Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 QDP

(0) Obligation:

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

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
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(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

ZEROS → CONS(0, n__zeros)
ZEROS → 01
U1011(tt, V1, V2) → U1021(isNatKind(activate(V1)), activate(V1), activate(V2))
U1011(tt, V1, V2) → ISNATKIND(activate(V1))
U1011(tt, V1, V2) → ACTIVATE(V1)
U1011(tt, V1, V2) → ACTIVATE(V2)
U1021(tt, V1, V2) → U1031(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U1021(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U1021(tt, V1, V2) → ACTIVATE(V2)
U1021(tt, V1, V2) → ACTIVATE(V1)
U1031(tt, V1, V2) → U1041(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U1031(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U1031(tt, V1, V2) → ACTIVATE(V2)
U1031(tt, V1, V2) → ACTIVATE(V1)
U1041(tt, V1, V2) → U1051(isNat(activate(V1)), activate(V2))
U1041(tt, V1, V2) → ISNAT(activate(V1))
U1041(tt, V1, V2) → ACTIVATE(V1)
U1041(tt, V1, V2) → ACTIVATE(V2)
U1051(tt, V2) → U1061(isNatIList(activate(V2)))
U1051(tt, V2) → ISNATILIST(activate(V2))
U1051(tt, V2) → ACTIVATE(V2)
U111(tt, V1) → U121(isNatIListKind(activate(V1)), activate(V1))
U111(tt, V1) → ISNATILISTKIND(activate(V1))
U111(tt, V1) → ACTIVATE(V1)
U1111(tt, L, N) → U1121(isNatIListKind(activate(L)), activate(L), activate(N))
U1111(tt, L, N) → ISNATILISTKIND(activate(L))
U1111(tt, L, N) → ACTIVATE(L)
U1111(tt, L, N) → ACTIVATE(N)
U1121(tt, L, N) → U1131(isNat(activate(N)), activate(L), activate(N))
U1121(tt, L, N) → ISNAT(activate(N))
U1121(tt, L, N) → ACTIVATE(N)
U1121(tt, L, N) → ACTIVATE(L)
U1131(tt, L, N) → U1141(isNatKind(activate(N)), activate(L))
U1131(tt, L, N) → ISNATKIND(activate(N))
U1131(tt, L, N) → ACTIVATE(N)
U1131(tt, L, N) → ACTIVATE(L)
U1141(tt, L) → S(length(activate(L)))
U1141(tt, L) → LENGTH(activate(L))
U1141(tt, L) → ACTIVATE(L)
U121(tt, V1) → U131(isNatList(activate(V1)))
U121(tt, V1) → ISNATLIST(activate(V1))
U121(tt, V1) → ACTIVATE(V1)
U1211(tt, IL) → U1221(isNatIListKind(activate(IL)))
U1211(tt, IL) → ISNATILISTKIND(activate(IL))
U1211(tt, IL) → ACTIVATE(IL)
U1221(tt) → NIL
U1311(tt, IL, M, N) → U1321(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U1311(tt, IL, M, N) → ISNATILISTKIND(activate(IL))
U1311(tt, IL, M, N) → ACTIVATE(IL)
U1311(tt, IL, M, N) → ACTIVATE(M)
U1311(tt, IL, M, N) → ACTIVATE(N)
U1321(tt, IL, M, N) → U1331(isNat(activate(M)), activate(IL), activate(M), activate(N))
U1321(tt, IL, M, N) → ISNAT(activate(M))
U1321(tt, IL, M, N) → ACTIVATE(M)
U1321(tt, IL, M, N) → ACTIVATE(IL)
U1321(tt, IL, M, N) → ACTIVATE(N)
U1331(tt, IL, M, N) → U1341(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U1331(tt, IL, M, N) → ISNATKIND(activate(M))
U1331(tt, IL, M, N) → ACTIVATE(M)
U1331(tt, IL, M, N) → ACTIVATE(IL)
U1331(tt, IL, M, N) → ACTIVATE(N)
U1341(tt, IL, M, N) → U1351(isNat(activate(N)), activate(IL), activate(M), activate(N))
U1341(tt, IL, M, N) → ISNAT(activate(N))
U1341(tt, IL, M, N) → ACTIVATE(N)
U1341(tt, IL, M, N) → ACTIVATE(IL)
U1341(tt, IL, M, N) → ACTIVATE(M)
U1351(tt, IL, M, N) → U1361(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U1351(tt, IL, M, N) → ISNATKIND(activate(N))
U1351(tt, IL, M, N) → ACTIVATE(N)
U1351(tt, IL, M, N) → ACTIVATE(IL)
U1351(tt, IL, M, N) → ACTIVATE(M)
U1361(tt, IL, M, N) → CONS(activate(N), n__take(activate(M), activate(IL)))
U1361(tt, IL, M, N) → ACTIVATE(N)
U1361(tt, IL, M, N) → ACTIVATE(M)
U1361(tt, IL, M, N) → ACTIVATE(IL)
U211(tt, V1) → U221(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → ISNATKIND(activate(V1))
U211(tt, V1) → ACTIVATE(V1)
U221(tt, V1) → U231(isNat(activate(V1)))
U221(tt, V1) → ISNAT(activate(V1))
U221(tt, V1) → ACTIVATE(V1)
