Termination w.r.t. Q proof of /tmp/aprove_benchmark_example

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__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
MARK(zeros) → A__ZEROS
A__LENGTH(cons(N, L)) → A__ISNAT(N)
MARK(s(X)) → MARK(X)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(uTake1(X)) → A__UTAKE1(mark(X))
A__AND(tt, T) → MARK(T)
MARK(length(X)) → MARK(X)
A__UTAKE2(tt, M, N, IL) → MARK(N)
MARK(uLength(X1, X2)) → A__ULENGTH(mark(X1), X2)
MARK(uTake2(X1, X2, X3, X4)) → MARK(X1)
A__LENGTH(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
MARK(and(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X1)
A__ISNATLIST(cons(N, L)) → A__ISNAT(N)
A__ULENGTH(tt, L) → A__LENGTH(mark(L))
MARK(and(X1, X2)) → MARK(X2)
A__ISNATILIST(IL) → A__ISNATLIST(IL)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__TAKE(0, IL) → A__UTAKE1(a__isNatIList(IL))
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
MARK(uTake2(X1, X2, X3, X4)) → A__UTAKE2(mark(X1), X2, X3, X4)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL)))
A__ISNAT(s(N)) → A__ISNAT(N)
A__TAKE(0, IL) → A__ISNATILIST(IL)
MARK(uLength(X1, X2)) → MARK(X1)
A__ISNATLIST(take(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__TAKE(s(M), cons(N, IL)) → A__ISNAT(N)
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__TAKE(s(M), cons(N, IL)) → A__UTAKE2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
MARK(uTake1(X)) → MARK(X)
A__TAKE(s(M), cons(N, IL)) → A__ISNAT(M)
A__LENGTH(cons(N, L)) → A__ULENGTH(a__and(a__isNat(N), a__isNatList(L)), L)
A__ULENGTH(tt, L) → MARK(L)
A__ISNAT(length(L)) → A__ISNATLIST(L)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNATLIST(take(N, IL)) → A__ISNAT(N)
A__ISNATILIST(cons(N, IL)) → A__ISNAT(N)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
MARK(zeros) → A__ZEROS
A__LENGTH(cons(N, L)) → A__ISNAT(N)
MARK(s(X)) → MARK(X)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(uTake1(X)) → A__UTAKE1(mark(X))
A__AND(tt, T) → MARK(T)
MARK(length(X)) → MARK(X)
A__UTAKE2(tt, M, N, IL) → MARK(N)
MARK(uLength(X1, X2)) → A__ULENGTH(mark(X1), X2)
MARK(uTake2(X1, X2, X3, X4)) → MARK(X1)
A__LENGTH(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
MARK(and(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X1)
A__ISNATLIST(cons(N, L)) → A__ISNAT(N)
A__ULENGTH(tt, L) → A__LENGTH(mark(L))
MARK(and(X1, X2)) → MARK(X2)
A__ISNATILIST(IL) → A__ISNATLIST(IL)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__TAKE(0, IL) → A__UTAKE1(a__isNatIList(IL))
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
MARK(uTake2(X1, X2, X3, X4)) → A__UTAKE2(mark(X1), X2, X3, X4)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL)))
A__ISNAT(s(N)) → A__ISNAT(N)
A__TAKE(0, IL) → A__ISNATILIST(IL)
MARK(uLength(X1, X2)) → MARK(X1)
A__ISNATLIST(take(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__TAKE(s(M), cons(N, IL)) → A__ISNAT(N)
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__TAKE(s(M), cons(N, IL)) → A__UTAKE2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
MARK(uTake1(X)) → MARK(X)
A__TAKE(s(M), cons(N, IL)) → A__ISNAT(M)
A__LENGTH(cons(N, L)) → A__ULENGTH(a__and(a__isNat(N), a__isNatList(L)), L)
A__ULENGTH(tt, L) → MARK(L)
A__ISNAT(length(L)) → A__ISNATLIST(L)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNATLIST(take(N, IL)) → A__ISNAT(N)
A__ISNATILIST(cons(N, IL)) → A__ISNAT(N)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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)
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__LENGTH(cons(N, L)) → A__ISNAT(N)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)
MARK(s(X)) → MARK(X)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
A__AND(tt, T) → MARK(T)
A__UTAKE2(tt, M, N, IL) → MARK(N)
MARK(length(X)) → MARK(X)
MARK(uLength(X1, X2)) → A__ULENGTH(mark(X1), X2)
MARK(uTake2(X1, X2, X3, X4)) → MARK(X1)
A__LENGTH(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
MARK(and(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X1)
A__ISNATLIST(cons(N, L)) → A__ISNAT(N)
A__ULENGTH(tt, L) → A__LENGTH(mark(L))
A__ISNATILIST(IL) → A__ISNATLIST(IL)
MARK(and(X1, X2)) → MARK(X2)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
MARK(uTake2(X1, X2, X3, X4)) → A__UTAKE2(mark(X1), X2, X3, X4)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL)))
A__TAKE(0, IL) → A__ISNATILIST(IL)
A__ISNAT(s(N)) → A__ISNAT(N)
MARK(uLength(X1, X2)) → MARK(X1)
A__ISNATLIST(take(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__TAKE(s(M), cons(N, IL)) → A__ISNAT(N)
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__TAKE(s(M), cons(N, IL)) → A__UTAKE2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
MARK(uTake1(X)) → MARK(X)
A__TAKE(s(M), cons(N, IL)) → A__ISNAT(M)
A__LENGTH(cons(N, L)) → A__ULENGTH(a__and(a__isNat(N), a__isNatList(L)), L)
A__ULENGTH(tt, L) → MARK(L)
A__ISNAT(length(L)) → A__ISNATLIST(L)
A__ISNATLIST(take(N, IL)) → A__ISNAT(N)
A__ISNATILIST(cons(N, IL)) → A__ISNAT(N)
MARK(isNat(X)) → A__ISNAT(X)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
MARK(isNatList(X)) → A__ISNATLIST(X)
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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(uTake2(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X1)
MARK(uTake2(X1, X2, X3, X4)) → A__UTAKE2(mark(X1), X2, X3, X4)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
MARK(uTake1(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.

A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__LENGTH(cons(N, L)) → A__ISNAT(N)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)
MARK(s(X)) → MARK(X)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
A__AND(tt, T) → MARK(T)
A__UTAKE2(tt, M, N, IL) → MARK(N)
MARK(length(X)) → MARK(X)
MARK(uLength(X1, X2)) → A__ULENGTH(mark(X1), X2)
A__LENGTH(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
MARK(and(X1, X2)) → MARK(X1)
A__ISNATLIST(cons(N, L)) → A__ISNAT(N)
A__ULENGTH(tt, L) → A__LENGTH(mark(L))
A__ISNATILIST(IL) → A__ISNATLIST(IL)
MARK(and(X1, X2)) → MARK(X2)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL)))
A__TAKE(0, IL) → A__ISNATILIST(IL)
A__ISNAT(s(N)) → A__ISNAT(N)
MARK(uLength(X1, X2)) → MARK(X1)
A__ISNATLIST(take(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__TAKE(s(M), cons(N, IL)) → A__ISNAT(N)
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__TAKE(s(M), cons(N, IL)) → A__UTAKE2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
A__TAKE(s(M), cons(N, IL)) → A__ISNAT(M)
A__LENGTH(cons(N, L)) → A__ULENGTH(a__and(a__isNat(N), a__isNatList(L)), L)
A__ULENGTH(tt, L) → MARK(L)
A__ISNAT(length(L)) → A__ISNATLIST(L)
A__ISNATLIST(take(N, IL)) → A__ISNAT(N)
A__ISNATILIST(cons(N, IL)) → A__ISNAT(N)
MARK(isNat(X)) → A__ISNAT(X)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
MARK(isNatList(X)) → A__ISNATLIST(X)
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)

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__ULENGTH(x₁, x₂)) = x₂
POL(A__UTAKE2(x₁, x₂, x₃, x₄)) = x₃
POL(MARK(x₁)) = x₁
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__uLength(x₁, x₂)) = x₁ + x₂
POL(a__uTake1(x₁)) = 1 + x₁
POL(a__uTake2(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₂ + x₃ + x₄
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(uLength(x₁, x₂)) = x₁ + x₂
POL(uTake1(x₁)) = 1 + x₁
POL(uTake2(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₂ + x₃ + x₄
POL(zeros) = 0

The following usable rules [17] were oriented:

mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__isNatIList(X) → isNatIList(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__zeros → zeros
a__length(X) → length(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zeros) → a__zeros
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uTake1(X)) → a__uTake1(mark(X))
mark(tt) → tt
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)
a__isNatIList(zeros) → tt
a__isNat(0) → tt
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake1(tt) → nil
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__isNatList(nil) → tt
mark(length(X)) → a__length(mark(X))
a__isNatIList(IL) → a__isNatList(IL)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(N)) → a__isNat(N)
a__and(tt, T) → mark(T)
mark(isNatList(X)) → a__isNatList(X)
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
mark(isNatIList(X)) → a__isNatIList(X)
a__isNat(length(L)) → a__isNatList(L)
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__uLength(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

