Termination w.r.t. Q proof of /tmp/tmpduB5n_/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:

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__U21(X) → U21(X)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__U21(X) → U21(X)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(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:

MARK(take(X1, X2)) → MARK(X2)
A__U11(tt, L) → MARK(L)
MARK(zeros) → A__ZEROS
MARK(U11(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U21(X)) → A__U21(mark(X))
A__AND(tt, X) → MARK(X)
MARK(length(X)) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNatIList(IL), and(isNat(M), isNat(N)))
A__TAKE(s(M), cons(N, IL)) → A__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
MARK(take(X1, X2)) → MARK(X1)
MARK(U31(X1, X2, X3, X4)) → MARK(X1)
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
A__U11(tt, L) → A__LENGTH(mark(L))
MARK(U31(X1, X2, X3, X4)) → A__U31(mark(X1), X2, X3, X4)
A__ISNATILIST(V) → A__ISNATLIST(V)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__TAKE(0, IL) → A__ISNATILIST(IL)
MARK(U21(X)) → MARK(X)
A__TAKE(0, IL) → A__U21(a__isNatIList(IL))
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNat(N))
A__ISNATLIST(take(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__U31(tt, IL, M, N) → MARK(N)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatList(V2))
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNATLIST(take(V1, V2)) → A__ISNAT(V1)
A__ISNATILIST(cons(V1, V2)) → A__ISNAT(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__U21(X) → U21(X)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(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:

MARK(take(X1, X2)) → MARK(X2)
A__U11(tt, L) → MARK(L)
MARK(zeros) → A__ZEROS
MARK(U11(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U21(X)) → A__U21(mark(X))
A__AND(tt, X) → MARK(X)
MARK(length(X)) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNatIList(IL), and(isNat(M), isNat(N)))
A__TAKE(s(M), cons(N, IL)) → A__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
MARK(take(X1, X2)) → MARK(X1)
MARK(U31(X1, X2, X3, X4)) → MARK(X1)
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
A__U11(tt, L) → A__LENGTH(mark(L))
MARK(U31(X1, X2, X3, X4)) → A__U31(mark(X1), X2, X3, X4)
A__ISNATILIST(V) → A__ISNATLIST(V)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__TAKE(0, IL) → A__ISNATILIST(IL)
MARK(U21(X)) → MARK(X)
A__TAKE(0, IL) → A__U21(a__isNatIList(IL))
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNat(N))
A__ISNATLIST(take(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__U31(tt, IL, M, N) → MARK(N)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatList(V2))
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNATLIST(take(V1, V2)) → A__ISNAT(V1)
A__ISNATILIST(cons(V1, V2)) → A__ISNAT(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__U21(X) → U21(X)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(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 3 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

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

MARK(take(X1, X2)) → MARK(X2)
MARK(U11(X1, X2)) → MARK(X1)
A__U11(tt, L) → MARK(L)
MARK(s(X)) → MARK(X)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
A__AND(tt, X) → MARK(X)
MARK(length(X)) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNatIList(IL), and(isNat(M), isNat(N)))
A__TAKE(s(M), cons(N, IL)) → A__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
MARK(take(X1, X2)) → MARK(X1)
MARK(U31(X1, X2, X3, X4)) → MARK(X1)
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
A__U11(tt, L) → A__LENGTH(mark(L))
MARK(U31(X1, X2, X3, X4)) → A__U31(mark(X1), X2, X3, X4)
A__ISNATILIST(V) → A__ISNATLIST(V)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__TAKE(0, IL) → A__ISNATILIST(IL)
MARK(U21(X)) → MARK(X)
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNat(N))
A__ISNATLIST(take(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__U31(tt, IL, M, N) → MARK(N)
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatList(V2))
MARK(isNat(X)) → A__ISNAT(X)
A__ISNATLIST(take(V1, V2)) → A__ISNAT(V1)
A__ISNATILIST(cons(V1, V2)) → A__ISNAT(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__U21(X) → U21(X)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(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.

