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, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))), 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)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(n__isNatKind(X)) → isNatKind(X)
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, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))), 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)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(n__isNatKind(X)) → isNatKind(X)
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)
ISNATLIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__take(V1, V2)) → ISNATKIND(activate(V1))
U621(tt, V2) → U631(isNatIList(activate(V2)))
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ACTIVATE(n__0) → 01
ISNATKIND(n__length(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__length(V1)) → U111(isNatIListKind(activate(V1)), activate(V1))
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → S(activate(X))
U111(tt, V1) → U121(isNatList(activate(V1)))
TAKE(0, IL) → ISNATILIST(IL)
ACTIVATE(n__cons(X1, X2)) → CONS(activate(X1), X2)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
U611(tt, V1, V2) → ACTIVATE(V2)
U421(tt, V2) → ACTIVATE(V2)
U521(tt, V2) → ACTIVATE(V2)
TAKE(s(M), cons(N, IL)) → U911(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))), activate(IL), M, N)
U411(tt, V1, V2) → ISNAT(activate(V1))
U211(tt, V1) → ISNAT(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__isNatIListKind(activate(IL)))
TAKE(0, IL) → AND(isNatIList(IL), n__isNatIListKind(IL))
U511(tt, V1, V2) → ACTIVATE(V1)
U911(tt, IL, M, N) → CONS(activate(N), n__take(activate(M), activate(IL)))
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATILIST(n__cons(V1, V2)) → U411(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U911(tt, IL, M, N) → ACTIVATE(M)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
U111(tt, V1) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
U411(tt, V1, V2) → ACTIVATE(V1)
U511(tt, V1, V2) → U521(isNat(activate(V1)), activate(V2))
U211(tt, V1) → ACTIVATE(V1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
U421(tt, V2) → U431(isNatIList(activate(V2)))
U711(tt, L) → ACTIVATE(L)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
LENGTH(cons(N, L)) → AND(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N)))
ZEROS → CONS(0, n__zeros)
U911(tt, IL, M, N) → ACTIVATE(IL)
U111(tt, V1) → ISNATLIST(activate(V1))
U511(tt, V1, V2) → ACTIVATE(V2)
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
U711(tt, L) → LENGTH(activate(L))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U611(tt, V1, V2) → ISNAT(activate(V1))
U521(tt, V2) → ISNATLIST(activate(V2))
ACTIVATE(n__s(X)) → ACTIVATE(X)
U621(tt, V2) → ISNATILIST(activate(V2))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
U911(tt, IL, M, N) → ACTIVATE(N)
U421(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__take(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
LENGTH(nil) → 01
ACTIVATE(n__zeros) → ZEROS
U811(tt) → NIL
U521(tt, V2) → U531(isNatList(activate(V2)))
LENGTH(cons(N, L)) → U711(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__nil) → NIL
U611(tt, V1, V2) → ACTIVATE(V1)
U611(tt, V1, V2) → U621(isNat(activate(V1)), activate(V2))
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNATILIST(V) → ISNATILISTKIND(activate(V))
ISNATLIST(n__take(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U311(tt, V) → ISNATLIST(activate(V))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNATLIST(n__take(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V2)
U411(tt, V1, V2) → U421(isNat(activate(V1)), activate(V2))
U211(tt, V1) → U221(isNat(activate(V1)))
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
U311(tt, V) → ACTIVATE(V)
TAKE(s(M), cons(N, IL)) → AND(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))))
U311(tt, V) → U321(isNatList(activate(V)))
TAKE(0, IL) → U811(and(isNatIList(IL), n__isNatIListKind(IL)))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
U411(tt, V1, V2) → ACTIVATE(V2)
U511(tt, V1, V2) → ISNAT(activate(V1))
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNatIListKind(activate(L)))
LENGTH(cons(N, L)) → ACTIVATE(L)
ISNATLIST(n__cons(V1, V2)) → U511(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ISNATLIST(n__take(V1, V2)) → U611(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATILIST(V) → U311(isNatIListKind(activate(V)), activate(V))
U711(tt, L) → S(length(activate(L)))
U621(tt, V2) → ACTIVATE(V2)
ISNATILIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNAT(n__length(V1)) → ACTIVATE(V1)
ZEROS → 01
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))), 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)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(n__isNatKind(X)) → isNatKind(X)
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)
ISNATLIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__take(V1, V2)) → ISNATKIND(activate(V1))
U621(tt, V2) → U631(isNatIList(activate(V2)))
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ACTIVATE(n__0) → 01
ISNATKIND(n__length(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__length(V1)) → U111(isNatIListKind(activate(V1)), activate(V1))
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → S(activate(X))
U111(tt, V1) → U121(isNatList(activate(V1)))
TAKE(0, IL) → ISNATILIST(IL)
ACTIVATE(n__cons(X1, X2)) → CONS(activate(X1), X2)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
U611(tt, V1, V2) → ACTIVATE(V2)
U421(tt, V2) → ACTIVATE(V2)
U521(tt, V2) → ACTIVATE(V2)
TAKE(s(M), cons(N, IL)) → U911(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))), activate(IL), M, N)
U411(tt, V1, V2) → ISNAT(activate(V1))
U211(tt, V1) → ISNAT(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__isNatIListKind(activate(IL)))
TAKE(0, IL) → AND(isNatIList(IL), n__isNatIListKind(IL))
U511(tt, V1, V2) → ACTIVATE(V1)
U911(tt, IL, M, N) → CONS(activate(N), n__take(activate(M), activate(IL)))
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATILIST(n__cons(V1, V2)) → U411(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U911(tt, IL, M, N) → ACTIVATE(M)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
U111(tt, V1) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
U411(tt, V1, V2) → ACTIVATE(V1)
U511(tt, V1, V2) → U521(isNat(activate(V1)), activate(V2))
U211(tt, V1) → ACTIVATE(V1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
U421(tt, V2) → U431(isNatIList(activate(V2)))
U711(tt, L) → ACTIVATE(L)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
LENGTH(cons(N, L)) → AND(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N)))
ZEROS → CONS(0, n__zeros)
U911(tt, IL, M, N) → ACTIVATE(IL)
U111(tt, V1) → ISNATLIST(activate(V1))
U511(tt, V1, V2) → ACTIVATE(V2)
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
U711(tt, L) → LENGTH(activate(L))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U611(tt, V1, V2) → ISNAT(activate(V1))
U521(tt, V2) → ISNATLIST(activate(V2))
ACTIVATE(n__s(X)) → ACTIVATE(X)
U621(tt, V2) → ISNATILIST(activate(V2))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
U911(tt, IL, M, N) → ACTIVATE(N)
U421(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__take(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
LENGTH(nil) → 01
ACTIVATE(n__zeros) → ZEROS
U811(tt) → NIL
U521(tt, V2) → U531(isNatList(activate(V2)))
LENGTH(cons(N, L)) → U711(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__nil) → NIL
U611(tt, V1, V2) → ACTIVATE(V1)
U611(tt, V1, V2) → U621(isNat(activate(V1)), activate(V2))
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNATILIST(V) → ISNATILISTKIND(activate(V))
ISNATLIST(n__take(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U311(tt, V) → ISNATLIST(activate(V))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNATLIST(n__take(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V2)
U411(tt, V1, V2) → U421(isNat(activate(V1)), activate(V2))
U211(tt, V1) → U221(isNat(activate(V1)))
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
U311(tt, V) → ACTIVATE(V)
TAKE(s(M), cons(N, IL)) → AND(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))))
U311(tt, V) → U321(isNatList(activate(V)))
TAKE(0, IL) → U811(and(isNatIList(IL), n__isNatIListKind(IL)))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
U411(tt, V1, V2) → ACTIVATE(V2)
U511(tt, V1, V2) → ISNAT(activate(V1))
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNatIListKind(activate(L)))
LENGTH(cons(N, L)) → ACTIVATE(L)
ISNATLIST(n__cons(V1, V2)) → U511(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ISNATLIST(n__take(V1, V2)) → U611(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATILIST(V) → U311(isNatIListKind(activate(V)), activate(V))
U711(tt, L) → S(length(activate(L)))
U621(tt, V2) → ACTIVATE(V2)
ISNATILIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNAT(n__length(V1)) → ACTIVATE(V1)
ZEROS → 01
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))), 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)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(n__isNatKind(X)) → isNatKind(X)
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:

U111(tt, V1) → ISNATLIST(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
U511(tt, V1, V2) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U711(tt, L) → LENGTH(activate(L))
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V1)
U611(tt, V1, V2) → ISNAT(activate(V1))
U521(tt, V2) → ISNATLIST(activate(V2))
ACTIVATE(n__s(X)) → ACTIVATE(X)
U621(tt, V2) → ISNATILIST(activate(V2))
ISNATILISTKIND(n__take(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
U911(tt, IL, M, N) → ACTIVATE(N)
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
U421(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__length(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__take(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__length(V1)) → U111(isNatIListKind(activate(V1)), activate(V1))
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
LENGTH(cons(N, L)) → U711(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
TAKE(0, IL) → ISNATILIST(IL)
U611(tt, V1, V2) → ACTIVATE(V1)
U611(tt, V1, V2) → U621(isNat(activate(V1)), activate(V2))
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
U611(tt, V1, V2) → ACTIVATE(V2)
U521(tt, V2) → ACTIVATE(V2)
U421(tt, V2) → ACTIVATE(V2)
ISNATILIST(V) → ISNATILISTKIND(activate(V))
TAKE(s(M), cons(N, IL)) → U911(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))), activate(IL), M, N)
ISNATLIST(n__take(V1, V2)) → ISNATKIND(activate(V1))
U411(tt, V1, V2) → ISNAT(activate(V1))
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U311(tt, V) → ISNATLIST(activate(V))
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__isNatIListKind(activate(IL)))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNATLIST(n__take(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V2)
TAKE(0, IL) → AND(isNatIList(IL), n__isNatIListKind(IL))
U411(tt, V1, V2) → U421(isNat(activate(V1)), activate(V2))
U511(tt, V1, V2) → ACTIVATE(V1)
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
TAKE(s(M), cons(N, IL)) → AND(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))))
U311(tt, V) → ACTIVATE(V)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATILIST(n__cons(V1, V2)) → U411(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
U911(tt, IL, M, N) → ACTIVATE(M)
U411(tt, V1, V2) → ACTIVATE(V2)
U511(tt, V1, V2) → ISNAT(activate(V1))
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNatIListKind(activate(L)))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
U111(tt, V1) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
LENGTH(cons(N, L)) → ACTIVATE(L)
U411(tt, V1, V2) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ISNATLIST(n__cons(V1, V2)) → U511(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U511(tt, V1, V2) → U521(isNat(activate(V1)), activate(V2))
U211(tt, V1) → ACTIVATE(V1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__take(V1, V2)) → U611(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U711(tt, L) → ACTIVATE(L)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
LENGTH(cons(N, L)) → AND(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N)))
ISNATILIST(V) → U311(isNatIListKind(activate(V)), activate(V))
U911(tt, IL, M, N) → ACTIVATE(IL)
U621(tt, V2) → ACTIVATE(V2)
ISNATILIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNAT(n__length(V1)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))), 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)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(n__isNatKind(X)) → isNatKind(X)
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.