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

A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__LENGTH(cons(N, L)) → A__ISNAT(N)
MARK(s(X)) → MARK(X)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
A__AND(tt, T) → MARK(T)
MARK(length(X)) → MARK(X)
A__UTAKE2(tt, M, N, IL) → MARK(N)
MARK(uLength(X1, X2)) → A__ULENGTH(mark(X1), X2)
A__LENGTH(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
MARK(and(X1, X2)) → MARK(X1)
A__ISNATLIST(cons(N, L)) → A__ISNAT(N)
A__ULENGTH(tt, L) → A__LENGTH(mark(L))
A__ISNATILIST(IL) → A__ISNATLIST(IL)
MARK(and(X1, X2)) → MARK(X2)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL)))
A__TAKE(0, IL) → A__ISNATILIST(IL)
A__ISNAT(s(N)) → A__ISNAT(N)
MARK(uLength(X1, X2)) → MARK(X1)
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATLIST(take(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__TAKE(s(M), cons(N, IL)) → A__ISNAT(N)
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__TAKE(s(M), cons(N, IL)) → A__UTAKE2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
A__TAKE(s(M), cons(N, IL)) → A__ISNAT(M)
A__LENGTH(cons(N, L)) → A__ULENGTH(a__and(a__isNat(N), a__isNatList(L)), L)
A__ISNAT(length(L)) → A__ISNATLIST(L)
A__ULENGTH(tt, L) → MARK(L)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNATILIST(cons(N, IL)) → A__ISNAT(N)
A__ISNATLIST(take(N, IL)) → A__ISNAT(N)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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 8 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

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

A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__LENGTH(cons(N, L)) → A__ISNAT(N)
MARK(s(X)) → MARK(X)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
A__AND(tt, T) → MARK(T)
MARK(length(X)) → MARK(X)
MARK(uLength(X1, X2)) → A__ULENGTH(mark(X1), X2)
A__LENGTH(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
MARK(and(X1, X2)) → MARK(X1)
A__ISNATLIST(cons(N, L)) → A__ISNAT(N)
A__ULENGTH(tt, L) → A__LENGTH(mark(L))
A__ISNATILIST(IL) → A__ISNATLIST(IL)
MARK(and(X1, X2)) → MARK(X2)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__ISNAT(s(N)) → A__ISNAT(N)
MARK(uLength(X1, X2)) → MARK(X1)
A__ISNATLIST(take(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__LENGTH(cons(N, L)) → A__ULENGTH(a__and(a__isNat(N), a__isNatList(L)), L)
A__ULENGTH(tt, L) → MARK(L)
A__ISNAT(length(L)) → A__ISNATLIST(L)
A__ISNATLIST(take(N, IL)) → A__ISNAT(N)
A__ISNATILIST(cons(N, IL)) → A__ISNAT(N)
MARK(isNat(X)) → A__ISNAT(X)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
MARK(isNatList(X)) → A__ISNATLIST(X)
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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(length(X)) → A__LENGTH(mark(X))
MARK(length(X)) → MARK(X)
MARK(uLength(X1, X2)) → A__ULENGTH(mark(X1), X2)
MARK(uLength(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.

A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__LENGTH(cons(N, L)) → A__ISNAT(N)
MARK(s(X)) → MARK(X)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
A__AND(tt, T) → MARK(T)
A__LENGTH(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
MARK(and(X1, X2)) → MARK(X1)
A__ISNATLIST(cons(N, L)) → A__ISNAT(N)
A__ULENGTH(tt, L) → A__LENGTH(mark(L))
A__ISNATILIST(IL) → A__ISNATLIST(IL)
MARK(and(X1, X2)) → MARK(X2)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__ISNAT(s(N)) → A__ISNAT(N)
A__ISNATLIST(take(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__LENGTH(cons(N, L)) → A__ULENGTH(a__and(a__isNat(N), a__isNatList(L)), L)
A__ULENGTH(tt, L) → MARK(L)
A__ISNAT(length(L)) → A__ISNATLIST(L)
A__ISNATLIST(take(N, IL)) → A__ISNAT(N)
A__ISNATILIST(cons(N, IL)) → A__ISNAT(N)
MARK(isNat(X)) → A__ISNAT(X)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
MARK(isNatList(X)) → A__ISNATLIST(X)
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)

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__ULENGTH(x₁, x₂)) = x₂
POL(MARK(x₁)) = x₁
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)) = x₂
POL(a__uLength(x₁, x₂)) = 1 + x₁ + x₂
POL(a__uTake1(x₁)) = 0
POL(a__uTake2(x₁, x₂, x₃, x₄)) = x₃ + x₄
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)) = x₂
POL(tt) = 0
POL(uLength(x₁, x₂)) = 1 + x₁ + x₂
POL(uTake1(x₁)) = 0
POL(uTake2(x₁, x₂, x₃, x₄)) = x₃ + x₄
POL(zeros) = 0

The following usable rules [17] were oriented:

mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__isNatIList(X) → isNatIList(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__zeros → zeros
a__length(X) → length(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zeros) → a__zeros
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uTake1(X)) → a__uTake1(mark(X))
mark(tt) → tt
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)
a__isNatIList(zeros) → tt
a__isNat(0) → tt
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake1(tt) → nil
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__isNatList(nil) → tt
mark(length(X)) → a__length(mark(X))
a__isNatIList(IL) → a__isNatList(IL)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(N)) → a__isNat(N)
a__and(tt, T) → mark(T)
mark(isNatList(X)) → a__isNatList(X)
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
mark(isNatIList(X)) → a__isNatIList(X)
a__isNat(length(L)) → a__isNatList(L)
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__uLength(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

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

A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__LENGTH(cons(N, L)) → A__ISNAT(N)
MARK(s(X)) → MARK(X)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
A__AND(tt, T) → MARK(T)
A__LENGTH(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
MARK(and(X1, X2)) → MARK(X1)
A__ISNATLIST(cons(N, L)) → A__ISNAT(N)
A__ULENGTH(tt, L) → A__LENGTH(mark(L))
MARK(and(X1, X2)) → MARK(X2)
A__ISNATILIST(IL) → A__ISNATLIST(IL)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__ISNAT(s(N)) → A__ISNAT(N)
A__ISNATLIST(take(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__LENGTH(cons(N, L)) → A__ULENGTH(a__and(a__isNat(N), a__isNatList(L)), L)
A__ULENGTH(tt, L) → MARK(L)
A__ISNAT(length(L)) → A__ISNATLIST(L)
A__ISNATLIST(take(N, IL)) → A__ISNAT(N)
A__ISNATILIST(cons(N, IL)) → A__ISNAT(N)
MARK(isNat(X)) → A__ISNAT(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 4 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                          ↳ QDP

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

A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
MARK(s(X)) → MARK(X)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
A__ISNAT(s(N)) → A__ISNAT(N)
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__AND(tt, T) → MARK(T)
A__ISNATLIST(take(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(length(L)) → A__ISNATLIST(L)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNATILIST(cons(N, IL)) → A__ISNAT(N)
A__ISNATLIST(cons(N, L)) → A__ISNAT(N)
A__ISNATLIST(take(N, IL)) → A__ISNAT(N)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)
MARK(and(X1, X2)) → MARK(X2)
A__ISNATILIST(IL) → A__ISNATLIST(IL)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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(cons(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.

A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
MARK(s(X)) → MARK(X)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__ISNAT(s(N)) → A__ISNAT(N)
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__AND(tt, T) → MARK(T)
A__ISNATLIST(take(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(length(L)) → A__ISNATLIST(L)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNATILIST(cons(N, IL)) → A__ISNAT(N)
A__ISNATLIST(cons(N, L)) → A__ISNAT(N)
A__ISNATLIST(take(N, IL)) → A__ISNAT(N)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)
MARK(and(X1, X2)) → MARK(X2)
A__ISNATILIST(IL) → A__ISNATLIST(IL)

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(MARK(x₁)) = x₁
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)) = 0
POL(a__take(x₁, x₂)) = x₁ + x₂
POL(a__uLength(x₁, x₂)) = 0
POL(a__uTake1(x₁)) = 0
POL(a__uTake2(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₃
POL(a__zeros) = 1
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 1 + x₁
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)) = x₁
POL(take(x₁, x₂)) = x₁ + x₂
POL(tt) = 0
POL(uLength(x₁, x₂)) = 0
POL(uTake1(x₁)) = 0
POL(uTake2(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₃
POL(zeros) = 1

The following usable rules [17] were oriented:

mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__isNatIList(X) → isNatIList(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__zeros → zeros
a__length(X) → length(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zeros) → a__zeros
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uTake1(X)) → a__uTake1(mark(X))
mark(tt) → tt
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)
a__isNatIList(zeros) → tt
a__isNat(0) → tt
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake1(tt) → nil
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__isNatList(nil) → tt
mark(length(X)) → a__length(mark(X))
a__isNatIList(IL) → a__isNatList(IL)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(N)) → a__isNat(N)
a__and(tt, T) → mark(T)
mark(isNatList(X)) → a__isNatList(X)
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
mark(isNatIList(X)) → a__isNatIList(X)
a__isNat(length(L)) → a__isNatList(L)
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__uLength(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                          ↳ QDP

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

A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
MARK(s(X)) → MARK(X)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__ISNAT(s(N)) → A__ISNAT(N)
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__AND(tt, T) → MARK(T)
A__ISNATLIST(take(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
MARK(and(X1, X2)) → MARK(X1)
A__ISNAT(length(L)) → A__ISNATLIST(L)
A__ISNATLIST(take(N, IL)) → A__ISNAT(N)
A__ISNATLIST(cons(N, L)) → A__ISNAT(N)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNATILIST(cons(N, IL)) → A__ISNAT(N)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
MARK(isNatList(X)) → A__ISNATLIST(X)
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)
A__ISNATILIST(IL) → A__ISNATLIST(IL)
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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__ISNAT(length(L)) → A__ISNATLIST(L)

The remaining pairs can at least be oriented weakly.