MARK(take(X1, X2)) → MARK(X2)
MARK(take(X1, X2)) → MARK(X1)
MARK(U31(X1, X2, X3, X4)) → MARK(X1)
MARK(U31(X1, X2, X3, X4)) → A__U31(mark(X1), X2, X3, X4)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
MARK(U21(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → MARK(X1)
A__U11(tt, L) → MARK(L)
MARK(s(X)) → MARK(X)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
A__AND(tt, X) → MARK(X)
MARK(length(X)) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNatIList(IL), and(isNat(M), isNat(N)))
A__TAKE(s(M), cons(N, IL)) → A__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
A__U11(tt, L) → A__LENGTH(mark(L))
A__ISNATILIST(V) → A__ISNATLIST(V)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__TAKE(0, IL) → A__ISNATILIST(IL)
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNat(N))
A__ISNATLIST(take(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__U31(tt, IL, M, N) → MARK(N)
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatList(V2))
MARK(isNat(X)) → A__ISNAT(X)
A__ISNATLIST(take(V1, V2)) → A__ISNAT(V1)
A__ISNATILIST(cons(V1, V2)) → A__ISNAT(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__AND(x₁, x₂)) = x₂
POL(A__ISNAT(x₁)) = 0
POL(A__ISNATILIST(x₁)) = 0
POL(A__ISNATLIST(x₁)) = 0
POL(A__LENGTH(x₁)) = x₁
POL(A__TAKE(x₁, x₂)) = x₂
POL(A__U11(x₁, x₂)) = x₂
POL(A__U31(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁ + x₂
POL(U21(x₁)) = 1 + x₁
POL(U31(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₂ + x₃ + x₄
POL(a__U11(x₁, x₂)) = x₁ + x₂
POL(a__U21(x₁)) = 1 + x₁
POL(a__U31(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₂ + x₃ + x₄
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isNat(x₁)) = 0
POL(a__isNatIList(x₁)) = 0
POL(a__isNatList(x₁)) = 0
POL(a__length(x₁)) = x₁
POL(a__take(x₁, x₂)) = 1 + x₁ + x₂
POL(a__zeros) = 0
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = x₁
POL(mark(x₁)) = x₁
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:

a__and(X1, X2) → and(X1, X2)
a__take(X1, X2) → take(X1, X2)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__isNatIList(X) → isNatIList(X)
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__zeros → zeros
mark(nil) → nil
a__length(X) → length(X)
a__U11(X1, X2) → U11(X1, X2)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__U21(X) → U21(X)
a__isNat(0) → tt
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__isNatList(nil) → tt
a__isNatIList(zeros) → tt
mark(U21(X)) → a__U21(mark(X))
mark(length(X)) → a__length(mark(X))
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(zeros) → a__zeros
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__length(nil) → 0
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__isNatIList(V) → a__isNatList(V)
mark(isNat(X)) → a__isNat(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNat(s(V1)) → a__isNat(V1)
a__and(tt, X) → mark(X)
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
a__isNat(length(V1)) → a__isNatList(V1)
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

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

A__U11(tt, L) → MARK(L)
MARK(U11(X1, X2)) → MARK(X1)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)
MARK(s(X)) → MARK(X)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
A__AND(tt, X) → MARK(X)
MARK(length(X)) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNatIList(IL), and(isNat(M), isNat(N)))
MARK(and(X1, X2)) → MARK(X1)
A__TAKE(s(M), cons(N, IL)) → A__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
A__U11(tt, L) → A__LENGTH(mark(L))
A__ISNATILIST(V) → A__ISNATLIST(V)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__TAKE(0, IL) → A__ISNATILIST(IL)
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNat(N))
A__ISNATLIST(take(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__U31(tt, IL, M, N) → MARK(N)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatList(V2))
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__ISNATILIST(cons(V1, V2)) → A__ISNAT(V1)
A__ISNATLIST(take(V1, V2)) → A__ISNAT(V1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(isNatList(X)) → A__ISNATLIST(X)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__U21(X) → U21(X)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(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 5 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → MARK(X1)
A__U11(tt, L) → MARK(L)
MARK(s(X)) → MARK(X)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
A__AND(tt, X) → MARK(X)
MARK(length(X)) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__U11(tt, L) → A__LENGTH(mark(L))
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
A__ISNATILIST(V) → A__ISNATLIST(V)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNat(N))
A__ISNATLIST(take(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatList(V2))
MARK(isNat(X)) → A__ISNAT(X)
A__ISNATLIST(take(V1, V2)) → A__ISNAT(V1)
A__ISNATILIST(cons(V1, V2)) → A__ISNAT(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__U21(X) → U21(X)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(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.

MARK(U11(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(length(X)) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)

The remaining pairs can at least be oriented weakly.

