Termination w.r.t. Q proof of /tmp/tmpavDjow/OvConsOS_nokinds-noand

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

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

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.

↳ QTRS

  ↳ DependencyPairsProof

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.

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

ISNATILIST(V) → ACTIVATE(V)
ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
U92¹(tt, IL, M, N) → ISNAT(activate(N))
U72¹(tt, L) → LENGTH(activate(L))
ISNATLIST(n__take(V1, V2)) → U61¹(isNat(activate(V1)), activate(V2))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U93¹(tt, IL, M, N) → ACTIVATE(M)
ISNATILIST(V) → U31¹(isNatList(activate(V)))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
U91¹(tt, IL, M, N) → ACTIVATE(N)
ACTIVATE(n__0) → 0¹
ISNATILIST(V) → ISNATLIST(activate(V))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
U61¹(tt, V2) → ACTIVATE(V2)
TAKE(s(M), cons(N, IL)) → U91¹(isNatIList(activate(IL)), activate(IL), M, N)
U72¹(tt, L) → ACTIVATE(L)
ISNAT(n__length(V1)) → U11¹(isNatList(activate(V1)))
ACTIVATE(n__zeros) → ZEROS
LENGTH(nil) → 0¹
U71¹(tt, L, N) → U72¹(isNat(activate(N)), activate(L))
U81¹(tt) → NIL
U41¹(tt, V2) → ACTIVATE(V2)
U71¹(tt, L, N) → ACTIVATE(L)
U92¹(tt, IL, M, N) → ACTIVATE(IL)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__nil) → NIL
ACTIVATE(n__s(X)) → S(activate(X))
TAKE(0, IL) → ISNATILIST(IL)
ACTIVATE(n__cons(X1, X2)) → CONS(activate(X1), X2)
U92¹(tt, IL, M, N) → ACTIVATE(M)
U93¹(tt, IL, M, N) → ACTIVATE(IL)
ISNAT(n__s(V1)) → ACTIVATE(V1)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
U71¹(tt, L, N) → ACTIVATE(N)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
LENGTH(cons(N, L)) → U71¹(isNatList(activate(L)), activate(L), N)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → U51¹(isNat(activate(V1)), activate(V2))
U71¹(tt, L, N) → ISNAT(activate(N))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
U93¹(tt, IL, M, N) → CONS(activate(N), n__take(activate(M), activate(IL)))
U51¹(tt, V2) → ISNATLIST(activate(V2))
U51¹(tt, V2) → U52¹(isNatList(activate(V2)))
U92¹(tt, IL, M, N) → ACTIVATE(N)
U51¹(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
TAKE(0, IL) → U81¹(isNatIList(IL))
U41¹(tt, V2) → ISNATILIST(activate(V2))
U93¹(tt, IL, M, N) → ACTIVATE(N)
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
U91¹(tt, IL, M, N) → ACTIVATE(M)
ISNAT(n__s(V1)) → U21¹(isNat(activate(V1)))
U41¹(tt, V2) → U42¹(isNatIList(activate(V2)))
U91¹(tt, IL, M, N) → U92¹(isNat(activate(M)), activate(IL), activate(M), activate(N))
LENGTH(cons(N, L)) → ACTIVATE(L)
U92¹(tt, IL, M, N) → U93¹(isNat(activate(N)), activate(IL), activate(M), activate(N))
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → U41¹(isNat(activate(V1)), activate(V2))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
U61¹(tt, V2) → ISNATILIST(activate(V2))
U91¹(tt, IL, M, N) → ISNAT(activate(M))
U72¹(tt, L) → S(length(activate(L)))
ZEROS → CONS(0, n__zeros)
U91¹(tt, IL, M, N) → ACTIVATE(IL)
ZEROS → 0¹
U61¹(tt, V2) → U62¹(isNatIList(activate(V2)))
ISNAT(n__length(V1)) → ACTIVATE(V1)
ACTIVATE(n__length(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.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 18 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

