Termination w.r.t. Q of the following Term Rewriting System could be disproven:

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.


ISNATLIST(n__take(N, IL)) → ACTIVATE(N)
ISNATILIST(IL) → ACTIVATE(IL)
ISNAT(n__length(L)) → ISNATLIST(activate(L))
ISNATILIST(n__cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__take(N, IL)) → ACTIVATE(IL)
ULENGTH(tt, L) → ACTIVATE(L)
ISNATILIST(n__cons(N, IL)) → ACTIVATE(N)
ISNATLIST(n__take(N, IL)) → ISNAT(activate(N))
ISNATLIST(n__take(N, IL)) → ISNATILIST(activate(IL))
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__take(X1, X2)) → ACTIVATE(X1)
ISNAT(n__s(N)) → ACTIVATE(N)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
ISNAT(n__length(L)) → ACTIVATE(L)
ACTIVATE(n__length(X)) → ACTIVATE(X)
The remaining pairs can at least be oriented weakly.

ISNATILIST(n__cons(N, IL)) → ISNAT(activate(N))
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))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATILIST(IL) → ISNATLIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ULENGTH(tt, L) → LENGTH(activate(L))
TAKE(s(M), cons(N, IL)) → ISNAT(N)
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))
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ISNAT(n__s(N)) → ISNAT(activate(N))
TAKE(0, IL) → ISNATILIST(IL)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
UTAKE2(tt, M, N, IL) → ACTIVATE(IL)
ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVATE(x1)) = x1   
POL(ISNAT(x1)) = 1 + x1   
POL(ISNATILIST(x1)) = 1 + x1   
POL(ISNATLIST(x1)) = 1 + x1   
POL(LENGTH(x1)) = 1 + x1   
POL(TAKE(x1, x2)) = 1 + x1 + x2   
POL(ULENGTH(x1, x2)) = 1 + x2   
POL(UTAKE2(x1, x2, x3, x4)) = x2 + x3 + x4   
POL(activate(x1)) = x1   
POL(and(x1, x2)) = x2   
POL(cons(x1, x2)) = x1 + x2   
POL(isNat(x1)) = 0   
POL(isNatIList(x1)) = 0   
POL(isNatList(x1)) = 0   
POL(length(x1)) = 1 + x1   
POL(n__0) = 0   
POL(n__cons(x1, x2)) = x1 + x2   
POL(n__length(x1)) = 1 + x1   
POL(n__nil) = 0   
POL(n__s(x1)) = x1   
POL(n__take(x1, x2)) = 1 + x1 + x2   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(take(x1, x2)) = 1 + x1 + x2   
POL(tt) = 0   
POL(uLength(x1, x2)) = 1 + 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:

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



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

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

ISNATILIST(n__cons(N, IL)) → ISNAT(activate(N))
UTAKE2(tt, M, N, IL) → ACTIVATE(M)
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), 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))
ACTIVATE(n__length(X)) → LENGTH(activate(X))
TAKE(0, IL) → ISNATILIST(IL)
ISNAT(n__s(N)) → ISNAT(activate(N))
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATILIST(IL) → ISNATLIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
UTAKE2(tt, M, N, IL) → ACTIVATE(IL)
ULENGTH(tt, L) → LENGTH(activate(L))
TAKE(s(M), cons(N, IL)) → ISNAT(N)
LENGTH(cons(N, L)) → ISNAT(N)
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 5 SCCs with 14 less nodes.

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

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

ISNAT(n__s(N)) → ISNAT(activate(N))

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__s(N)) → ISNAT(activate(N))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( n__length(x1) ) =
/0\
\0/
+
/10\
\01/
·x1

M( n__zeros ) =
/0\
\0/

M( n__cons(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/10\
\11/
·x2

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

M( n__nil ) =
/1\
\0/

M( n__s(x1) ) =
/0\
\1/
+
/10\
\01/
·x1

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

M( take(x1, x2) ) =
/1\
\0/
+
/00\
\01/
·x1+
/10\
\01/
·x2

M( 0 ) =
/1\
\0/

M( cons(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/10\
\11/
·x2

M( uTake2(x1, ..., x4) ) =
/1\
\1/
+
/00\
\00/
·x1+
/00\
\01/
·x2+
/00\
\00/
·x3+
/10\
\11/
·x4