A__U11(tt, L) → MARK(L)
MARK(s(X)) → MARK(X)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
A__AND(tt, X) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)
A__U11(tt, L) → A__LENGTH(mark(L))
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
A__ISNATILIST(V) → A__ISNATLIST(V)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNat(N))
A__ISNATLIST(take(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatList(V2))
MARK(isNat(X)) → A__ISNAT(X)
A__ISNATLIST(take(V1, V2)) → A__ISNAT(V1)
A__ISNATILIST(cons(V1, V2)) → A__ISNAT(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)

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

POL(0) = 0
POL(A__AND(x₁, x₂)) = x₂
POL(A__ISNAT(x₁)) = 0
POL(A__ISNATILIST(x₁)) = 0
POL(A__ISNATLIST(x₁)) = 0
POL(A__LENGTH(x₁)) = x₁
POL(A__U11(x₁, x₂)) = x₂
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = 1 + x₁ + x₂
POL(U21(x₁)) = 0
POL(U31(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₄
POL(a__U11(x₁, x₂)) = 1 + x₁ + x₂
POL(a__U21(x₁)) = 0
POL(a__U31(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₄
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isNat(x₁)) = 0
POL(a__isNatIList(x₁)) = 0
POL(a__isNatList(x₁)) = 0
POL(a__length(x₁)) = 1 + x₁
POL(a__take(x₁, x₂)) = 1 + x₂
POL(a__zeros) = 0
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 1 + x₁
POL(mark(x₁)) = x₁
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:

a__and(X1, X2) → and(X1, X2)
a__take(X1, X2) → take(X1, X2)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__isNatIList(X) → isNatIList(X)
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__zeros → zeros
mark(nil) → nil
a__length(X) → length(X)
a__U11(X1, X2) → U11(X1, X2)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__U21(X) → U21(X)
a__isNat(0) → tt
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__isNatList(nil) → tt
a__isNatIList(zeros) → tt
mark(U21(X)) → a__U21(mark(X))
mark(length(X)) → a__length(mark(X))
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(zeros) → a__zeros
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__length(nil) → 0
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__isNatIList(V) → a__isNatList(V)
mark(isNat(X)) → a__isNat(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNat(s(V1)) → a__isNat(V1)
a__and(tt, X) → mark(X)
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
a__isNat(length(V1)) → a__isNatList(V1)
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

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

A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__U11(tt, L) → MARK(L)
MARK(s(X)) → MARK(X)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNat(N))
A__AND(tt, X) → MARK(X)
A__ISNATLIST(take(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatList(V2))
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__ISNATILIST(cons(V1, V2)) → A__ISNAT(V1)
A__ISNATLIST(take(V1, V2)) → A__ISNAT(V1)
MARK(isNat(X)) → A__ISNAT(X)
A__U11(tt, L) → A__LENGTH(mark(L))
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
A__ISNATILIST(V) → A__ISNATLIST(V)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__U21(X) → U21(X)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(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 3 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                          ↳ QDP

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

MARK(s(X)) → MARK(X)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__AND(tt, X) → MARK(X)
A__ISNATLIST(take(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatList(V2))
A__ISNATILIST(cons(V1, V2)) → A__ISNAT(V1)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
A__ISNATLIST(take(V1, V2)) → A__ISNAT(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
A__ISNATILIST(V) → A__ISNATLIST(V)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__U21(X) → U21(X)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(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.

A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
MARK(cons(X1, X2)) → MARK(X1)
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNat(V1), isNatList(V2))
A__ISNATILIST(cons(V1, V2)) → A__ISNAT(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)

The remaining pairs can at least be oriented weakly.

MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__AND(tt, X) → MARK(X)
A__ISNATLIST(take(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNATLIST(take(V1, V2)) → A__ISNAT(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
A__ISNATILIST(V) → A__ISNATLIST(V)

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

POL(0) = 0
POL(A__AND(x₁, x₂)) = x₂
POL(A__ISNAT(x₁)) = x₁
POL(A__ISNATILIST(x₁)) = x₁
POL(A__ISNATLIST(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U21(x₁)) = 0
POL(U31(x₁, x₂, x₃, x₄)) = 0
POL(a__U11(x₁, x₂)) = 0
POL(a__U21(x₁)) = 0
POL(a__U31(x₁, x₂, x₃, x₄)) = 0
POL(a__and(x₁, x₂)) = 0
POL(a__isNat(x₁)) = 0
POL(a__isNatIList(x₁)) = 0
POL(a__isNatList(x₁)) = 0
POL(a__length(x₁)) = 0
POL(a__take(x₁, x₂)) = 0
POL(a__zeros) = 0
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 1 + x₁ + x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 1 + x₁
POL(mark(x₁)) = 0
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = x₁ + x₂
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                          ↳ QDP

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

A__ISNATLIST(take(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(s(X)) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__ISNATLIST(take(V1, V2)) → A__ISNAT(V1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
A__ISNATILIST(V) → A__ISNATLIST(V)
A__AND(tt, X) → MARK(X)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__U21(X) → U21(X)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(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 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ AND