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

ISNATILIST(V) → ACTIVATE(V)
ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
U92¹(tt, IL, M, N) → ISNAT(activate(N))
U72¹(tt, L) → LENGTH(activate(L))
ISNATLIST(n__take(V1, V2)) → U61¹(isNat(activate(V1)), activate(V2))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U93¹(tt, IL, M, N) → ACTIVATE(M)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
U91¹(tt, IL, M, N) → ACTIVATE(N)
ISNATILIST(V) → ISNATLIST(activate(V))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
U61¹(tt, V2) → ACTIVATE(V2)
TAKE(s(M), cons(N, IL)) → U91¹(isNatIList(activate(IL)), activate(IL), M, N)
U72¹(tt, L) → ACTIVATE(L)
U71¹(tt, L, N) → U72¹(isNat(activate(N)), activate(L))
U41¹(tt, V2) → ACTIVATE(V2)
U92¹(tt, IL, M, N) → ACTIVATE(IL)
U71¹(tt, L, N) → ACTIVATE(L)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
TAKE(0, IL) → ISNATILIST(IL)
U92¹(tt, IL, M, N) → ACTIVATE(M)
ISNAT(n__s(V1)) → ACTIVATE(V1)
U93¹(tt, IL, M, N) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
U71¹(tt, L, N) → ACTIVATE(N)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
LENGTH(cons(N, L)) → U71¹(isNatList(activate(L)), activate(L), N)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → U51¹(isNat(activate(V1)), activate(V2))
U71¹(tt, L, N) → ISNAT(activate(N))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
U51¹(tt, V2) → ISNATLIST(activate(V2))
U92¹(tt, IL, M, N) → ACTIVATE(N)
U51¹(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
U41¹(tt, V2) → ISNATILIST(activate(V2))
U93¹(tt, IL, M, N) → ACTIVATE(N)
U91¹(tt, IL, M, N) → ACTIVATE(M)
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
U91¹(tt, IL, M, N) → U92¹(isNat(activate(M)), activate(IL), activate(M), activate(N))
LENGTH(cons(N, L)) → ACTIVATE(L)
U92¹(tt, IL, M, N) → U93¹(isNat(activate(N)), activate(IL), activate(M), activate(N))
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ISNATILIST(n__cons(V1, V2)) → U41¹(isNat(activate(V1)), activate(V2))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
U61¹(tt, V2) → ISNATILIST(activate(V2))
U91¹(tt, IL, M, N) → ISNAT(activate(M))
U91¹(tt, IL, M, N) → ACTIVATE(IL)
ISNAT(n__length(V1)) → ACTIVATE(V1)
ACTIVATE(n__length(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.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.

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

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(ISNAT(x₁)) = x₁
POL(ISNATILIST(x₁)) = x₁
POL(ISNATLIST(x₁)) = x₁
POL(LENGTH(x₁)) = 1 + x₁
POL(TAKE(x₁, x₂)) = 1 + 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)) = 1 + x₂ + x₃
POL(U72(x₁, x₂)) = 1 + x₂
POL(U72¹(x₁, x₂)) = 1 + 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 [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

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

ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
U51¹(tt, V2) → ISNATLIST(activate(V2))
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
U92¹(tt, IL, M, N) → ISNAT(activate(N))
U72¹(tt, L) → LENGTH(activate(L))
U92¹(tt, IL, M, N) → ACTIVATE(N)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U51¹(tt, V2) → ACTIVATE(V2)
U93¹(tt, IL, M, N) → ACTIVATE(M)
ACTIVATE(n__s(X)) → ACTIVATE(X)
U91¹(tt, IL, M, N) → ACTIVATE(N)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(V) → ISNATLIST(activate(V))
U93¹(tt, IL, M, N) → ACTIVATE(N)
U41¹(tt, V2) → ISNATILIST(activate(V2))
U91¹(tt, IL, M, N) → ACTIVATE(M)
U61¹(tt, V2) → ACTIVATE(V2)
U71¹(tt, L, N) → U72¹(isNat(activate(N)), activate(L))
U41¹(tt, V2) → ACTIVATE(V2)
U91¹(tt, IL, M, N) → U92¹(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92¹(tt, IL, M, N) → ACTIVATE(IL)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
U92¹(tt, IL, M, N) → U93¹(isNat(activate(N)), activate(IL), activate(M), activate(N))
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → U41¹(isNat(activate(V1)), activate(V2))
U92¹(tt, IL, M, N) → ACTIVATE(M)
ISNAT(n__s(V1)) → ACTIVATE(V1)
U93¹(tt, IL, M, N) → ACTIVATE(IL)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
U61¹(tt, V2) → ISNATILIST(activate(V2))
U91¹(tt, IL, M, N) → ISNAT(activate(M))
LENGTH(cons(N, L)) → U71¹(isNatList(activate(L)), activate(L), N)
U91¹(tt, IL, M, N) → ACTIVATE(IL)
ISNATLIST(n__cons(V1, V2)) → 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.
The approximation of the Dependency Graph [15,17,22] contains 5 SCCs with 28 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

