(M), n__isNat.0(N))), activate.1(IL), M, N)
isNat.0(X) → n__isNat.0(X)
and.0-1(X1, X2) → n__and.0-1(X1, X2)
activate.0(n__take.0-0(X1, X2)) → take.0-0(X1, X2)
isNatList.0(n__take.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.1(activate.1(V2)))
isNatIList.0(n__cons.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.1(activate.1(V2)))
isNatList.0(n__take.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.0(activate.0(V2)))
isNatList.0(n__take.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.0(activate.0(V2)))
isNat.0(n__s.0(V1)) → isNat.0(activate.0(V1))
isNatIList.0(n__cons.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.0(activate.0(V2)))
isNat.0(n__length.0(V1)) → isNatList.0(activate.0(V1))
isNatList.0(n__take.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.1(activate.1(V2)))
activate.1(n__isNatList.0(X)) → isNatList.0(X)
and.1-1(tt., X) → activate.1(X)
isNatIList.0(n__cons.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.0(activate.0(V2)))
activate.1(n__and.1-1(X1, X2)) → and.1-1(X1, X2)
isNatList.0(n__cons.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatList.1(activate.1(V2)))
isNatIList.0(V) → isNatList.0(activate.0(V))
activate.1(n__isNatIList.0(X)) → isNatIList.0(X)
activate.1(n__isNat.0(X)) → isNat.0(X)
isNatList.0(n__cons.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatList.0(activate.0(V2)))
isNatList.0(n__cons.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatList.0(activate.0(V2)))
isNatList.0(n__cons.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatList.1(activate.1(V2)))
isNatIList.0(n__cons.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.1(activate.1(V2)))
cons.0-1(X1, X2) → n__cons.0-1(X1, X2)
U31.1-1-1-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.1-1(activate.1(M), activate.1(IL)))
activate.0(n__cons.0-0(X1, X2)) → cons.0-0(X1, X2)
s.0(X) → n__s.0(X)
and.1-0(tt., X) → activate.0(X)
activate.0(n__and.1-0(X1, X2)) → and.1-0(X1, X2)
activate.0(n__take.1-1(X1, X2)) → take.1-1(X1, X2)
U21.1(tt.) → nil.
activate.0(n__zeros.) → zeros.
and.0-0(X1, X2) → n__and.0-0(X1, X2)
U31.1-0-1-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.1-0(activate.1(M), activate.0(IL)))
take.0-0(s.1(M), cons.0-0(N, IL)) → U31.1-0-1-0(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.1(M), n__isNat.0(N))), activate.0(IL), M, N)
activate.0(n__take.0-1(X1, X2)) → take.0-1(X1, X2)
cons.1-0(X1, X2) → n__cons.1-0(X1, X2)
and.1-1(X1, X2) → n__and.1-1(X1, X2)
activate.0(n__s.0(X)) → s.0(X)
length.0(X) → n__length.0(X)
activate.1(n__isNat.1(X)) → isNat.1(X)
activate.0(n__length.1(X)) → length.1(X)
isNatList.0(X) → n__isNatList.0(X)
length.0(cons.0-0(N, L)) → U11.1-0(and.1-1(isNatList.0(activate.0(L)), n__isNat.0(N)), activate.0(L))
U11.1-1(tt., L) → s.0(length.1(activate.1(L)))
activate.0(n__cons.1-1(X1, X2)) → cons.1-1(X1, X2)
take.0-0(s.0(M), cons.0-1(N, IL)) → U31.1-1-0-0(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.0(M), n__isNat.0(N))), activate.1(IL), M, N)