                                    ↳ QDP

                                      ↳ UsableRulesProof

                                    ↳ QDP

                          ↳ QDP

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

A__ISNAT(s(V1)) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__U21(X) → U21(X)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ AND

                                    ↳ QDP

                                      ↳ UsableRulesProof

                                        ↳ QDP

                                          ↳ QDPSizeChangeProof

                                    ↳ QDP

                          ↳ QDP

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

A__ISNAT(s(V1)) → A__ISNAT(V1)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

A__ISNAT(s(V1)) → A__ISNAT(V1)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ AND

                                    ↳ QDP

                                    ↳ QDP

                                      ↳ QDPOrderProof

                          ↳ QDP

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

A__ISNATLIST(take(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
MARK(s(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
A__ISNATILIST(V) → A__ISNATLIST(V)
A__AND(tt, X) → MARK(X)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__U21(X) → U21(X)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(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.

MARK(s(X)) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)
A__AND(tt, X) → MARK(X)

The remaining pairs can at least be oriented weakly.

A__ISNATLIST(take(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
A__ISNATILIST(V) → A__ISNATLIST(V)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__AND(x₁, x₂)) = 1 + x₂
POL(A__ISNATILIST(x₁)) = x₁
POL(A__ISNATLIST(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U21(x₁)) = 0
POL(U31(x₁, x₂, x₃, x₄)) = 0
POL(a__U11(x₁, x₂)) = 0
POL(a__U21(x₁)) = 0
POL(a__U31(x₁, x₂, x₃, x₄)) = 0
POL(a__and(x₁, x₂)) = 0
POL(a__isNat(x₁)) = 0
POL(a__isNatIList(x₁)) = 0
POL(a__isNatList(x₁)) = 0
POL(a__length(x₁)) = 0
POL(a__take(x₁, x₂)) = 0
POL(a__zeros) = 0
POL(and(x₁, x₂)) = 1 + x₁ + x₂
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 0
POL(mark(x₁)) = 0
POL(nil) = 0
POL(s(x₁)) = 1 + x₁
POL(take(x₁, x₂)) = 1 + x₂
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ AND

                                    ↳ QDP

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                          ↳ QDP

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

A__ISNATLIST(take(V1, V2)) → A__AND(a__isNat(V1), isNatIList(V2))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
A__ISNATILIST(V) → A__ISNATLIST(V)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__U21(X) → U21(X)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 5 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ Narrowing

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

A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
A__U11(tt, L) → A__LENGTH(mark(L))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__U21(X) → U21(X)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A__U11(tt, L) → A__LENGTH(mark(L)) at position [0] we obtained the following new rules:

A__U11(tt, s(x0)) → A__LENGTH(s(mark(x0)))
A__U11(tt, U31(x0, x1, x2, x3)) → A__LENGTH(a__U31(mark(x0), x1, x2, x3))
A__U11(tt, U21(x0)) → A__LENGTH(a__U21(mark(x0)))
A__U11(tt, zeros) → A__LENGTH(a__zeros)
A__U11(tt, nil) → A__LENGTH(nil)
A__U11(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U11(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__U11(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U11(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U11(tt, and(x0, x1)) → A__LENGTH(a__and(mark(x0), x1))
A__U11(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))
A__U11(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U11(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U11(tt, tt) → A__LENGTH(tt)
A__U11(tt, 0) → A__LENGTH(0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

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

A__U11(tt, s(x0)) → A__LENGTH(s(mark(x0)))
A__U11(tt, U31(x0, x1, x2, x3)) → A__LENGTH(a__U31(mark(x0), x1, x2, x3))
A__U11(tt, U21(x0)) → A__LENGTH(a__U21(mark(x0)))
A__U11(tt, zeros) → A__LENGTH(a__zeros)
A__U11(tt, nil) → A__LENGTH(nil)
A__U11(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U11(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
A__U11(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U11(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U11(tt, and(x0, x1)) → A__LENGTH(a__and(mark(x0), x1))
A__U11(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))
A__U11(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U11(tt, tt) → A__LENGTH(tt)
A__U11(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U11(tt, 0) → A__LENGTH(0)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__U21(X) → U21(X)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(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 4 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

