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

(0) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0, IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U101(X1, X2, X3)) → active(U101(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U102(X1, X2, X3)) → active(U102(mark(X1), X2, X3))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U103(X1, X2, X3)) → active(U103(mark(X1), X2, X3))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U104(X1, X2, X3)) → active(U104(mark(X1), X2, X3))
mark(U105(X1, X2)) → active(U105(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(U106(X)) → active(U106(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(U111(X1, X2, X3)) → active(U111(mark(X1), X2, X3))
mark(U112(X1, X2, X3)) → active(U112(mark(X1), X2, X3))
mark(U113(X1, X2, X3)) → active(U113(mark(X1), X2, X3))
mark(U114(X1, X2)) → active(U114(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U121(X1, X2)) → active(U121(mark(X1), X2))
mark(U122(X)) → active(U122(mark(X)))
mark(nil) → active(nil)
mark(U131(X1, X2, X3, X4)) → active(U131(mark(X1), X2, X3, X4))
mark(U132(X1, X2, X3, X4)) → active(U132(mark(X1), X2, X3, X4))
mark(U133(X1, X2, X3, X4)) → active(U133(mark(X1), X2, X3, X4))
mark(U134(X1, X2, X3, X4)) → active(U134(mark(X1), X2, X3, X4))
mark(U135(X1, X2, X3, X4)) → active(U135(mark(X1), X2, X3, X4))
mark(U136(X1, X2, X3, X4)) → active(U136(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U23(X)) → active(U23(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2, X3)) → active(U94(mark(X1), X2, X3))
mark(U95(X1, X2)) → active(U95(mark(X1), X2))
mark(U96(X)) → active(U96(mark(X)))
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U101(mark(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, mark(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, mark(X3)) → U101(X1, X2, X3)
U101(active(X1), X2, X3) → U101(X1, X2, X3)
U101(X1, active(X2), X3) → U101(X1, X2, X3)
U101(X1, X2, active(X3)) → U101(X1, X2, X3)
U102(mark(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, mark(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, mark(X3)) → U102(X1, X2, X3)
U102(active(X1), X2, X3) → U102(X1, X2, X3)
U102(X1, active(X2), X3) → U102(X1, X2, X3)
U102(X1, X2, active(X3)) → U102(X1, X2, X3)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U103(mark(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, mark(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, mark(X3)) → U103(X1, X2, X3)
U103(active(X1), X2, X3) → U103(X1, X2, X3)
U103(X1, active(X2), X3) → U103(X1, X2, X3)
U103(X1, X2, active(X3)) → U103(X1, X2, X3)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U104(mark(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, mark(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, mark(X3)) → U104(X1, X2, X3)
U104(active(X1), X2, X3) → U104(X1, X2, X3)
U104(X1, active(X2), X3) → U104(X1, X2, X3)
U104(X1, X2, active(X3)) → U104(X1, X2, X3)
U105(mark(X1), X2) → U105(X1, X2)
U105(X1, mark(X2)) → U105(X1, X2)
U105(active(X1), X2) → U105(X1, X2)
U105(X1, active(X2)) → U105(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U106(mark(X)) → U106(X)
U106(active(X)) → U106(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U111(mark(X1), X2, X3) → U111(X1, X2, X3)
U111(X1, mark(X2), X3) → U111(X1, X2, X3)
U111(X1, X2, mark(X3)) → U111(X1, X2, X3)
U111(active(X1), X2, X3) → U111(X1, X2, X3)
U111(X1, active(X2), X3) → U111(X1, X2, X3)
U111(X1, X2, active(X3)) → U111(X1, X2, X3)
U112(mark(X1), X2, X3) → U112(X1, X2, X3)
U112(X1, mark(X2), X3) → U112(X1, X2, X3)
U112(X1, X2, mark(X3)) → U112(X1, X2, X3)
U112(active(X1), X2, X3) → U112(X1, X2, X3)
U112(X1, active(X2), X3) → U112(X1, X2, X3)
