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:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
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:

ZEROS → CONS(0, n__zeros)
ZEROS → 0¹
U41¹(tt, V2) → U42¹(isNatIList(activate(V2)))
U41¹(tt, V2) → ISNATILIST(activate(V2))
U41¹(tt, V2) → ACTIVATE(V2)
U51¹(tt, V2) → U52¹(isNatList(activate(V2)))
U51¹(tt, V2) → ISNATLIST(activate(V2))
U51¹(tt, V2) → ACTIVATE(V2)
U61¹(tt, V2) → U62¹(isNatIList(activate(V2)))
U61¹(tt, V2) → ISNATILIST(activate(V2))
U61¹(tt, V2) → ACTIVATE(V2)
U71¹(tt, L, N) → U72¹(isNat(activate(N)), activate(L))
U71¹(tt, L, N) → ISNAT(activate(N))
U71¹(tt, L, N) → ACTIVATE(N)
U71¹(tt, L, N) → ACTIVATE(L)
U72¹(tt, L) → S(length(activate(L)))
U72¹(tt, L) → LENGTH(activate(L))
U72¹(tt, L) → ACTIVATE(L)
U81¹(tt) → NIL
U91¹(tt, IL, M, N) → U92¹(isNat(activate(M)), activate(IL), activate(M), activate(N))
U91¹(tt, IL, M, N) → ISNAT(activate(M))
U91¹(tt, IL, M, N) → ACTIVATE(M)
U91¹(tt, IL, M, N) → ACTIVATE(IL)
U91¹(tt, IL, M, N) → ACTIVATE(N)
U92¹(tt, IL, M, N) → U93¹(isNat(activate(N)), activate(IL), activate(M), activate(N))
U92¹(tt, IL, M, N) → ISNAT(activate(N))
U92¹(tt, IL, M, N) → ACTIVATE(N)
U92¹(tt, IL, M, N) → ACTIVATE(IL)
U92¹(tt, IL, M, N) → ACTIVATE(M)
U93¹(tt, IL, M, N) → CONS(activate(N), n__take(activate(M), activate(IL)))
U93¹(tt, IL, M, N) → ACTIVATE(N)
U93¹(tt, IL, M, N) → ACTIVATE(M)
U93¹(tt, IL, M, N) → ACTIVATE(IL)
ISNAT(n__length(V1)) → U11¹(isNatList(activate(V1)))
ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
ISNAT(n__length(V1)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → U21¹(isNat(activate(V1)))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNATILIST(V) → U31¹(isNatList(activate(V)))
ISNATILIST(V) → ISNATLIST(activate(V))
ISNATILIST(V) → ACTIVATE(V)
ISNATILIST(n__cons(V1, V2)) → U41¹(isNat(activate(V1)), activate(V2))
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → U51¹(isNat(activate(V1)), activate(V2))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → U61¹(isNat(activate(V1)), activate(V2))
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
LENGTH(nil) → 0¹
LENGTH(cons(N, L)) → U71¹(isNatList(activate(L)), activate(L), N)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
LENGTH(cons(N, L)) → ACTIVATE(L)
TAKE(0, IL) → U81¹(isNatIList(IL))
TAKE(0, IL) → ISNATILIST(IL)
TAKE(s(M), cons(N, IL)) → U91¹(isNatIList(activate(IL)), activate(IL), M, N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ACTIVATE(n__zeros) → ZEROS
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__0) → 0¹
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ACTIVATE(n__length(X)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → S(activate(X))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → CONS(activate(X1), X2)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__nil) → NIL

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
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 18 less nodes.

(4) Obligation:

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

ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
TAKE(0, IL) → ISNATILIST(IL)
ISNATILIST(V) → ISNATLIST(activate(V))
ISNATLIST(n__cons(V1, V2)) → U51¹(isNat(activate(V1)), activate(V2))
U51¹(tt, V2) → ISNATLIST(activate(V2))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__length(X)) → LENGTH(activate(X))
LENGTH(cons(N, L)) → U71¹(isNatList(activate(L)), activate(L), N)
U71¹(tt, L, N) → U72¹(isNat(activate(N)), activate(L))
U72¹(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ACTIVATE(n__length(X)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATLIST(n__take(V1, V2)) → U61¹(isNat(activate(V1)), activate(V2))
U61¹(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(V) → ACTIVATE(V)
ISNATILIST(n__cons(V1, V2)) → U41¹(isNat(activate(V1)), activate(V2))
U41¹(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__length(V1)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U41¹(tt, V2) → ACTIVATE(V2)
U61¹(tt, V2) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
LENGTH(cons(N, L)) → ACTIVATE(L)
U72¹(tt, L) → ACTIVATE(L)
U71¹(tt, L, N) → ISNAT(activate(N))
U71¹(tt, L, N) → ACTIVATE(N)
U71¹(tt, L, N) → ACTIVATE(L)
U51¹(tt, V2) → ACTIVATE(V2)
TAKE(s(M), cons(N, IL)) → U91¹(isNatIList(activate(IL)), activate(IL), M, N)
U91¹(tt, IL, M, N) → U92¹(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92¹(tt, IL, M, N) → U93¹(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93¹(tt, IL, M, N) → ACTIVATE(N)
U93¹(tt, IL, M, N) → ACTIVATE(M)
U93¹(tt, IL, M, N) → ACTIVATE(IL)
U92¹(tt, IL, M, N) → ISNAT(activate(N))
U92¹(tt, IL, M, N) → ACTIVATE(N)
U92¹(tt, IL, M, N) → ACTIVATE(IL)
U92¹(tt, IL, M, N) → ACTIVATE(M)
U91¹(tt, IL, M, N) → ISNAT(activate(M))
U91¹(tt, IL, M, N) → ACTIVATE(M)
U91¹(tt, IL, M, N) → ACTIVATE(IL)
U91¹(tt, IL, M, N) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
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))
ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ACTIVATE(n__length(X)) → ACTIVATE(X)
ISNATLIST(n__take(V1, V2)) → U61¹(isNat(activate(V1)), activate(V2))
ISNAT(n__length(V1)) → ACTIVATE(V1)
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)

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(U11(x₁)) = 0   
POL(U21(x₁)) = 0   
POL(U31(x₁)) = 0   
POL(U41(x₁, x₂)) = 0   
POL(U41¹(x₁, x₂)) = x₂   
POL(U42(x₁)) = 0   
POL(U51(x₁, x₂)) = 0   
POL(U51¹(x₁, x₂)) = x₂   
POL(U52(x₁)) = 0   
POL(U61(x₁, x₂)) = 0   
POL(U61¹(x₁, x₂)) = x₂   
POL(U62(x₁)) = 0   
POL(U71(x₁, x₂, x₃)) = 1 + x₂ + x₃   
POL(U71¹(x₁, x₂, x₃)) = x₂ + x₃   
POL(U72(x₁, x₂)) = 1 + x₂   
POL(U72¹(x₁, x₂)) = x₂   
POL(U81(x₁)) = 1   
POL(U91(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₃ + x₄   
POL(U91¹(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄   
POL(U92(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₃ + x₄   
POL(U92¹(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄   
POL(U93(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₃ + x₄   
POL(U93¹(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄   
POL(activate(x₁)) = x₁   
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(zeros) = 0   