U311(tt, V) → U321(isNatIListKind(activate(V)), activate(V))
U311(tt, V) → ISNATILISTKIND(activate(V))
U311(tt, V) → ACTIVATE(V)
U321(tt, V) → U331(isNatList(activate(V)))
U321(tt, V) → ISNATLIST(activate(V))
U321(tt, V) → ACTIVATE(V)
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U411(tt, V1, V2) → ISNATKIND(activate(V1))
U411(tt, V1, V2) → ACTIVATE(V1)
U411(tt, V1, V2) → ACTIVATE(V2)
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U421(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U421(tt, V1, V2) → ACTIVATE(V2)
U421(tt, V1, V2) → ACTIVATE(V1)
U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U431(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U431(tt, V1, V2) → ACTIVATE(V2)
U431(tt, V1, V2) → ACTIVATE(V1)
U441(tt, V1, V2) → U451(isNat(activate(V1)), activate(V2))
U441(tt, V1, V2) → ISNAT(activate(V1))
U441(tt, V1, V2) → ACTIVATE(V1)
U441(tt, V1, V2) → ACTIVATE(V2)
U451(tt, V2) → U461(isNatIList(activate(V2)))
U451(tt, V2) → ISNATILIST(activate(V2))
U451(tt, V2) → ACTIVATE(V2)
U511(tt, V2) → U521(isNatIListKind(activate(V2)))
U511(tt, V2) → ISNATILISTKIND(activate(V2))
U511(tt, V2) → ACTIVATE(V2)
U611(tt, V2) → U621(isNatIListKind(activate(V2)))
U611(tt, V2) → ISNATILISTKIND(activate(V2))
U611(tt, V2) → ACTIVATE(V2)
U911(tt, V1, V2) → U921(isNatKind(activate(V1)), activate(V1), activate(V2))
U911(tt, V1, V2) → ISNATKIND(activate(V1))
U911(tt, V1, V2) → ACTIVATE(V1)
U911(tt, V1, V2) → ACTIVATE(V2)
U921(tt, V1, V2) → U931(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U921(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U921(tt, V1, V2) → ACTIVATE(V2)
U921(tt, V1, V2) → ACTIVATE(V1)
U931(tt, V1, V2) → U941(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U931(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U931(tt, V1, V2) → ACTIVATE(V2)
U931(tt, V1, V2) → ACTIVATE(V1)
U941(tt, V1, V2) → U951(isNat(activate(V1)), activate(V2))
U941(tt, V1, V2) → ISNAT(activate(V1))
U941(tt, V1, V2) → ACTIVATE(V1)
U941(tt, V1, V2) → ACTIVATE(V2)
U951(tt, V2) → U961(isNatList(activate(V2)))
U951(tt, V2) → ISNATLIST(activate(V2))
U951(tt, V2) → ACTIVATE(V2)
ISNAT(n__length(V1)) → U111(isNatIListKind(activate(V1)), activate(V1))
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNAT(n__length(V1)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
ISNATILIST(V) → U311(isNatIListKind(activate(V)), activate(V))
ISNATILIST(V) → ISNATILISTKIND(activate(V))
ISNATILIST(V) → ACTIVATE(V)
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILISTKIND(n__cons(V1, V2)) → U511(isNatKind(activate(V1)), activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILISTKIND(n__take(V1, V2)) → U611(isNatKind(activate(V1)), activate(V2))
ISNATILISTKIND(n__take(V1, V2)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V2)
ISNATKIND(n__length(V1)) → U711(isNatIListKind(activate(V1)))
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNATKIND(n__length(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(V1)) → U811(isNatKind(activate(V1)))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → U911(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → U1011(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATLIST(n__take(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
LENGTH(nil) → 01
LENGTH(cons(N, L)) → U1111(isNatList(activate(L)), activate(L), N)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
LENGTH(cons(N, L)) → ACTIVATE(L)
TAKE(0, IL) → U1211(isNatIList(IL), IL)
TAKE(0, IL) → ISNATILIST(IL)
TAKE(s(M), cons(N, IL)) → U1311(isNatIList(activate(IL)), activate(IL), M, N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ACTIVATE(n__zeros) → ZEROS
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
ACTIVATE(n__0) → 01
ACTIVATE(n__length(X)) → LENGTH(X)
ACTIVATE(n__s(X)) → S(X)
ACTIVATE(n__cons(X1, X2)) → CONS(X1, X2)
ACTIVATE(n__nil) → NIL