LENGTH(cons(N, L)) → U71¹(isNatList(activate(L)), activate(L), N)
U72¹(tt, L) → LENGTH(activate(L))
U71¹(tt, L, N) → U72¹(isNat(activate(N)), 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.
We use the reduction pair processor [15].

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.

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

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

M( U81(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U92(x₁, ..., x₄) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	1	/

x₂

/	0	0	\
\	0	1	/

x₃

/	0	0	\
\	0	0	/

x₄

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( n__s(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	1	1	/

x₁

M( n__nil ) =

/	1	\
\	0	/

M( take(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	0	0	\
\	0	1	/

x₂

M( U51(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

M( tt ) =

/	1	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( U52(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	1	1	/

x₁

M( U11(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( nil ) =

/	1	\
\	0	/

M( U91(x₁, ..., x₄) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	1	/

x₂

/	0	0	\
\	1	1	/

x₃

/	0	0	\
\	0	0	/

x₄

M( n__length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( U93(x₁, ..., x₄) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	1	/

x₂

/	0	0	\
\	0	1	/

x₃

/	0	0	\
\	0	0	/

x₄

M( n__zeros ) =

/	0	\
\	0	/

M( n__cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	1	1	/

x₂

M( U72(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	1	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( U62(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	1	1	/

x₂

M( U61(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( U31(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U41(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( U42(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__take(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	0	0	\
\	0	1	/

x₂

M( length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( U21(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U71(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

/	0	0	\
\	0	1	/

x₂

/	0	0	\
\	0	0	/

x₃

Tuple symbols:

M( U71¹(x₁, ..., x₃) ) =

[

]

x₁

[

]

x₂

[

]

x₃

M( LENGTH(x₁) ) =

[

]

x₁

M( U72¹(x₁, x₂) ) =

[

]

x₁

[

]

x₂

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

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

The TRS R consists of the following rules:

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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                  ↳ QDP

                  ↳ QDP

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.
We use the reduction pair processor [15].

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.
none
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( U81(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U92(x₁, ..., x₄) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	1	0	/

x₃

/	0	0	\
\	0	0	/

x₄

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( n__s(x₁) ) =

/	1	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( n__nil ) =

/	1	\
\	0	/

M( take(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	1	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( U51(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	0	/

x₂

M( tt ) =

/	0	\
\	1	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( U52(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	1	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( U11(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( nil ) =

/	1	\
\	0	/

M( U91(x₁, ..., x₄) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	1	0	/

x₃

/	0	0	\
\	0	0	/

x₄

M( n__length(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

M( U93(x₁, ..., x₄) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	1	0	/

x₃

/	0	0	\
\	0	0	/

x₄

M( n__zeros ) =

/	0	\
\	0	/

M( n__cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	1	1	/

x₂

M( U72(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	1	\
\	0	0	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( U62(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	1	1	/

x₂

M( U61(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( U31(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( U41(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( U42(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( n__take(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	1	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( length(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

M( U21(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( U71(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

/	0	1	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

Tuple symbols:

M( ISNAT(x₁) ) =

[

]

x₁

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ PisEmptyProof

                  ↳ QDP

                  ↳ QDP

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                  ↳ QDP

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.
By narrowing [15] the rule ISNATLIST(n__cons(V1, V2)) → U51¹(isNat(activate(V1)), activate(V2)) at position [0] we obtained the following new rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                  ↳ QDP

