Termination w.r.t. Q of the given QTRS could be disproven:

(0) Obligation:

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.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

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

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.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 17 less nodes.

(4) Obligation:

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

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

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ISNATLIST(n__take(N, IL)) → ISNAT(activate(N))
ISNATLIST(n__take(N, IL)) → ACTIVATE(N)
ISNATLIST(n__take(N, IL)) → ISNATILIST(activate(IL))
ISNATLIST(n__take(N, IL)) → ACTIVATE(IL)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(ACTIVATE(x₁)) = x₁   
POL(ISNAT(x₁)) = x₁   
POL(ISNATILIST(x₁)) = x₁   
POL(ISNATLIST(x₁)) = x₁   
POL(LENGTH(x₁)) = x₁   
POL(TAKE(x₁, x₂)) = x₁ + x₂   
POL(ULENGTH(x₁, x₂)) = 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)) = x₁   
POL(n__0) = 0   
POL(n__cons(x₁, x₂)) = x₁ + x₂   
POL(n__length(x₁)) = x₁   
POL(n__nil) = 0   
POL(n__s(x₁)) = x₁   
POL(n__take(x₁, x₂)) = 1 + x₁ + x₂   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x₁)) = x₁   
POL(take(x₁, x₂)) = 1 + x₁ + x₂   
POL(tt) = 0   
POL(uLength(x₁, x₂)) = x₂   
POL(uTake1(x₁)) = 0   
POL(uTake2(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₃ + x₄   
POL(zeros) = 0   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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

(6) Obligation:

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

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 14 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

ISNAT(n__s(N)) → ISNAT(activate(N))
ISNAT(n__s(N)) → ACTIVATE(N)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__length(X)) → LENGTH(activate(X))
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
ULENGTH(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → ISNAT(N)
ISNAT(n__length(L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
ISNAT(n__length(L)) → ACTIVATE(L)
ACTIVATE(n__length(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
LENGTH(cons(N, L)) → ACTIVATE(L)
ULENGTH(tt, L) → 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.

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__length(X)) → LENGTH(activate(X))
ISNAT(n__length(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.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(ACTIVATE(x₁)) = x₁   
POL(ISNAT(x₁)) = x₁   
POL(ISNATLIST(x₁)) = x₁   
POL(LENGTH(x₁)) = x₁   
POL(ULENGTH(x₁, x₂)) = x₂   
POL(activate(x₁)) = x₁   
POL(and(x₁, x₂)) = x₂   
POL(cons(x₁, x₂)) = x₁ + x₂   
POL(isNat(x₁)) = 1   
POL(isNatIList(x₁)) = 1   
POL(isNatList(x₁)) = 1   
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)) = x₂   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x₁)) = x₁   
POL(take(x₁, x₂)) = x₂   
POL(tt) = 1   
POL(uLength(x₁, x₂)) = x₁ + x₂   
POL(uTake1(x₁)) = 0   
POL(uTake2(x₁, x₂, x₃, x₄)) = x₃ + x₄   
POL(zeros) = 0   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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

(11) Obligation:

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

ISNAT(n__s(N)) → ISNAT(activate(N))
ISNAT(n__s(N)) → ACTIVATE(N)
ACTIVATE(n__s(X)) → ACTIVATE(X)
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(activate(L))), activate(L))
ULENGTH(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → ISNAT(N)
ISNATLIST(n__cons(N, L)) → ISNAT(activate(N))
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATLIST(n__cons(N, L)) → ACTIVATE(N)
ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(N, L)) → ACTIVATE(L)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
LENGTH(cons(N, L)) → ACTIVATE(L)
ULENGTH(tt, L) → 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.

(12) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 8 less nodes.

(13) Complex Obligation (AND)

(14) Obligation:

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.

(15) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(16) Obligation:

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

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

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

(17) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

• ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
  The graph contains the following edges 1 > 1

• ACTIVATE(n__s(X)) → ACTIVATE(X)
  The graph contains the following edges 1 > 1

(18) YES

(19) Obligation:

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.