ISNATILIST(V) → ACTIVATE(V)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V1)
U611(tt, V1, V2) → ISNAT(activate(V1))
ISNATILISTKIND(n__take(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__take(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
U611(tt, V1, V2) → ACTIVATE(V1)
U611(tt, V1, V2) → ACTIVATE(V2)
U421(tt, V2) → ACTIVATE(V2)
ISNATILIST(V) → ISNATILISTKIND(activate(V))
TAKE(s(M), cons(N, IL)) → U911(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))), activate(IL), M, N)
ISNATLIST(n__take(V1, V2)) → ISNATKIND(activate(V1))
U411(tt, V1, V2) → ISNAT(activate(V1))
U311(tt, V) → ISNATLIST(activate(V))
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__isNatIListKind(activate(IL)))
ISNATLIST(n__take(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V2)
TAKE(0, IL) → AND(isNatIList(IL), n__isNatIListKind(IL))
TAKE(s(M), cons(N, IL)) → AND(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))))
U311(tt, V) → ACTIVATE(V)
U411(tt, V1, V2) → ACTIVATE(V2)
U411(tt, V1, V2) → ACTIVATE(V1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ISNATLIST(n__take(V1, V2)) → U611(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U621(tt, V2) → ACTIVATE(V2)
ISNATILIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
The remaining pairs can at least be oriented weakly.

U111(tt, V1) → ISNATLIST(activate(V1))
U511(tt, V1, V2) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
U711(tt, L) → LENGTH(activate(L))
U521(tt, V2) → ISNATLIST(activate(V2))
ACTIVATE(n__s(X)) → ACTIVATE(X)
U621(tt, V2) → ISNATILIST(activate(V2))
U911(tt, IL, M, N) → ACTIVATE(N)
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
U421(tt, V2) → ISNATILIST(activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__length(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__length(V1)) → U111(isNatIListKind(activate(V1)), activate(V1))
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
LENGTH(cons(N, L)) → U711(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
TAKE(0, IL) → ISNATILIST(IL)
U611(tt, V1, V2) → U621(isNat(activate(V1)), activate(V2))
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
U521(tt, V2) → ACTIVATE(V2)
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U411(tt, V1, V2) → U421(isNat(activate(V1)), activate(V2))
U511(tt, V1, V2) → ACTIVATE(V1)
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATILIST(n__cons(V1, V2)) → U411(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
U911(tt, IL, M, N) → ACTIVATE(M)
U511(tt, V1, V2) → ISNAT(activate(V1))
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNatIListKind(activate(L)))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
U111(tt, V1) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
LENGTH(cons(N, L)) → ACTIVATE(L)
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ISNATLIST(n__cons(V1, V2)) → U511(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U511(tt, V1, V2) → U521(isNat(activate(V1)), activate(V2))
U211(tt, V1) → ACTIVATE(V1)
U711(tt, L) → ACTIVATE(L)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
LENGTH(cons(N, L)) → AND(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N)))
ISNATILIST(V) → U311(isNatIListKind(activate(V)), activate(V))
U911(tt, IL, M, N) → ACTIVATE(IL)
ISNAT(n__length(V1)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → ACTIVATE(X)
Used ordering: Polynomial interpretation [25,35]:

POL(U62(x1, x2)) = 0   
POL(ISNATILISTKIND(x1)) = (4)x_1   
POL(n__isNat(x1)) = x_1   
POL(U81(x1)) = 4   
POL(U41(x1, x2, x3)) = 0   
POL(U711(x1, x2)) = (4)x_2   
POL(n__s(x1)) = x_1   
POL(n__nil) = 0   
POL(take(x1, x2)) = 4 + (4)x_1 + (4)x_2   
POL(U71(x1, x2)) = (4)x_2   
POL(TAKE(x1, x2)) = 4 + (2)x_1 + (4)x_2   
POL(tt) = 0   
POL(U611(x1, x2, x3)) = 4 + (2)x_2 + (4)x_3   
POL(U311(x1, x2)) = 4 + (4)x_2   
POL(AND(x1, x2)) = x_2   
POL(nil) = 0   
POL(ACTIVATE(x1)) = x_1   
POL(LENGTH(x1)) = (4)x_1   
POL(U91(x1, x2, x3, x4)) = 4 + (4)x_2 + (4)x_3 + (4)x_4   
POL(U31(x1, x2)) = 0   
POL(n__length(x1)) = (4)x_1   
POL(isNatKind(x1)) = x_1   
POL(U411(x1, x2, x3)) = 4 + (4)x_2 + (4)x_3   
POL(ISNAT(x1)) = x_1   
POL(U511(x1, x2, x3)) = (4)x_2 + (4)x_3   
POL(U22(x1)) = x_1   
POL(U53(x1)) = 0   
POL(U32(x1)) = 0   
POL(n__take(x1, x2)) = 4 + (4)x_1 + (4)x_2   
POL(length(x1)) = (4)x_1   
POL(U52(x1, x2)) = 0   
POL(U521(x1, x2)) = (4)x_2   
POL(U61(x1, x2, x3)) = 0   
POL(ISNATILIST(x1)) = 4 + (4)x_1   
POL(U211(x1, x2)) = x_2   
POL(activate(x1)) = x_1   
POL(and(x1, x2)) = x_1 + x_2   
POL(isNatIListKind(x1)) = (4)x_1   
POL(U21(x1, x2)) = x_2   
POL(ISNATLIST(x1)) = (4)x_1   
POL(U111(x1, x2)) = (4)x_2   
POL(isNatList(x1)) = 0   
POL(zeros) = 0   
POL(isNatIList(x1)) = 0   
POL(U12(x1)) = 0   
POL(s(x1)) = x_1   
POL(isNat(x1)) = x_1   
POL(U51(x1, x2, x3)) = 0   
POL(n__isNatKind(x1)) = x_1   
POL(ISNATKIND(x1)) = x_1   
POL(U42(x1, x2)) = 0   
POL(n__zeros) = 0   
POL(n__cons(x1, x2)) = (4)x_1 + x_2   
POL(U11(x1, x2)) = x_2   
POL(U621(x1, x2)) = 4 + (4)x_2   
POL(U63(x1)) = 0   
POL(0) = 0   
POL(U43(x1)) = 0   
POL(n__and(x1, x2)) = x_1 + x_2   
POL(cons(x1, x2)) = (4)x_1 + x_2   
POL(U911(x1, x2, x3, x4)) = x_2 + x_3 + x_4   
POL(U421(x1, x2)) = 4 + (4)x_2   
POL(n__0) = 0   
POL(n__isNatIListKind(x1)) = (4)x_1   
The value of delta used in the strict ordering is 4.
The following usable rules [17] were oriented:

isNatIListKind(n__zeros) → tt
isNatKind(n__0) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIListKind(n__nil) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
isNatKind(X) → n__isNatKind(X)
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
activate(X) → X
activate(n__isNat(X)) → isNat(X)
activate(n__length(X)) → length(activate(X))
activate(n__0) → 0
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zeros) → zeros
activate(n__nil) → nil
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__isNatKind(X)) → isNatKind(X)
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__s(X)) → s(activate(X))
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
zeros → cons(0, n__zeros)
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U43(tt) → tt
U42(tt, V2) → U43(isNatIList(activate(V2)))
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U32(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U22(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U81(tt) → nil
U71(tt, L) → s(length(activate(L)))
U63(tt) → tt
U62(tt, V2) → U63(isNatIList(activate(V2)))
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U53(tt) → tt
U52(tt, V2) → U53(isNatList(activate(V2)))