A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
MARK(s(X)) → MARK(X)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__ISNAT(s(N)) → A__ISNAT(N)
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__AND(tt, T) → MARK(T)
A__ISNATLIST(take(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
MARK(and(X1, X2)) → MARK(X1)
A__ISNATLIST(take(N, IL)) → A__ISNAT(N)
A__ISNATLIST(cons(N, L)) → A__ISNAT(N)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNATILIST(cons(N, IL)) → A__ISNAT(N)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
MARK(isNatList(X)) → A__ISNATLIST(X)
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)
A__ISNATILIST(IL) → A__ISNATLIST(IL)
MARK(and(X1, X2)) → MARK(X2)

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

M( a__zeros ) =

/	0	\
\	0	/

M( mark(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( a__isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	0	0	\
\	0	1	/

x₂

M( a__uTake1(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	0	0	\
\	0	1	/

x₂

M( a__uLength(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	1	/

x₂

M( a__length(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	1	/

x₁

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( a__isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	1	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( a__isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( nil ) =

/	0	\
\	0	/

M( a__uTake2(x₁, ..., x₄) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	1	/

x₂

/	0	0	\
\	0	1	/

x₃

/	0	0	\
\	0	1	/

x₄

M( a__and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	0	0	\
\	0	1	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( a__take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	0	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	0	0	\
\	0	1	/

x₂

M( uTake2(x₁, ..., x₄) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	1	/

x₂

/	0	0	\
\	0	1	/

x₃

/	0	0	\
\	0	1	/

x₄

M( uTake1(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( length(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	1	/

x₁

M( uLength(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	1	/

x₂

Tuple symbols:

M( A__AND(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( MARK(x₁) ) =

[

]

x₁

M( A__ISNATILIST(x₁) ) =

[

]

x₁

M( A__ISNAT(x₁) ) =

[

]

x₁

M( A__ISNATLIST(x₁) ) =

[

]

x₁

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

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

mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__isNatIList(X) → isNatIList(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__zeros → zeros
a__length(X) → length(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zeros) → a__zeros
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uTake1(X)) → a__uTake1(mark(X))
mark(tt) → tt
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)
a__isNatIList(zeros) → tt
a__isNat(0) → tt
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake1(tt) → nil
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__isNatList(nil) → tt
mark(length(X)) → a__length(mark(X))
a__isNatIList(IL) → a__isNatList(IL)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(N)) → a__isNat(N)
a__and(tt, T) → mark(T)
mark(isNatList(X)) → a__isNatList(X)
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
mark(isNatIList(X)) → a__isNatIList(X)
a__isNat(length(L)) → a__isNatList(L)
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__uLength(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                          ↳ QDP

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

A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
MARK(s(X)) → MARK(X)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__ISNAT(s(N)) → A__ISNAT(N)
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__AND(tt, T) → MARK(T)
A__ISNATLIST(take(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
MARK(and(X1, X2)) → MARK(X1)
A__ISNATILIST(cons(N, IL)) → A__ISNAT(N)
A__ISNATLIST(take(N, IL)) → A__ISNAT(N)
A__ISNATLIST(cons(N, L)) → A__ISNAT(N)
MARK(isNat(X)) → A__ISNAT(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)
MARK(and(X1, X2)) → MARK(X2)
A__ISNATILIST(IL) → A__ISNATLIST(IL)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 4 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ UsableRulesProof

                                        ↳ QDP

                          ↳ QDP

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

A__ISNAT(s(N)) → A__ISNAT(N)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ UsableRulesProof

                                            ↳ QDP

                                              ↳ QDPSizeChangeProof

                                        ↳ QDP

                          ↳ QDP

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

A__ISNAT(s(N)) → A__ISNAT(N)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ QDPOrderProof

                          ↳ QDP

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

A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
MARK(s(X)) → MARK(X)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__AND(tt, T) → MARK(T)
A__ISNATLIST(take(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
MARK(isNatList(X)) → A__ISNATLIST(X)
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)
MARK(and(X1, X2)) → MARK(X2)
A__ISNATILIST(IL) → A__ISNATLIST(IL)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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(IL) → A__ISNATLIST(IL)

The remaining pairs can at least be oriented weakly.

A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
MARK(s(X)) → MARK(X)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__AND(tt, T) → MARK(T)
A__ISNATLIST(take(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
MARK(isNatList(X)) → A__ISNATLIST(X)
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)
MARK(and(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__AND(x₁, x₂)) = x₂
POL(A__ISNATILIST(x₁)) = 1 + x₁
POL(A__ISNATLIST(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isNat(x₁)) = x₁
POL(a__isNatIList(x₁)) = 1 + x₁
POL(a__isNatList(x₁)) = x₁
POL(a__length(x₁)) = x₁
POL(a__take(x₁, x₂)) = 1 + x₁ + x₂
POL(a__uLength(x₁, x₂)) = x₂
POL(a__uTake1(x₁)) = 0
POL(a__uTake2(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₃ + x₄
POL(a__zeros) = 0
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 1 + x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 1 + x₁ + x₂
POL(tt) = 0
POL(uLength(x₁, x₂)) = x₂
POL(uTake1(x₁)) = 0
POL(uTake2(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₃ + x₄
POL(zeros) = 0

The following usable rules [17] were oriented:

mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__isNatIList(X) → isNatIList(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__zeros → zeros
a__length(X) → length(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zeros) → a__zeros
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uTake1(X)) → a__uTake1(mark(X))
mark(tt) → tt
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)
a__isNatIList(zeros) → tt
a__isNat(0) → tt
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake1(tt) → nil
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__isNatList(nil) → tt
mark(length(X)) → a__length(mark(X))
a__isNatIList(IL) → a__isNatList(IL)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(N)) → a__isNat(N)
a__and(tt, T) → mark(T)
mark(isNatList(X)) → a__isNatList(X)
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
mark(isNatIList(X)) → a__isNatIList(X)
a__isNat(length(L)) → a__isNatList(L)
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__uLength(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                          ↳ QDP

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

A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
MARK(s(X)) → MARK(X)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__AND(tt, T) → MARK(T)
A__ISNATLIST(take(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
MARK(isNatList(X)) → A__ISNATLIST(X)
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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__ISNATLIST(take(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATLIST(take(N, IL)) → A__ISNATILIST(IL)

The remaining pairs can at least be oriented weakly.

A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
MARK(s(X)) → MARK(X)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__AND(tt, T) → MARK(T)
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__AND(x₁, x₂)) = x₂
POL(A__ISNATILIST(x₁)) = x₁
POL(A__ISNATLIST(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isNat(x₁)) = x₁
POL(a__isNatIList(x₁)) = x₁
POL(a__isNatList(x₁)) = x₁
POL(a__length(x₁)) = x₁
POL(a__take(x₁, x₂)) = 1 + x₁ + x₂
POL(a__uLength(x₁, x₂)) = x₂
POL(a__uTake1(x₁)) = 1
POL(a__uTake2(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₃ + x₄
POL(a__zeros) = 0
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 1
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 1 + x₁ + x₂
POL(tt) = 0
POL(uLength(x₁, x₂)) = x₂
POL(uTake1(x₁)) = 1
POL(uTake2(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₃ + x₄
POL(zeros) = 0

The following usable rules [17] were oriented:

mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__isNatIList(X) → isNatIList(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__zeros → zeros
a__length(X) → length(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zeros) → a__zeros
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uTake1(X)) → a__uTake1(mark(X))
mark(tt) → tt
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)
a__isNatIList(zeros) → tt
a__isNat(0) → tt
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake1(tt) → nil
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__isNatList(nil) → tt
mark(length(X)) → a__length(mark(X))
a__isNatIList(IL) → a__isNatList(IL)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(N)) → a__isNat(N)
a__and(tt, T) → mark(T)
mark(isNatList(X)) → a__isNatList(X)
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
mark(isNatIList(X)) → a__isNatIList(X)
a__isNat(length(L)) → a__isNatList(L)
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__uLength(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                          ↳ QDP

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

A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
MARK(s(X)) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
MARK(and(X1, X2)) → MARK(X2)
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__AND(tt, T) → MARK(T)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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)

The remaining pairs can at least be oriented weakly.

