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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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))
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)))
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)
ACTIVATE(n__cons(X1, X2)) → CONS(X1, X2)
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)
ACTIVATE(n__length(X)) → LENGTH(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__nil) → NIL
TAKE(0, IL) → ISNATILIST(IL)
ACTIVATE(n__s(X)) → S(X)
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)
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))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → U411(isNat(activate(V1)), activate(V2))
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)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
U911(tt, IL, M, N) → ACTIVATE(IL)
ZEROS → 01
U611(tt, V2) → U621(isNatIList(activate(V2)))
ISNAT(n__length(V1)) → 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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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))
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)))
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)
ACTIVATE(n__cons(X1, X2)) → CONS(X1, X2)
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)
ACTIVATE(n__length(X)) → LENGTH(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__nil) → NIL
TAKE(0, IL) → ISNATILIST(IL)
ACTIVATE(n__s(X)) → S(X)
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)
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))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → U411(isNat(activate(V1)), activate(V2))
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)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
U911(tt, IL, M, N) → ACTIVATE(IL)
ZEROS → 01
U611(tt, V2) → U621(isNatIList(activate(V2)))
ISNAT(n__length(V1)) → 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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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))
U921(tt, IL, M, N) → ISNAT(activate(N))
U721(tt, L) → LENGTH(activate(L))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → U611(isNat(activate(V1)), activate(V2))
U931(tt, IL, M, N) → ACTIVATE(M)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
U911(tt, IL, M, N) → ACTIVATE(N)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(V) → ISNATLIST(activate(V))
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)
U711(tt, L, N) → ACTIVATE(L)
U921(tt, IL, M, N) → ACTIVATE(IL)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → LENGTH(X)
TAKE(0, IL) → ISNATILIST(IL)
ISNAT(n__s(V1)) → ACTIVATE(V1)
U931(tt, IL, M, N) → ACTIVATE(IL)
U921(tt, IL, M, N) → ACTIVATE(M)
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)
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)
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))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → U411(isNat(activate(V1)), activate(V2))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
U611(tt, V2) → ISNATILIST(activate(V2))
U911(tt, IL, M, N) → ISNAT(activate(M))
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
U911(tt, IL, M, N) → ACTIVATE(IL)
ISNAT(n__length(V1)) → 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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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__take(V1, V2)) → U611(isNat(activate(V1)), activate(V2))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__take(V1, V2)) → ISNAT(activate(V1))
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
The remaining pairs can at least be oriented weakly.

ISNATILIST(V) → ACTIVATE(V)
ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
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)
U911(tt, IL, M, N) → ACTIVATE(N)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(V) → ISNATLIST(activate(V))
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)
U711(tt, L, N) → ACTIVATE(L)
U921(tt, IL, M, N) → ACTIVATE(IL)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → LENGTH(X)
TAKE(0, IL) → ISNATILIST(IL)
ISNAT(n__s(V1)) → ACTIVATE(V1)
U931(tt, IL, M, N) → ACTIVATE(IL)
U921(tt, IL, M, N) → ACTIVATE(M)
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))
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)
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))
LENGTH(cons(N, L)) → ACTIVATE(L)
U921(tt, IL, M, N) → U931(isNat(activate(N)), activate(IL), activate(M), activate(N))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → U411(isNat(activate(V1)), activate(V2))
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)
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)) = x1   
POL(TAKE(x1, x2)) = x1 + x2   
POL(U11(x1)) = 0   
POL(U21(x1)) = 0   
POL(U31(x1)) = 0   
POL(U41(x1, x2)) = 0   
POL(U411(x1, x2)) = x2   
POL(U42(x1)) = 0   
POL(U51(x1, x2)) = 0   
POL(U511(x1, x2)) = x2   
POL(U52(x1)) = 0   
POL(U61(x1, x2)) = 0   
POL(U611(x1, x2)) = x2   
POL(U62(x1)) = 0   
POL(U71(x1, x2, x3)) = x2 + x3   
POL(U711(x1, x2, x3)) = x2 + x3   
POL(U72(x1, x2)) = x2   
POL(U721(x1, x2)) = x2   
POL(U81(x1)) = 0   
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)) = x1   
POL(isNatIList(x1)) = 0   
POL(isNatList(x1)) = x1   
POL(length(x1)) = x1   
POL(n__0) = 0   
POL(n__cons(x1, x2)) = x1 + x2   
POL(n__length(x1)) = x1   
POL(n__nil) = 0   
POL(n__s(x1)) = 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:

U31(tt) → tt
U21(tt) → tt
U42(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U52(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U62(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U11(tt) → tt
zeros → cons(0, n__zeros)
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
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
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__length(V1)) → U11(isNatList(activate(V1)))
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__zeros) → zeros
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__0) → 0
zeros → n__zeros
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)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
take(X1, X2) → n__take(X1, X2)
activate(n__nil) → nil
activate(X) → X



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

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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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 2 SCCs with 25 less nodes.

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

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

U711(tt, L, N) → ACTIVATE(L)
ISNAT(n__length(V1)) → ISNATLIST(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → LENGTH(X)
LENGTH(cons(N, L)) → ACTIVATE(L)
U511(tt, V2) → ISNATLIST(activate(V2))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
U721(tt, L) → LENGTH(activate(L))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U511(tt, V2) → ACTIVATE(V2)
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))
U721(tt, L) → ACTIVATE(L)
ISNAT(n__length(V1)) → ACTIVATE(V1)
U711(tt, L, N) → U721(isNat(activate(N)), activate(L))
U711(tt, L, N) → ISNAT(activate(N))

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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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))
ACTIVATE(n__length(X)) → LENGTH(X)
ISNAT(n__length(V1)) → ACTIVATE(V1)
The remaining pairs can at least be oriented weakly.

U711(tt, L, N) → ACTIVATE(L)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ACTIVATE(L)
U511(tt, V2) → ISNATLIST(activate(V2))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
U721(tt, L) → LENGTH(activate(L))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U511(tt, V2) → ACTIVATE(V2)
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))
U721(tt, L) → ACTIVATE(L)
U711(tt, L, N) → U721(isNat(activate(N)), activate(L))
U711(tt, L, N) → ISNAT(activate(N))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVATE(x1)) = x1   
POL(ISNAT(x1)) = x1   
POL(ISNATLIST(x1)) = x1   
POL(LENGTH(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 + x2 + x3   
POL(U711(x1, x2, x3)) = x2 + x3   
POL(U72(x1, x2)) = 1 + x2   
POL(U721(x1, x2)) = x2   
POL(U81(x1)) = 0   
POL(U91(x1, x2, x3, x4)) = x2 + x3 + x4   
POL(U92(x1, x2, x3, x4)) = x2 + x3 + x4   
POL(U93(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)) = x1 + x2   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(take(x1, x2)) = x1 + x2   
POL(tt) = 0   
POL(zeros) = 0   

The following usable rules [17] were oriented:

U31(tt) → tt
U21(tt) → tt
U42(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U52(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U62(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U11(tt) → tt
zeros → cons(0, n__zeros)
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
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
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__length(V1)) → U11(isNatList(activate(V1)))
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__zeros) → zeros
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__0) → 0
zeros → n__zeros
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)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
take(X1, X2) → n__take(X1, X2)
activate(n__nil) → nil
activate(X) → X



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

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

U711(tt, L, N) → ACTIVATE(L)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ACTIVATE(L)
U511(tt, V2) → ISNATLIST(activate(V2))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
U721(tt, L) → LENGTH(activate(L))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U511(tt, V2) → ACTIVATE(V2)
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)
U721(tt, L) → ACTIVATE(L)
ISNATLIST(n__cons(V1, V2)) → U511(isNat(activate(V1)), activate(V2))
U711(tt, L, N) → U721(isNat(activate(N)), activate(L))
U711(tt, L, N) → ISNAT(activate(N))

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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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 3 SCCs with 11 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ AND
                            ↳ QDP
                              ↳ QDPOrderProof
                            ↳ QDP
                            ↳ 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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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/
+
/00\
\00/
·x1

M( U92(x1, ..., x4) ) =
/1\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\01/
·x3+
/00\
\00/
·x4