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
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(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 22 less nodes.

(4) Obligation:

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

ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
TAKE(0, IL) → U1211(isNatIList(IL), IL)
U1211(tt, IL) → ISNATILISTKIND(activate(IL))
ISNATILISTKIND(n__cons(V1, V2)) → U511(isNatKind(activate(V1)), activate(V2))
U511(tt, V2) → ISNATILISTKIND(activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → LENGTH(X)
LENGTH(cons(N, L)) → U1111(isNatList(activate(L)), activate(L), N)
U1111(tt, L, N) → U1121(isNatIListKind(activate(L)), activate(L), activate(N))
U1121(tt, L, N) → U1131(isNat(activate(N)), activate(L), activate(N))
U1131(tt, L, N) → U1141(isNatKind(activate(N)), activate(L))
U1141(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(V1, V2)) → U911(isNatKind(activate(V1)), activate(V1), activate(V2))
U911(tt, V1, V2) → U921(isNatKind(activate(V1)), activate(V1), activate(V2))
U921(tt, V1, V2) → U931(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U931(tt, V1, V2) → U941(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U941(tt, V1, V2) → U951(isNat(activate(V1)), activate(V2))
U951(tt, V2) → ISNATLIST(activate(V2))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__length(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → U1011(isNatKind(activate(V1)), activate(V1), activate(V2))
U1011(tt, V1, V2) → U1021(isNatKind(activate(V1)), activate(V1), activate(V2))
U1021(tt, V1, V2) → U1031(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U1031(tt, V1, V2) → U1041(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U1041(tt, V1, V2) → U1051(isNat(activate(V1)), activate(V2))
U1051(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(V) → U311(isNatIListKind(activate(V)), activate(V))
U311(tt, V) → U321(isNatIListKind(activate(V)), activate(V))
U321(tt, V) → ISNATLIST(activate(V))
ISNATLIST(n__take(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
U321(tt, V) → ACTIVATE(V)
U311(tt, V) → ISNATILISTKIND(activate(V))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILISTKIND(n__take(V1, V2)) → U611(isNatKind(activate(V1)), activate(V2))
U611(tt, V2) → ISNATILISTKIND(activate(V2))
ISNATILISTKIND(n__take(V1, V2)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V2)
U611(tt, V2) → ACTIVATE(V2)
U311(tt, V) → ACTIVATE(V)
ISNATILIST(V) → ISNATILISTKIND(activate(V))
ISNATILIST(V) → ACTIVATE(V)
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, V1, V2) → U451(isNat(activate(V1)), activate(V2))
U451(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U451(tt, V2) → ACTIVATE(V2)
U441(tt, V1, V2) → ISNAT(activate(V1))
ISNAT(n__length(V1)) → U111(isNatIListKind(activate(V1)), activate(V1))
U111(tt, V1) → U121(isNatIListKind(activate(V1)), activate(V1))
U121(tt, V1) → ISNATLIST(activate(V1))
U121(tt, V1) → ACTIVATE(V1)
U111(tt, V1) → ISNATILISTKIND(activate(V1))
U111(tt, V1) → ACTIVATE(V1)
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNAT(n__length(V1)) → ACTIVATE(V1)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → U221(isNatKind(activate(V1)), activate(V1))
U221(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
U221(tt, V1) → ACTIVATE(V1)
U211(tt, V1) → ISNATKIND(activate(V1))
U211(tt, V1) → ACTIVATE(V1)