(20) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

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.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ISNAT(x1)) =

0A

+

0A

·

x1

POL(n__s(x1)) =

2A

+

1A

·

x1

POL(activate(x1)) =

1A

+

0A

·

x1

POL(n__0) =

0A

POL(0) =

1A

POL(s(x1)) =

2A

+

1A

·

x1

POL(n__length(x1)) =

2A

+

0A

·

x1

POL(length(x1)) =

2A

+

0A

·

x1

POL(n__zeros) =

0A

POL(zeros) =

1A

POL(n__cons(x1, x2)) =

-I

+

-I

·

x1

+

1A

·

x2

POL(cons(x1, x2)) =

-I

+

-I

·

x1

+

1A

·

x2

POL(n__nil) =

2A

POL(nil) =

2A

POL(n__take(x1, x2)) =

2A

+

1A

·

x1

+

1A

·

x2

POL(take(x1, x2)) =

2A

+

1A

·

x1

+

1A

·

x2

POL(uTake1(x1)) =

2A

+

0A

·

x1

POL(isNatIList(x1)) =

2A

+

0A

·

x1

POL(tt) =

2A

POL(isNatList(x1)) =

1A

+

0A

·

x1

POL(and(x1, x2)) =

0A

+

-I

·

x1

+

0A

·

x2

POL(isNat(x1)) =

5A

+

0A

·

x1

POL(uTake2(x1, x2, x3, x4)) =

3A

+

-I

·

x1

+

2A

·

x2

+

-I

·

x3

+

2A

·

x4

POL(uLength(x1, x2)) =

-I

+

1A

·

x1

+

1A

·

x2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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

(21) Obligation:

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.

(22) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(23) YES

(24) Obligation:

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.

(25) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATLIST(n__cons(N, L)) → ISNATLIST(activate(L)) at position [0] we obtained the following new rules [LPAR04]:

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

(26) Obligation:

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

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

(27) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATLIST(n__cons(y0, n__0)) → ISNATLIST(0) at position [0] we obtained the following new rules [LPAR04]:

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

(28) Obligation:

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

ISNATLIST(n__cons(y0, n__s(x0))) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(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__take(x0, x1))) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__0)) → ISNATLIST(n__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.

(29) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(30) Obligation:

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

ISNATLIST(n__cons(y0, n__s(x0))) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(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__take(x0, x1))) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(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.

(31) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(zeros) at position [0] we obtained the following new rules [LPAR04]:

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

(32) Obligation:

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

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

(33) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(34) Obligation:

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

ISNATLIST(n__cons(y0, n__s(x0))) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(y0, n__length(x0))) → ISNATLIST(length(activate(x0)))
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__take(x0, x1))) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(y0, x0)) → ISNATLIST(x0)
ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(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.

(35) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATLIST(n__cons(y0, n__nil)) → ISNATLIST(nil) at position [0] we obtained the following new rules [LPAR04]:

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

(36) Obligation:

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

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

(37) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(38) Obligation:

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

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

(39) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]:

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

(40) Obligation:

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

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

(41) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATLIST(n__cons(y0, n__zeros)) → ISNATLIST(cons(n__0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]:

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

(42) Obligation:

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

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

(43) Obligation:

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.