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

M( n__nil ) =
/1\
\0/

M( n__s(x1) ) =
/0\
\1/
+
/00\
\01/
·x1

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

M( U51(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/10\
\00/
·x2

M( tt ) =
/1\
\0/

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

M( zeros ) =
/0\
\1/

M( U52(x1) ) =
/0\
\0/
+
/10\
\01/
·x1

M( isNatIList(x1) ) =
/0\
\0/
+
/11\
\10/
·x1

M( s(x1) ) =
/0\
\1/
+
/00\
\01/
·x1

M( U11(x1) ) =
/1\
\0/
+
/00\
\00/
·x1

M( isNat(x1) ) =
/0\
\0/
+
/11\
\00/
·x1

M( nil ) =
/1\
\0/

M( U91(x1, ..., x4) ) =
/1\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\01/
·x3+
/00\
\00/
·x4

M( n__length(x1) ) =
/1\
\0/
+
/00\
\01/
·x1

M( U93(x1, ..., x4) ) =
/1\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\01/
·x3+
/00\
\00/
·x4

M( n__zeros ) =
/0\
\1/

M( U72(x1, x2) ) =
/0\
\1/
+
/00\
\00/
·x1+
/00\
\01/
·x2

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

M( 0 ) =
/0\
\0/

M( U62(x1) ) =
/1\
\0/
+
/00\
\00/
·x1

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

M( U31(x1) ) =
/0\
\0/
+
/10\
\00/
·x1

M( U61(x1, x2) ) =
/1\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( U41(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/11\
\10/
·x2

M( U42(x1) ) =
/0\
\0/
+
/10\
\00/
·x1

M( n__0 ) =
/0\
\0/

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

M( length(x1) ) =
/1\
\0/
+
/00\
\01/
·x1

M( U71(x1, ..., x3) ) =
/0\
\0/
+
/00\
\10/
·x1+
/00\
\01/
·x2+
/00\
\00/
·x3

M( U21(x1) ) =
/1\
\0/
+
/00\
\00/
·x1

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


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


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

U31(tt) → tt
U21(tt) → tt
U42(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U52(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U62(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U11(tt) → tt
zeros → cons(0, n__zeros)
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
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__length(V1)) → U11(isNatList(activate(V1)))
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__zeros) → zeros
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__0) → 0
zeros → n__zeros
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)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
take(X1, X2) → n__take(X1, X2)
activate(n__nil) → nil
activate(X) → X



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ AND
                            ↳ QDP
                              ↳ QDPOrderProof
                                ↳ QDP
                                  ↳ PisEmptyProof
                            ↳ QDP
                            ↳ 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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ AND
                            ↳ QDP
                            ↳ QDP
                              ↳ Narrowing
                            ↳ QDP
                  ↳ 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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(x0, x1)), 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__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(x0)), activate(y1))



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

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

ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U511(isNat(cons(x0, x1)), 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(x0, x1)), activate(y1))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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__s(x0)) → ISNATLIST(s(x0))
U511(tt, n__zeros) → ISNATLIST(zeros)
U511(tt, n__0) → ISNATLIST(0)
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
U511(tt, x0) → ISNATLIST(x0)
U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))



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

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

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

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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__cons(x0, x1), y1)) → U511(isNat(cons(x0, x1)), activate(y1)) at position [0] we obtained the following new rules:

ISNATLIST(n__cons(n__cons(x0, x1), y2)) → U511(isNat(n__cons(x0, x1)), activate(y2))



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

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

ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(x0)), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U511(tt, n__nil) → ISNATLIST(nil)
U511(tt, n__s(x0)) → ISNATLIST(s(x0))
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(x0))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__cons(x0, x1), y2)) → U511(isNat(n__cons(x0, x1)), activate(y2))

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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ AND
                            ↳ QDP
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                            ↳ QDP
                  ↳ QDP

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

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

The TRS R consists of the following rules:

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

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

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

The TRS R consists of the following rules:

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

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

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

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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__s(x0)) → ISNATLIST(s(x0)) at position [0] we obtained the following new rules:

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



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

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

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

The TRS R consists of the following rules:

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

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

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

The TRS R consists of the following rules:

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

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

ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
ISNATLIST(n__cons(n__nil, y1)) → U511(isNat(nil), activate(y1))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(x0)), activate(y1))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
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(x0))
U511(tt, n__0) → ISNATLIST(0)
U511(tt, x0) → ISNATLIST(x0)
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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ AND
                            ↳ QDP
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                            ↳ QDP
                  ↳ QDP

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

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

The TRS R consists of the following rules:

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

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

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

The TRS R consists of the following rules:

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

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

ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), 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, n__length(x0)) → ISNATLIST(length(x0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(x0)), activate(y1))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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__cons(x0, x1)) → ISNATLIST(cons(x0, x1)) at position [0] we obtained the following new rules:

U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))



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

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

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

The TRS R consists of the following rules:

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

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

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

The TRS R consists of the following rules:

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

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

ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(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(x0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(x0)), activate(y1))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ AND
                            ↳ 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
                                                                                      ↳ DependencyGraphProof
                                                                                        ↳ QDP
                                                                                          ↳ Narrowing
                                                                                            ↳ QDP
                                                                                              ↳ DependencyGraphProof
                            ↳ QDP
                  ↳ QDP

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

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

The TRS R consists of the following rules:

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

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

ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(x0)), activate(y1))
U511(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ AND
                            ↳ 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
                                                                                      ↳ DependencyGraphProof
                                                                                        ↳ QDP
                                                                                          ↳ Narrowing
                                                                                            ↳ QDP
                                                                                              ↳ DependencyGraphProof
                                                                                                ↳ QDP
                                                                                                  ↳ Narrowing
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                            ↳ QDP
                  ↳ QDP

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

ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y1)) → U511(isNat(0), activate(y1))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(s(x0)), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U511(tt, x0) → ISNATLIST(x0)
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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ AND
                            ↳ 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
                                                                                      ↳ DependencyGraphProof
                                                                                        ↳ QDP
                                                                                          ↳ Narrowing
                                                                                            ↳ QDP
                                                                                              ↳ DependencyGraphProof
                                                                                                ↳ QDP
                                                                                                  ↳ Narrowing
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ Narrowing
                            ↳ QDP
                  ↳ QDP

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

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

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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__s(x0), y1)) → U511(isNat(s(x0)), activate(y1)) at position [0] we obtained the following new rules:

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



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

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

ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(0, n__zeros)), activate(y0))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(n__s(x0)), activate(y1))
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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ AND
                            ↳ 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
                                                                                      ↳ DependencyGraphProof
                                                                                        ↳ QDP
                                                                                          ↳ Narrowing
                                                                                            ↳ QDP
                                                                                              ↳ DependencyGraphProof
                                                                                                ↳ QDP
                                                                                                  ↳ Narrowing
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ Narrowing
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                            ↳ QDP
                  ↳ QDP

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

ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
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(x0))
U511(tt, x0) → ISNATLIST(x0)
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(n__0, n__zeros)), activate(y0))
U511(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(n__s(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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ AND
                            ↳ 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
                                                                                      ↳ DependencyGraphProof
                                                                                        ↳ QDP
                                                                                          ↳ Narrowing
                                                                                            ↳ QDP
                                                                                              ↳ DependencyGraphProof
                                                                                                ↳ QDP
                                                                                                  ↳ Narrowing
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ Narrowing
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                            ↳ QDP
                  ↳ QDP

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

ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
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(x0, x1)), activate(y1))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(n__0, n__zeros)), activate(y0))
U511(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(n__s(x0)), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ AND
                            ↳ 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
                                                                                      ↳ DependencyGraphProof
                                                                                        ↳ QDP
                                                                                          ↳ Narrowing
                                                                                            ↳ QDP
                                                                                              ↳ DependencyGraphProof
                                                                                                ↳ QDP
                                                                                                  ↳ Narrowing
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ Narrowing
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ Narrowing
                            ↳ QDP
                  ↳ QDP

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

ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(x0, y1)) → U511(isNat(x0), activate(y1))
U511(tt, n__length(x0)) → ISNATLIST(length(x0))
U511(tt, x0) → ISNATLIST(x0)
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__zeros, y0)) → U511(isNat(cons(n__0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(n__s(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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ AND
                            ↳ 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
                                                                                      ↳ DependencyGraphProof
                                                                                        ↳ QDP
                                                                                          ↳ Narrowing
                                                                                            ↳ QDP
                                                                                              ↳ DependencyGraphProof
                                                                                                ↳ QDP
                                                                                                  ↳ Narrowing
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ Narrowing
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ Narrowing
                                                                                                                            ↳ QDP
                                                                                                                              ↳ DependencyGraphProof
                            ↳ QDP
                  ↳ QDP