U441(tt, V1, V2) → ACTIVATE(V1)
U441(tt, V1, V2) → ACTIVATE(V2)
U431(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U431(tt, V1, V2) → ACTIVATE(V2)
U431(tt, V1, V2) → ACTIVATE(V1)
U421(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U421(tt, V1, V2) → ACTIVATE(V2)
U421(tt, V1, V2) → ACTIVATE(V1)
U411(tt, V1, V2) → ISNATKIND(activate(V1))
U411(tt, V1, V2) → ACTIVATE(V1)
U411(tt, V1, V2) → ACTIVATE(V2)
U1051(tt, V2) → ACTIVATE(V2)
U1041(tt, V1, V2) → ISNAT(activate(V1))
U1041(tt, V1, V2) → ACTIVATE(V1)
U1041(tt, V1, V2) → ACTIVATE(V2)
U1031(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U1031(tt, V1, V2) → ACTIVATE(V2)
U1031(tt, V1, V2) → ACTIVATE(V1)
U1021(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U1021(tt, V1, V2) → ACTIVATE(V2)
U1021(tt, V1, V2) → ACTIVATE(V1)
U1011(tt, V1, V2) → ISNATKIND(activate(V1))
U1011(tt, V1, V2) → ACTIVATE(V1)
U1011(tt, V1, V2) → ACTIVATE(V2)
U951(tt, V2) → ACTIVATE(V2)
U941(tt, V1, V2) → ISNAT(activate(V1))
U941(tt, V1, V2) → ACTIVATE(V1)
U941(tt, V1, V2) → ACTIVATE(V2)
U931(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U931(tt, V1, V2) → ACTIVATE(V2)
U931(tt, V1, V2) → ACTIVATE(V1)
U921(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U921(tt, V1, V2) → ACTIVATE(V2)
U921(tt, V1, V2) → ACTIVATE(V1)
U911(tt, V1, V2) → ISNATKIND(activate(V1))
U911(tt, V1, V2) → ACTIVATE(V1)
U911(tt, V1, V2) → ACTIVATE(V2)
LENGTH(cons(N, L)) → ACTIVATE(L)
U1141(tt, L) → ACTIVATE(L)
U1131(tt, L, N) → ISNATKIND(activate(N))
U1131(tt, L, N) → ACTIVATE(N)
U1131(tt, L, N) → ACTIVATE(L)
U1121(tt, L, N) → ISNAT(activate(N))
U1121(tt, L, N) → ACTIVATE(N)
U1121(tt, L, N) → ACTIVATE(L)
U1111(tt, L, N) → ISNATILISTKIND(activate(L))
U1111(tt, L, N) → ACTIVATE(L)
U1111(tt, L, N) → ACTIVATE(N)
U511(tt, V2) → ACTIVATE(V2)
U1211(tt, IL) → ACTIVATE(IL)
TAKE(0, IL) → ISNATILIST(IL)
TAKE(s(M), cons(N, IL)) → U1311(isNatIList(activate(IL)), activate(IL), M, N)
U1311(tt, IL, M, N) → U1321(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U1321(tt, IL, M, N) → U1331(isNat(activate(M)), activate(IL), activate(M), activate(N))
U1331(tt, IL, M, N) → U1341(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U1341(tt, IL, M, N) → U1351(isNat(activate(N)), activate(IL), activate(M), activate(N))
U1351(tt, IL, M, N) → U1361(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U1361(tt, IL, M, N) → ACTIVATE(N)
U1361(tt, IL, M, N) → ACTIVATE(M)
U1361(tt, IL, M, N) → ACTIVATE(IL)
U1351(tt, IL, M, N) → ISNATKIND(activate(N))
U1351(tt, IL, M, N) → ACTIVATE(N)
U1351(tt, IL, M, N) → ACTIVATE(IL)
U1351(tt, IL, M, N) → ACTIVATE(M)
U1341(tt, IL, M, N) → ISNAT(activate(N))
U1341(tt, IL, M, N) → ACTIVATE(N)
U1341(tt, IL, M, N) → ACTIVATE(IL)
U1341(tt, IL, M, N) → ACTIVATE(M)
U1331(tt, IL, M, N) → ISNATKIND(activate(M))
U1331(tt, IL, M, N) → ACTIVATE(M)
U1331(tt, IL, M, N) → ACTIVATE(IL)
U1331(tt, IL, M, N) → ACTIVATE(N)
U1321(tt, IL, M, N) → ISNAT(activate(M))
U1321(tt, IL, M, N) → ACTIVATE(M)
U1321(tt, IL, M, N) → ACTIVATE(IL)
U1321(tt, IL, M, N) → ACTIVATE(N)
U1311(tt, IL, M, N) → ISNATILISTKIND(activate(IL))
U1311(tt, IL, M, N) → ACTIVATE(IL)
U1311(tt, IL, M, N) → ACTIVATE(M)
U1311(tt, IL, M, N) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
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(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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