A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
MARK(and(X1, X2)) → MARK(X1)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
MARK(and(X1, X2)) → MARK(X2)
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__AND(tt, T) → MARK(T)

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

M( a__zeros ) =

/	0	\
\	0	/

M( mark(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( a__isNatIList(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	1	0	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( a__uTake1(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( take(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	1	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( a__uLength(x₁, x₂) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

/	1	1	\
\	0	0	/

x₂

M( a__length(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	0	0	/

x₁

M( tt ) =

/	0	\
\	1	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	1	0	/

x₁

M( a__isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( s(x₁) ) =

/	1	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( a__isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( nil ) =

/	1	\
\	0	/

M( a__uTake2(x₁, ..., x₄) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	0	/

x₂

/	0	0	\
\	0	0	/

x₃

/	1	0	\
\	1	1	/

x₄

M( a__and(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( 0 ) =

/	1	\
\	0	/

M( a__take(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	1	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	1	1	/

x₂

M( uTake2(x₁, ..., x₄) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	0	/

x₂

/	0	0	\
\	0	0	/

x₃

/	1	0	\
\	1	1	/

x₄

M( uTake1(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	0	0	/

x₁

M( uLength(x₁, x₂) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

/	1	1	\
\	0	0	/

x₂

Tuple symbols:

M( A__AND(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( MARK(x₁) ) =

[

]

x₁

M( A__ISNATILIST(x₁) ) =

[

]

x₁

M( A__ISNATLIST(x₁) ) =

[

]

x₁

mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__isNatIList(X) → isNatIList(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__zeros → zeros
a__length(X) → length(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zeros) → a__zeros
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uTake1(X)) → a__uTake1(mark(X))
mark(tt) → tt
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)
a__isNatIList(zeros) → tt
a__isNat(0) → tt
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake1(tt) → nil
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__isNatList(nil) → tt
mark(length(X)) → a__length(mark(X))
a__isNatIList(IL) → a__isNatList(IL)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(N)) → a__isNat(N)
a__and(tt, T) → mark(T)
mark(isNatList(X)) → a__isNatList(X)
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
mark(isNatIList(X)) → a__isNatIList(X)
a__isNat(length(L)) → a__isNatList(L)
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__uLength(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                          ↳ QDP

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

A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L))
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(X1, X2)) → MARK(X2)
A__AND(tt, T) → MARK(T)
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A__ISNATLIST(cons(N, L)) → A__AND(a__isNat(N), a__isNatList(L)) at position [0] we obtained the following new rules:

A__ISNATLIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatList(y1))
A__ISNATLIST(cons(0, y1)) → A__AND(tt, a__isNatList(y1))
A__ISNATLIST(cons(x0, y1)) → A__AND(isNat(x0), a__isNatList(y1))
A__ISNATLIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatList(y1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                          ↳ QDP

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

A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__ISNATLIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatList(y1))
A__ISNATLIST(cons(x0, y1)) → A__AND(isNat(x0), a__isNatList(y1))
A__AND(tt, T) → MARK(T)
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATLIST(cons(0, y1)) → A__AND(tt, a__isNatList(y1))
A__ISNATLIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatList(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                          ↳ QDP

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

A__ISNATLIST(cons(0, y1)) → A__AND(tt, a__isNatList(y1))
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATLIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatList(y1))
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__ISNATLIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatList(y1))
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2))
MARK(and(X1, X2)) → MARK(X2)
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__AND(tt, T) → MARK(T)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(and(X1, X2)) → A__AND(mark(X1), mark(X2)) at position [0] we obtained the following new rules:

MARK(and(nil, y1)) → A__AND(nil, mark(y1))
MARK(and(uLength(x0, x1), y1)) → A__AND(a__uLength(mark(x0), x1), mark(y1))
MARK(and(length(x0), y1)) → A__AND(a__length(mark(x0)), mark(y1))
MARK(and(tt, y1)) → A__AND(tt, mark(y1))
MARK(and(cons(x0, x1), y1)) → A__AND(cons(mark(x0), x1), mark(y1))
MARK(and(uTake1(x0), y1)) → A__AND(a__uTake1(mark(x0)), mark(y1))
MARK(and(0, y1)) → A__AND(0, mark(y1))
MARK(and(isNatList(x0), y1)) → A__AND(a__isNatList(x0), mark(y1))
MARK(and(zeros, y1)) → A__AND(a__zeros, mark(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), mark(x1)), mark(y1))
MARK(and(s(x0), y1)) → A__AND(s(mark(x0)), mark(y1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), mark(y1))
MARK(and(uTake2(x0, x1, x2, x3), y1)) → A__AND(a__uTake2(mark(x0), x1, x2, x3), mark(y1))
MARK(and(take(x0, x1), y1)) → A__AND(a__take(mark(x0), mark(x1)), mark(y1))
MARK(and(isNatIList(x0), y1)) → A__AND(a__isNatIList(x0), mark(y1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                          ↳ QDP

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

MARK(and(nil, y1)) → A__AND(nil, mark(y1))
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(and(uLength(x0, x1), y1)) → A__AND(a__uLength(mark(x0), x1), mark(y1))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(and(length(x0), y1)) → A__AND(a__length(mark(x0)), mark(y1))
MARK(and(tt, y1)) → A__AND(tt, mark(y1))
MARK(and(uTake1(x0), y1)) → A__AND(a__uTake1(mark(x0)), mark(y1))
A__AND(tt, T) → MARK(T)
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), mark(x1)), mark(y1))
A__ISNATLIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatList(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(uTake2(x0, x1, x2, x3), y1)) → A__AND(a__uTake2(mark(x0), x1, x2, x3), mark(y1))
MARK(and(isNatIList(x0), y1)) → A__AND(a__isNatIList(x0), mark(y1))
MARK(and(X1, X2)) → MARK(X2)
A__ISNATLIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatList(y1))
MARK(and(cons(x0, x1), y1)) → A__AND(cons(mark(x0), x1), mark(y1))
MARK(and(0, y1)) → A__AND(0, mark(y1))
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
A__ISNATLIST(cons(0, y1)) → A__AND(tt, a__isNatList(y1))
MARK(and(isNatList(x0), y1)) → A__AND(a__isNatList(x0), mark(y1))
MARK(and(zeros, y1)) → A__AND(a__zeros, mark(y1))
MARK(and(s(x0), y1)) → A__AND(s(mark(x0)), mark(y1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), mark(y1))
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(take(x0, x1), y1)) → A__AND(a__take(mark(x0), mark(x1)), mark(y1))

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                          ↳ QDP

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

A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(and(uLength(x0, x1), y1)) → A__AND(a__uLength(mark(x0), x1), mark(y1))
A__ISNATLIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatList(y1))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(and(tt, y1)) → A__AND(tt, mark(y1))
MARK(and(length(x0), y1)) → A__AND(a__length(mark(x0)), mark(y1))
MARK(and(uTake1(x0), y1)) → A__AND(a__uTake1(mark(x0)), mark(y1))
A__AND(tt, T) → MARK(T)
A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL))
MARK(and(isNatList(x0), y1)) → A__AND(a__isNatList(x0), mark(y1))
A__ISNATLIST(cons(0, y1)) → A__AND(tt, a__isNatList(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), mark(x1)), mark(y1))
MARK(and(zeros, y1)) → A__AND(a__zeros, mark(y1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), mark(y1))
A__ISNATLIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatList(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(uTake2(x0, x1, x2, x3), y1)) → A__AND(a__uTake2(mark(x0), x1, x2, x3), mark(y1))
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(isNatIList(x0), y1)) → A__AND(a__isNatIList(x0), mark(y1))
MARK(and(take(x0, x1), y1)) → A__AND(a__take(mark(x0), mark(x1)), mark(y1))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A__ISNATILIST(cons(N, IL)) → A__AND(a__isNat(N), a__isNatIList(IL)) at position [0] we obtained the following new rules:

A__ISNATILIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatIList(y1))
A__ISNATILIST(cons(x0, y1)) → A__AND(isNat(x0), a__isNatIList(y1))
A__ISNATILIST(cons(0, y1)) → A__AND(tt, a__isNatIList(y1))
A__ISNATILIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatIList(y1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                          ↳ QDP