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

ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), activate(y1))
U511(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
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(x0))
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(n__s(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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ AND
                            ↳ 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
                                                                                      ↳ DependencyGraphProof
                                                                                        ↳ QDP
                                                                                          ↳ Narrowing
                                                                                            ↳ QDP
                                                                                              ↳ DependencyGraphProof
                                                                                                ↳ QDP
                                                                                                  ↳ Narrowing
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ Narrowing
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ Narrowing
                                                                                                                            ↳ QDP
                                                                                                                              ↳ DependencyGraphProof
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ QDPOrderProof
                            ↳ QDP
                  ↳ QDP

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

ISNATLIST(n__cons(n__length(x0), y1)) → U511(isNat(length(x0)), 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(x0))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
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(x0, x1)), activate(y1))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(n__s(x0)), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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(x0)), 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(x0))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
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(x0, x1)), activate(y1))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(n__s(x0)), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(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)) = 1 + x2 + x3   
POL(U72(x1, x2)) = 1 + x2   
POL(U81(x1)) = 0   
POL(U91(x1, x2, x3, x4)) = x2 + x4   
POL(U92(x1, x2, x3, x4)) = x2 + x4   
POL(U93(x1, x2, x3, x4)) = x2 + x4   
POL(activate(x1)) = x1   
POL(cons(x1, x2)) = x1 + x2   
POL(isNat(x1)) = 0   
POL(isNatIList(x1)) = 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)) = 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:

U31(tt) → tt
U21(tt) → tt
U42(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U52(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U62(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U11(tt) → tt
zeros → cons(0, n__zeros)
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
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
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__length(V1)) → U11(isNatList(activate(V1)))
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__zeros) → zeros
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__0) → 0
zeros → n__zeros
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)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
take(X1, X2) → n__take(X1, X2)
activate(n__nil) → nil
activate(X) → X



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ AND
                            ↳ 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
                                                                                      ↳ DependencyGraphProof
                                                                                        ↳ QDP
                                                                                          ↳ Narrowing
                                                                                            ↳ QDP
                                                                                              ↳ DependencyGraphProof
                                                                                                ↳ QDP
                                                                                                  ↳ Narrowing
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ Narrowing
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ Narrowing
                                                                                                                            ↳ QDP
                                                                                                                              ↳ DependencyGraphProof
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ QDPOrderProof
                            ↳ QDP
                  ↳ 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(x0))
U511(tt, x0) → ISNATLIST(x0)
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
ISNATLIST(n__cons(n__s(x0), y1)) → U511(isNat(n__s(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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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(n__s(x0)), 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(x0))
U511(tt, x0) → ISNATLIST(x0)
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__take(x0, x1), y1)) → U511(isNat(take(x0, x1)), activate(y1))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(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)) = x2   
POL(U511(x1, x2)) = x2   
POL(U52(x1)) = x1   
POL(U61(x1, x2)) = x2   
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 + x1   
POL(n__0) = 0   
POL(n__cons(x1, x2)) = x1 + x2   
POL(n__length(x1)) = 1 + x1   
POL(n__nil) = 0   
POL(n__s(x1)) = 1   
POL(n__take(x1, x2)) = x2   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x1)) = 1   
POL(take(x1, x2)) = x2   
POL(tt) = 0   
POL(zeros) = 0   

The following usable rules [17] were oriented:

U31(tt) → tt
U21(tt) → tt
U42(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U52(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U62(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U11(tt) → tt
zeros → cons(0, n__zeros)
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
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
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__length(V1)) → U11(isNatList(activate(V1)))
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__zeros) → zeros
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__0) → 0
zeros → n__zeros
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)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
take(X1, X2) → n__take(X1, X2)
activate(n__nil) → nil
activate(X) → X