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

activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
take(X1, X2) → n__take(X1, X2)
length(nil) → 0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
U11(tt) → tt
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
take(0, IL) → U81(isNatIList(IL))
U81(tt) → nil
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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:

TAKE(0, IL) → ISNATILIST(IL)
ISNATILIST(V) → ISNATLIST(activate(V))
ISNATLIST(n__cons(V1, V2)) → U51¹(isNat(activate(V1)), activate(V2))
U51¹(tt, V2) → ISNATLIST(activate(V2))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → U71¹(isNatList(activate(L)), activate(L), N)
U71¹(tt, L, N) → U72¹(isNat(activate(N)), activate(L))
U72¹(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
U61¹(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(V) → ACTIVATE(V)
ISNATILIST(n__cons(V1, V2)) → U41¹(isNat(activate(V1)), activate(V2))
U41¹(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U41¹(tt, V2) → ACTIVATE(V2)
U61¹(tt, V2) → ACTIVATE(V2)
LENGTH(cons(N, L)) → ACTIVATE(L)
U72¹(tt, L) → ACTIVATE(L)
U71¹(tt, L, N) → ISNAT(activate(N))
U71¹(tt, L, N) → ACTIVATE(N)
U71¹(tt, L, N) → ACTIVATE(L)
U51¹(tt, V2) → ACTIVATE(V2)
TAKE(s(M), cons(N, IL)) → U91¹(isNatIList(activate(IL)), activate(IL), M, N)
U91¹(tt, IL, M, N) → U92¹(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92¹(tt, IL, M, N) → U93¹(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93¹(tt, IL, M, N) → ACTIVATE(N)
U93¹(tt, IL, M, N) → ACTIVATE(M)
U93¹(tt, IL, M, N) → ACTIVATE(IL)
U92¹(tt, IL, M, N) → ISNAT(activate(N))
U92¹(tt, IL, M, N) → ACTIVATE(N)
U92¹(tt, IL, M, N) → ACTIVATE(IL)
U92¹(tt, IL, M, N) → ACTIVATE(M)
U91¹(tt, IL, M, N) → ISNAT(activate(M))
U91¹(tt, IL, M, N) → ACTIVATE(M)
U91¹(tt, IL, M, N) → ACTIVATE(IL)
U91¹(tt, IL, M, N) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
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 5 SCCs with 36 less nodes.

(8) Complex Obligation (AND)

(9) 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:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(10) 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.

(11) 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.

(12) 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

(13) YES

(14) Obligation:

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

ISNAT(n__s(V1)) → ISNAT(activate(V1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(15) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

ISNAT(n__s(V1)) → ISNAT(activate(V1))

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

POL(ISNAT(x1)) =

4A

+

2A

·

x1

POL(n__s(x1)) =

3A

+

1A

·

x1

POL(activate(x1)) =

1A

+

0A

·

x1

POL(n__zeros) =

0A

POL(zeros) =

1A

POL(n__take(x1, x2)) =

2A

+

0A

·

x1

+

1A

·

x2

POL(take(x1, x2)) =

2A

+

0A

·

x1

+

1A

·

x2

POL(n__0) =

1A

POL(0) =

1A

POL(n__length(x1)) =

2A

+

0A

·

x1

POL(length(x1)) =

2A

+

0A

·

x1

POL(s(x1)) =

3A

+

1A

·

x1

POL(n__cons(x1, x2)) =

1A

+

0A

·

x1

+

1A

·

x2

POL(cons(x1, x2)) =

1A

+

0A

·

x1

+

1A

·

x2

POL(n__nil) =

2A

POL(nil) =

2A

POL(U71(x1, x2, x3)) =

-I

+

1A

·

x1

+

1A

·

x2

+

0A

·

x3

POL(isNatList(x1)) =

-I

+

0A

·

x1

POL(tt) =

2A

POL(U51(x1, x2)) =

1A

+

-I

·

x1

+

0A

·

x2

POL(isNat(x1)) =

-I

+

1A

·

x1

POL(U11(x1)) =

-I

+

1A

·

x1

POL(U61(x1, x2)) =

2A

+

-I

·

x1

+

1A

·

x2

POL(U21(x1)) =

4A

+

-I

·

x1

POL(U62(x1)) =

-I

+

0A

·

x1

POL(isNatIList(x1)) =

2A

+

1A

·

x1

POL(U31(x1)) =

2A

+

-I

·

x1

POL(U41(x1, x2)) =

0A

+

0A

·

x1

+

1A

·

x2

POL(U42(x1)) =

-I

+

0A

·

x1

POL(U52(x1)) =

-I

+

0A

·

x1

POL(U72(x1, x2)) =

3A

+

-I

·

x1

+

1A

·

x2

POL(U81(x1)) =

-I

+

0A

·

x1

POL(U91(x1, x2, x3, x4)) =

3A

+

0A

·

x1

+

2A

·

x2

+

1A

·

x3

+

0A

·

x4

POL(U92(x1, x2, x3, x4)) =

3A

+

-I

·

x1

+

2A

·

x2

+

1A

·

x3

+

0A

·

x4

POL(U93(x1, x2, x3, x4)) =

3A

+

-I

·

x1

+

2A

·

x2

+

1A

·

x3

+

0A

·

x4

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

activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
take(X1, X2) → n__take(X1, X2)
length(nil) → 0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
U11(tt) → tt
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
isNatIList(n__zeros) → tt
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(tt) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
take(0, IL) → U81(isNatIList(IL))
U81(tt) → nil
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
zeros → cons(0, n__zeros)
zeros → n__zeros
0 → n__0
nil → n__nil

(16) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(17) PisEmptyProof (EQUIVALENT transformation)

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

(18) YES

(19) Obligation:

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

U51¹(tt, V2) → ISNATLIST(activate(V2))
ISNATLIST(n__cons(V1, V2)) → U51¹(isNat(activate(V1)), activate(V2))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(20) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U51¹(tt, V2) → ISNATLIST(activate(V2)) at position [0] we obtained the following new rules [LPAR04]:

U51¹(tt, n__zeros) → ISNATLIST(zeros)
U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
U51¹(tt, n__0) → ISNATLIST(0)
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
U51¹(tt, n__nil) → ISNATLIST(nil)
U51¹(tt, x0) → ISNATLIST(x0)

(21) Obligation:

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

ISNATLIST(n__cons(V1, V2)) → U51¹(isNat(activate(V1)), activate(V2))
U51¹(tt, n__zeros) → ISNATLIST(zeros)
U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
U51¹(tt, n__0) → ISNATLIST(0)
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
U51¹(tt, n__nil) → ISNATLIST(nil)
U51¹(tt, x0) → ISNATLIST(x0)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(22) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATLIST(n__cons(V1, V2)) → U51¹(isNat(activate(V1)), activate(V2)) at position [0] we obtained the following new rules [LPAR04]:

ISNATLIST(n__cons(n__zeros, y1)) → U51¹(isNat(zeros), activate(y1))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U51¹(isNat(take(activate(x0), activate(x1))), activate(y1))
ISNATLIST(n__cons(n__0, y1)) → U51¹(isNat(0), activate(y1))
ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(activate(x0))), activate(y1))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(n__nil, y1)) → U51¹(isNat(nil), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))

(23) Obligation:

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

U51¹(tt, n__zeros) → ISNATLIST(zeros)
U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
U51¹(tt, n__0) → ISNATLIST(0)
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
U51¹(tt, n__nil) → ISNATLIST(nil)
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__zeros, y1)) → U51¹(isNat(zeros), activate(y1))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U51¹(isNat(take(activate(x0), activate(x1))), activate(y1))
ISNATLIST(n__cons(n__0, y1)) → U51¹(isNat(0), activate(y1))
ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(activate(x0))), activate(y1))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(n__nil, y1)) → U51¹(isNat(nil), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(24) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U51¹(tt, n__zeros) → ISNATLIST(zeros) at position [0] we obtained the following new rules [LPAR04]:

U51¹(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
U51¹(tt, n__zeros) → ISNATLIST(n__zeros)

(25) Obligation:

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

U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
U51¹(tt, n__0) → ISNATLIST(0)
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
U51¹(tt, n__nil) → ISNATLIST(nil)
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__zeros, y1)) → U51¹(isNat(zeros), activate(y1))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U51¹(isNat(take(activate(x0), activate(x1))), activate(y1))
ISNATLIST(n__cons(n__0, y1)) → U51¹(isNat(0), activate(y1))
ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(activate(x0))), activate(y1))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(n__nil, y1)) → U51¹(isNat(nil), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
U51¹(tt, n__zeros) → ISNATLIST(n__zeros)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(26) DependencyGraphProof (EQUIVALENT transformation)

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

(27) Obligation:

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

ISNATLIST(n__cons(n__zeros, y1)) → U51¹(isNat(zeros), activate(y1))
U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U51¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__0) → ISNATLIST(0)
ISNATLIST(n__cons(n__0, y1)) → U51¹(isNat(0), activate(y1))
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(activate(x0))), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__nil) → ISNATLIST(nil)
ISNATLIST(n__cons(n__nil, y1)) → U51¹(isNat(nil), activate(y1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(28) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATLIST(n__cons(n__zeros, y1)) → U51¹(isNat(zeros), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(cons(0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(n__zeros), activate(y0))

(29) Obligation:

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

U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U51¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__0) → ISNATLIST(0)
ISNATLIST(n__cons(n__0, y1)) → U51¹(isNat(0), activate(y1))
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(activate(x0))), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__nil) → ISNATLIST(nil)
ISNATLIST(n__cons(n__nil, y1)) → U51¹(isNat(nil), activate(y1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(cons(0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(n__zeros), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(30) DependencyGraphProof (EQUIVALENT transformation)

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

(31) Obligation:

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

ISNATLIST(n__cons(n__take(x0, x1), y1)) → U51¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__0, y1)) → U51¹(isNat(0), activate(y1))
U51¹(tt, n__0) → ISNATLIST(0)
ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(activate(x0))), activate(y1))
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__nil, y1)) → U51¹(isNat(nil), activate(y1))
U51¹(tt, n__nil) → ISNATLIST(nil)
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(cons(0, n__zeros)), activate(y0))
U51¹(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(32) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATLIST(n__cons(n__0, y1)) → U51¹(isNat(0), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))

(33) Obligation:

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

ISNATLIST(n__cons(n__take(x0, x1), y1)) → U51¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
U51¹(tt, n__0) → ISNATLIST(0)
ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(activate(x0))), activate(y1))
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__nil, y1)) → U51¹(isNat(nil), activate(y1))
U51¹(tt, n__nil) → ISNATLIST(nil)
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(cons(0, n__zeros)), activate(y0))
U51¹(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(34) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U51¹(tt, n__0) → ISNATLIST(0) at position [0] we obtained the following new rules [LPAR04]:

U51¹(tt, n__0) → ISNATLIST(n__0)

(35) Obligation:

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

ISNATLIST(n__cons(n__take(x0, x1), y1)) → U51¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(activate(x0))), activate(y1))
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__nil, y1)) → U51¹(isNat(nil), activate(y1))
U51¹(tt, n__nil) → ISNATLIST(nil)
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(cons(0, n__zeros)), activate(y0))
U51¹(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
U51¹(tt, n__0) → ISNATLIST(n__0)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(36) DependencyGraphProof (EQUIVALENT transformation)

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

(37) Obligation:

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

U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U51¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(activate(x0))), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__nil) → ISNATLIST(nil)
ISNATLIST(n__cons(n__nil, y1)) → U51¹(isNat(nil), activate(y1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(cons(0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(38) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U51¹(tt, n__nil) → ISNATLIST(nil) at position [0] we obtained the following new rules [LPAR04]:

U51¹(tt, n__nil) → ISNATLIST(n__nil)

(39) Obligation:

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

U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U51¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(activate(x0))), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(n__nil, y1)) → U51¹(isNat(nil), activate(y1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(cons(0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
U51¹(tt, n__nil) → ISNATLIST(n__nil)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(40) DependencyGraphProof (EQUIVALENT transformation)

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

(41) Obligation:

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

ISNATLIST(n__cons(n__take(x0, x1), y1)) → U51¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(activate(x0))), activate(y1))
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__nil, y1)) → U51¹(isNat(nil), activate(y1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(cons(0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(42) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATLIST(n__cons(n__nil, y1)) → U51¹(isNat(nil), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

ISNATLIST(n__cons(n__nil, y0)) → U51¹(isNat(n__nil), activate(y0))

(43) Obligation:

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

ISNATLIST(n__cons(n__take(x0, x1), y1)) → U51¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(activate(x0))), activate(y1))
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(cons(0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__nil, y0)) → U51¹(isNat(n__nil), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(44) DependencyGraphProof (EQUIVALENT transformation)

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

(45) Obligation:

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

U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U51¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(activate(x0))), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(cons(0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(46) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U51¹(tt, n__zeros) → ISNATLIST(cons(0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]:

U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros))

(47) Obligation:

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

U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U51¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(activate(x0))), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(cons(0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(48) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(cons(0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules [LPAR04]:

ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(n__cons(0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(cons(n__0, n__zeros)), activate(y0))

(49) Obligation:

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

U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U51¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(activate(x0))), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros))
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(n__cons(0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(cons(n__0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(50) DependencyGraphProof (EQUIVALENT transformation)

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

(51) Obligation:

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

ISNATLIST(n__cons(n__take(x0, x1), y1)) → U51¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(activate(x0))), activate(y1))
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(cons(n__0, n__zeros)), activate(y0))
U51¹(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(52) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(cons(n__0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules [LPAR04]:

ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(n__cons(n__0, n__zeros)), activate(y0))

(53) Obligation:

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

ISNATLIST(n__cons(n__take(x0, x1), y1)) → U51¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(activate(x0))), activate(y1))
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros))
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(n__cons(n__0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(54) DependencyGraphProof (EQUIVALENT transformation)

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

(55) Obligation:

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

U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U51¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(activate(x0))), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
U51¹(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(56) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U51¹(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]:

U51¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))

(57) Obligation:

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

U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U51¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(activate(x0))), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(58) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

ISNATLIST(n__cons(n__take(x0, x1), y1)) → U51¹(isNat(take(activate(x0), activate(x1))), activate(y1))