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

A__U11(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U11(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U11(tt, U31(x0, x1, x2, x3)) → A__LENGTH(a__U31(mark(x0), x1, x2, x3))
A__U11(tt, zeros) → A__LENGTH(a__zeros)
A__U11(tt, U21(x0)) → A__LENGTH(a__U21(mark(x0)))
A__U11(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__U11(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U11(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
A__U11(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U11(tt, and(x0, x1)) → A__LENGTH(a__and(mark(x0), x1))
A__U11(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__U21(X) → U21(X)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A__U11(tt, zeros) → A__LENGTH(a__zeros) at position [0] we obtained the following new rules:

A__U11(tt, zeros) → A__LENGTH(zeros)
A__U11(tt, zeros) → A__LENGTH(cons(0, zeros))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

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

A__U11(tt, U31(x0, x1, x2, x3)) → A__LENGTH(a__U31(mark(x0), x1, x2, x3))
A__U11(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U11(tt, U21(x0)) → A__LENGTH(a__U21(mark(x0)))
A__U11(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U11(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
A__U11(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U11(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U11(tt, and(x0, x1)) → A__LENGTH(a__and(mark(x0), x1))
A__U11(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))
A__U11(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U11(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U11(tt, zeros) → A__LENGTH(zeros)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__U21(X) → U21(X)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(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

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                            ↳ QDPOrderProof

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

A__U11(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U11(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U11(tt, U31(x0, x1, x2, x3)) → A__LENGTH(a__U31(mark(x0), x1, x2, x3))
A__U11(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U11(tt, U21(x0)) → A__LENGTH(a__U21(mark(x0)))
A__U11(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__U11(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U11(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
A__U11(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U11(tt, and(x0, x1)) → A__LENGTH(a__and(mark(x0), x1))
A__U11(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__U21(X) → U21(X)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(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.

A__U11(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U11(tt, U21(x0)) → A__LENGTH(a__U21(mark(x0)))
A__U11(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))

The remaining pairs can at least be oriented weakly.

A__U11(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U11(tt, U31(x0, x1, x2, x3)) → A__LENGTH(a__U31(mark(x0), x1, x2, x3))
A__U11(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U11(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__U11(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U11(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
A__U11(tt, and(x0, x1)) → A__LENGTH(a__and(mark(x0), x1))
A__U11(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))

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

POL(0) = 0
POL(A__LENGTH(x₁)) = x₁
POL(A__U11(x₁, x₂)) = 1
POL(U11(x₁, x₂)) = 0
POL(U21(x₁)) = 0
POL(U31(x₁, x₂, x₃, x₄)) = 0
POL(a__U11(x₁, x₂)) = 0
POL(a__U21(x₁)) = 0
POL(a__U31(x₁, x₂, x₃, x₄)) = 1
POL(a__and(x₁, x₂)) = 1
POL(a__isNat(x₁)) = 1
POL(a__isNatIList(x₁)) = 1
POL(a__isNatList(x₁)) = 1
POL(a__length(x₁)) = 0
POL(a__take(x₁, x₂)) = 1
POL(a__zeros) = 1
POL(and(x₁, x₂)) = 0
POL(cons(x₁, x₂)) = 1
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = 1
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:

a__and(X1, X2) → and(X1, X2)
a__take(X1, X2) → take(X1, X2)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__isNatIList(X) → isNatIList(X)
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__zeros → zeros
mark(nil) → nil
a__length(X) → length(X)
a__U11(X1, X2) → U11(X1, X2)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__U21(X) → U21(X)
a__isNat(0) → tt
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__isNatList(nil) → tt
a__isNatIList(zeros) → tt
mark(U21(X)) → a__U21(mark(X))
mark(length(X)) → a__length(mark(X))
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(zeros) → a__zeros
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__length(nil) → 0
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__isNatIList(V) → a__isNatList(V)
mark(isNat(X)) → a__isNat(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNat(s(V1)) → a__isNat(V1)
a__and(tt, X) → mark(X)
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
a__isNat(length(V1)) → a__isNatList(V1)
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                            ↳ QDPOrderProof

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

A__U11(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U11(tt, U31(x0, x1, x2, x3)) → A__LENGTH(a__U31(mark(x0), x1, x2, x3))
A__U11(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U11(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U11(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
A__U11(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U11(tt, and(x0, x1)) → A__LENGTH(a__and(mark(x0), x1))
A__U11(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__U21(X) → U21(X)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(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.

A__U11(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))

The remaining pairs can at least be oriented weakly.