isNatList.0(n__nil.) → tt.
U11.1-0(tt., L) → s.0(length.0(activate.0(L)))
take.0-0(s.1(M), cons.1-1(N, IL)) → U31.1-1-1-1(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.1(M), n__isNat.1(N))), activate.1(IL), M, N)
zeros. → cons.0-0(0., n__zeros.)
length.0(cons.1-0(N, L)) → U11.1-0(and.1-1(isNatList.0(activate.0(L)), n__isNat.1(N)), activate.0(L))
isNatIList.0(n__zeros.) → tt.
activate.0(n__cons.1-0(X1, X2)) → cons.1-0(X1, X2)
activate.1(n__isNatList.1(X)) → isNatList.1(X)
activate.0(n__0.) → 0.
nil. → n__nil.
isNat.0(n__0.) → tt.
take.0-0(s.0(M), cons.1-0(N, IL)) → U31.1-0-0-1(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.0(M), n__isNat.1(N))), activate.0(IL), M, N)
take.0-1(X1, X2) → n__take.0-1(X1, X2)
and.1-0(X1, X2) → n__and.1-0(X1, X2)
length.1(X) → n__length.1(X)
cons.1-1(X1, X2) → n__cons.1-1(X1, X2)
activate.0(n__nil.) → nil.
activate.0(X) → X
activate.0(n__cons.0-1(X1, X2)) → cons.0-1(X1, X2)
activate.1(X) → X
take.0-0(s.0(M), cons.0-0(N, IL)) → U31.1-0-0-0(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.0(M), n__isNat.0(N))), activate.0(IL), M, N)
isNatIList.1(V) → isNatList.1(activate.1(V))
take.0-1(0., IL) → U21.1(isNatIList.1(IL))
isNat.0(n__s.1(V1)) → isNat.1(activate.1(V1))
U31.1-0-0-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.0-0(activate.0(M), activate.0(IL)))
0. → n__0.
U31.1-0-1-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.1-0(activate.1(M), activate.0(IL)))
s.1(X) → n__s.1(X)
take.1-0(X1, X2) → n__take.1-0(X1, X2)
take.0-0(0., IL) → U21.1(isNatIList.0(IL))
activate.0(n__length.0(X)) → length.0(X)
activate.0(n__and.0-0(X1, X2)) → and.0-0(X1, X2)
take.0-0(s.1(M), cons.1-0(N, IL)) → U31.1-0-1-1(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.1(M), n__isNat.1(N))), activate.0(IL), M, N)
U31.1-1-0-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.0-1(activate.0(M), activate.1(IL)))
U31.1-1-1-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.1-1(activate.1(M), activate.1(IL)))
isNatList.1(X) → n__isNatList.1(X)
isNat.0(n__length.1(V1)) → isNatList.1(activate.1(V1))
activate.1(n__isNatIList.1(X)) → isNatIList.1(X)
length.0(cons.1-1(N, L)) → U11.1-1(and.1-1(isNatList.1(activate.1(L)), n__isNat.1(N)), activate.1(L))
isNatIList.1(X) → n__isNatIList.1(X)
take.0-0(X1, X2) → n__take.0-0(X1, X2)
zeros. → n__zeros.
length.0(nil.) → 0.
activate.1(n__and.0-1(X1, X2)) → and.0-1(X1, X2)
isNat.1(X) → n__isNat.1(X)
activate.0(n__take.1-0(X1, X2)) → take.1-0(X1, X2)
take.0-0(s.0(M), cons.1-1(N, IL)) → U31.1-1-0-1(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.0(M), n__isNat.1(N))), activate.1(IL), M, N)
take.1-1(X1, X2) → n__take.1-1(X1, X2)
U31.1-0-0-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.0-0(activate.0(M), activate.0(IL)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ SemLabProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                ↳ SemLabProof2