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

POL(0) = 0   
POL(ISNATLIST(x₁)) = x₁   
POL(U11(x₁)) = 0   
POL(U21(x₁)) = 0   
POL(U31(x₁)) = 0   
POL(U41(x₁, x₂)) = x₂   
POL(U42(x₁)) = 0   
POL(U51(x₁, x₂)) = 0   
POL(U51¹(x₁, x₂)) = x₂   
POL(U52(x₁)) = 0   
POL(U61(x₁, x₂)) = 1 + x₂   
POL(U62(x₁)) = 1   
POL(U71(x₁, x₂, x₃)) = 0   
POL(U72(x₁, x₂)) = 0   
POL(U81(x₁)) = 0   
POL(U91(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₄   
POL(U92(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₄   
POL(U93(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₄   
POL(activate(x₁)) = x₁   
POL(cons(x₁, x₂)) = x₁ + x₂   
POL(isNat(x₁)) = 0   
POL(isNatIList(x₁)) = x₁   
POL(isNatList(x₁)) = x₁   
POL(length(x₁)) = 0   
POL(n__0) = 0   
POL(n__cons(x₁, x₂)) = x₁ + x₂   
POL(n__length(x₁)) = 0   
POL(n__nil) = 0   
POL(n__s(x₁)) = 0   
POL(n__take(x₁, x₂)) = 1 + x₂   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x₁)) = 0   
POL(take(x₁, x₂)) = 1 + x₂   
POL(tt) = 0   
POL(zeros) = 0   

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

activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
take(X1, X2) → n__take(X1, X2)
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
0 → n__0
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U11(tt) → tt
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(tt) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(59) Obligation:

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

U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(activate(x0))), activate(y1))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(activate(x0))), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(60) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(activate(x0))), activate(y1))

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

POL(0) = 0   
POL(ISNATLIST(x₁)) = x₁   
POL(U11(x₁)) = 0   
POL(U21(x₁)) = 0   
POL(U31(x₁)) = 0   
POL(U41(x₁, x₂)) = x₂   
POL(U42(x₁)) = 0   
POL(U51(x₁, x₂)) = 0   
POL(U51¹(x₁, x₂)) = x₂   
POL(U52(x₁)) = 0   
POL(U61(x₁, x₂)) = 1   
POL(U62(x₁)) = 1   
POL(U71(x₁, x₂, x₃)) = 1   
POL(U72(x₁, x₂)) = 0   
POL(U81(x₁)) = 0   
POL(U91(x₁, x₂, x₃, x₄)) = x₂ + x₄   
POL(U92(x₁, x₂, x₃, x₄)) = x₂ + x₄   
POL(U93(x₁, x₂, x₃, x₄)) = x₂ + x₄   
POL(activate(x₁)) = x₁   
POL(cons(x₁, x₂)) = x₁ + x₂   
POL(isNat(x₁)) = 0   
POL(isNatIList(x₁)) = x₁   
POL(isNatList(x₁)) = 1   
POL(length(x₁)) = 1   
POL(n__0) = 0   
POL(n__cons(x₁, x₂)) = x₁ + x₂   
POL(n__length(x₁)) = 1   
POL(n__nil) = 0   
POL(n__s(x₁)) = 0   
POL(n__take(x₁, x₂)) = x₂   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x₁)) = 0   
POL(take(x₁, x₂)) = x₂   
POL(tt) = 0   
POL(zeros) = 0   

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

activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
take(X1, X2) → n__take(X1, X2)
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
0 → n__0
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U11(tt) → tt
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(tt) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(61) Obligation:

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

U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(activate(x0))), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(62) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(activate(x0))), activate(y1))

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

POL(0) = 0   
POL(ISNATLIST(x₁)) = x₁   
POL(U11(x₁)) = x₁   
POL(U21(x₁)) = 1   
POL(U31(x₁)) = 1 + x₁   
POL(U41(x₁, x₂)) = 1   
POL(U42(x₁)) = 1   
POL(U51(x₁, x₂)) = x₂   
POL(U51¹(x₁, x₂)) = x₂   
POL(U52(x₁)) = x₁   
POL(U61(x₁, x₂)) = 1   
POL(U62(x₁)) = 1   
POL(U71(x₁, x₂, x₃)) = x₁ + x₃   
POL(U72(x₁, x₂)) = 1   
POL(U81(x₁)) = 1   
POL(U91(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₄   
POL(U92(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₄   
POL(U93(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₄   
POL(activate(x₁)) = x₁   
POL(cons(x₁, x₂)) = x₁ + x₂   
POL(isNat(x₁)) = x₁   
POL(isNatIList(x₁)) = 1 + x₁   
POL(isNatList(x₁)) = x₁   
POL(length(x₁)) = x₁   
POL(n__0) = 0   
POL(n__cons(x₁, x₂)) = x₁ + x₂   
POL(n__length(x₁)) = x₁   
POL(n__nil) = 1   
POL(n__s(x₁)) = 1   
POL(n__take(x₁, x₂)) = 1 + x₂   
POL(n__zeros) = 0   
POL(nil) = 1   
POL(s(x₁)) = 1   
POL(take(x₁, x₂)) = 1 + x₂   
POL(tt) = 1   
POL(zeros) = 0   

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

activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
take(X1, X2) → n__take(X1, X2)
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
s(X) → n__s(X)
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
cons(X1, X2) → n__cons(X1, X2)
0 → n__0
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U11(tt) → tt
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(tt) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(63) Obligation:

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

U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(64) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(x0)))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U51¹(x₁, x₂) ) = 2x₂ + 2

POL( ISNATLIST(x₁) ) = x₁ + 2

POL( U41(x₁, x₂) ) = 2

POL( U51(x₁, x₂) ) = 1

POL( U61(x₁, x₂) ) = max{0, -1}

POL( U71(x₁, ..., x₃) ) = 1

POL( U72(x₁, x₂) ) = 1

POL( U91(x₁, ..., x₄) ) = max{0, -2}

POL( U92(x₁, ..., x₄) ) = max{0, -2}

POL( U93(x₁, ..., x₄) ) = max{0, -1}

POL( cons(x₁, x₂) ) = 2x₂

POL( length(x₁) ) = 1

POL( s(x₁) ) = 1

POL( take(x₁, x₂) ) = max{0, -2}

POL( U11(x₁) ) = max{0, -2}

POL( isNatList(x₁) ) = 2

POL( U21(x₁) ) = max{0, -2}

POL( isNat(x₁) ) = max{0, 2x₁ - 1}

POL( U31(x₁) ) = max{0, -1}

POL( U42(x₁) ) = 2

POL( isNatIList(x₁) ) = 2

POL( U52(x₁) ) = max{0, -2}

POL( U62(x₁) ) = max{0, -2}

POL( n__take(x₁, x₂) ) = 0

POL( activate(x₁) ) = x₁

POL( n__zeros ) = 0

POL( zeros ) = 0

POL( n__0 ) = 0

POL( 0 ) = 0

POL( n__length(x₁) ) = 1

POL( n__s(x₁) ) = 1

POL( n__cons(x₁, x₂) ) = 2x₂

POL( n__nil ) = 0

POL( nil ) = 0

POL( U81(x₁) ) = max{0, -2}

POL( tt ) = 0

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

activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
take(X1, X2) → n__take(X1, X2)
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
0 → n__0
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U11(tt) → tt
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(tt) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(65) Obligation:

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

U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(66) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))