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

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

U111(tt, V1) → ISNATLIST(activate(V1))
U511(tt, V1, V2) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
U711(tt, L) → LENGTH(activate(L))
U521(tt, V2) → ISNATLIST(activate(V2))
ACTIVATE(n__s(X)) → ACTIVATE(X)
U621(tt, V2) → ISNATILIST(activate(V2))
U911(tt, IL, M, N) → ACTIVATE(N)
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
U421(tt, V2) → ISNATILIST(activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__length(V1)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__length(V1)) → U111(isNatIListKind(activate(V1)), activate(V1))
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → U711(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
U611(tt, V1, V2) → U621(isNat(activate(V1)), activate(V2))
TAKE(0, IL) → ISNATILIST(IL)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
U521(tt, V2) → ACTIVATE(V2)
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U411(tt, V1, V2) → U421(isNat(activate(V1)), activate(V2))
U511(tt, V1, V2) → ACTIVATE(V1)
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATILIST(n__cons(V1, V2)) → U411(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
U911(tt, IL, M, N) → ACTIVATE(M)
U511(tt, V1, V2) → ISNAT(activate(V1))
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNatIListKind(activate(L)))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
U111(tt, V1) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
LENGTH(cons(N, L)) → ACTIVATE(L)
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ISNATLIST(n__cons(V1, V2)) → U511(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U511(tt, V1, V2) → U521(isNat(activate(V1)), activate(V2))
U211(tt, V1) → ACTIVATE(V1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
U711(tt, L) → ACTIVATE(L)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ACTIVATE(n__isNat(X)) → ISNAT(X)
LENGTH(cons(N, L)) → AND(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N)))
ISNATILIST(V) → U311(isNatIListKind(activate(V)), activate(V))
U911(tt, IL, M, N) → ACTIVATE(IL)
ISNAT(n__length(V1)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))), 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)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(n__isNatKind(X)) → isNatKind(X)
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 9 less nodes.

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

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

U421(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → U411(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U411(tt, V1, V2) → U421(isNat(activate(V1)), activate(V2))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))), 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)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(n__isNatKind(X)) → isNatKind(X)
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
                    ↳ QDPOrderProof

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

ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U111(tt, V1) → ISNATLIST(activate(V1))
U511(tt, V1, V2) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
U511(tt, V1, V2) → ACTIVATE(V1)
U711(tt, L) → LENGTH(activate(L))
U521(tt, V2) → ISNATLIST(activate(V2))
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__length(V1)) → ACTIVATE(V1)
U511(tt, V1, V2) → ISNAT(activate(V1))
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNatIListKind(activate(L)))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
U111(tt, V1) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__length(V1)) → U111(isNatIListKind(activate(V1)), activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → U711(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
LENGTH(cons(N, L)) → ACTIVATE(L)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ISNATLIST(n__cons(V1, V2)) → U511(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U511(tt, V1, V2) → U521(isNat(activate(V1)), activate(V2))
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
U211(tt, V1) → ACTIVATE(V1)
U521(tt, V2) → ACTIVATE(V2)
U711(tt, L) → ACTIVATE(L)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ACTIVATE(n__isNat(X)) → ISNAT(X)
LENGTH(cons(N, L)) → AND(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N)))
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__length(V1)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))), 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)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(n__isNatKind(X)) → isNatKind(X)
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)) → ISNATILISTKIND(activate(V1))
ISNATKIND(n__length(V1)) → ACTIVATE(V1)
ISNAT(n__length(V1)) → U111(isNatIListKind(activate(V1)), activate(V1))
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ISNAT(n__length(V1)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → ACTIVATE(X)
The remaining pairs can at least be oriented weakly.

ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U111(tt, V1) → ISNATLIST(activate(V1))
U511(tt, V1, V2) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
U511(tt, V1, V2) → ACTIVATE(V1)
U711(tt, L) → LENGTH(activate(L))
U521(tt, V2) → ISNATLIST(activate(V2))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
U511(tt, V1, V2) → ISNAT(activate(V1))
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNatIListKind(activate(L)))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
U111(tt, V1) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → U711(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
LENGTH(cons(N, L)) → ACTIVATE(L)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ISNATLIST(n__cons(V1, V2)) → U511(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U511(tt, V1, V2) → U521(isNat(activate(V1)), activate(V2))
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
U211(tt, V1) → ACTIVATE(V1)
U521(tt, V2) → ACTIVATE(V2)
U711(tt, L) → ACTIVATE(L)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ACTIVATE(n__isNat(X)) → ISNAT(X)
LENGTH(cons(N, L)) → AND(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N)))
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
Used ordering: Polynomial interpretation [25,35]:

POL(U62(x1, x2)) = 0   
POL(ISNATILISTKIND(x1)) = (4)x_1   
POL(n__isNat(x1)) = x_1   
POL(U41(x1, x2, x3)) = 2 + (3)x_2 + x_3   
POL(U81(x1)) = 0   
POL(U711(x1, x2)) = (4)x_2   
POL(n__s(x1)) = x_1   
POL(n__nil) = 0   
POL(take(x1, x2)) = (2)x_1 + x_2   
POL(U71(x1, x2)) = 2 + x_2   
POL(tt) = 0   
POL(AND(x1, x2)) = (4)x_1 + (4)x_2   
POL(nil) = 0   
POL(ACTIVATE(x1)) = (4)x_1   
POL(LENGTH(x1)) = (4)x_1   
POL(U91(x1, x2, x3, x4)) = x_2 + (2)x_3 + (4)x_4   
POL(U31(x1, x2)) = x_2   
POL(n__length(x1)) = 2 + x_1   
POL(isNatKind(x1)) = x_1   
POL(ISNAT(x1)) = (4)x_1   
POL(U511(x1, x2, x3)) = (4)x_2 + (4)x_3   
POL(U22(x1)) = x_1   
POL(U53(x1)) = 0   
POL(U32(x1)) = 0   
POL(n__take(x1, x2)) = (2)x_1 + x_2   
POL(length(x1)) = 2 + x_1   
POL(U521(x1, x2)) = (4)x_2   
POL(U52(x1, x2)) = 0   
POL(U61(x1, x2, x3)) = 0   
POL(U211(x1, x2)) = (4)x_2   
POL(activate(x1)) = x_1   
POL(and(x1, x2)) = (2)x_1 + x_2   
POL(isNatIListKind(x1)) = x_1   
POL(U21(x1, x2)) = x_2   
POL(ISNATLIST(x1)) = (4)x_1   
POL(U111(x1, x2)) = (4)x_2   
POL(isNatList(x1)) = 0   
POL(zeros) = 0   
POL(isNatIList(x1)) = 2 + (2)x_1   
POL(U12(x1)) = 0   
POL(s(x1)) = x_1   
POL(isNat(x1)) = x_1   
POL(U51(x1, x2, x3)) = 0   
POL(U42(x1, x2)) = 2   
POL(n__isNatKind(x1)) = x_1   
POL(ISNATKIND(x1)) = (4)x_1   
POL(n__zeros) = 0   
POL(n__cons(x1, x2)) = (4)x_1 + x_2   
POL(U11(x1, x2)) = 2   
POL(U63(x1)) = 0   
POL(0) = 0   
POL(U43(x1)) = 1   
POL(n__and(x1, x2)) = (2)x_1 + x_2   
POL(cons(x1, x2)) = (4)x_1 + x_2   
POL(n__0) = 0   
POL(n__isNatIListKind(x1)) = x_1   
The value of delta used in the strict ordering is 8.
The following usable rules [17] were oriented:

isNatIListKind(n__zeros) → tt
isNatKind(n__0) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__nil) → tt
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIListKind(n__nil) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
isNatKind(X) → n__isNatKind(X)
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
activate(X) → X
activate(n__isNat(X)) → isNat(X)
activate(n__length(X)) → length(activate(X))
activate(n__0) → 0
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zeros) → zeros
activate(n__nil) → nil
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__isNatKind(X)) → isNatKind(X)
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__s(X)) → s(activate(X))
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
zeros → cons(0, n__zeros)
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U43(tt) → tt
U42(tt, V2) → U43(isNatIList(activate(V2)))
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U32(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U22(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U81(tt) → nil
U71(tt, L) → s(length(activate(L)))
U63(tt) → tt
U62(tt, V2) → U63(isNatIList(activate(V2)))
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U53(tt) → tt
U52(tt, V2) → U53(isNatList(activate(V2)))



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

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

ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U111(tt, V1) → ISNATLIST(activate(V1))
ISNATLIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
U511(tt, V1, V2) → ACTIVATE(V2)
U511(tt, V1, V2) → ACTIVATE(V1)
U711(tt, L) → LENGTH(activate(L))
U521(tt, V2) → ISNATLIST(activate(V2))
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
U511(tt, V1, V2) → ISNAT(activate(V1))
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNatIListKind(activate(L)))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
U111(tt, V1) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → U711(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
LENGTH(cons(N, L)) → ACTIVATE(L)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ISNATLIST(n__cons(V1, V2)) → U511(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U511(tt, V1, V2) → U521(isNat(activate(V1)), activate(V2))
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
U211(tt, V1) → ACTIVATE(V1)
U521(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
U711(tt, L) → ACTIVATE(L)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
LENGTH(cons(N, L)) → AND(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N)))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → ISNAT(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))), 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)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(n__isNatKind(X)) → isNatKind(X)
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 15 less nodes.