                                                                            ↳ QDPOrderProof

                                                                          ↳ QDP

                                            ↳ QDP

                  ↳ QDP

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

AND.1-1(tt., n__isNatList.0(y_4)) → ACTIVATE.1(n__isNatList.0(y_4))
ISNATLIST.0(n__cons.0-0(n__0., y1)) → AND.1-1(isNat.0(0.), n__isNatList.0(activate.0(y1)))
ISNATLIST.0(n__cons.0-0(x0, y1)) → AND.1-1(isNat.0(x0), n__isNatList.0(activate.0(y1)))
ACTIVATE.1(n__isNatList.0(X)) → ISNATLIST.0(X)

The TRS R consists of the following rules:

take.0-0(s.1(M), cons.0-0(N, IL)) → U31.1-0-1-0(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.1(M), n__isNat.0(N))), activate.0(IL), M, N)
activate.0(n__take.0-1(X1, X2)) → take.0-1(X1, X2)
U31.1-0-1-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.1-0(activate.1(M), activate.0(IL)))
and.1-0(tt., X) → activate.0(X)
activate.0(n__zeros.) → zeros.
and.0-0(X1, X2) → n__and.0-0(X1, X2)
activate.0(n__take.1-1(X1, X2)) → take.1-1(X1, X2)
U21.1(tt.) → nil.
activate.1(n__isNat.1(X)) → isNat.1(X)
activate.0(n__length.1(X)) → length.1(X)
activate.0(n__s.0(X)) → s.0(X)
length.0(X) → n__length.0(X)
isNatList.0(n__take.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.0(activate.0(V2)))
isNat.0(n__s.0(V1)) → isNat.0(activate.0(V1))
cons.1-0(X1, X2) → n__cons.1-0(X1, X2)
and.1-1(X1, X2) → n__and.1-1(X1, X2)
isNat.0(X) → n__isNat.0(X)
take.0-0(s.1(M), cons.0-1(N, IL)) → U31.1-1-1-0(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.1(M), n__isNat.0(N))), activate.1(IL), M, N)
isNatList.0(n__cons.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatList.0(activate.0(V2)))
isNatIList.0(X) → n__isNatIList.0(X)
length.0(cons.0-1(N, L)) → U11.1-1(and.1-1(isNatList.1(activate.1(L)), n__isNat.0(N)), activate.1(L))
U31.1-1-0-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.0-1(activate.0(M), activate.1(IL)))
cons.0-0(X1, X2) → n__cons.0-0(X1, X2)
activate.0(n__s.1(X)) → s.1(X)
activate.0(n__and.1-0(X1, X2)) → and.1-0(X1, X2)
s.0(X) → n__s.0(X)
activate.0(n__cons.0-0(X1, X2)) → cons.0-0(X1, X2)
U31.1-1-1-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.1-1(activate.1(M), activate.1(IL)))
cons.0-1(X1, X2) → n__cons.0-1(X1, X2)
activate.1(n__isNatList.0(X)) → isNatList.0(X)
activate.0(n__take.0-0(X1, X2)) → take.0-0(X1, X2)
and.0-1(X1, X2) → n__and.0-1(X1, X2)
isNat.0(n__length.1(V1)) → isNatList.1(activate.1(V1))
activate.1(n__isNatIList.1(X)) → isNatIList.1(X)
isNatIList.0(n__cons.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.1(activate.1(V2)))
length.0(cons.1-1(N, L)) → U11.1-1(and.1-1(isNatList.1(activate.1(L)), n__isNat.1(N)), activate.1(L))
isNatIList.1(X) → n__isNatIList.1(X)
take.0-0(X1, X2) → n__take.0-0(X1, X2)
zeros. → n__zeros.
length.0(nil.) → 0.
activate.1(n__and.0-1(X1, X2)) → and.0-1(X1, X2)
isNat.1(X) → n__isNat.1(X)
activate.0(n__take.1-0(X1, X2)) → take.1-0(X1, X2)
take.0-0(s.0(M), cons.1-1(N, IL)) → U31.1-1-0-1(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.0(M), n__isNat.1(N))), activate.1(IL), M, N)
isNatList.0(n__take.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.0(activate.0(V2)))
take.1-1(X1, X2) → n__take.1-1(X1, X2)
U31.1-0-0-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.0-0(activate.0(M), activate.0(IL)))
isNat.0(n__length.0(V1)) → isNatList.0(activate.0(V1))
isNatIList.0(n__cons.1-0(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.0(activate.0(V2)))
s.1(X) → n__s.1(X)