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

POL(U511(x1, x2)) =

-I

+

0A

·

x1

+

1A

·

x2

POL(tt) =

2A

POL(n__take(x1, x2)) =

3A

+

0A

·

x1

+

2A

·

x2

POL(ISNATLIST(x1)) =

2A

+

0A

·

x1

POL(take(x1, x2)) =

3A

+

0A

·

x1

+

2A

·

x2

POL(activate(x1)) =

1A

+

0A

·

x1

POL(n__cons(x1, x2)) =

0A

+

0A

·

x1

+

1A

·

x2

POL(cons(x1, x2)) =

0A

+

0A

·

x1

+

1A

·

x2

POL(isNat(x1)) =

2A

+

0A

·

x1

POL(n__zeros) =

0A

POL(0) =

0A

POL(n__0) =

0A

POL(zeros) =

1A

POL(n__length(x1)) =

3A

+

1A

·

x1

POL(length(x1)) =

3A

+

1A

·

x1

POL(n__s(x1)) =

4A

+

1A

·

x1

POL(s(x1)) =

4A

+

1A

·

x1

POL(n__nil) =

2A

POL(nil) =

2A

POL(U81(x1)) =

-I

+

0A

·

x1

POL(isNatIList(x1)) =

3A

+

0A

·

x1

POL(U91(x1, x2, x3, x4)) =

4A

+

0A

·

x1

+

3A

·

x2

+

1A

·

x3

+

2A

·

x4

POL(U11(x1)) =

3A

+

0A

·

x1

POL(isNatList(x1)) =

1A

+

0A

·

x1

POL(U21(x1)) =

-I

+

0A

·

x1

POL(U71(x1, x2, x3)) =

-I

+

2A

·

x1

+

2A

·

x2

+

0A

·

x3

POL(U51(x1, x2)) =

1A

+

-I

·

x1

+

0A

·

x2

POL(U61(x1, x2)) =

3A

+

-I

·

x1

+

-I

·

x2

POL(U62(x1)) =

3A

+

-I

·

x1

POL(U31(x1)) =

2A

+

-I

·

x1

POL(U41(x1, x2)) =

-I

+

0A

·

x1

+

0A

·

x2

POL(U42(x1)) =

2A

+

-I

·

x1

POL(U52(x1)) =

0A

+

0A

·

x1

POL(U72(x1, x2)) =

4A

+

0A

·

x1

+

2A

·

x2

POL(U92(x1, x2, x3, x4)) =

4A

+

0A

·

x1

+

3A

·

x2

+

1A

·

x3

+

1A

·

x4

POL(U93(x1, x2, x3, x4)) =

4A

+

-I

·

x1

+

3A

·

x2

+

1A

·

x3

+

1A

·

x4

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

activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
take(X1, X2) → n__take(X1, X2)
cons(X1, X2) → n__cons(X1, X2)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
0 → n__0
length(nil) → 0
length(X) → n__length(X)
s(X) → n__s(X)
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U11(tt) → tt
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
isNatIList(n__zeros) → tt
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(tt) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(67) Obligation:

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

U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(68) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

POL(U511(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 1, 0 ]

·

x2

POL(tt) =

/ 0 \ \ 0 /

POL(n__cons(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 1 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

POL(ISNATLIST(x1)) =

0

+

[ 1, 0 ]

·

x1

POL(cons(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 1 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

POL(activate(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 1 /

·

x1

POL(isNat(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(n__zeros) =

/ 0 \ \ 1 /

POL(0) =

/ 0 \ \ 0 /

POL(n__0) =

/ 0 \ \ 0 /

POL(zeros) =

/ 0 \ \ 1 /

POL(n__take(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 1 /

·

x2

POL(take(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 1 /

·

x2

POL(n__length(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(length(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(n__s(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(s(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(n__nil) =

/ 0 \ \ 0 /

POL(nil) =

/ 0 \ \ 0 /

POL(U11(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(isNatList(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 1 0 /

·

x1

POL(U21(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U71(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U51(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U61(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U62(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(isNatIList(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

POL(U31(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U41(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U42(x1)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U52(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U72(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U81(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U91(x1, x2, x3, x4)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 1 \ \ 0 0 /

·

x4

POL(U92(x1, x2, x3, x4)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 1 \ \ 0 0 /

·

x4

POL(U93(x1, x2, x3, x4)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 1 \ \ 0 0 /

·

x4

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

activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
cons(X1, X2) → n__cons(X1, X2)
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
0 → n__0
take(X1, X2) → n__take(X1, X2)
length(nil) → 0
length(X) → n__length(X)
s(X) → n__s(X)
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U11(tt) → tt
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(tt) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
take(0, IL) → U81(isNatIList(IL))
U81(tt) → nil
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(69) Obligation:

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

U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(70) NonTerminationLoopProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = U51¹(isNat(n__0), activate(n__zeros)) evaluates to t =U51¹(isNat(n__0), activate(n__zeros))

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

U51¹(isNat(n__0), activate(n__zeros)) → U51¹(isNat(n__0), n__zeros)
with rule activate(X) → X at position [1] and matcher [X / n__zeros]

U51¹(isNat(n__0), n__zeros) → U51¹(tt, n__zeros)
with rule isNat(n__0) → tt at position [0] and matcher [ ]

U51¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
with rule U51¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros)) at position [] and matcher [ ]

ISNATLIST(n__cons(n__0, n__zeros)) → U51¹(isNat(n__0), activate(n__zeros))
with rule ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.

(71) NO

(72) Obligation:

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

LENGTH(cons(N, L)) → U71¹(isNatList(activate(L)), activate(L), N)
U71¹(tt, L, N) → U72¹(isNat(activate(N)), activate(L))
U72¹(tt, L) → LENGTH(activate(L))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(73) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

U71¹(tt, L, N) → U72¹(isNat(activate(N)), activate(L))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( LENGTH(x₁) ) = 2x₁

POL( U71¹(x₁, ..., x₃) ) = max{0, 2x₁ + 2x₂ - 1}

POL( U72¹(x₁, x₂) ) = 2x₂

POL( U41(x₁, x₂) ) = 2

POL( U51(x₁, x₂) ) = 2x₂

POL( U61(x₁, x₂) ) = x₁ + x₂

POL( U71(x₁, ..., x₃) ) = 2x₂

POL( U72(x₁, x₂) ) = 2x₂

POL( U91(x₁, ..., x₄) ) = 2x₂ + 2x₃

POL( U92(x₁, ..., x₄) ) = 2x₂ + 2x₃

POL( U93(x₁, ..., x₄) ) = 2x₂ + 2x₃

POL( cons(x₁, x₂) ) = 2x₂

POL( isNatList(x₁) ) = x₁

POL( length(x₁) ) = x₁

POL( s(x₁) ) = 2x₁

POL( take(x₁, x₂) ) = x₁ + x₂

POL( isNat(x₁) ) = x₁

POL( U11(x₁) ) = x₁

POL( U21(x₁) ) = max{0, 2x₁ - 1}

POL( U31(x₁) ) = 1

POL( U42(x₁) ) = 2

POL( isNatIList(x₁) ) = 2

POL( U52(x₁) ) = 2x₁

POL( U62(x₁) ) = 1

POL( n__take(x₁, x₂) ) = x₁ + x₂

POL( activate(x₁) ) = x₁

POL( n__zeros ) = 0

POL( zeros ) = 0

POL( n__0 ) = 2

POL( 0 ) = 2

POL( n__length(x₁) ) = x₁

POL( n__s(x₁) ) = 2x₁

POL( n__cons(x₁, x₂) ) = 2x₂

POL( n__nil ) = 2

POL( nil ) = 2

POL( tt ) = 1

POL( U81(x₁) ) = 2

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

activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
take(X1, X2) → n__take(X1, X2)
length(nil) → 0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
U11(tt) → tt
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(tt) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
take(0, IL) → U81(isNatIList(IL))
U81(tt) → nil
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
zeros → cons(0, n__zeros)
zeros → n__zeros
0 → n__0
nil → n__nil