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ AND
                            ↳ 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
                                                                                      ↳ DependencyGraphProof
                                                                                        ↳ QDP
                                                                                          ↳ Narrowing
                                                                                            ↳ QDP
                                                                                              ↳ DependencyGraphProof
                                                                                                ↳ QDP
                                                                                                  ↳ Narrowing
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ Narrowing
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ Narrowing
                                                                                                                            ↳ QDP
                                                                                                                              ↳ DependencyGraphProof
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ QDPOrderProof
                            ↳ QDP
                  ↳ 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(x0))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
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(x0, x1)), activate(y1))
U511(tt, n__take(x0, x1)) → ISNATLIST(take(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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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(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(x0))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
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(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)) = 1 + x2   
POL(U511(x1, x2)) = x2   
POL(U52(x1)) = 0   
POL(U61(x1, x2)) = 1   
POL(U62(x1)) = 0   
POL(U71(x1, x2, x3)) = 0   
POL(U72(x1, x2)) = 0   
POL(U81(x1)) = 1   
POL(U91(x1, x2, x3, x4)) = 1 + x2 + x4   
POL(U92(x1, x2, x3, x4)) = 1 + x2 + x4   
POL(U93(x1, x2, x3, x4)) = 1 + x2 + x4   
POL(activate(x1)) = x1   
POL(cons(x1, x2)) = x1 + x2   
POL(isNat(x1)) = 0   
POL(isNatIList(x1)) = 1   
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:

U31(tt) → tt
U21(tt) → tt
U42(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U52(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U62(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U11(tt) → tt
zeros → cons(0, n__zeros)
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
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
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__length(V1)) → U11(isNatList(activate(V1)))
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__zeros) → zeros
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__0) → 0
zeros → n__zeros
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)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
take(X1, X2) → n__take(X1, X2)
activate(n__nil) → nil
activate(X) → X



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ AND
                            ↳ 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
                                                                                      ↳ DependencyGraphProof
                                                                                        ↳ QDP
                                                                                          ↳ Narrowing
                                                                                            ↳ QDP
                                                                                              ↳ DependencyGraphProof
                                                                                                ↳ QDP
                                                                                                  ↳ Narrowing
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ Narrowing
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ Narrowing
                                                                                                                            ↳ QDP
                                                                                                                              ↳ DependencyGraphProof
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ QDPOrderProof
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ QDPOrderProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ QDPOrderProof
                            ↳ QDP
                  ↳ 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(x0))
U511(tt, x0) → ISNATLIST(x0)
U511(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
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__take(x0, x1)) → ISNATLIST(take(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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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(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__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
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__take(x0, x1)) → ISNATLIST(take(x0, x1))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( U81(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U92(x1, ..., x4) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3+
/00\
\00/
·x4

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

M( n__nil ) =
/0\
\0/

M( n__s(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

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

M( U51(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/11\
\11/
·x2

M( tt ) =
/0\
\0/

M( isNatList(x1) ) =
/0\
\0/
+
/01\
\01/
·x1

M( zeros ) =
/0\
\0/

M( U52(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( isNatIList(x1) ) =
/1\
\1/
+
/11\
\11/
·x1

M( s(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U11(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

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

M( nil ) =
/0\
\0/

M( U91(x1, ..., x4) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3+
/00\
\00/
·x4

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

M( U93(x1, ..., x4) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3+
/00\
\00/
·x4

M( n__zeros ) =
/0\
\0/

M( U72(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

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

M( 0 ) =
/0\
\0/

M( U62(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

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

M( U31(x1) ) =
/1\
\1/
+
/00\
\00/
·x1

M( U61(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( U41(x1, x2) ) =
/1\
\1/
+
/00\
\00/
·x1+
/11\
\11/
·x2

M( U42(x1) ) =
/1\
\1/
+
/00\
\00/
·x1

M( n__0 ) =
/0\
\0/

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

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

M( U71(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( U21(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

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

M( U511(x1, x2) ) = 0+
[0,0]
·x1+
[1,1]
·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:

U31(tt) → tt
U21(tt) → tt
U42(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U52(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U62(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U11(tt) → tt
zeros → cons(0, n__zeros)
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__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
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__length(V1)) → U11(isNatList(activate(V1)))
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__zeros) → zeros
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__0) → 0
zeros → n__zeros
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)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
take(X1, X2) → n__take(X1, X2)
activate(n__nil) → nil
activate(X) → X