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

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))
ISNATLIST(n__cons(n__zeros, y1)) → U51¹(isNat(zeros), activate(y1))
U51¹(tt, V2) → ISNATLIST(activate(V2))
ISNATLIST(n__cons(n__0, y1)) → U51¹(isNat(0), activate(y1))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U51¹(isNat(take(activate(x0), activate(x1))), activate(y1))
ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(activate(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.
By narrowing [15] the rule U51¹(tt, V2) → ISNATLIST(activate(V2)) at position [0] we obtained the following new rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                  ↳ QDP

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

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

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.
By narrowing [15] the rule ISNATLIST(n__cons(n__nil, y1)) → U51¹(isNat(nil), activate(y1)) at position [0] we obtained the following new rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                  ↳ QDP

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

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

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.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                  ↳ QDP

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

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

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.
By narrowing [15] the rule U51¹(tt, n__nil) → ISNATLIST(nil) at position [0] we obtained the following new rules:

U51¹(tt, n__nil) → ISNATLIST(n__nil)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                  ↳ QDP

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

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

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.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                  ↳ QDP

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

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

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.
By narrowing [15] the rule U51¹(tt, n__zeros) → ISNATLIST(zeros) at position [0] we obtained the following new rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                  ↳ QDP

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

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(n__0, y1)) → U51¹(isNat(0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
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__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(n__zeros, y1)) → U51¹(isNat(zeros), activate(y1))
U51¹(tt, n__0) → ISNATLIST(0)
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U51¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
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.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Narrowing

                  ↳ QDP

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

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

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.
By narrowing [15] the rule U51¹(tt, n__0) → ISNATLIST(0) at position [0] we obtained the following new rules:

U51¹(tt, n__0) → ISNATLIST(n__0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                  ↳ QDP

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

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

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.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                  ↳ QDP

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

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

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.
By narrowing [15] the rule ISNATLIST(n__cons(n__zeros, y1)) → U51¹(isNat(zeros), activate(y1)) at position [0] we obtained the following new rules:

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))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                  ↳ QDP

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

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

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.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                  ↳ QDP

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

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

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.
By narrowing [15] the rule ISNATLIST(n__cons(n__0, y1)) → U51¹(isNat(0), activate(y1)) at position [0] we obtained the following new rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                  ↳ QDP

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

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

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.
By narrowing [15] the rule U51¹(tt, n__zeros) → ISNATLIST(cons(0, n__zeros)) at position [0] we obtained the following new rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                  ↳ QDP

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

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(cons(0, n__zeros)), activate(y0))
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__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U51¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
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.
By narrowing [15] 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:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                  ↳ QDP

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

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
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__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(n__cons(0, n__zeros)), activate(y0))
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U51¹(isNat(take(activate(x0), activate(x1))), activate(y1))
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(cons(n__0, n__zeros)), activate(y0))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
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.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                  ↳ QDP

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

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
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__s(x0)) → ISNATLIST(s(activate(x0)))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(activate(x0))), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U51¹(isNat(take(activate(x0), activate(x1))), activate(y1))
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(cons(n__0, n__zeros)), activate(y0))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
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.
By narrowing [15] the rule U51¹(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros)) at position [0] we obtained the following new rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                  ↳ QDP

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

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

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.
By narrowing [15] 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:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                  ↳ QDP

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

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

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.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                  ↳ QDP

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

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

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.
We use the reduction pair processor [15].

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.

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

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ISNATLIST(x₁)) = x₁
POL(U11(x₁)) = 0
POL(U21(x₁)) = 0
POL(U31(x₁)) = 0
POL(U41(x₁, x₂)) = 0
POL(U42(x₁)) = 0
POL(U51(x₁, x₂)) = 0
POL(U51¹(x₁, x₂)) = x₂
POL(U52(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁, x₂, x₃)) = 0
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)) = 0
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 [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                  ↳ QDP

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

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(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U51¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U51¹(isNat(take(activate(x0), activate(x1))), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(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.
We use the reduction pair processor [15].

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.