A__U11(tt, U31(x0, x1, x2, x3)) → A__LENGTH(a__U31(mark(x0), x1, x2, x3))
A__U11(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U11(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U11(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
A__U11(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U11(tt, and(x0, x1)) → A__LENGTH(a__and(mark(x0), x1))
A__U11(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__LENGTH(x₁)) = x₁
POL(A__U11(x₁, x₂)) = x₁ + x₂
POL(U11(x₁, x₂)) = 0
POL(U21(x₁)) = 0
POL(U31(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₂
POL(a__U11(x₁, x₂)) = 0
POL(a__U21(x₁)) = 1
POL(a__U31(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₂
POL(a__and(x₁, x₂)) = 1 + x₂
POL(a__isNat(x₁)) = 1
POL(a__isNatIList(x₁)) = 1
POL(a__isNatList(x₁)) = 1
POL(a__length(x₁)) = 0
POL(a__take(x₁, x₂)) = 1 + x₂
POL(a__zeros) = 1
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = 1 + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = 1 + x₁
POL(nil) = 1
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 1 + x₂
POL(tt) = 1
POL(zeros) = 0

The following usable rules [17] were oriented:

a__and(X1, X2) → and(X1, X2)
a__take(X1, X2) → take(X1, X2)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__isNatIList(X) → isNatIList(X)
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__zeros → zeros
mark(nil) → nil
a__length(X) → length(X)
a__U11(X1, X2) → U11(X1, X2)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__U21(X) → U21(X)
a__isNat(0) → tt
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__isNatList(nil) → tt
a__isNatIList(zeros) → tt
mark(U21(X)) → a__U21(mark(X))
mark(length(X)) → a__length(mark(X))
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(zeros) → a__zeros
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__length(nil) → 0
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__isNatIList(V) → a__isNatList(V)
mark(isNat(X)) → a__isNat(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNat(s(V1)) → a__isNat(V1)
a__and(tt, X) → mark(X)
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
a__isNat(length(V1)) → a__isNatList(V1)
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                            ↳ QDPOrderProof

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

A__U11(tt, U31(x0, x1, x2, x3)) → A__LENGTH(a__U31(mark(x0), x1, x2, x3))
A__U11(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U11(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__U11(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U11(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
A__U11(tt, and(x0, x1)) → A__LENGTH(a__and(mark(x0), x1))
A__U11(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__U21(X) → U21(X)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(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.

A__U11(tt, U31(x0, x1, x2, x3)) → A__LENGTH(a__U31(mark(x0), x1, x2, x3))

The remaining pairs can at least be oriented weakly.

A__U11(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U11(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__U11(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U11(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
A__U11(tt, and(x0, x1)) → A__LENGTH(a__and(mark(x0), x1))
A__U11(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__LENGTH(x₁)) = x₁
POL(A__U11(x₁, x₂)) = x₁ + x₂
POL(U11(x₁, x₂)) = 1
POL(U21(x₁)) = 1
POL(U31(x₁, x₂, x₃, x₄)) = 1
POL(a__U11(x₁, x₂)) = 1
POL(a__U21(x₁)) = 1
POL(a__U31(x₁, x₂, x₃, x₄)) = 1
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isNat(x₁)) = 1
POL(a__isNatIList(x₁)) = 1
POL(a__isNatList(x₁)) = 1
POL(a__length(x₁)) = 1
POL(a__take(x₁, x₂)) = 1
POL(a__zeros) = 1
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 1 + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 1
POL(mark(x₁)) = 1 + x₁
POL(nil) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 1
POL(zeros) = 0

The following usable rules [17] were oriented:

a__and(X1, X2) → and(X1, X2)
a__take(X1, X2) → take(X1, X2)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__isNatIList(X) → isNatIList(X)
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__zeros → zeros
mark(nil) → nil
a__length(X) → length(X)
a__U11(X1, X2) → U11(X1, X2)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__U21(X) → U21(X)
a__isNat(0) → tt
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__isNatList(nil) → tt
a__isNatIList(zeros) → tt
mark(U21(X)) → a__U21(mark(X))
mark(length(X)) → a__length(mark(X))
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(zeros) → a__zeros
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__length(nil) → 0
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__isNatIList(V) → a__isNatList(V)
mark(isNat(X)) → a__isNat(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNat(s(V1)) → a__isNat(V1)
a__and(tt, X) → mark(X)
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
a__isNat(length(V1)) → a__isNatList(V1)
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                            ↳ QDPOrderProof

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

A__U11(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U11(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U11(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
A__U11(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U11(tt, and(x0, x1)) → A__LENGTH(a__and(mark(x0), x1))
A__U11(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__U21(X) → U21(X)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(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.

A__U11(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))

The remaining pairs can at least be oriented weakly.

