Termination w.r.t. Q proof of /tmp/tmpmVOtAy/Ex4_DLMMU04

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:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__length(X)) → length(activate(X))
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__length(X)) → length(activate(X))
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X

Q is empty.

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

ISNATILIST(IL) → ACTIVATE(IL)
ISNAT(n__length(L)) → ISNATLIST(activate(L))
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(IL)
UTAKE1(tt) → NIL
ISNATLIST(n__take(N, IL)) → AND(isNat(activate(N)), isNatIList(activate(IL)))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATLIST(n__take(N, IL)) → ACTIVATE(IL)
ULENGTH(tt, L) → ACTIVATE(L)
ISNATILIST(IL) → ISNATLIST(activate(IL))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(N)
ACTIVATE(n__0) → 0¹
ULENGTH(tt, L) → S(length(activate(L)))
ULENGTH(tt, L) → LENGTH(activate(L))
ISNATLIST(n__take(N, IL)) → ISNATILIST(activate(IL))
LENGTH(cons(N, L)) → ISNAT(N)
ACTIVATE(n__zeros) → ZEROS
UTAKE2(tt, M, N, IL) → ACTIVATE(M)
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__nil) → NIL
TAKE(0, IL) → ISNATILIST(IL)
ACTIVATE(n__s(X)) → S(activate(X))
ACTIVATE(n__cons(X1, X2)) → CONS(activate(X1), X2)
ISNAT(n__s(N)) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
UTAKE2(tt, M, N, IL) → CONS(activate(N), n__take(activate(M), activate(IL)))
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
ISNATLIST(n__cons(N, L)) → AND(isNat(activate(N)), isNatList(activate(L)))
TAKE(s(M), cons(N, IL)) → AND(isNat(N), isNatIList(activate(IL)))
UTAKE2(tt, M, N, IL) → ACTIVATE(IL)
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
ISNATLIST(n__take(N, IL)) → ACTIVATE(N)
ISNATILIST(n__cons(N, IL)) → ISNAT(activate(N))
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
UTAKE2(tt, M, N, IL) → ACTIVATE(N)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNATLIST(n__take(N, IL)) → ISNAT(activate(N))
TAKE(s(M), cons(N, IL)) → ISNAT(N)
TAKE(s(M), cons(N, IL)) → AND(isNat(M), and(isNat(N), isNatIList(activate(IL))))
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
TAKE(0, IL) → UTAKE1(isNatIList(IL))
LENGTH(cons(N, L)) → AND(isNat(N), isNatList(activate(L)))
LENGTH(cons(N, L)) → ACTIVATE(L)
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ISNAT(n__s(N)) → ISNAT(activate(N))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ISNATILIST(n__cons(N, IL)) → AND(isNat(activate(N)), isNatIList(activate(IL)))
ZEROS → CONS(0, n__zeros)
ZEROS → 0¹
ISNAT(n__length(L)) → ACTIVATE(L)
ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__length(X)) → length(activate(X))
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

ISNATILIST(IL) → ACTIVATE(IL)
ISNAT(n__length(L)) → ISNATLIST(activate(L))
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(IL)
UTAKE1(tt) → NIL
ISNATLIST(n__take(N, IL)) → AND(isNat(activate(N)), isNatIList(activate(IL)))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATLIST(n__take(N, IL)) → ACTIVATE(IL)
ULENGTH(tt, L) → ACTIVATE(L)
ISNATILIST(IL) → ISNATLIST(activate(IL))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(N)
ACTIVATE(n__0) → 0¹
ULENGTH(tt, L) → S(length(activate(L)))
ULENGTH(tt, L) → LENGTH(activate(L))
ISNATLIST(n__take(N, IL)) → ISNATILIST(activate(IL))
LENGTH(cons(N, L)) → ISNAT(N)
ACTIVATE(n__zeros) → ZEROS
UTAKE2(tt, M, N, IL) → ACTIVATE(M)
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__nil) → NIL
TAKE(0, IL) → ISNATILIST(IL)
ACTIVATE(n__s(X)) → S(activate(X))
ACTIVATE(n__cons(X1, X2)) → CONS(activate(X1), X2)
ISNAT(n__s(N)) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
UTAKE2(tt, M, N, IL) → CONS(activate(N), n__take(activate(M), activate(IL)))
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
ISNATLIST(n__cons(N, L)) → AND(isNat(activate(N)), isNatList(activate(L)))
TAKE(s(M), cons(N, IL)) → AND(isNat(N), isNatIList(activate(IL)))
UTAKE2(tt, M, N, IL) → ACTIVATE(IL)
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
ISNATLIST(n__take(N, IL)) → ACTIVATE(N)
ISNATILIST(n__cons(N, IL)) → ISNAT(activate(N))
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
UTAKE2(tt, M, N, IL) → ACTIVATE(N)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNATLIST(n__take(N, IL)) → ISNAT(activate(N))
TAKE(s(M), cons(N, IL)) → ISNAT(N)
TAKE(s(M), cons(N, IL)) → AND(isNat(M), and(isNat(N), isNatIList(activate(IL))))
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
TAKE(0, IL) → UTAKE1(isNatIList(IL))
LENGTH(cons(N, L)) → AND(isNat(N), isNatList(activate(L)))
LENGTH(cons(N, L)) → ACTIVATE(L)
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ISNAT(n__s(N)) → ISNAT(activate(N))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ISNATILIST(n__cons(N, IL)) → AND(isNat(activate(N)), isNatIList(activate(IL)))
ZEROS → CONS(0, n__zeros)
ZEROS → 0¹
ISNAT(n__length(L)) → ACTIVATE(L)
ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__length(X)) → length(activate(X))
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