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ AND
                            ↳ 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
                                                                                      ↳ DependencyGraphProof
                                                                                        ↳ QDP
                                                                                          ↳ Narrowing
                                                                                            ↳ QDP
                                                                                              ↳ DependencyGraphProof
                                                                                                ↳ QDP
                                                                                                  ↳ Narrowing
                                                                                                    ↳ 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
                  ↳ 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, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
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(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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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(x0, 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, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U511(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( U81(x1) ) =
/0\
\0/
+
/01\
\00/
·x1

M( U92(x1, ..., x4) ) =
/1\
\1/
+
/00\
\00/
·x1+
/00\
\01/
·x2+
/00\
\01/
·x3+
/00\
\10/
·x4

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

M( n__nil ) =
/1\
\0/

M( n__s(x1) ) =
/0\
\1/
+
/00\
\01/
·x1

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

M( U51(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\10/
·x2

M( tt ) =
/0\
\1/

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

M( zeros ) =
/0\
\0/

M( U52(x1) ) =
/0\
\0/
+
/00\
\01/
·x1

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

M( s(x1) ) =
/0\
\1/
+
/00\
\01/
·x1

M( U11(x1) ) =
/0\
\0/
+
/00\
\01/
·x1

M( isNat(x1) ) =
/0\
\0/
+
/00\
\01/
·x1

M( nil ) =
/1\
\0/

M( U91(x1, ..., x4) ) =
/1\
\0/
+
/00\
\01/
·x1+
/00\
\01/
·x2+
/00\
\01/
·x3+
/00\
\10/
·x4

M( n__length(x1) ) =
/0\
\0/
+
/01\
\11/
·x1

M( U93(x1, ..., x4) ) =
/1\
\1/
+
/00\
\00/
·x1+
/00\
\01/
·x2+
/00\
\01/
·x3+
/00\
\10/
·x4

M( n__zeros ) =
/0\
\0/

M( U72(x1, x2) ) =
/0\
\1/
+
/00\
\00/
·x1+
/00\
\11/
·x2

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

M( 0 ) =
/0\
\0/

M( U62(x1) ) =
/0\
\0/
+
/00\
\11/
·x1

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

M( U31(x1) ) =
/0\
\1/
+
/00\
\00/
·x1

M( U61(x1, x2) ) =
/0\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( U41(x1, x2) ) =
/0\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( U42(x1) ) =
/0\
\1/
+
/00\
\00/
·x1

M( n__0 ) =
/0\
\0/

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

M( length(x1) ) =
/0\
\0/
+
/01\
\11/
·x1

M( U71(x1, ..., x3) ) =
/0\
\0/
+
/00\
\01/
·x1+
/00\
\11/
·x2+
/10\
\10/
·x3

M( U21(x1) ) =
/0\
\1/
+
/00\
\00/
·x1

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

M( U511(x1, x2) ) = 0+
[0,0]
·x1+
[1,1]
·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:

U31(tt) → tt
U21(tt) → tt
U42(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U52(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U62(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U11(tt) → tt
zeros → cons(0, n__zeros)
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
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__length(V1)) → U11(isNatList(activate(V1)))
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__zeros) → zeros
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__s(X)) → s(X)
activate(n__length(X)) → length(X)
activate(n__0) → 0
zeros → n__zeros
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)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
take(X1, X2) → n__take(X1, X2)
activate(n__nil) → nil
activate(X) → X



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ AND
                            ↳ 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
                                                                                      ↳ DependencyGraphProof
                                                                                        ↳ QDP
                                                                                          ↳ Narrowing
                                                                                            ↳ QDP
                                                                                              ↳ DependencyGraphProof
                                                                                                ↳ QDP
                                                                                                  ↳ Narrowing
                                                                                                    ↳ 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
                                                                                                                                                      ↳ NonTerminationProof
                            ↳ QDP
                  ↳ 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__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U511(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U511(isNat(n__0), activate(y0))

The TRS R consists of the following rules:

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



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
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ AND
                            ↳ 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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ 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(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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