(74) Obligation:

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

LENGTH(cons(N, L)) → U71¹(isNatList(activate(L)), activate(L), N)
U72¹(tt, L) → LENGTH(activate(L))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(75) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(76) TRUE

(77) Obligation:

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

ISNATILIST(n__cons(V1, V2)) → U41¹(isNat(activate(V1)), activate(V2))
U41¹(tt, V2) → ISNATILIST(activate(V2))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(78) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(V1, V2)) → U41¹(isNat(activate(V1)), activate(V2)) at position [0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(n__zeros, y1)) → U41¹(isNat(zeros), activate(y1))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U41¹(isNat(take(activate(x0), activate(x1))), activate(y1))
ISNATILIST(n__cons(n__0, y1)) → U41¹(isNat(0), activate(y1))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))

(79) Obligation:

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

U41¹(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(n__zeros, y1)) → U41¹(isNat(zeros), activate(y1))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U41¹(isNat(take(activate(x0), activate(x1))), activate(y1))
ISNATILIST(n__cons(n__0, y1)) → U41¹(isNat(0), activate(y1))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(80) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U41¹(tt, V2) → ISNATILIST(activate(V2)) at position [0] we obtained the following new rules [LPAR04]:

U41¹(tt, n__zeros) → ISNATILIST(zeros)
U41¹(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
U41¹(tt, n__0) → ISNATILIST(0)
U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U41¹(tt, n__nil) → ISNATILIST(nil)
U41¹(tt, x0) → ISNATILIST(x0)

(81) Obligation:

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

ISNATILIST(n__cons(n__zeros, y1)) → U41¹(isNat(zeros), activate(y1))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U41¹(isNat(take(activate(x0), activate(x1))), activate(y1))
ISNATILIST(n__cons(n__0, y1)) → U41¹(isNat(0), activate(y1))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, n__zeros) → ISNATILIST(zeros)
U41¹(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
U41¹(tt, n__0) → ISNATILIST(0)
U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U41¹(tt, n__nil) → ISNATILIST(nil)
U41¹(tt, x0) → ISNATILIST(x0)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(82) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(n__zeros, y1)) → U41¹(isNat(zeros), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(n__zeros), activate(y0))

(83) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U41¹(isNat(take(activate(x0), activate(x1))), activate(y1))
ISNATILIST(n__cons(n__0, y1)) → U41¹(isNat(0), activate(y1))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, n__zeros) → ISNATILIST(zeros)
U41¹(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
U41¹(tt, n__0) → ISNATILIST(0)
U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U41¹(tt, n__nil) → ISNATILIST(nil)
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(n__zeros), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(84) DependencyGraphProof (EQUIVALENT transformation)

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

(85) Obligation:

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

U41¹(tt, n__zeros) → ISNATILIST(zeros)
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U41¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U41¹(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__0, y1)) → U41¹(isNat(0), activate(y1))
U41¹(tt, n__0) → ISNATILIST(0)
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
U41¹(tt, n__nil) → ISNATILIST(nil)
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(86) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U41¹(tt, n__zeros) → ISNATILIST(zeros) at position [0] we obtained the following new rules [LPAR04]:

U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
U41¹(tt, n__zeros) → ISNATILIST(n__zeros)

(87) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U41¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U41¹(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__0, y1)) → U41¹(isNat(0), activate(y1))
U41¹(tt, n__0) → ISNATILIST(0)
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
U41¹(tt, n__nil) → ISNATILIST(nil)
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(0, n__zeros)), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
U41¹(tt, n__zeros) → ISNATILIST(n__zeros)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(88) DependencyGraphProof (EQUIVALENT transformation)

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

(89) Obligation:

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

U41¹(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U41¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U41¹(tt, n__0) → ISNATILIST(0)
ISNATILIST(n__cons(n__0, y1)) → U41¹(isNat(0), activate(y1))
U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, n__nil) → ISNATILIST(nil)
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(90) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U41¹(tt, n__0) → ISNATILIST(0) at position [0] we obtained the following new rules [LPAR04]:

U41¹(tt, n__0) → ISNATILIST(n__0)

(91) Obligation:

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

U41¹(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U41¹(isNat(take(activate(x0), activate(x1))), activate(y1))
ISNATILIST(n__cons(n__0, y1)) → U41¹(isNat(0), activate(y1))
U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, n__nil) → ISNATILIST(nil)
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(0, n__zeros)), activate(y0))
U41¹(tt, n__0) → ISNATILIST(n__0)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(92) DependencyGraphProof (EQUIVALENT transformation)

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

(93) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U41¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U41¹(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__0, y1)) → U41¹(isNat(0), activate(y1))
U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, n__nil) → ISNATILIST(nil)
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(94) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(n__0, y1)) → U41¹(isNat(0), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))

(95) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U41¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U41¹(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, n__nil) → ISNATILIST(nil)
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(96) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U41¹(tt, n__nil) → ISNATILIST(nil) at position [0] we obtained the following new rules [LPAR04]:

U41¹(tt, n__nil) → ISNATILIST(n__nil)

(97) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U41¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U41¹(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__nil) → ISNATILIST(n__nil)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(98) DependencyGraphProof (EQUIVALENT transformation)

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

(99) Obligation:

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

U41¹(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U41¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(100) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1)) at position [0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(n__nil, y0)) → U41¹(isNat(n__nil), activate(y0))

(101) Obligation:

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

U41¹(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U41¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
ISNATILIST(n__cons(n__nil, y0)) → U41¹(isNat(n__nil), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(102) DependencyGraphProof (EQUIVALENT transformation)

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

(103) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U41¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U41¹(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(0, n__zeros)), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(104) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(n__cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(n__0, n__zeros)), activate(y0))

(105) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U41¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U41¹(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(n__cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(n__0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(106) DependencyGraphProof (EQUIVALENT transformation)

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

(107) Obligation:

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

U41¹(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U41¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(n__0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(108) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]:

U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros))

(109) Obligation:

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

U41¹(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U41¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(n__0, n__zeros)), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(110) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(n__0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules [LPAR04]:

ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(n__cons(n__0, n__zeros)), activate(y0))

(111) Obligation:

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

U41¹(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__take(x0, x1), y1)) → U41¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros))
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(n__cons(n__0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(112) DependencyGraphProof (EQUIVALENT transformation)

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

(113) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U41¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U41¹(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(114) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U41¹(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]:

U41¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))

(115) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U41¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U41¹(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))
U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(116) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(activate(x0))), activate(y1))