M( tt ) =
/0\
\1/

M( isNatList(x1) ) =
/0\
\0/
+
/00\
\10/
·x1

M( n__0 ) =
/1\
\0/

M( zeros ) =
/0\
\0/

M( isNatIList(x1) ) =
/0\
\1/
+
/00\
\10/
·x1

M( n__take(x1, x2) ) =
/1\
\0/
+
/00\
\01/
·x1+
/10\
\01/
·x2

M( uTake1(x1) ) =
/1\
\0/
+
/00\
\00/
·x1

M( s(x1) ) =
/0\
\1/
+
/10\
\01/
·x1

M( isNat(x1) ) =
/0\
\0/
+
/00\
\10/
·x1

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

M( nil ) =
/1\
\0/

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

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


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


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

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



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

Q DP problem:
P is empty.
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 TRS P is empty. Hence, there is no (P,Q,R) chain.

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

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

ISNATLIST(n__cons(N, L)) → ISNATLIST(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.
By narrowing [15] the rule ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L)) at position [0] we obtained the following new rules:

ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(zeros)
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(y0, n__0)) → ISNATLIST(0)
ISNATLIST(n__cons(y0, n__s(x0))) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(nil)



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

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

ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(zeros)
ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(nil)
ISNATLIST(n__cons(y0, n__s(x0))) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(y0, n__0)) → ISNATLIST(0)

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.
By narrowing [15] the rule ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(zeros) at position [0] we obtained the following new rules:

ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__zeros)



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

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

ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__zeros)
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(y0, n__0)) → ISNATLIST(0)
ISNATLIST(n__cons(y0, n__s(x0))) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(nil)

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 1 less node.

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

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

ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(nil)
ISNATLIST(n__cons(y0, n__s(x0))) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(y0, n__0)) → ISNATLIST(0)

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.
By narrowing [15] the rule ISNATLIST(n__cons(y0, n__0)) → ISNATLIST(0) at position [0] we obtained the following new rules:

ISNATLIST(n__cons(y0, n__0)) → ISNATLIST(n__0)



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

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

ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(y0, n__0)) → ISNATLIST(n__0)
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(y0, n__s(x0))) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(nil)

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 1 less node.

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

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

ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(nil)
ISNATLIST(n__cons(y0, n__s(x0))) → ISNATLIST(s(activate(x0)))

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.
By narrowing [15] the rule ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(nil) at position [0] we obtained the following new rules:

ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(n__nil)



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

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

ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(n__nil)
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(y0, n__s(x0))) → ISNATLIST(s(activate(x0)))

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 1 less node.

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

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

ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(y0, n__s(x0))) → ISNATLIST(s(activate(x0)))

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.
By narrowing [15] the rule ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(0, n__zeros)) at position [0] we obtained the following new rules:

ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(n__0, n__zeros))



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

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

ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(n__0, n__zeros))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(y0, n__s(x0))) → ISNATLIST(s(activate(x0)))

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.
By narrowing [15] the rule ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(n__0, n__zeros)) at position [0] we obtained the following new rules:

ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))



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

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

ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))
ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(y0, n__s(x0))) → ISNATLIST(s(activate(x0)))

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.


ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__cons(x0, x1))) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(y0, n__s(x0))) → ISNATLIST(s(activate(x0)))
The remaining pairs can at least be oriented weakly.

ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(0, n__zeros))
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0   
POL(ISNATLIST(x1)) = x1   
POL(activate(x1)) = 1 + x1   
POL(and(x1, x2)) = x2   
POL(cons(x1, x2)) = 1 + x2   
POL(isNat(x1)) = 0   
POL(isNatIList(x1)) = 0   
POL(isNatList(x1)) = 0   
POL(length(x1)) = 0   
POL(n__0) = 0   
POL(n__cons(x1, x2)) = 1 + x2   
POL(n__length(x1)) = 0   
POL(n__nil) = 0   
POL(n__s(x1)) = 0   
POL(n__take(x1, x2)) = 0   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x1)) = 0   
POL(take(x1, x2)) = 1   
POL(tt) = 0   
POL(uLength(x1, x2)) = 0   
POL(uTake1(x1)) = 0   
POL(uTake2(x1, x2, x3, x4)) = 1   
POL(zeros) = 0   