A__U11(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U11(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
A__U11(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U11(tt, and(x0, x1)) → A__LENGTH(a__and(mark(x0), x1))
A__U11(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))

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

POL(0) = 0
POL(A__LENGTH(x₁)) = x₁
POL(A__U11(x₁, x₂)) = 1 + x₂
POL(U11(x₁, x₂)) = 1
POL(U21(x₁)) = 0
POL(U31(x₁, x₂, x₃, x₄)) = 0
POL(a__U11(x₁, x₂)) = 1
POL(a__U21(x₁)) = 0
POL(a__U31(x₁, x₂, x₃, x₄)) = 1
POL(a__and(x₁, x₂)) = 1 + x₂
POL(a__isNat(x₁)) = x₁
POL(a__isNatIList(x₁)) = 1
POL(a__isNatList(x₁)) = 1
POL(a__length(x₁)) = 1
POL(a__take(x₁, x₂)) = 1
POL(a__zeros) = 1
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = 1 + x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 1
POL(mark(x₁)) = 1 + x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

a__and(X1, X2) → and(X1, X2)
a__take(X1, X2) → take(X1, X2)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__isNatIList(X) → isNatIList(X)
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__zeros → zeros
mark(nil) → nil
a__length(X) → length(X)
a__U11(X1, X2) → U11(X1, X2)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__U21(X) → U21(X)
a__isNat(0) → tt
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__isNatList(nil) → tt
a__isNatIList(zeros) → tt
mark(U21(X)) → a__U21(mark(X))
mark(length(X)) → a__length(mark(X))
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(zeros) → a__zeros
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__length(nil) → 0
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__isNatIList(V) → a__isNatList(V)
mark(isNat(X)) → a__isNat(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNat(s(V1)) → a__isNat(V1)
a__and(tt, X) → mark(X)
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
a__isNat(length(V1)) → a__isNatList(V1)
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                            ↳ QDPOrderProof

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

A__U11(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U11(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U11(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
A__U11(tt, and(x0, x1)) → A__LENGTH(a__and(mark(x0), x1))
A__U11(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__U21(X) → U21(X)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(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.

A__U11(tt, and(x0, x1)) → A__LENGTH(a__and(mark(x0), x1))

The remaining pairs can at least be oriented weakly.

A__U11(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U11(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U11(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
A__U11(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))

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

POL(0) = 0
POL(A__LENGTH(x₁)) = x₁
POL(A__U11(x₁, x₂)) = x₁ + x₂
POL(U11(x₁, x₂)) = 0
POL(U21(x₁)) = 0
POL(U31(x₁, x₂, x₃, x₄)) = 0
POL(a__U11(x₁, x₂)) = 0
POL(a__U21(x₁)) = 0
POL(a__U31(x₁, x₂, x₃, x₄)) = 1
POL(a__and(x₁, x₂)) = 1 + x₂
POL(a__isNat(x₁)) = 1
POL(a__isNatIList(x₁)) = 1
POL(a__isNatList(x₁)) = 1
POL(a__length(x₁)) = 0
POL(a__take(x₁, x₂)) = 1
POL(a__zeros) = 1
POL(and(x₁, x₂)) = 1 + x₂
POL(cons(x₁, x₂)) = 1 + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = 1 + x₁
POL(nil) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 1
POL(zeros) = 0

The following usable rules [17] were oriented:

a__and(X1, X2) → and(X1, X2)
a__take(X1, X2) → take(X1, X2)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__isNatIList(X) → isNatIList(X)
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__zeros → zeros
mark(nil) → nil
a__length(X) → length(X)
a__U11(X1, X2) → U11(X1, X2)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__U21(X) → U21(X)
a__isNat(0) → tt
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__isNatList(nil) → tt
a__isNatIList(zeros) → tt
mark(U21(X)) → a__U21(mark(X))
mark(length(X)) → a__length(mark(X))
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(zeros) → a__zeros
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__length(nil) → 0
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__isNatIList(V) → a__isNatList(V)
mark(isNat(X)) → a__isNat(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNat(s(V1)) → a__isNat(V1)
a__and(tt, X) → mark(X)
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
a__isNat(length(V1)) → a__isNatList(V1)
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                            ↳ QDPOrderProof

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

A__U11(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U11(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
A__U11(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U11(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__U21(X) → U21(X)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(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.

A__U11(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U11(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))

The remaining pairs can at least be oriented weakly.