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

POL(0) = 0   
POL(ISNATILIST(x₁)) = x₁   
POL(U11(x₁)) = 0   
POL(U21(x₁)) = 0   
POL(U31(x₁)) = 0   
POL(U41(x₁, x₂)) = x₂   
POL(U41¹(x₁, x₂)) = x₂   
POL(U42(x₁)) = 0   
POL(U51(x₁, x₂)) = x₂   
POL(U52(x₁)) = 0   
POL(U61(x₁, x₂)) = x₂   
POL(U62(x₁)) = 0   
POL(U71(x₁, x₂, x₃)) = 1   
POL(U72(x₁, x₂)) = 0   
POL(U81(x₁)) = 0   
POL(U91(x₁, x₂, x₃, x₄)) = x₂ + x₄   
POL(U92(x₁, x₂, x₃, x₄)) = x₂ + x₄   
POL(U93(x₁, x₂, x₃, x₄)) = x₂ + x₄   
POL(activate(x₁)) = x₁   
POL(cons(x₁, x₂)) = x₁ + x₂   
POL(isNat(x₁)) = 0   
POL(isNatIList(x₁)) = x₁   
POL(isNatList(x₁)) = x₁   
POL(length(x₁)) = 1   
POL(n__0) = 0   
POL(n__cons(x₁, x₂)) = x₁ + x₂   
POL(n__length(x₁)) = 1   
POL(n__nil) = 0   
POL(n__s(x₁)) = 0   
POL(n__take(x₁, x₂)) = x₂   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x₁)) = 0   
POL(take(x₁, x₂)) = x₂   
POL(tt) = 0   
POL(zeros) = 0   

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

activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
take(X1, X2) → n__take(X1, X2)
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
0 → n__0
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U11(tt) → tt
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(tt) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(117) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U41¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U41¹(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(118) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(activate(x0))), activate(y1))

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

POL(0) = 0   
POL(ISNATILIST(x₁)) = x₁   
POL(U11(x₁)) = 0   
POL(U21(x₁)) = 0   
POL(U31(x₁)) = 0   
POL(U41(x₁, x₂)) = x₂   
POL(U41¹(x₁, x₂)) = x₂   
POL(U42(x₁)) = 0   
POL(U51(x₁, x₂)) = x₂   
POL(U52(x₁)) = 0   
POL(U61(x₁, x₂)) = x₂   
POL(U62(x₁)) = 0   
POL(U71(x₁, x₂, x₃)) = 1 + x₂ + x₃   
POL(U72(x₁, x₂)) = 1 + x₂   
POL(U81(x₁)) = 0   
POL(U91(x₁, x₂, x₃, x₄)) = x₂ + x₄   
POL(U92(x₁, x₂, x₃, x₄)) = x₂ + x₄   
POL(U93(x₁, x₂, x₃, x₄)) = x₂ + x₄   
POL(activate(x₁)) = x₁   
POL(cons(x₁, x₂)) = x₁ + x₂   
POL(isNat(x₁)) = 0   
POL(isNatIList(x₁)) = x₁   
POL(isNatList(x₁)) = x₁   
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)) = 1   
POL(n__take(x₁, x₂)) = x₂   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x₁)) = 1   
POL(take(x₁, x₂)) = x₂   
POL(tt) = 0   
POL(zeros) = 0   

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

activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
take(X1, X2) → n__take(X1, X2)
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
0 → n__0
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U11(tt) → tt
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(tt) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(119) Obligation:

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

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U41¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U41¹(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(120) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

ISNATILIST(n__cons(n__take(x0, x1), y1)) → U41¹(isNat(take(activate(x0), activate(x1))), activate(y1))

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

POL(0) = 0   
POL(ISNATILIST(x₁)) = x₁   
POL(U11(x₁)) = 0   
POL(U21(x₁)) = 0   
POL(U31(x₁)) = 1   
POL(U41(x₁, x₂)) = 1 + x₂   
POL(U41¹(x₁, x₂)) = x₂   
POL(U42(x₁)) = 1   
POL(U51(x₁, x₂)) = x₂   
POL(U52(x₁)) = 0   
POL(U61(x₁, x₂)) = 1 + x₂   
POL(U62(x₁)) = 1   
POL(U71(x₁, x₂, x₃)) = x₂ + x₃   
POL(U72(x₁, x₂)) = x₂   
POL(U81(x₁)) = 1   
POL(U91(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₄   
POL(U92(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₄   
POL(U93(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₄   
POL(activate(x₁)) = x₁   
POL(cons(x₁, x₂)) = x₁ + x₂   
POL(isNat(x₁)) = 0   
POL(isNatIList(x₁)) = 1 + x₁   
POL(isNatList(x₁)) = x₁   
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)) = 0   
POL(n__take(x₁, x₂)) = 1 + x₂   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x₁)) = 0   
POL(take(x₁, x₂)) = 1 + x₂   
POL(tt) = 0   
POL(zeros) = 0   

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

activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
take(X1, X2) → n__take(X1, X2)
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
0 → n__0
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U11(tt) → tt
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(tt) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(121) Obligation:

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

U41¹(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(122) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

U41¹(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U41¹(x₁, x₂) ) = 2x₂

POL( ISNATILIST(x₁) ) = x₁

POL( U41(x₁, x₂) ) = 2x₁

POL( U51(x₁, x₂) ) = 2

POL( U61(x₁, x₂) ) = max{0, -2}

POL( U71(x₁, ..., x₃) ) = 2x₂ + 1

POL( U72(x₁, x₂) ) = 2x₂ + 1

POL( U91(x₁, ..., x₄) ) = max{0, -1}

POL( U92(x₁, ..., x₄) ) = max{0, -2}

POL( U93(x₁, ..., x₄) ) = max{0, -2}

POL( cons(x₁, x₂) ) = 2x₂

POL( length(x₁) ) = 2x₁ + 2

POL( s(x₁) ) = max{0, -1}

POL( take(x₁, x₂) ) = max{0, -1}

POL( U11(x₁) ) = max{0, -2}

POL( isNatList(x₁) ) = 2

POL( U21(x₁) ) = max{0, -2}

POL( isNat(x₁) ) = 0

POL( U31(x₁) ) = max{0, -2}

POL( U42(x₁) ) = max{0, -1}

POL( isNatIList(x₁) ) = max{0, -2}

POL( U52(x₁) ) = max{0, -2}

POL( U62(x₁) ) = max{0, -2}

POL( n__take(x₁, x₂) ) = max{0, -1}

POL( activate(x₁) ) = x₁

POL( n__zeros ) = 0

POL( zeros ) = 0

POL( n__0 ) = 0

POL( 0 ) = 0

POL( n__length(x₁) ) = 2x₁ + 2

POL( n__s(x₁) ) = 0

POL( n__cons(x₁, x₂) ) = 2x₂

POL( n__nil ) = 0

POL( nil ) = 0

POL( U81(x₁) ) = 0

POL( tt ) = 0

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

activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
take(X1, X2) → n__take(X1, X2)
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
0 → n__0
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U11(tt) → tt
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(tt) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(123) Obligation:

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

U41¹(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(124) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

U41¹(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U41¹(x₁, x₂) ) = max{0, x₁ + 2x₂ - 2}