isNatList.0(n__cons.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatList.1(activate.1(V2)))
take.1-0(X1, X2) → n__take.1-0(X1, X2)
isNatIList.0(V) → isNatList.0(activate.0(V))
take.0-0(0., IL) → U21.1(isNatIList.0(IL))
activate.0(n__length.0(X)) → length.0(X)
activate.1(n__isNatIList.0(X)) → isNatIList.0(X)
activate.0(n__and.0-0(X1, X2)) → and.0-0(X1, X2)
isNatList.0(n__cons.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatList.0(activate.0(V2)))
isNatList.0(n__cons.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatList.1(activate.1(V2)))
take.0-0(s.1(M), cons.1-0(N, IL)) → U31.1-0-1-1(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.1(M), n__isNat.1(N))), activate.0(IL), M, N)
isNatList.0(n__take.1-1(V1, V2)) → and.1-1(isNat.1(activate.1(V1)), n__isNatIList.1(activate.1(V2)))
U31.1-1-0-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.0-1(activate.0(M), activate.1(IL)))
isNatList.1(X) → n__isNatList.1(X)
U31.1-1-1-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.1-1(activate.1(M), activate.1(IL)))
length.1(X) → n__length.1(X)
cons.1-1(X1, X2) → n__cons.1-1(X1, X2)
take.0-1(X1, X2) → n__take.0-1(X1, X2)
and.1-0(X1, X2) → n__and.1-0(X1, X2)
activate.0(X) → X
activate.0(n__cons.0-1(X1, X2)) → cons.0-1(X1, X2)
activate.0(n__nil.) → nil.
and.1-1(tt., X) → activate.1(X)
isNatIList.1(V) → isNatList.1(activate.1(V))
take.0-1(0., IL) → U21.1(isNatIList.1(IL))
activate.1(X) → X
take.0-0(s.0(M), cons.0-0(N, IL)) → U31.1-0-0-0(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.0(M), n__isNat.0(N))), activate.0(IL), M, N)
0. → n__0.
U31.1-0-1-0(tt., IL, M, N) → cons.0-0(activate.0(N), n__take.1-0(activate.1(M), activate.0(IL)))
isNat.0(n__s.1(V1)) → isNat.1(activate.1(V1))
U31.1-0-0-1(tt., IL, M, N) → cons.1-0(activate.1(N), n__take.0-0(activate.0(M), activate.0(IL)))
take.0-0(s.1(M), cons.1-1(N, IL)) → U31.1-1-1-1(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.1(M), n__isNat.1(N))), activate.1(IL), M, N)
U11.1-0(tt., L) → s.0(length.0(activate.0(L)))
isNatList.0(n__nil.) → tt.
take.0-0(s.0(M), cons.0-1(N, IL)) → U31.1-1-0-0(and.1-1(isNatIList.1(activate.1(IL)), n__and.1-1(isNat.0(M), n__isNat.0(N))), activate.1(IL), M, N)
isNatList.0(n__take.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.1(activate.1(V2)))
isNatIList.0(n__zeros.) → tt.
length.0(cons.1-0(N, L)) → U11.1-0(and.1-1(isNatList.0(activate.0(L)), n__isNat.1(N)), activate.0(L))
zeros. → cons.0-0(0., n__zeros.)
isNatIList.0(n__cons.0-0(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.0(activate.0(V2)))
activate.0(n__0.) → 0.
activate.1(n__isNatList.1(X)) → isNatList.1(X)
activate.0(n__cons.1-0(X1, X2)) → cons.1-0(X1, X2)
activate.1(n__and.1-1(X1, X2)) → and.1-1(X1, X2)
take.0-0(s.0(M), cons.1-0(N, IL)) → U31.1-0-0-1(and.1-1(isNatIList.0(activate.0(IL)), n__and.1-1(isNat.0(M), n__isNat.1(N))), activate.0(IL), M, N)
isNat.0(n__0.) → tt.
nil. → n__nil.
U11.1-1(tt., L) → s.0(length.1(activate.1(L)))
length.0(cons.0-0(N, L)) → U11.1-0(and.1-1(isNatList.0(activate.0(L)), n__isNat.0(N)), activate.0(L))
activate.1(n__isNat.0(X)) → isNat.0(X)
activate.0(n__cons.1-1(X1, X2)) → cons.1-1(X1, X2)
isNatList.0(X) → n__isNatList.0(X)
isNatIList.0(n__cons.0-1(V1, V2)) → and.1-1(isNat.0(activate.0(V1)), n__isNatIList.1(activate.1(V2)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
As can be seen after transforming the QDP problem by semantic labelling [33] and then some rule deleting processors, only certain labelled rules and pairs can be used. Hence, we only have to consider all unlabelled pairs and rules (without the decreasing rules for quasi-models).