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

A__ISNATILIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatIList(y1))
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(and(uLength(x0, x1), y1)) → A__AND(a__uLength(mark(x0), x1), mark(y1))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(and(tt, y1)) → A__AND(tt, mark(y1))
MARK(and(length(x0), y1)) → A__AND(a__length(mark(x0)), mark(y1))
A__ISNATILIST(cons(x0, y1)) → A__AND(isNat(x0), a__isNatIList(y1))
A__ISNATILIST(cons(0, y1)) → A__AND(tt, a__isNatIList(y1))
MARK(and(uTake1(x0), y1)) → A__AND(a__uTake1(mark(x0)), mark(y1))
A__AND(tt, T) → MARK(T)
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), mark(x1)), mark(y1))
A__ISNATLIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatList(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(uTake2(x0, x1, x2, x3), y1)) → A__AND(a__uTake2(mark(x0), x1, x2, x3), mark(y1))
MARK(and(isNatIList(x0), y1)) → A__AND(a__isNatIList(x0), mark(y1))
MARK(and(X1, X2)) → MARK(X2)
A__ISNATILIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatIList(y1))
A__ISNATLIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatList(y1))
MARK(and(isNatList(x0), y1)) → A__AND(a__isNatList(x0), mark(y1))
A__ISNATLIST(cons(0, y1)) → A__AND(tt, a__isNatList(y1))
MARK(and(zeros, y1)) → A__AND(a__zeros, mark(y1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), mark(y1))
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(take(x0, x1), y1)) → A__AND(a__take(mark(x0), mark(x1)), mark(y1))

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                          ↳ QDP

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

A__ISNATILIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatIList(y1))
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(and(uLength(x0, x1), y1)) → A__AND(a__uLength(mark(x0), x1), mark(y1))
A__ISNATILIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatIList(y1))
A__ISNATLIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatList(y1))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(and(tt, y1)) → A__AND(tt, mark(y1))
MARK(and(length(x0), y1)) → A__AND(a__length(mark(x0)), mark(y1))
A__ISNATILIST(cons(0, y1)) → A__AND(tt, a__isNatIList(y1))
MARK(and(uTake1(x0), y1)) → A__AND(a__uTake1(mark(x0)), mark(y1))
A__AND(tt, T) → MARK(T)
MARK(and(isNatList(x0), y1)) → A__AND(a__isNatList(x0), mark(y1))
A__ISNATLIST(cons(0, y1)) → A__AND(tt, a__isNatList(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), mark(x1)), mark(y1))
MARK(and(zeros, y1)) → A__AND(a__zeros, mark(y1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), mark(y1))
A__ISNATLIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatList(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(uTake2(x0, x1, x2, x3), y1)) → A__AND(a__uTake2(mark(x0), x1, x2, x3), mark(y1))
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(isNatIList(x0), y1)) → A__AND(a__isNatIList(x0), mark(y1))
MARK(and(take(x0, x1), y1)) → A__AND(a__take(mark(x0), mark(x1)), mark(y1))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(and(zeros, y1)) → A__AND(a__zeros, mark(y1)) at position [0] we obtained the following new rules:

MARK(and(zeros, y0)) → A__AND(zeros, mark(y0))
MARK(and(zeros, y0)) → A__AND(cons(0, zeros), mark(y0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                          ↳ QDP

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

A__ISNATILIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatIList(y1))
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(and(uLength(x0, x1), y1)) → A__AND(a__uLength(mark(x0), x1), mark(y1))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(and(length(x0), y1)) → A__AND(a__length(mark(x0)), mark(y1))
MARK(and(tt, y1)) → A__AND(tt, mark(y1))
A__ISNATILIST(cons(0, y1)) → A__AND(tt, a__isNatIList(y1))
MARK(and(uTake1(x0), y1)) → A__AND(a__uTake1(mark(x0)), mark(y1))
A__AND(tt, T) → MARK(T)
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), mark(x1)), mark(y1))
A__ISNATLIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatList(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(uTake2(x0, x1, x2, x3), y1)) → A__AND(a__uTake2(mark(x0), x1, x2, x3), mark(y1))
MARK(and(isNatIList(x0), y1)) → A__AND(a__isNatIList(x0), mark(y1))
MARK(and(X1, X2)) → MARK(X2)
MARK(and(zeros, y0)) → A__AND(cons(0, zeros), mark(y0))
A__ISNATILIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatIList(y1))
A__ISNATLIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatList(y1))
A__ISNATLIST(cons(0, y1)) → A__AND(tt, a__isNatList(y1))
MARK(and(isNatList(x0), y1)) → A__AND(a__isNatList(x0), mark(y1))
MARK(and(zeros, y0)) → A__AND(zeros, mark(y0))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), mark(y1))
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(take(x0, x1), y1)) → A__AND(a__take(mark(x0), mark(x1)), mark(y1))

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ QDPOrderProof

                          ↳ QDP

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

A__ISNATILIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatIList(y1))
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(and(uLength(x0, x1), y1)) → A__AND(a__uLength(mark(x0), x1), mark(y1))
A__ISNATILIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatIList(y1))
A__ISNATLIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatList(y1))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(and(tt, y1)) → A__AND(tt, mark(y1))
MARK(and(length(x0), y1)) → A__AND(a__length(mark(x0)), mark(y1))
A__ISNATILIST(cons(0, y1)) → A__AND(tt, a__isNatIList(y1))
MARK(and(uTake1(x0), y1)) → A__AND(a__uTake1(mark(x0)), mark(y1))
A__AND(tt, T) → MARK(T)
MARK(and(isNatList(x0), y1)) → A__AND(a__isNatList(x0), mark(y1))
A__ISNATLIST(cons(0, y1)) → A__AND(tt, a__isNatList(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), mark(x1)), mark(y1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), mark(y1))
A__ISNATLIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatList(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(uTake2(x0, x1, x2, x3), y1)) → A__AND(a__uTake2(mark(x0), x1, x2, x3), mark(y1))
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(isNatIList(x0), y1)) → A__AND(a__isNatIList(x0), mark(y1))
MARK(and(take(x0, x1), y1)) → A__AND(a__take(mark(x0), mark(x1)), mark(y1))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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(and(take(x0, x1), y1)) → A__AND(a__take(mark(x0), mark(x1)), mark(y1))

The remaining pairs can at least be oriented weakly.

A__ISNATILIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatIList(y1))
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(and(uLength(x0, x1), y1)) → A__AND(a__uLength(mark(x0), x1), mark(y1))
A__ISNATILIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatIList(y1))
A__ISNATLIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatList(y1))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(and(tt, y1)) → A__AND(tt, mark(y1))
MARK(and(length(x0), y1)) → A__AND(a__length(mark(x0)), mark(y1))
A__ISNATILIST(cons(0, y1)) → A__AND(tt, a__isNatIList(y1))
MARK(and(uTake1(x0), y1)) → A__AND(a__uTake1(mark(x0)), mark(y1))
A__AND(tt, T) → MARK(T)
MARK(and(isNatList(x0), y1)) → A__AND(a__isNatList(x0), mark(y1))
A__ISNATLIST(cons(0, y1)) → A__AND(tt, a__isNatList(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), mark(x1)), mark(y1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), mark(y1))
A__ISNATLIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatList(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(uTake2(x0, x1, x2, x3), y1)) → A__AND(a__uTake2(mark(x0), x1, x2, x3), mark(y1))
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(isNatIList(x0), y1)) → A__AND(a__isNatIList(x0), mark(y1))
MARK(and(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__AND(x₁, x₂)) = x₂
POL(A__ISNATILIST(x₁)) = 0
POL(A__ISNATLIST(x₁)) = 0
POL(MARK(x₁)) = x₁
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
POL(a__uLength(x₁, x₂)) = x₁
POL(a__uTake1(x₁)) = 0
POL(a__uTake2(x₁, x₂, x₃, x₄)) = x₁
POL(a__zeros) = 1
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 0
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)) = 0
POL(take(x₁, x₂)) = 1
POL(tt) = 0
POL(uLength(x₁, x₂)) = x₁
POL(uTake1(x₁)) = 0
POL(uTake2(x₁, x₂, x₃, x₄)) = x₁
POL(zeros) = 1

The following usable rules [17] were oriented:

mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__isNatIList(X) → isNatIList(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__zeros → zeros
a__length(X) → length(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zeros) → a__zeros
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uTake1(X)) → a__uTake1(mark(X))
mark(tt) → tt
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)
a__isNatIList(zeros) → tt
a__isNat(0) → tt
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake1(tt) → nil
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__isNatList(nil) → tt
mark(length(X)) → a__length(mark(X))
a__isNatIList(IL) → a__isNatList(IL)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(N)) → a__isNat(N)
a__and(tt, T) → mark(T)
mark(isNatList(X)) → a__isNatList(X)
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
mark(isNatIList(X)) → a__isNatIList(X)
a__isNat(length(L)) → a__isNatList(L)
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__uLength(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ QDPOrderProof

                                                                                        ↳ QDP

                                                                                          ↳ QDPOrderProof

                          ↳ QDP

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

A__ISNATILIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatIList(y1))
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(and(uLength(x0, x1), y1)) → A__AND(a__uLength(mark(x0), x1), mark(y1))
A__ISNATILIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatIList(y1))
A__ISNATLIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatList(y1))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(and(tt, y1)) → A__AND(tt, mark(y1))
MARK(and(length(x0), y1)) → A__AND(a__length(mark(x0)), mark(y1))
A__ISNATILIST(cons(0, y1)) → A__AND(tt, a__isNatIList(y1))
MARK(and(uTake1(x0), y1)) → A__AND(a__uTake1(mark(x0)), mark(y1))
A__AND(tt, T) → MARK(T)
A__ISNATLIST(cons(0, y1)) → A__AND(tt, a__isNatList(y1))
MARK(and(isNatList(x0), y1)) → A__AND(a__isNatList(x0), mark(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), mark(x1)), mark(y1))
A__ISNATLIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatList(y1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), mark(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(uTake2(x0, x1, x2, x3), y1)) → A__AND(a__uTake2(mark(x0), x1, x2, x3), mark(y1))
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(isNatIList(x0), y1)) → A__AND(a__isNatIList(x0), mark(y1))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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(and(length(x0), y1)) → A__AND(a__length(mark(x0)), mark(y1))

The remaining pairs can at least be oriented weakly.