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

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ISNATLIST(x₁)) = x₁
POL(U11(x₁)) = 0
POL(U21(x₁)) = 0
POL(U31(x₁)) = 1
POL(U41(x₁, x₂)) = 0
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)) = 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)) = 1
POL(isNatList(x₁)) = 1 + 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 [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                  ↳ QDP

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

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(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(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.
We use the reduction pair processor [15].

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.

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

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ISNATLIST(x₁)) = x₁
POL(U11(x₁)) = 0
POL(U21(x₁)) = 0
POL(U31(x₁)) = 0
POL(U41(x₁, x₂)) = 0
POL(U42(x₁)) = 0
POL(U51(x₁, x₂)) = 0
POL(U51¹(x₁, x₂)) = x₂
POL(U52(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁, x₂, x₃)) = 1
POL(U72(x₁, x₂)) = 1
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)) = 0
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)) = 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 [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                  ↳ QDP

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

ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(activate(x0), x1)), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U51¹(tt, x0) → ISNATLIST(x0)
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__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(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.
We use the reduction pair processor [15].

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.

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

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

M( U81(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U92(x₁, ..., x₄) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	1	/

x₂

/	0	0	\
\	1	1	/

x₃

/	0	0	\
\	1	0	/

x₄

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( n__s(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	0	0	/

x₁

M( n__nil ) =

/	0	\
\	0	/

M( take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( U51(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U52(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( zeros ) =

/	1	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	0	0	/

x₁

M( U11(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( nil ) =

/	0	\
\	0	/

M( U91(x₁, ..., x₄) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	1	/

x₂

/	0	0	\
\	1	1	/

x₃

/	0	0	\
\	1	0	/

x₄

M( n__length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U93(x₁, ..., x₄) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	1	/

x₂

/	0	0	\
\	1	1	/

x₃

/	0	0	\
\	1	0	/

x₄

M( n__zeros ) =

/	1	\
\	0	/

M( n__cons(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	1	0	/

x₁

/	0	0	\
\	0	1	/

x₂

M( U72(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( U62(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( cons(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	1	0	/

x₁

/	0	0	\
\	0	1	/

x₂

M( U61(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( U31(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U41(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( U42(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U21(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U71(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

Tuple symbols:

M( ISNATLIST(x₁) ) =

[

]

x₁

M( U51¹(x₁, x₂) ) =

[

]

x₁

[

]

x₂

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                  ↳ QDP

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

ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U51¹(tt, n__length(x0)) → ISNATLIST(length(activate(x0)))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(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.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.

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

Used ordering: Polynomial interpretation [25,35]:

POL(U51¹(x₁, x₂)) = (4)x_2
POL(U81(x₁)) = 0
POL(U92(x₁, x₂, x₃, x₄)) = 0
POL(activate(x₁)) = x_1
POL(n__s(x₁)) = 0
POL(n__nil) = 0
POL(take(x₁, x₂)) = 0
POL(U51(x₁, x₂)) = x_2
POL(ISNATLIST(x₁)) = x_1
POL(tt) = 0
POL(isNatList(x₁)) = x_1
POL(zeros) = 0
POL(U52(x₁)) = 0
POL(isNatIList(x₁)) = 4 + (4)x_1
POL(s(x₁)) = 0
POL(U11(x₁)) = 0
POL(isNat(x₁)) = 0
POL(nil) = 0
POL(U91(x₁, x₂, x₃, x₄)) = 0
POL(n__length(x₁)) = 2
POL(U93(x₁, x₂, x₃, x₄)) = 0
POL(n__zeros) = 0
POL(U72(x₁, x₂)) = 0
POL(n__cons(x₁, x₂)) = (4)x_2
POL(0) = 0
POL(U62(x₁)) = 0
POL(cons(x₁, x₂)) = (4)x_2
POL(U31(x₁)) = 1 + (3)x_1
POL(U61(x₁, x₂)) = 0
POL(U41(x₁, x₂)) = 3 + x_2
POL(U42(x₁)) = 3
POL(n__0) = 0
POL(n__take(x₁, x₂)) = 0
POL(length(x₁)) = 2
POL(U21(x₁)) = 0
POL(U71(x₁, x₂, x₃)) = 1

The value of delta used in the strict ordering is 6.
The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                  ↳ QDP

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

ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U51¹(tt, x0) → ISNATLIST(x0)
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__cons(x0, x1)) → ISNATLIST(cons(activate(x0), x1))
U51¹(tt, n__take(x0, x1)) → ISNATLIST(take(activate(x0), activate(x1)))
U51¹(tt, n__s(x0)) → ISNATLIST(s(activate(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.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.