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

                                                                                            ↳ QDPOrderProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ SemLabProof

                                                                                                                ↳ SemLabProof2

                                                                                                                  ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                          ↳ QDP

                                            ↳ QDP

                  ↳ QDP

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

ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
AND(tt, n__isNatList(y_4)) → ACTIVATE(n__isNatList(y_4))

The TRS R consists of the following rules:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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.

ISNATLIST(n__cons(n__take(x0, x1), y1)) → AND(isNat(take(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatList(activate(y1)))

The remaining pairs can at least be oriented weakly.

ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
AND(tt, n__isNatList(y_4)) → ACTIVATE(n__isNatList(y_4))
ISNATLIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatList(activate(y1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(ISNATLIST(x₁)) = x₁
POL(U11(x₁, x₂)) = x₂
POL(U21(x₁)) = 1
POL(U31(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₄
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 1 + x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 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)) = 1 + x₁
POL(n__isNatList(x₁)) = x₁
POL(n__length(x₁)) = x₁
POL(n__nil) = 1
POL(n__s(x₁)) = x₁
POL(n__take(x₁, x₂)) = 1 + x₂
POL(n__zeros) = 0
POL(nil) = 1
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 1 + 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)
0 → n__0
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
nil → n__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)))
zeros → cons(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)
zeros → n__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

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                          ↳ QDP

                                            ↳ QDP

                  ↳ QDP

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

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatList(activate(y1)))
ACTIVATE(n__isNatList(X)) → ISNATLIST(X)
ISNATLIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatList(activate(y1)))
AND(tt, n__isNatList(y_4)) → ACTIVATE(n__isNatList(y_4))
ISNATLIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))
ISNATLIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Instantiation

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QDP

                                                                          ↳ QDP

                                            ↳ QDP

                  ↳ QDP

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

ISNATILIST(n__cons(n__and(x0, x1), y1)) → AND(isNat(and(x0, x1)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatList(x0), y1)) → AND(isNat(isNatList(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__nil, y1)) → AND(isNat(nil), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__isNatIList(x0), y1)) → AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__length(x0), y1)) → AND(isNat(length(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__zeros, y1)) → AND(isNat(zeros), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → AND(isNat(take(x0, x1)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(x0, y1)) → AND(isNat(x0), n__isNatIList(activate(y1)))
AND(tt, n__isNatIList(y_5)) → ACTIVATE(n__isNatIList(y_5))
ISNATILIST(n__cons(n__isNat(x0), y1)) → AND(isNat(isNat(x0)), n__isNatIList(activate(y1)))
ACTIVATE(n__isNatIList(X)) → ISNATILIST(X)
ISNATILIST(n__cons(n__0, y1)) → AND(isNat(0), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__s(x0), y1)) → AND(isNat(s(x0)), n__isNatIList(activate(y1)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1)))

The TRS R consists of the following rules:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                  ↳ QDP

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

ISNAT(n__s(V1)) → ISNAT(activate(V1))

The TRS R consists of the following rules:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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:

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:

zeros → cons(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)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIList(X) → n__isNatIList(X)
nil → n__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.