A__ISNATILIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatIList(y1))
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(and(uLength(x0, x1), y1)) → A__AND(a__uLength(mark(x0), x1), mark(y1))
A__ISNATILIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatIList(y1))
A__ISNATLIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatList(y1))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(and(tt, y1)) → A__AND(tt, mark(y1))
A__ISNATILIST(cons(0, y1)) → A__AND(tt, a__isNatIList(y1))
MARK(and(uTake1(x0), y1)) → A__AND(a__uTake1(mark(x0)), mark(y1))
A__AND(tt, T) → MARK(T)
A__ISNATLIST(cons(0, y1)) → A__AND(tt, a__isNatList(y1))
MARK(and(isNatList(x0), y1)) → A__AND(a__isNatList(x0), mark(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), mark(x1)), mark(y1))
A__ISNATLIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatList(y1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), mark(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(uTake2(x0, x1, x2, x3), y1)) → A__AND(a__uTake2(mark(x0), x1, x2, x3), mark(y1))
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(isNatIList(x0), y1)) → A__AND(a__isNatIList(x0), mark(y1))
MARK(and(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__AND(x₁, x₂)) = x₂
POL(A__ISNATILIST(x₁)) = 0
POL(A__ISNATLIST(x₁)) = 0
POL(MARK(x₁)) = x₁
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
POL(a__take(x₁, x₂)) = 1
POL(a__uLength(x₁, x₂)) = 0
POL(a__uTake1(x₁)) = 0
POL(a__uTake2(x₁, x₂, x₃, x₄)) = 0
POL(a__zeros) = 1
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 1
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 1
POL(tt) = 0
POL(uLength(x₁, x₂)) = 0
POL(uTake1(x₁)) = 0
POL(uTake2(x₁, x₂, x₃, x₄)) = 0
POL(zeros) = 1

The following usable rules [17] were oriented:

mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__isNatIList(X) → isNatIList(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__zeros → zeros
a__length(X) → length(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zeros) → a__zeros
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uTake1(X)) → a__uTake1(mark(X))
mark(tt) → tt
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)
a__isNatIList(zeros) → tt
a__isNat(0) → tt
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake1(tt) → nil
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__isNatList(nil) → tt
mark(length(X)) → a__length(mark(X))
a__isNatIList(IL) → a__isNatList(IL)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(N)) → a__isNat(N)
a__and(tt, T) → mark(T)
mark(isNatList(X)) → a__isNatList(X)
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
mark(isNatIList(X)) → a__isNatIList(X)
a__isNat(length(L)) → a__isNatList(L)
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__uLength(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ QDPOrderProof

                                                                                        ↳ QDP

                                                                                          ↳ QDPOrderProof

                                                                                            ↳ QDP

                                                                                              ↳ QDPOrderProof

                          ↳ QDP

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

A__ISNATILIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatIList(y1))
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(and(uLength(x0, x1), y1)) → A__AND(a__uLength(mark(x0), x1), mark(y1))
A__ISNATILIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatIList(y1))
A__ISNATLIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatList(y1))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(and(tt, y1)) → A__AND(tt, mark(y1))
A__ISNATILIST(cons(0, y1)) → A__AND(tt, a__isNatIList(y1))
MARK(and(uTake1(x0), y1)) → A__AND(a__uTake1(mark(x0)), mark(y1))
A__AND(tt, T) → MARK(T)
A__ISNATLIST(cons(0, y1)) → A__AND(tt, a__isNatList(y1))
MARK(and(isNatList(x0), y1)) → A__AND(a__isNatList(x0), mark(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), mark(x1)), mark(y1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), mark(y1))
A__ISNATLIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatList(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(uTake2(x0, x1, x2, x3), y1)) → A__AND(a__uTake2(mark(x0), x1, x2, x3), mark(y1))
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(isNatIList(x0), y1)) → A__AND(a__isNatIList(x0), mark(y1))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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(and(uTake1(x0), y1)) → A__AND(a__uTake1(mark(x0)), mark(y1))

The remaining pairs can at least be oriented weakly.

A__ISNATILIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatIList(y1))
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(and(uLength(x0, x1), y1)) → A__AND(a__uLength(mark(x0), x1), mark(y1))
A__ISNATILIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatIList(y1))
A__ISNATLIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatList(y1))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(and(tt, y1)) → A__AND(tt, mark(y1))
A__ISNATILIST(cons(0, y1)) → A__AND(tt, a__isNatIList(y1))
A__AND(tt, T) → MARK(T)
A__ISNATLIST(cons(0, y1)) → A__AND(tt, a__isNatList(y1))
MARK(and(isNatList(x0), y1)) → A__AND(a__isNatList(x0), mark(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), mark(x1)), mark(y1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), mark(y1))
A__ISNATLIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatList(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(uTake2(x0, x1, x2, x3), y1)) → A__AND(a__uTake2(mark(x0), x1, x2, x3), mark(y1))
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(isNatIList(x0), y1)) → A__AND(a__isNatIList(x0), mark(y1))
MARK(and(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__AND(x₁, x₂)) = x₂
POL(A__ISNATILIST(x₁)) = 0
POL(A__ISNATLIST(x₁)) = 0
POL(MARK(x₁)) = x₁
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
POL(a__uLength(x₁, x₂)) = 0
POL(a__uTake1(x₁)) = 1
POL(a__uTake2(x₁, x₂, x₃, x₄)) = 0
POL(a__zeros) = 0
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 0
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)) = 0
POL(take(x₁, x₂)) = 1
POL(tt) = 0
POL(uLength(x₁, x₂)) = 0
POL(uTake1(x₁)) = 1
POL(uTake2(x₁, x₂, x₃, x₄)) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__isNatIList(X) → isNatIList(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__zeros → zeros
a__length(X) → length(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zeros) → a__zeros
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uTake1(X)) → a__uTake1(mark(X))
mark(tt) → tt
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)
a__isNatIList(zeros) → tt
a__isNat(0) → tt
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake1(tt) → nil
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__isNatList(nil) → tt
mark(length(X)) → a__length(mark(X))
a__isNatIList(IL) → a__isNatList(IL)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(N)) → a__isNat(N)
a__and(tt, T) → mark(T)
mark(isNatList(X)) → a__isNatList(X)
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
mark(isNatIList(X)) → a__isNatIList(X)
a__isNat(length(L)) → a__isNatList(L)
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__uLength(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ QDPOrderProof

                                                                                        ↳ QDP

                                                                                          ↳ QDPOrderProof

                                                                                            ↳ QDP

                                                                                              ↳ QDPOrderProof

                                                                                                ↳ QDP

                                                                                                  ↳ QDPOrderProof

                          ↳ QDP

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

A__ISNATILIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatIList(y1))
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(and(uLength(x0, x1), y1)) → A__AND(a__uLength(mark(x0), x1), mark(y1))
A__ISNATILIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatIList(y1))
A__ISNATLIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatList(y1))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(and(tt, y1)) → A__AND(tt, mark(y1))
A__ISNATILIST(cons(0, y1)) → A__AND(tt, a__isNatIList(y1))
A__AND(tt, T) → MARK(T)
A__ISNATLIST(cons(0, y1)) → A__AND(tt, a__isNatList(y1))
MARK(and(isNatList(x0), y1)) → A__AND(a__isNatList(x0), mark(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), mark(x1)), mark(y1))
A__ISNATLIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatList(y1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), mark(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(and(uTake2(x0, x1, x2, x3), y1)) → A__AND(a__uTake2(mark(x0), x1, x2, x3), mark(y1))
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(isNatIList(x0), y1)) → A__AND(a__isNatIList(x0), mark(y1))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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(and(uTake2(x0, x1, x2, x3), y1)) → A__AND(a__uTake2(mark(x0), x1, x2, x3), mark(y1))

The remaining pairs can at least be oriented weakly.