A__U11(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U11(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__LENGTH(x₁)) = x₁
POL(A__U11(x₁, x₂)) = 1
POL(U11(x₁, x₂)) = 0
POL(U21(x₁)) = 0
POL(U31(x₁, x₂, x₃, x₄)) = 1
POL(a__U11(x₁, x₂)) = 0
POL(a__U21(x₁)) = 0
POL(a__U31(x₁, x₂, x₃, x₄)) = 1
POL(a__and(x₁, x₂)) = x₂
POL(a__isNat(x₁)) = 0
POL(a__isNatIList(x₁)) = 0
POL(a__isNatList(x₁)) = 0
POL(a__length(x₁)) = 0
POL(a__take(x₁, x₂)) = 1
POL(a__zeros) = 1
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = 1
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 1
POL(tt) = 0
POL(zeros) = 1

The following usable rules [17] were oriented:

a__and(X1, X2) → and(X1, X2)
a__take(X1, X2) → take(X1, X2)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__isNatIList(X) → isNatIList(X)
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__zeros → zeros
mark(nil) → nil
a__length(X) → length(X)
a__U11(X1, X2) → U11(X1, X2)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__U21(X) → U21(X)
a__isNat(0) → tt
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__isNatList(nil) → tt
a__isNatIList(zeros) → tt
mark(U21(X)) → a__U21(mark(X))
mark(length(X)) → a__length(mark(X))
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(zeros) → a__zeros
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__length(nil) → 0
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__isNatIList(V) → a__isNatList(V)
mark(isNat(X)) → a__isNat(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNat(s(V1)) → a__isNat(V1)
a__and(tt, X) → mark(X)
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
a__isNat(length(V1)) → a__isNatList(V1)
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                            ↳ QDPOrderProof

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

A__U11(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U11(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__U21(X) → U21(X)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(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.

A__U11(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U11(tt, U21(x0)) → A__LENGTH(a__U21(mark(x0)))
A__U11(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))

The remaining pairs can at least be oriented weakly.

A__U11(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U11(tt, U31(x0, x1, x2, x3)) → A__LENGTH(a__U31(mark(x0), x1, x2, x3))
A__U11(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U11(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__U11(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U11(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
A__U11(tt, and(x0, x1)) → A__LENGTH(a__and(mark(x0), x1))
A__U11(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__LENGTH(x₁)) = 1 + x₁
POL(A__U11(x₁, x₂)) = 1 + x₁
POL(U11(x₁, x₂)) = 0
POL(U21(x₁)) = 0
POL(U31(x₁, x₂, x₃, x₄)) = 0
POL(a__U11(x₁, x₂)) = 0
POL(a__U21(x₁)) = 0
POL(a__U31(x₁, x₂, x₃, x₄)) = x₁
POL(a__and(x₁, x₂)) = x₁
POL(a__isNat(x₁)) = 1
POL(a__isNatIList(x₁)) = 1
POL(a__isNatList(x₁)) = 1
POL(a__length(x₁)) = 0
POL(a__take(x₁, x₂)) = 1
POL(a__zeros) = 1
POL(and(x₁, x₂)) = 0
POL(cons(x₁, x₂)) = 1
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = 1
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 0
POL(tt) = 1
POL(zeros) = 0

The following usable rules [17] were oriented:

a__and(X1, X2) → and(X1, X2)
a__take(X1, X2) → take(X1, X2)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__isNatIList(X) → isNatIList(X)
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__zeros → zeros
mark(nil) → nil
a__length(X) → length(X)
a__U11(X1, X2) → U11(X1, X2)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__U21(X) → U21(X)
a__isNat(0) → tt
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__isNatList(nil) → tt
a__isNatIList(zeros) → tt
mark(U21(X)) → a__U21(mark(X))
mark(length(X)) → a__length(mark(X))
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(zeros) → a__zeros
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__length(nil) → 0
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__isNatIList(V) → a__isNatList(V)
mark(isNat(X)) → a__isNat(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNat(s(V1)) → a__isNat(V1)
a__and(tt, X) → mark(X)
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
a__isNat(length(V1)) → a__isNatList(V1)
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                            ↳ QDPOrderProof

                                              ↳ QDP

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

A__U11(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U11(tt, U31(x0, x1, x2, x3)) → A__LENGTH(a__U31(mark(x0), x1, x2, x3))
A__U11(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U11(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U11(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__LENGTH(cons(N, L)) → A__U11(a__and(a__isNatList(L), isNat(N)), L)
A__U11(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U11(tt, and(x0, x1)) → A__LENGTH(a__and(mark(x0), x1))
A__U11(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__U21(tt) → nil
a__U31(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__isNatList(take(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
a__take(0, IL) → a__U21(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U31(a__and(a__isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2, X3, X4)) → a__U31(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__U21(X) → U21(X)
a__U31(X1, X2, X3, X4) → U31(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)

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