U112(X1, X2, active(X3)) → U112(X1, X2, X3)
U113(mark(X1), X2, X3) → U113(X1, X2, X3)
U113(X1, mark(X2), X3) → U113(X1, X2, X3)
U113(X1, X2, mark(X3)) → U113(X1, X2, X3)
U113(active(X1), X2, X3) → U113(X1, X2, X3)
U113(X1, active(X2), X3) → U113(X1, X2, X3)
U113(X1, X2, active(X3)) → U113(X1, X2, X3)
U114(mark(X1), X2) → U114(X1, X2)
U114(X1, mark(X2)) → U114(X1, X2)
U114(active(X1), X2) → U114(X1, X2)
U114(X1, active(X2)) → U114(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U121(mark(X1), X2) → U121(X1, X2)
U121(X1, mark(X2)) → U121(X1, X2)
U121(active(X1), X2) → U121(X1, X2)
U121(X1, active(X2)) → U121(X1, X2)
U122(mark(X)) → U122(X)
U122(active(X)) → U122(X)
U131(mark(X1), X2, X3, X4) → U131(X1, X2, X3, X4)
U131(X1, mark(X2), X3, X4) → U131(X1, X2, X3, X4)
U131(X1, X2, mark(X3), X4) → U131(X1, X2, X3, X4)
U131(X1, X2, X3, mark(X4)) → U131(X1, X2, X3, X4)
U131(active(X1), X2, X3, X4) → U131(X1, X2, X3, X4)
U131(X1, active(X2), X3, X4) → U131(X1, X2, X3, X4)
U131(X1, X2, active(X3), X4) → U131(X1, X2, X3, X4)
U131(X1, X2, X3, active(X4)) → U131(X1, X2, X3, X4)
U132(mark(X1), X2, X3, X4) → U132(X1, X2, X3, X4)
U132(X1, mark(X2), X3, X4) → U132(X1, X2, X3, X4)
U132(X1, X2, mark(X3), X4) → U132(X1, X2, X3, X4)
U132(X1, X2, X3, mark(X4)) → U132(X1, X2, X3, X4)
U132(active(X1), X2, X3, X4) → U132(X1, X2, X3, X4)
U132(X1, active(X2), X3, X4) → U132(X1, X2, X3, X4)
U132(X1, X2, active(X3), X4) → U132(X1, X2, X3, X4)
U132(X1, X2, X3, active(X4)) → U132(X1, X2, X3, X4)
U133(mark(X1), X2, X3, X4) → U133(X1, X2, X3, X4)
U133(X1, mark(X2), X3, X4) → U133(X1, X2, X3, X4)
U133(X1, X2, mark(X3), X4) → U133(X1, X2, X3, X4)
U133(X1, X2, X3, mark(X4)) → U133(X1, X2, X3, X4)
U133(active(X1), X2, X3, X4) → U133(X1, X2, X3, X4)
U133(X1, active(X2), X3, X4) → U133(X1, X2, X3, X4)
U133(X1, X2, active(X3), X4) → U133(X1, X2, X3, X4)
U133(X1, X2, X3, active(X4)) → U133(X1, X2, X3, X4)
U134(mark(X1), X2, X3, X4) → U134(X1, X2, X3, X4)
U134(X1, mark(X2), X3, X4) → U134(X1, X2, X3, X4)
U134(X1, X2, mark(X3), X4) → U134(X1, X2, X3, X4)
U134(X1, X2, X3, mark(X4)) → U134(X1, X2, X3, X4)
U134(active(X1), X2, X3, X4) → U134(X1, X2, X3, X4)
U134(X1, active(X2), X3, X4) → U134(X1, X2, X3, X4)
U134(X1, X2, active(X3), X4) → U134(X1, X2, X3, X4)
U134(X1, X2, X3, active(X4)) → U134(X1, X2, X3, X4)
U135(mark(X1), X2, X3, X4) → U135(X1, X2, X3, X4)
U135(X1, mark(X2), X3, X4) → U135(X1, X2, X3, X4)
U135(X1, X2, mark(X3), X4) → U135(X1, X2, X3, X4)
U135(X1, X2, X3, mark(X4)) → U135(X1, X2, X3, X4)
U135(active(X1), X2, X3, X4) → U135(X1, X2, X3, X4)
U135(X1, active(X2), X3, X4) → U135(X1, X2, X3, X4)
U135(X1, X2, active(X3), X4) → U135(X1, X2, X3, X4)
U135(X1, X2, X3, active(X4)) → U135(X1, X2, X3, X4)
U136(mark(X1), X2, X3, X4) → U136(X1, X2, X3, X4)
U136(X1, mark(X2), X3, X4) → U136(X1, X2, X3, X4)
U136(X1, X2, mark(X3), X4) → U136(X1, X2, X3, X4)
U136(X1, X2, X3, mark(X4)) → U136(X1, X2, X3, X4)
U136(active(X1), X2, X3, X4) → U136(X1, X2, X3, X4)
U136(X1, active(X2), X3, X4) → U136(X1, X2, X3, X4)
U136(X1, X2, active(X3), X4) → U136(X1, X2, X3, X4)
U136(X1, X2, X3, active(X4)) → U136(X1, X2, X3, X4)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U62(mark(X)) → U62(X)
U62(active(X)) → U62(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2, X3) → U94(X1, X2, X3)
U94(X1, mark(X2), X3) → U94(X1, X2, X3)
U94(X1, X2, mark(X3)) → U94(X1, X2, X3)
U94(active(X1), X2, X3) → U94(X1, X2, X3)
U94(X1, active(X2), X3) → U94(X1, X2, X3)
U94(X1, X2, active(X3)) → U94(X1, X2, X3)
U95(mark(X1), X2) → U95(X1, X2)
U95(X1, mark(X2)) → U95(X1, X2)
U95(active(X1), X2) → U95(X1, X2)
U95(X1, active(X2)) → U95(X1, X2)
U96(mark(X)) → U96(X)
U96(active(X)) → U96(X)

Q is empty.