A__ISNATILIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatIList(y1))
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(and(uLength(x0, x1), y1)) → A__AND(a__uLength(mark(x0), x1), mark(y1))
A__ISNATILIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatIList(y1))
A__ISNATLIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatList(y1))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(and(tt, y1)) → A__AND(tt, mark(y1))
A__ISNATILIST(cons(0, y1)) → A__AND(tt, a__isNatIList(y1))
A__AND(tt, T) → MARK(T)
A__ISNATLIST(cons(0, y1)) → A__AND(tt, a__isNatList(y1))
MARK(and(isNatList(x0), y1)) → A__AND(a__isNatList(x0), mark(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), mark(x1)), mark(y1))
A__ISNATLIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatList(y1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), mark(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(isNatIList(x0), y1)) → A__AND(a__isNatIList(x0), mark(y1))
MARK(and(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__AND(x₁, x₂)) = x₂
POL(A__ISNATILIST(x₁)) = 0
POL(A__ISNATLIST(x₁)) = 0
POL(MARK(x₁)) = x₁
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
POL(a__uLength(x₁, x₂)) = 0
POL(a__uTake1(x₁)) = 1
POL(a__uTake2(x₁, x₂, x₃, x₄)) = 1 + x₁
POL(a__zeros) = 0
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 0
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)) = 0
POL(take(x₁, x₂)) = 1 + x₁
POL(tt) = 0
POL(uLength(x₁, x₂)) = 0
POL(uTake1(x₁)) = 1
POL(uTake2(x₁, x₂, x₃, x₄)) = 1 + x₁
POL(zeros) = 0

The following usable rules [17] were oriented:

mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__isNatIList(X) → isNatIList(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__zeros → zeros
a__length(X) → length(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zeros) → a__zeros
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uTake1(X)) → a__uTake1(mark(X))
mark(tt) → tt
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)
a__isNatIList(zeros) → tt
a__isNat(0) → tt
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake1(tt) → nil
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__isNatList(nil) → tt
mark(length(X)) → a__length(mark(X))
a__isNatIList(IL) → a__isNatList(IL)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(N)) → a__isNat(N)
a__and(tt, T) → mark(T)
mark(isNatList(X)) → a__isNatList(X)
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
mark(isNatIList(X)) → a__isNatIList(X)
a__isNat(length(L)) → a__isNatList(L)
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__uLength(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ QDPOrderProof

                                                                                        ↳ QDP

                                                                                          ↳ QDPOrderProof

                                                                                            ↳ QDP

                                                                                              ↳ QDPOrderProof

                                                                                                ↳ QDP

                                                                                                  ↳ QDPOrderProof

                                                                                                    ↳ QDP

                                                                                                      ↳ QDPOrderProof

                          ↳ QDP

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

A__ISNATILIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatIList(y1))
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(and(uLength(x0, x1), y1)) → A__AND(a__uLength(mark(x0), x1), mark(y1))
A__ISNATILIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatIList(y1))
A__ISNATLIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatList(y1))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(and(tt, y1)) → A__AND(tt, mark(y1))
A__ISNATILIST(cons(0, y1)) → A__AND(tt, a__isNatIList(y1))
A__AND(tt, T) → MARK(T)
A__ISNATLIST(cons(0, y1)) → A__AND(tt, a__isNatList(y1))
MARK(and(isNatList(x0), y1)) → A__AND(a__isNatList(x0), mark(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), mark(x1)), mark(y1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), mark(y1))
A__ISNATLIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatList(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(isNatIList(x0), y1)) → A__AND(a__isNatIList(x0), mark(y1))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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(and(uLength(x0, x1), y1)) → A__AND(a__uLength(mark(x0), x1), mark(y1))

The remaining pairs can at least be oriented weakly.

A__ISNATILIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatIList(y1))
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
A__ISNATILIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatIList(y1))
A__ISNATLIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatList(y1))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(and(tt, y1)) → A__AND(tt, mark(y1))
A__ISNATILIST(cons(0, y1)) → A__AND(tt, a__isNatIList(y1))
A__AND(tt, T) → MARK(T)
A__ISNATLIST(cons(0, y1)) → A__AND(tt, a__isNatList(y1))
MARK(and(isNatList(x0), y1)) → A__AND(a__isNatList(x0), mark(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), mark(x1)), mark(y1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), mark(y1))
A__ISNATLIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatList(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(isNatIList(x0), y1)) → A__AND(a__isNatIList(x0), mark(y1))
MARK(and(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__AND(x₁, x₂)) = x₂
POL(A__ISNATILIST(x₁)) = 0
POL(A__ISNATLIST(x₁)) = 0
POL(MARK(x₁)) = x₁
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
POL(a__take(x₁, x₂)) = x₁ + x₂
POL(a__uLength(x₁, x₂)) = 1 + x₁
POL(a__uTake1(x₁)) = x₁
POL(a__uTake2(x₁, x₂, x₃, x₄)) = 0
POL(a__zeros) = 0
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 1
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = x₁ + x₂
POL(tt) = 0
POL(uLength(x₁, x₂)) = 1 + x₁
POL(uTake1(x₁)) = x₁
POL(uTake2(x₁, x₂, x₃, x₄)) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__isNatIList(X) → isNatIList(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__zeros → zeros
a__length(X) → length(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zeros) → a__zeros
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uTake1(X)) → a__uTake1(mark(X))
mark(tt) → tt
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)
a__isNatIList(zeros) → tt
a__isNat(0) → tt
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake1(tt) → nil
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__isNatList(nil) → tt
mark(length(X)) → a__length(mark(X))
a__isNatIList(IL) → a__isNatList(IL)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(N)) → a__isNat(N)
a__and(tt, T) → mark(T)
mark(isNatList(X)) → a__isNatList(X)
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
mark(isNatIList(X)) → a__isNatIList(X)
a__isNat(length(L)) → a__isNatList(L)
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__uLength(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ QDPOrderProof

                                                                                        ↳ QDP

                                                                                          ↳ QDPOrderProof

                                                                                            ↳ QDP

                                                                                              ↳ QDPOrderProof

                                                                                                ↳ QDP

                                                                                                  ↳ QDPOrderProof

                                                                                                    ↳ QDP

                                                                                                      ↳ QDPOrderProof

                                                                                                        ↳ QDP

                                                                                                          ↳ QDPOrderProof

                          ↳ QDP

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

A__ISNATILIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatIList(y1))
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
A__ISNATILIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatIList(y1))
A__ISNATLIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatList(y1))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(and(tt, y1)) → A__AND(tt, mark(y1))
A__ISNATILIST(cons(0, y1)) → A__AND(tt, a__isNatIList(y1))
A__AND(tt, T) → MARK(T)
A__ISNATLIST(cons(0, y1)) → A__AND(tt, a__isNatList(y1))
MARK(and(isNatList(x0), y1)) → A__AND(a__isNatList(x0), mark(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), mark(x1)), mark(y1))
A__ISNATLIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatList(y1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), mark(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(isNatIList(x0), y1)) → A__AND(a__isNatIList(x0), mark(y1))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatIList(y1))
A__ISNATLIST(cons(length(x0), y1)) → A__AND(a__isNatList(x0), a__isNatList(y1))

The remaining pairs can at least be oriented weakly.

A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
A__ISNATILIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatIList(y1))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(and(tt, y1)) → A__AND(tt, mark(y1))
A__ISNATILIST(cons(0, y1)) → A__AND(tt, a__isNatIList(y1))
A__AND(tt, T) → MARK(T)
A__ISNATLIST(cons(0, y1)) → A__AND(tt, a__isNatList(y1))
MARK(and(isNatList(x0), y1)) → A__AND(a__isNatList(x0), mark(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), mark(x1)), mark(y1))
A__ISNATLIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatList(y1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), mark(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(isNatIList(x0), y1)) → A__AND(a__isNatIList(x0), mark(y1))
MARK(and(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__AND(x₁, x₂)) = x₂
POL(A__ISNATILIST(x₁)) = x₁
POL(A__ISNATLIST(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isNat(x₁)) = x₁
POL(a__isNatIList(x₁)) = x₁
POL(a__isNatList(x₁)) = x₁
POL(a__length(x₁)) = 1 + x₁
POL(a__take(x₁, x₂)) = 1 + x₁ + x₂
POL(a__uLength(x₁, x₂)) = 1 + x₂
POL(a__uTake1(x₁)) = 1
POL(a__uTake2(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₃ + x₄
POL(a__zeros) = 0
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 1 + x₁
POL(mark(x₁)) = x₁
POL(nil) = 1
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 1 + x₁ + x₂
POL(tt) = 0
POL(uLength(x₁, x₂)) = 1 + x₂
POL(uTake1(x₁)) = 1
POL(uTake2(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₃ + x₄
POL(zeros) = 0

The following usable rules [17] were oriented:

mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__isNatIList(X) → isNatIList(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__zeros → zeros
a__length(X) → length(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zeros) → a__zeros
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uTake1(X)) → a__uTake1(mark(X))
mark(tt) → tt
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)
a__isNatIList(zeros) → tt
a__isNat(0) → tt
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake1(tt) → nil
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__isNatList(nil) → tt
mark(length(X)) → a__length(mark(X))
a__isNatIList(IL) → a__isNatList(IL)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(N)) → a__isNat(N)
a__and(tt, T) → mark(T)
mark(isNatList(X)) → a__isNatList(X)
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
mark(isNatIList(X)) → a__isNatIList(X)
a__isNat(length(L)) → a__isNatList(L)
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__uLength(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ QDPOrderProof