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

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

ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
U211(tt, V1) → ACTIVATE(V1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → ISNAT(activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))), 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)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(n__isNatKind(X)) → isNatKind(X)
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.


ACTIVATE(n__isNat(X)) → ISNAT(X)
The remaining pairs can at least be oriented weakly.

ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
U211(tt, V1) → ACTIVATE(V1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → ISNAT(activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
Used ordering: Polynomial interpretation [25,35]:

POL(U62(x1, x2)) = 1   
POL(ISNATILISTKIND(x1)) = 1 + x_1   
POL(n__isNat(x1)) = 1 + (2)x_1   
POL(U41(x1, x2, x3)) = 1   
POL(U81(x1)) = 0   
POL(n__nil) = 0   
POL(n__s(x1)) = (4)x_1   
POL(take(x1, x2)) = x_1 + x_2   
POL(U71(x1, x2)) = (4)x_2   
POL(tt) = 0   
POL(AND(x1, x2)) = 1 + x_2   
POL(ACTIVATE(x1)) = 1 + x_1   
POL(nil) = 0   
POL(U91(x1, x2, x3, x4)) = (4)x_2 + (4)x_3 + (2)x_4   
POL(U31(x1, x2)) = 2   
POL(n__length(x1)) = x_1   
POL(isNatKind(x1)) = x_1   
POL(ISNAT(x1)) = 1 + x_1   
POL(U22(x1)) = 0   
POL(U53(x1)) = x_1   
POL(U32(x1)) = 2   
POL(n__take(x1, x2)) = x_1 + x_2   
POL(length(x1)) = x_1   
POL(U52(x1, x2)) = 4 + x_2   
POL(U61(x1, x2, x3)) = 1 + x_1   
POL(U211(x1, x2)) = 1 + x_2   
POL(activate(x1)) = x_1   
POL(and(x1, x2)) = x_1 + x_2   
POL(U21(x1, x2)) = 1 + (4)x_1 + x_2   
POL(isNatIListKind(x1)) = x_1   
POL(isNatList(x1)) = 4 + x_1   
POL(zeros) = 0   
POL(isNatIList(x1)) = 2   
POL(U12(x1)) = 0   
POL(s(x1)) = (4)x_1   
POL(isNat(x1)) = 1 + (2)x_1   
POL(U51(x1, x2, x3)) = 4 + (2)x_2 + x_3   
POL(n__isNatKind(x1)) = x_1   
POL(ISNATKIND(x1)) = 1 + x_1   
POL(U42(x1, x2)) = 1   
POL(n__zeros) = 0   
POL(n__cons(x1, x2)) = (2)x_1 + (4)x_2   
POL(U11(x1, x2)) = 0   
POL(U63(x1)) = 1   
POL(0) = 0   
POL(U43(x1)) = 1   
POL(n__and(x1, x2)) = x_1 + x_2   
POL(cons(x1, x2)) = (2)x_1 + (4)x_2   
POL(n__0) = 0   
POL(n__isNatIListKind(x1)) = x_1   
The value of delta used in the strict ordering is 1.
The following usable rules [17] were oriented:

isNatIListKind(n__zeros) → tt
isNatKind(n__0) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__nil) → tt
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIListKind(n__nil) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
isNatKind(X) → n__isNatKind(X)
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
activate(X) → X
activate(n__isNat(X)) → isNat(X)
activate(n__length(X)) → length(activate(X))
activate(n__0) → 0
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zeros) → zeros
activate(n__nil) → nil
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__isNatKind(X)) → isNatKind(X)
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__s(X)) → s(activate(X))
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
zeros → cons(0, n__zeros)
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U43(tt) → tt
U42(tt, V2) → U43(isNatIList(activate(V2)))
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U32(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U22(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U81(tt) → nil
U71(tt, L) → s(length(activate(L)))
U63(tt) → tt
U62(tt, V2) → U63(isNatIList(activate(V2)))
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U53(tt) → tt
U52(tt, V2) → U53(isNatList(activate(V2)))



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

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

ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
U211(tt, V1) → ACTIVATE(V1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))), 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)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(n__isNatKind(X)) → isNatKind(X)
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 3 less nodes.

