Termination w.r.t. Q proof of /tmp/tmp2dnS4d/OvConsOS_nokinds-noand

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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 [25,35]:

POL(U51¹(x₁, x₂)) = x_2
POL(ISNATILIST(x₁)) = x_1
POL(U81(x₁)) = 1
POL(U61¹(x₁, x₂)) = (4)x_2
POL(U92(x₁, x₂, x₃, x₄)) = 1 + (4)x_2 + x_3 + (3)x_4
POL(activate(x₁)) = x_1
POL(n__s(x₁)) = x_1
POL(n__nil) = 0
POL(take(x₁, x₂)) = 1 + x_1 + (4)x_2
POL(U51(x₁, x₂)) = 0
POL(U71¹(x₁, x₂, x₃)) = x_2 + x_3
POL(TAKE(x₁, x₂)) = x_1 + (4)x_2
POL(ISNATLIST(x₁)) = x_1
POL(tt) = 0
POL(isNatList(x₁)) = 4
POL(U52(x₁)) = 0
POL(zeros) = 0
POL(isNatIList(x₁)) = (4)x_1
POL(U11(x₁)) = 0
POL(s(x₁)) = x_1
POL(isNat(x₁)) = 0
POL(nil) = 0
POL(ACTIVATE(x₁)) = x_1
POL(LENGTH(x₁)) = x_1
POL(U91(x₁, x₂, x₃, x₄)) = 1 + (4)x_2 + x_3 + (4)x_4
POL(n__length(x₁)) = (4)x_1
POL(U93(x₁, x₂, x₃, x₄)) = 1 + (4)x_2 + x_3 + (2)x_4
POL(n__zeros) = 0
POL(n__cons(x₁, x₂)) = x_1 + x_2
POL(U72(x₁, x₂)) = (4)x_2
POL(U93¹(x₁, x₂, x₃, x₄)) = (2)x_2 + x_3 + (4)x_4
POL(0) = 0
POL(ISNAT(x₁)) = x_1
POL(U62(x₁)) = 2
POL(cons(x₁, x₂)) = x_1 + x_2
POL(U91¹(x₁, x₂, x₃, x₄)) = (2)x_2 + x_3 + (4)x_4
POL(U41¹(x₁, x₂)) = x_2
POL(U61(x₁, x₂)) = 2
POL(U31(x₁)) = 0
POL(U41(x₁, x₂)) = (4)x_2
POL(U42(x₁)) = 0
POL(n__0) = 0
POL(n__take(x₁, x₂)) = 1 + x_1 + (4)x_2
POL(length(x₁)) = (4)x_1
POL(U92¹(x₁, x₂, x₃, x₄)) = (2)x_2 + x_3 + (4)x_4
POL(U72¹(x₁, x₂)) = x_2
POL(U21(x₁)) = 0
POL(U71(x₁, x₂, x₃)) = (4)x_2 + (3)x_3

The value of delta used in the strict ordering is 1.
The following usable rules [17] were oriented:

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

↳ 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.

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) → ACTIVATE(V2)
U71¹(tt, L, N) → ACTIVATE(N)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U72¹(tt, L) → ACTIVATE(L)
ISNAT(n__length(V1)) → ACTIVATE(V1)
U71¹(tt, L, N) → ISNAT(activate(N))

The remaining pairs can at least be oriented weakly.

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

Used ordering: Polynomial interpretation [25,35]:

POL(U51¹(x₁, x₂)) = 4 + (4)x_2
POL(U81(x₁)) = 0
POL(U92(x₁, x₂, x₃, x₄)) = (4)x_2 + (2)x_4
POL(activate(x₁)) = x_1
POL(n__s(x₁)) = x_1
POL(n__nil) = 0
POL(take(x₁, x₂)) = x_2
POL(U51(x₁, x₂)) = 0
POL(U71¹(x₁, x₂, x₃)) = 4 + (4)x_2 + (4)x_3
POL(ISNATLIST(x₁)) = 4 + (2)x_1
POL(tt) = 0
POL(isNatList(x₁)) = (2)x_1
POL(zeros) = 0
POL(U52(x₁)) = 0
POL(isNatIList(x₁)) = 2 + (3)x_1
POL(s(x₁)) = x_1
POL(U11(x₁)) = 0
POL(isNat(x₁)) = 0
POL(nil) = 0
POL(ACTIVATE(x₁)) = 2 + (4)x_1
POL(LENGTH(x₁)) = 4 + (4)x_1
POL(U91(x₁, x₂, x₃, x₄)) = (4)x_2 + (2)x_4
POL(n__length(x₁)) = 2 + (2)x_1
POL(U93(x₁, x₂, x₃, x₄)) = (4)x_2 + (2)x_4
POL(n__zeros) = 0
POL(n__cons(x₁, x₂)) = (2)x_1 + (4)x_2
POL(U72(x₁, x₂)) = 2 + (3)x_2
POL(0) = 0
POL(ISNAT(x₁)) = 2 + (4)x_1
POL(U62(x₁)) = 0
POL(cons(x₁, x₂)) = (2)x_1 + (4)x_2
POL(U61(x₁, x₂)) = x_2
POL(U31(x₁)) = 0
POL(U41(x₁, x₂)) = 1
POL(U42(x₁)) = 0
POL(n__0) = 0
POL(n__take(x₁, x₂)) = x_2
POL(length(x₁)) = 2 + (2)x_1
POL(U72¹(x₁, x₂)) = 4 + (4)x_2
POL(U21(x₁)) = 0
POL(U71(x₁, x₂, x₃)) = 2 + x_1 + (4)x_2 + (4)x_3

The value of delta used in the strict ordering is 2.
The following usable rules [17] were oriented:

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

↳ 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:

LENGTH(cons(N, L)) → U71¹(isNatList(activate(L)), activate(L), N)
U51¹(tt, V2) → ISNATLIST(activate(V2))
U72¹(tt, L) → LENGTH(activate(L))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(V1, V2)) → U51¹(isNat(activate(V1)), activate(V2))
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.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs with 2 less nodes.

↳ 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:

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.

↳ 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:

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.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                              ↳ QDPOrderProof

                  ↳ 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.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

U72¹(tt, L) → LENGTH(activate(L))

The remaining pairs can at least be oriented weakly.

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

Used ordering: Polynomial interpretation [25,35]:

POL(U81(x₁)) = 2
POL(U92(x₁, x₂, x₃, x₄)) = (4)x_3
POL(activate(x₁)) = x_1
POL(n__nil) = 2
POL(n__s(x₁)) = (4)x_1
POL(take(x₁, x₂)) = x_1
POL(U51(x₁, x₂)) = x_2
POL(U71¹(x₁, x₂, x₃)) = (2)x_1 + x_2
POL(tt) = 1
POL(isNatList(x₁)) = x_1
POL(U52(x₁)) = x_1
POL(zeros) = 0
POL(isNatIList(x₁)) = 4 + (2)x_1
POL(s(x₁)) = (4)x_1
POL(U11(x₁)) = x_1
POL(isNat(x₁)) = x_1
POL(nil) = 2
POL(LENGTH(x₁)) = x_1
POL(U91(x₁, x₂, x₃, x₄)) = (4)x_3
POL(n__length(x₁)) = x_1
POL(U93(x₁, x₂, x₃, x₄)) = (4)x_3
POL(n__zeros) = 0
POL(n__cons(x₁, x₂)) = (4)x_2
POL(U72(x₁, x₂)) = (4)x_2
POL(0) = 2
POL(U62(x₁)) = 1
POL(cons(x₁, x₂)) = (4)x_2
POL(U31(x₁)) = 2
POL(U61(x₁, x₂)) = x_1
POL(U41(x₁, x₂)) = 2
POL(n__0) = 2
POL(U42(x₁)) = 1
POL(n__take(x₁, x₂)) = x_1
POL(length(x₁)) = x_1
POL(U21(x₁)) = (4)x_1
POL(U72¹(x₁, x₂)) = 2 + x_2
POL(U71(x₁, x₂, x₃)) = (4)x_2

The value of delta used in the strict ordering is 2.
The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

                  ↳ QDP

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

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

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 0 SCCs with 2 less nodes.

↳ 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.