The following usable rules [17] were oriented:

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



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

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

ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))
ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(0, n__zeros))

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
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ DependencyGraphProof
                                                              ↳ AND
                                                                ↳ QDP
                                                                  ↳ UsableRulesProof
                                                                ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP

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

ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(0, n__zeros))

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 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
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ DependencyGraphProof
                                                              ↳ AND
                                                                ↳ QDP
                                                                  ↳ UsableRulesProof
                                                                    ↳ QDP
                                                                      ↳ MNOCProof
                                                                ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP

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

ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(0, n__zeros))

The TRS R consists of the following rules:

0 → n__0

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the modular non-overlap check [15] to enlarge Q to all left-hand sides of R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ DependencyGraphProof
                                                              ↳ AND
                                                                ↳ QDP
                                                                  ↳ UsableRulesProof
                                                                    ↳ QDP
                                                                      ↳ MNOCProof
                                                                        ↳ QDP
                                                                          ↳ Rewriting
                                                                ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP

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

ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(0, n__zeros))

The TRS R consists of the following rules:

0 → n__0

The set Q consists of the following terms:

0

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(0, n__zeros)) at position [0,0] we obtained the following new rules:

ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ DependencyGraphProof
                                                              ↳ AND
                                                                ↳ QDP
                                                                  ↳ UsableRulesProof
                                                                    ↳ QDP
                                                                      ↳ MNOCProof
                                                                        ↳ QDP
                                                                          ↳ Rewriting
                                                                            ↳ QDP
                                                                              ↳ UsableRulesProof
                                                                ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP

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

ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

0 → n__0

The set Q consists of the following terms:

0

We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ DependencyGraphProof
                                                              ↳ AND
                                                                ↳ QDP
                                                                  ↳ UsableRulesProof
                                                                    ↳ QDP
                                                                      ↳ MNOCProof
                                                                        ↳ QDP
                                                                          ↳ Rewriting
                                                                            ↳ QDP
                                                                              ↳ UsableRulesProof
                                                                                ↳ QDP
                                                                                  ↳ QReductionProof
                                                                ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP

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

ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))

R is empty.
The set Q consists of the following terms:

0

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

0



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ DependencyGraphProof
                                                              ↳ AND
                                                                ↳ QDP
                                                                  ↳ UsableRulesProof
                                                                    ↳ QDP
                                                                      ↳ MNOCProof
                                                                        ↳ QDP
                                                                          ↳ Rewriting
                                                                            ↳ QDP
                                                                              ↳ UsableRulesProof
                                                                                ↳ QDP
                                                                                  ↳ QReductionProof
                                                                                    ↳ QDP
                                                                                      ↳ Instantiation
                                                                ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP

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

ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros)) we obtained the following new rules:

ISNATLIST(n__cons(n__0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ DependencyGraphProof
                                                              ↳ AND
                                                                ↳ QDP
                                                                  ↳ UsableRulesProof
                                                                    ↳ QDP
                                                                      ↳ MNOCProof
                                                                        ↳ QDP
                                                                          ↳ Rewriting
                                                                            ↳ QDP
                                                                              ↳ UsableRulesProof
                                                                                ↳ QDP
                                                                                  ↳ QReductionProof
                                                                                    ↳ QDP
                                                                                      ↳ Instantiation
                                                                                        ↳ QDP
                                                                                          ↳ NonTerminationProof
                                                                ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP

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

ISNATLIST(n__cons(n__0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

ISNATLIST(n__cons(n__0, n__zeros)) → ISNATLIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:none


s = ISNATLIST(n__cons(n__0, n__zeros)) evaluates to t =ISNATLIST(n__cons(n__0, n__zeros))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from ISNATLIST(n__cons(n__0, n__zeros)) to ISNATLIST(n__cons(n__0, n__zeros)).





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

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

ISNATLIST(n__cons(y0, n__take(x0, x1))) → ISNATLIST(take(activate(x0), activate(x1)))

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
                  ↳ QDP
                  ↳ 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
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP

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

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.

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

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

ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(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.