POL( ISNATILIST(x₁) ) = x₁

POL( U41(x₁, x₂) ) = 2

POL( U51(x₁, x₂) ) = 2

POL( U61(x₁, x₂) ) = x₁

POL( U71(x₁, ..., x₃) ) = 1

POL( U72(x₁, x₂) ) = 1

POL( U91(x₁, ..., x₄) ) = max{0, -2}

POL( U92(x₁, ..., x₄) ) = max{0, -1}

POL( U93(x₁, ..., x₄) ) = max{0, -2}

POL( cons(x₁, x₂) ) = 2x₂

POL( length(x₁) ) = 1

POL( s(x₁) ) = 1

POL( take(x₁, x₂) ) = max{0, -1}

POL( U11(x₁) ) = x₁

POL( isNatList(x₁) ) = 2

POL( U21(x₁) ) = max{0, 2x₁ - 2}

POL( isNat(x₁) ) = 2

POL( U31(x₁) ) = x₁

POL( U42(x₁) ) = max{0, 2x₁ - 2}

POL( isNatIList(x₁) ) = 2

POL( U52(x₁) ) = 2

POL( U62(x₁) ) = x₁

POL( n__take(x₁, x₂) ) = 0

POL( activate(x₁) ) = x₁

POL( n__zeros ) = 0

POL( zeros ) = 0

POL( n__0 ) = 0

POL( 0 ) = 0

POL( n__length(x₁) ) = 1

POL( n__s(x₁) ) = 1

POL( n__cons(x₁, x₂) ) = 2x₂

POL( n__nil ) = 0

POL( nil ) = 0

POL( U81(x₁) ) = max{0, -2}

POL( tt ) = 2

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

activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
take(X1, X2) → n__take(X1, X2)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
0 → n__0
length(nil) → 0
length(X) → n__length(X)
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U11(tt) → tt
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
isNatIList(n__zeros) → tt
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(tt) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(125) Obligation:

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

U41¹(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(126) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

U41¹(tt, n__take(x0, x1)) → ISNATILIST(take(activate(x0), activate(x1)))

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

POL(U411(x1, x2)) =

2A

+

-I

·

x1

+

1A

·

x2

POL(tt) =

2A

POL(n__take(x1, x2)) =

2A

+

1A

·

x1

+

-I

·

x2

POL(ISNATILIST(x1)) =

2A

+

0A

·

x1

POL(take(x1, x2)) =

2A

+

1A

·

x1

+

-I

·

x2

POL(activate(x1)) =

1A

+

0A

·

x1

POL(n__cons(x1, x2)) =

0A

+

-I

·

x1

+

1A

·

x2

POL(isNat(x1)) =

0A

+

0A

·

x1

POL(cons(x1, x2)) =

0A

+

-I

·

x1

+

1A

·

x2

POL(n__0) =

4A

POL(n__zeros) =

0A

POL(0) =

4A

POL(zeros) =

1A

POL(n__length(x1)) =

2A

+

0A

·

x1

POL(length(x1)) =

2A

+

0A

·

x1

POL(n__s(x1)) =

2A

+

1A

·

x1

POL(s(x1)) =

2A

+

1A

·

x1

POL(n__nil) =

5A

POL(nil) =

5A

POL(U81(x1)) =

5A

+

-I

·

x1

POL(isNatIList(x1)) =

2A

+

-I

·

x1

POL(U91(x1, x2, x3, x4)) =

3A

+

-I

·

x1

+

-I

·

x2

+

2A

·

x3

+

-I

·

x4

POL(U11(x1)) =

-I

+

0A

·

x1

POL(isNatList(x1)) =

1A

+

0A

·

x1

POL(U21(x1)) =

-I

+

0A

·

x1

POL(U71(x1, x2, x3)) =

-I

+

1A

·

x1

+

1A

·

x2

+

-I

·

x3

POL(U51(x1, x2)) =

1A

+

-I

·

x1

+

0A

·

x2

POL(U61(x1, x2)) =

0A

+

1A

·

x1

+

-I

·

x2

POL(U62(x1)) =

3A

+

0A

·

x1

POL(U31(x1)) =

2A

+

-I

·

x1

POL(U41(x1, x2)) =

2A

+

-I

·

x1

+

-I

·

x2

POL(U42(x1)) =

2A

+

-I

·

x1

POL(U52(x1)) =

0A

+

0A

·

x1

POL(U72(x1, x2)) =

3A

+

-I

·

x1

+

1A

·

x2

POL(U92(x1, x2, x3, x4)) =

-I

+

1A

·

x1

+

-I

·

x2

+

2A

·

x3

+

-I

·

x4

POL(U93(x1, x2, x3, x4)) =

3A

+

-I

·

x1

+

-I

·

x2

+

2A

·

x3

+

-I

·

x4

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

activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
take(X1, X2) → n__take(X1, X2)
cons(X1, X2) → n__cons(X1, X2)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
0 → n__0
length(nil) → 0
length(X) → n__length(X)
s(X) → n__s(X)
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U11(tt) → tt
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
isNatIList(n__zeros) → tt
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(tt) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(127) Obligation:

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

ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

(128) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(activate(x0), x1)), activate(y1))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

POL(ISNATILIST(x1)) =

0

+

[ 0, 1 ]

·

x1

POL(n__cons(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(U411(x1, x2)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 1 ]

·

x2

POL(isNat(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(cons(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(activate(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 1 /

·

x1

POL(tt) =

/ 0 \ \ 0 /

POL(n__0) =

/ 0 \ \ 0 /

POL(n__zeros) =

/ 1 \ \ 0 /

POL(0) =

/ 0 \ \ 0 /

POL(zeros) =

/ 1 \ \ 0 /

POL(n__take(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(take(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

POL(n__length(x1)) =

/ 1 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(length(x1)) =

/ 1 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(n__s(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(s(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(n__nil) =

/ 0 \ \ 0 /

POL(nil) =

/ 0 \ \ 0 /

POL(U11(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(isNatList(x1)) =

/ 1 \ \ 1 /

+

/ 1 1 \ \ 1 1 /

·

x1

POL(U21(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

POL(U71(x1, x2, x3)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

POL(U51(x1, x2)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 1 /

·

x2

POL(U61(x1, x2)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 0 1 /

·

x2

POL(U62(x1)) =

/ 1 \ \ 1 /

+

/ 0 1 \ \ 0 0 /

·

x1

POL(isNatIList(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U31(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U41(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U42(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U52(x1)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U72(x1, x2)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U81(x1)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

POL(U91(x1, x2, x3, x4)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 1 0 /

·

x4

POL(U92(x1, x2, x3, x4)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 1 0 /

·

x4

POL(U93(x1, x2, x3, x4)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 1 0 /

·

x4

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

activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
cons(X1, X2) → n__cons(X1, X2)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
0 → n__0
take(X1, X2) → n__take(X1, X2)
length(nil) → 0
length(X) → n__length(X)
s(X) → n__s(X)
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U11(tt) → tt
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
U21(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
isNatIList(n__zeros) → tt
U62(tt) → tt
isNatIList(V) → U31(isNatList(activate(V)))
U31(tt) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
take(0, IL) → U81(isNatIList(IL))
U81(tt) → nil
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
zeros → cons(0, n__zeros)
zeros → n__zeros
nil → n__nil

(129) Obligation:

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

U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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