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

ISNATLIST(n__take(N, IL)) → ACTIVATE(N)
ISNATILIST(n__cons(N, IL)) → ISNAT(activate(N))
ISNATILIST(IL) → ACTIVATE(IL)
ISNAT(n__length(L)) → ISNATLIST(activate(L))
UTAKE2(tt, M, N, IL) → ACTIVATE(N)
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(IL)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATLIST(n__take(N, IL)) → ACTIVATE(IL)
ULENGTH(tt, L) → ACTIVATE(L)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATILIST(n__cons(N, IL)) → ACTIVATE(N)
ISNATILIST(IL) → ISNATLIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ULENGTH(tt, L) → LENGTH(activate(L))
ISNATLIST(n__take(N, IL)) → ISNAT(activate(N))
TAKE(s(M), cons(N, IL)) → ISNAT(N)
ISNATLIST(n__take(N, IL)) → ISNATILIST(activate(IL))
LENGTH(cons(N, L)) → ISNAT(N)
UTAKE2(tt, M, N, IL) → ACTIVATE(M)
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
LENGTH(cons(N, L)) → ACTIVATE(L)
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ISNAT(n__s(N)) → ISNAT(activate(N))
TAKE(0, IL) → ISNATILIST(IL)
ISNAT(n__s(N)) → ACTIVATE(N)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
UTAKE2(tt, M, N, IL) → ACTIVATE(IL)
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))
ISNAT(n__length(L)) → ACTIVATE(L)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__length(X)) → length(activate(X))
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X

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

The following pairs can be oriented strictly and are deleted.

ISNAT(n__length(L)) → ISNATLIST(activate(L))
ULENGTH(tt, L) → ACTIVATE(L)
LENGTH(cons(N, L)) → ISNAT(N)
LENGTH(cons(N, L)) → ACTIVATE(L)
ACTIVATE(n__length(X)) → LENGTH(activate(X))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNAT(n__length(L)) → ACTIVATE(L)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The remaining pairs can at least be oriented weakly.

ISNATLIST(n__take(N, IL)) → ACTIVATE(N)
ISNATILIST(n__cons(N, IL)) → ISNAT(activate(N))
ISNATILIST(IL) → ACTIVATE(IL)
UTAKE2(tt, M, N, IL) → ACTIVATE(N)
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(IL)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATLIST(n__take(N, IL)) → ACTIVATE(IL)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATILIST(n__cons(N, IL)) → ACTIVATE(N)
ISNATILIST(IL) → ISNATLIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ULENGTH(tt, L) → LENGTH(activate(L))
ISNATLIST(n__take(N, IL)) → ISNAT(activate(N))
TAKE(s(M), cons(N, IL)) → ISNAT(N)
ISNATLIST(n__take(N, IL)) → ISNATILIST(activate(IL))
UTAKE2(tt, M, N, IL) → ACTIVATE(M)
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ISNAT(n__s(N)) → ISNAT(activate(N))
TAKE(0, IL) → ISNATILIST(IL)
ISNAT(n__s(N)) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
UTAKE2(tt, M, N, IL) → ACTIVATE(IL)
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))

Used ordering: Polynomial interpretation [25,35]:

POL(UTAKE2(x₁, x₂, x₃, x₄)) = (9/4)x_2 + (13/4)x_3 + (4)x_4
POL(ISNATILIST(x₁)) = (5/2)x_1
POL(activate(x₁)) = x_1
POL(and(x₁, x₂)) = (2)x_2
POL(n__s(x₁)) = x_1
POL(n__nil) = 0
POL(take(x₁, x₂)) = (15/4)x_1 + (4)x_2
POL(ISNATLIST(x₁)) = (9/4)x_1
POL(TAKE(x₁, x₂)) = (4)x_1 + (4)x_2
POL(tt) = 0
POL(isNatList(x₁)) = 0
POL(zeros) = 0
POL(ULENGTH(x₁, x₂)) = 4 + (3)x_2
POL(isNatIList(x₁)) = 0
POL(s(x₁)) = x_1
POL(isNat(x₁)) = 0
POL(nil) = 0
POL(ACTIVATE(x₁)) = (9/4)x_1
POL(LENGTH(x₁)) = 4 + (3)x_1
POL(n__length(x₁)) = 15/4 + (9/4)x_1
POL(n__zeros) = 0
POL(n__cons(x₁, x₂)) = (2)x_1 + x_2
POL(0) = 0
POL(ISNAT(x₁)) = (4)x_1
POL(cons(x₁, x₂)) = (2)x_1 + x_2
POL(uTake2(x₁, x₂, x₃, x₄)) = (4)x_1 + (15/4)x_2 + (4)x_3 + (4)x_4
POL(n__0) = 0
POL(uTake1(x₁)) = 0
POL(n__take(x₁, x₂)) = (15/4)x_1 + (4)x_2
POL(length(x₁)) = 15/4 + (9/4)x_1
POL(uLength(x₁, x₂)) = 15/4 + (9/4)x_2

The value of delta used in the strict ordering is 4.
The following usable rules [17] were oriented:

zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
activate(n__zeros) → zeros
activate(n__length(X)) → length(activate(X))
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
take(X1, X2) → n__take(X1, X2)
nil → n__nil
activate(n__s(X)) → s(activate(X))
activate(n__0) → 0
activate(X) → X
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
isNatIList(IL) → isNatList(activate(IL))
and(tt, T) → T
isNatIList(n__zeros) → tt
isNat(n__length(L)) → isNatList(activate(L))
isNat(n__s(N)) → isNat(activate(N))
isNat(n__0) → tt

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

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

ISNATLIST(n__take(N, IL)) → ACTIVATE(N)
ISNATILIST(n__cons(N, IL)) → ISNAT(activate(N))
ISNATILIST(IL) → ACTIVATE(IL)
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
UTAKE2(tt, M, N, IL) → ACTIVATE(N)
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(IL)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATLIST(n__take(N, IL)) → ACTIVATE(IL)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATILIST(IL) → ISNATLIST(activate(IL))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ULENGTH(tt, L) → LENGTH(activate(L))
ISNATLIST(n__take(N, IL)) → ISNAT(activate(N))
TAKE(s(M), cons(N, IL)) → ISNAT(N)
ISNATLIST(n__take(N, IL)) → ISNATILIST(activate(IL))
UTAKE2(tt, M, N, IL) → ACTIVATE(M)
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ISNAT(n__s(N)) → ISNAT(activate(N))
TAKE(0, IL) → ISNATILIST(IL)
ISNAT(n__s(N)) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
UTAKE2(tt, M, N, IL) → ACTIVATE(IL)
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__length(X)) → length(activate(X))
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

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

LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
ULENGTH(tt, L) → LENGTH(activate(L))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__length(X)) → length(activate(X))
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

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

ISNATLIST(n__take(N, IL)) → ACTIVATE(N)
ISNATILIST(n__cons(N, IL)) → ISNAT(activate(N))
ISNATILIST(IL) → ACTIVATE(IL)
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
UTAKE2(tt, M, N, IL) → ACTIVATE(N)
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(IL)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATLIST(n__take(N, IL)) → ACTIVATE(IL)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATILIST(IL) → ISNATLIST(activate(IL))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNATLIST(n__take(N, IL)) → ISNAT(activate(N))
TAKE(s(M), cons(N, IL)) → ISNAT(N)
ISNATLIST(n__take(N, IL)) → ISNATILIST(activate(IL))
UTAKE2(tt, M, N, IL) → ACTIVATE(M)
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ISNAT(n__s(N)) → ISNAT(activate(N))
TAKE(0, IL) → ISNATILIST(IL)
ISNAT(n__s(N)) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
UTAKE2(tt, M, N, IL) → ACTIVATE(IL)
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))

The TRS R consists of the following rules:

and(tt, T) → T
isNatIList(IL) → isNatList(activate(IL))
isNat(n__0) → tt
isNat(n__s(N)) → isNat(activate(N))
isNat(n__length(L)) → isNatList(activate(L))
isNatIList(n__zeros) → tt
isNatIList(n__cons(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
isNatList(n__nil) → tt
isNatList(n__cons(N, L)) → and(isNat(activate(N)), isNatList(activate(L)))
isNatList(n__take(N, IL)) → and(isNat(activate(N)), isNatIList(activate(IL)))
zeros → cons(0, n__zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(activate(IL)))), M, N, activate(IL))
uTake2(tt, M, N, IL) → cons(activate(N), n__take(activate(M), activate(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(activate(L))), activate(L))
uLength(tt, L) → s(length(activate(L)))
0 → n__0
s(X) → n__s(X)
length(X) → n__length(X)
zeros → n__zeros
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
take(X1, X2) → n__take(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__length(X)) → length(activate(X))
activate(n__zeros) → zeros
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X

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