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

ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))), 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)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(n__isNatKind(X)) → isNatKind(X)
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.


ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
The remaining pairs can at least be oriented weakly.

ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
Used ordering: Polynomial interpretation [25,35]:

POL(U52(x1, x2)) = x_1   
POL(U62(x1, x2)) = 1 + (4)x_1   
POL(ISNATILISTKIND(x1)) = 1 + (2)x_1   
POL(n__isNat(x1)) = 0   
POL(U41(x1, x2, x3)) = 3   
POL(U61(x1, x2, x3)) = 1   
POL(U81(x1)) = 3   
POL(activate(x1)) = x_1   
POL(n__nil) = 3   
POL(and(x1, x2)) = (4)x_1 + x_2   
POL(n__s(x1)) = x_1   
POL(take(x1, x2)) = 3 + (3)x_1 + x_2   
POL(U71(x1, x2)) = 1 + (4)x_2   
POL(isNatIListKind(x1)) = 1 + (2)x_1   
POL(U21(x1, x2)) = 0   
POL(tt) = 0   
POL(AND(x1, x2)) = x_1 + x_2   
POL(isNatList(x1)) = 1   
POL(zeros) = 0   
POL(isNatIList(x1)) = 3   
POL(U12(x1)) = 0   
POL(s(x1)) = x_1   
POL(isNat(x1)) = 0   
POL(U51(x1, x2, x3)) = 0   
POL(ACTIVATE(x1)) = x_1   
POL(nil) = 3   
POL(U91(x1, x2, x3, x4)) = 3 + x_2 + (3)x_3 + (4)x_4   
POL(U31(x1, x2)) = 0   
POL(n__length(x1)) = 1 + (4)x_1   
POL(ISNATKIND(x1)) = x_1   
POL(n__isNatKind(x1)) = x_1   
POL(U42(x1, x2)) = 3   
POL(n__zeros) = 0   
POL(n__cons(x1, x2)) = (4)x_1 + x_2   
POL(U11(x1, x2)) = 0   
POL(isNatKind(x1)) = x_1   
POL(U63(x1)) = 1   
POL(0) = 0   
POL(U43(x1)) = 3   
POL(n__and(x1, x2)) = (4)x_1 + x_2   
POL(cons(x1, x2)) = (4)x_1 + x_2   
POL(U22(x1)) = (2)x_1   
POL(U53(x1)) = 0   
POL(n__0) = 0   
POL(U32(x1)) = 0   
POL(n__isNatIListKind(x1)) = 1 + (2)x_1   
POL(n__take(x1, x2)) = 3 + (3)x_1 + x_2   
POL(length(x1)) = 1 + (4)x_1   
The value of delta used in the strict ordering is 1.
The following usable rules [17] were oriented:

isNatIListKind(n__zeros) → tt
isNatKind(n__0) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIListKind(n__nil) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
isNatKind(X) → n__isNatKind(X)
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
activate(X) → X
activate(n__isNat(X)) → isNat(X)
activate(n__length(X)) → length(activate(X))
activate(n__0) → 0
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zeros) → zeros
activate(n__nil) → nil
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__isNatKind(X)) → isNatKind(X)
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__s(X)) → s(activate(X))
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
zeros → cons(0, n__zeros)
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U43(tt) → tt
U42(tt, V2) → U43(isNatIList(activate(V2)))
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U32(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U22(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U81(tt) → nil
U71(tt, L) → s(length(activate(L)))
U63(tt) → tt
U62(tt, V2) → U63(isNatIList(activate(V2)))
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U53(tt) → tt
U52(tt, V2) → U53(isNatList(activate(V2)))



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

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

ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))), 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)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(n__isNatKind(X)) → isNatKind(X)
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
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ AND
                            ↳ QDP
                              ↳ QDPOrderProof
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ AND
                                      ↳ QDP
                                      ↳ QDP
                            ↳ QDP
                            ↳ QDP

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

U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))), 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)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(n__isNatKind(X)) → isNatKind(X)
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
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ AND
                            ↳ QDP
                            ↳ QDP
                            ↳ QDP

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

ISNATLIST(n__cons(V1, V2)) → U511(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U511(tt, V1, V2) → U521(isNat(activate(V1)), activate(V2))
U521(tt, V2) → ISNATLIST(activate(V2))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))), 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)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(n__isNatKind(X)) → isNatKind(X)
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
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ AND
                            ↳ QDP
                            ↳ QDP
                            ↳ QDP

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

LENGTH(cons(N, L)) → U711(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
U711(tt, L) → LENGTH(activate(L))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(n__and(n__isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N)))), 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)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(activate(X1), X2)
activate(n__isNat(X)) → isNat(X)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

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