(44) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(LENGTH(x1)) =

2A

+

0A

·

x1

POL(cons(x1, x2)) =

1A

+

0A

·

x1

+

1A

·

x2

POL(ULENGTH(x1, x2)) =

0A

+

0A

·

x1

+

0A

·

x2

POL(and(x1, x2)) =

-I

+

-I

·

x1

+

0A

·

x2

POL(isNat(x1)) =

3A

+

0A

·

x1

POL(isNatList(x1)) =

-I

+

0A

·

x1

POL(activate(x1)) =

1A

+

0A

·

x1

POL(tt) =

3A

POL(n__0) =

0A

POL(n__s(x1)) =

2A

+

1A

·

x1

POL(n__length(x1)) =

2A

+

0A

·

x1

POL(0) =

0A

POL(s(x1)) =

2A

+

1A

·

x1

POL(length(x1)) =

2A

+

0A

·

x1

POL(n__zeros) =

0A

POL(zeros) =

1A

POL(n__cons(x1, x2)) =

1A

+

0A

·

x1

+

1A

·

x2

POL(n__nil) =

3A

POL(nil) =

3A

POL(n__take(x1, x2)) =

3A

+

2A

·

x1

+

0A

·

x2

POL(take(x1, x2)) =

3A

+

2A

·

x1

+

0A

·

x2

POL(isNatIList(x1)) =

3A

+

0A

·

x1

POL(uTake1(x1)) =

3A

+

-I

·

x1

POL(uTake2(x1, x2, x3, x4)) =

4A

+

-I

·

x1

+

3A

·

x2

+

0A

·

x3

+

1A

·

x4

POL(uLength(x1, x2)) =

-I

+

0A

·

x1

+

1A

·

x2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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

(45) Obligation:

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

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.

(46) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(47) TRUE

(48) Obligation:

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.

(49) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(N, IL)) → ISNATILIST(activate(IL)) at position [0] we obtained the following new rules [LPAR04]:

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

(50) Obligation:

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

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

(51) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(y0, n__0)) → ISNATILIST(0) at position [0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(y0, n__0)) → ISNATILIST(n__0)

(52) Obligation:

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

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

(53) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(54) Obligation:

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

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

(55) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(zeros) at position [0] we obtained the following new rules [LPAR04]:

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

(56) Obligation:

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

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

(57) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(58) Obligation:

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

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

(59) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(nil) at position [0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(y0, n__nil)) → ISNATILIST(n__nil)

(60) Obligation:

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

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

(61) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(62) Obligation:

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

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

(63) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(cons(0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]:

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

(64) Obligation:

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

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

(65) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(cons(n__0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]:

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

(66) Obligation:

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

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

(67) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

ISNATILIST(n__cons(y0, n__s(x0))) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(y0, n__length(x0))) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(y0, n__cons(x0, x1))) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(y0, x0)) → ISNATILIST(x0)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 1   
POL(ISNATILIST(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)) = 1   
POL(uTake2(x₁, x₂, x₃, x₄)) = 1   
POL(zeros) = 0   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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

(68) Obligation:

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

ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__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.

(69) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.

(70) Complex Obligation (AND)

(71) Obligation:

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

ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__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.

(72) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(73) Obligation:

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

ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__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.

(74) MNOCProof (EQUIVALENT transformation)

We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.

(75) Obligation:

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

ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(0, n__zeros))
ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(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.

(76) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(0, n__zeros)) at position [0,0] we obtained the following new rules [LPAR04]:

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

(77) Obligation:

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

ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(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.

(78) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(79) Obligation:

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

ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(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.

(80) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

0

(81) Obligation:

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

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

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

(82) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule ISNATILIST(n__cons(y0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros)) we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(n__0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros))

(83) Obligation:

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

ISNATILIST(n__cons(n__0, n__zeros)) → ISNATILIST(n__cons(n__0, n__zeros))

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

(84) InfRuleLoopProof (EQUIVALENT transformation)

By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP.
We apply the theorem with m = 0, b = 0,
σ' = [ ], and μ' = [ ] on the rule
ISNATILIST(n__cons(n__0, n__zeros))[ ]ⁿ[ ] → ISNATILIST(n__cons(n__0, n__zeros))[ ]ⁿ[ ]
This rule is correct for the QDP as the following derivation shows:

ISNATILIST(n__cons(n__0, n__zeros))[ ]ⁿ[ ] → ISNATILIST(n__cons(n__0, n__zeros))[ ]ⁿ[ ]
    by OriginalRule from TRS P

(85) NO

(86) Obligation:

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

ISNATILIST(n__cons(y0, n__take(x0, x1))) → ISNATILIST(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.