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

Used ordering: Polynomial interpretation [25,35]:

POL(U51¹(x₁, x₂)) = (2)x_2
POL(U81(x₁)) = 0
POL(U92(x₁, x₂, x₃, x₄)) = 0
POL(activate(x₁)) = (2)x_1
POL(n__s(x₁)) = 4
POL(n__nil) = 0
POL(take(x₁, x₂)) = 0
POL(U51(x₁, x₂)) = 2 + (2)x_2
POL(ISNATLIST(x₁)) = x_1
POL(tt) = 0
POL(isNatList(x₁)) = 2 + x_1
POL(zeros) = 0
POL(U52(x₁)) = 2
POL(isNatIList(x₁)) = 0
POL(s(x₁)) = 4
POL(U11(x₁)) = 0
POL(isNat(x₁)) = 0
POL(nil) = 0
POL(U91(x₁, x₂, x₃, x₄)) = 0
POL(n__length(x₁)) = 4 + (4)x_1
POL(U93(x₁, x₂, x₃, x₄)) = 0
POL(n__zeros) = 0
POL(U72(x₁, x₂)) = 4
POL(n__cons(x₁, x₂)) = (4)x_2
POL(0) = 0
POL(U62(x₁)) = 0
POL(cons(x₁, x₂)) = (4)x_2
POL(U31(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U41(x₁, x₂)) = 0
POL(U42(x₁)) = 0
POL(n__0) = 0
POL(n__take(x₁, x₂)) = 0
POL(length(x₁)) = 4 + (4)x_1
POL(U21(x₁)) = 0
POL(U71(x₁, x₂, x₃)) = 4 + (4)x_2

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                  ↳ QDP

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

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

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.
We use the reduction pair processor [15].

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.

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

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

M( U81(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( U92(x₁, ..., x₄) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	1	\
\	0	0	/

x₃

/	0	0	\
\	0	0	/

x₄

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( n__s(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	1	/

x₁

M( n__nil ) =

/	0	\
\	1	/

M( take(x₁, x₂) ) =

/	0	\
\	1	/

/	0	1	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( U51(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	1	/

x₂

M( tt ) =

/	0	\
\	1	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( U52(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	1	0	/

x₁

M( s(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	1	/

x₁

M( U11(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	1	/

x₁

M( nil ) =

/	0	\
\	1	/

M( U91(x₁, ..., x₄) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	1	\
\	0	0	/

x₃

/	0	0	\
\	0	0	/

x₄

M( n__length(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	1	0	/

x₁

M( U93(x₁, ..., x₄) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	1	\
\	0	0	/

x₃

/	0	0	\
\	0	0	/

x₄

M( n__zeros ) =

/	0	\
\	0	/

M( n__cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	0	1	/

x₂

M( U72(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	0	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( U62(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	0	1	/

x₂

M( U61(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( U31(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( U41(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( U42(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( n__take(x₁, x₂) ) =

/	0	\
\	1	/

/	0	1	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( length(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	1	0	/

x₁

M( U21(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( U71(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	0	0	\
\	1	0	/

x₂

/	0	0	\
\	0	0	/

x₃

Tuple symbols:

M( ISNATLIST(x₁) ) =

[

]

x₁

M( U51¹(x₁, x₂) ) =

[

]

x₁

[

]

x₂

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ QDPOrderProof

                                                                                                  ↳ QDP

                                                                                                    ↳ QDPOrderProof

                                                                                                      ↳ QDP

                                                                                                        ↳ QDPOrderProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ NonTerminationProof

                  ↳ QDP

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

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

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.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

The TRS P consists of the following rules:

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

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

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:

Semiunifier: [ ]
Matcher: [ ]

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.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

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