                                                                                        ↳ QDP

                                                                                          ↳ QDPOrderProof

                                                                                            ↳ QDP

                                                                                              ↳ QDPOrderProof

                                                                                                ↳ QDP

                                                                                                  ↳ QDPOrderProof

                                                                                                    ↳ QDP

                                                                                                      ↳ QDPOrderProof

                                                                                                        ↳ QDP

                                                                                                          ↳ QDPOrderProof

                                                                                                            ↳ QDP

                                                                                                              ↳ QDPOrderProof

                          ↳ QDP

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

A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
A__ISNATILIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatIList(y1))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(and(tt, y1)) → A__AND(tt, mark(y1))
A__ISNATILIST(cons(0, y1)) → A__AND(tt, a__isNatIList(y1))
A__AND(tt, T) → MARK(T)
A__ISNATLIST(cons(0, y1)) → A__AND(tt, a__isNatList(y1))
MARK(and(isNatList(x0), y1)) → A__AND(a__isNatList(x0), mark(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), mark(x1)), mark(y1))
A__ISNATLIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatList(y1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), mark(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(isNatIList(x0), y1)) → A__AND(a__isNatIList(x0), mark(y1))
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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(and(isNatList(x0), y1)) → A__AND(a__isNatList(x0), mark(y1))
MARK(and(isNatIList(x0), y1)) → A__AND(a__isNatIList(x0), mark(y1))

The remaining pairs can at least be oriented weakly.

A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
A__ISNATILIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatIList(y1))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(and(tt, y1)) → A__AND(tt, mark(y1))
A__ISNATILIST(cons(0, y1)) → A__AND(tt, a__isNatIList(y1))
A__AND(tt, T) → MARK(T)
A__ISNATLIST(cons(0, y1)) → A__AND(tt, a__isNatList(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), mark(x1)), mark(y1))
A__ISNATLIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatList(y1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), mark(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__AND(x₁, x₂)) = x₂
POL(A__ISNATILIST(x₁)) = 1 + x₁
POL(A__ISNATLIST(x₁)) = 1 + x₁
POL(MARK(x₁)) = x₁
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isNat(x₁)) = x₁
POL(a__isNatIList(x₁)) = 1 + x₁
POL(a__isNatList(x₁)) = 1 + x₁
POL(a__length(x₁)) = 1 + x₁
POL(a__take(x₁, x₂)) = x₁ + x₂
POL(a__uLength(x₁, x₂)) = 1 + x₂
POL(a__uTake1(x₁)) = 0
POL(a__uTake2(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄
POL(a__zeros) = 0
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 1 + x₁
POL(isNatList(x₁)) = 1 + x₁
POL(length(x₁)) = 1 + x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = x₁ + x₂
POL(tt) = 0
POL(uLength(x₁, x₂)) = 1 + x₂
POL(uTake1(x₁)) = 0
POL(uTake2(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄
POL(zeros) = 0

The following usable rules [17] were oriented:

mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__isNatIList(X) → isNatIList(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__zeros → zeros
a__length(X) → length(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zeros) → a__zeros
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uTake1(X)) → a__uTake1(mark(X))
mark(tt) → tt
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)
a__isNatIList(zeros) → tt
a__isNat(0) → tt
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake1(tt) → nil
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__isNatList(nil) → tt
mark(length(X)) → a__length(mark(X))
a__isNatIList(IL) → a__isNatList(IL)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(N)) → a__isNat(N)
a__and(tt, T) → mark(T)
mark(isNatList(X)) → a__isNatList(X)
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
mark(isNatIList(X)) → a__isNatIList(X)
a__isNat(length(L)) → a__isNatList(L)
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__uLength(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ QDPOrderProof

                                                                                        ↳ QDP

                                                                                          ↳ QDPOrderProof

                                                                                            ↳ QDP

                                                                                              ↳ QDPOrderProof

                                                                                                ↳ QDP

                                                                                                  ↳ QDPOrderProof

                                                                                                    ↳ QDP

                                                                                                      ↳ QDPOrderProof

                                                                                                        ↳ QDP

                                                                                                          ↳ QDPOrderProof

                                                                                                            ↳ QDP

                                                                                                              ↳ QDPOrderProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ QDPOrderProof

                          ↳ QDP

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

A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
A__ISNATILIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatIList(y1))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(and(tt, y1)) → A__AND(tt, mark(y1))
A__ISNATILIST(cons(0, y1)) → A__AND(tt, a__isNatIList(y1))
A__AND(tt, T) → MARK(T)
A__ISNATLIST(cons(0, y1)) → A__AND(tt, a__isNatList(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), mark(x1)), mark(y1))
A__ISNATLIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatList(y1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), mark(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatIList(y1))
A__ISNATLIST(cons(s(x0), y1)) → A__AND(a__isNat(x0), a__isNatList(y1))

The remaining pairs can at least be oriented weakly.

A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(and(tt, y1)) → A__AND(tt, mark(y1))
A__ISNATILIST(cons(0, y1)) → A__AND(tt, a__isNatIList(y1))
A__AND(tt, T) → MARK(T)
A__ISNATLIST(cons(0, y1)) → A__AND(tt, a__isNatList(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), mark(x1)), mark(y1))
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), mark(y1))
MARK(and(X1, X2)) → MARK(X1)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(and(X1, X2)) → MARK(X2)

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

M( a__zeros ) =

/	0	\
\	0	/

M( mark(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( a__isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	1	0	\
\	0	1	/

x₂

M( a__uTake1(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

M( a__uLength(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	0	/

x₂

M( a__length(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( a__isNat(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	1	/

x₁

M( s(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	1	/

x₁

M( a__isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( nil ) =

/	0	\
\	0	/

M( a__uTake2(x₁, ..., x₄) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	1	\
\	0	0	/

x₂

/	1	1	\
\	0	0	/

x₃

/	1	0	\
\	0	0	/

x₄

M( a__and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	1	0	\
\	0	1	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( a__take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	1	1	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

M( uTake2(x₁, ..., x₄) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	1	\
\	0	0	/

x₂

/	1	1	\
\	0	0	/

x₃

/	1	0	\
\	0	0	/

x₄

M( uTake1(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( uLength(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	0	/

x₂

Tuple symbols:

M( A__AND(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( MARK(x₁) ) =

[

]

x₁

M( A__ISNATILIST(x₁) ) =

[

]

x₁

M( A__ISNATLIST(x₁) ) =

[

]

x₁

mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__isNatIList(X) → isNatIList(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__zeros → zeros
a__length(X) → length(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(zeros) → a__zeros
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uTake1(X)) → a__uTake1(mark(X))
mark(tt) → tt
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)
a__isNatIList(zeros) → tt
a__isNat(0) → tt
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake1(tt) → nil
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__isNatList(nil) → tt
mark(length(X)) → a__length(mark(X))
a__isNatIList(IL) → a__isNatList(IL)
mark(isNat(X)) → a__isNat(X)
a__isNat(s(N)) → a__isNat(N)
a__and(tt, T) → mark(T)
mark(isNatList(X)) → a__isNatList(X)
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
mark(isNatIList(X)) → a__isNatIList(X)
a__isNat(length(L)) → a__isNatList(L)
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__uLength(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ QDPOrderProof

                                                                                        ↳ QDP

                                                                                          ↳ QDPOrderProof

                                                                                            ↳ QDP

                                                                                              ↳ QDPOrderProof

                                                                                                ↳ QDP

                                                                                                  ↳ QDPOrderProof

                                                                                                    ↳ QDP

                                                                                                      ↳ QDPOrderProof

                                                                                                        ↳ QDP

                                                                                                          ↳ QDPOrderProof

                                                                                                            ↳ QDP

                                                                                                              ↳ QDPOrderProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ QDPOrderProof

                                                                                                                    ↳ QDP

                          ↳ QDP

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

A__ISNATLIST(cons(0, y1)) → A__AND(tt, a__isNatList(y1))
MARK(and(and(x0, x1), y1)) → A__AND(a__and(mark(x0), mark(x1)), mark(y1))
A__ISNATLIST(cons(N, L)) → A__ISNATLIST(L)
MARK(and(isNat(x0), y1)) → A__AND(a__isNat(x0), mark(y1))
MARK(and(X1, X2)) → MARK(X1)
A__ISNATILIST(cons(N, IL)) → A__ISNATILIST(IL)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(and(tt, y1)) → A__AND(tt, mark(y1))
MARK(isNatList(X)) → A__ISNATLIST(X)
A__ISNATILIST(cons(0, y1)) → A__AND(tt, a__isNatIList(y1))
MARK(and(X1, X2)) → MARK(X2)
A__AND(tt, T) → MARK(T)

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

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

A__LENGTH(cons(N, L)) → A__ULENGTH(a__and(a__isNat(N), a__isNatList(L)), L)
A__ULENGTH(tt, L) → A__LENGTH(mark(L))

The TRS R consists of the following rules:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

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