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.

(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:

ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(zeros) → CONS(0, zeros)
ACTIVE(U101(tt, V1, V2)) → MARK(U102(isNatKind(V1), V1, V2))
ACTIVE(U101(tt, V1, V2)) → U102¹(isNatKind(V1), V1, V2)
ACTIVE(U101(tt, V1, V2)) → ISNATKIND(V1)
ACTIVE(U102(tt, V1, V2)) → MARK(U103(isNatIListKind(V2), V1, V2))
ACTIVE(U102(tt, V1, V2)) → U103¹(isNatIListKind(V2), V1, V2)
ACTIVE(U102(tt, V1, V2)) → ISNATILISTKIND(V2)
ACTIVE(U103(tt, V1, V2)) → MARK(U104(isNatIListKind(V2), V1, V2))
ACTIVE(U103(tt, V1, V2)) → U104¹(isNatIListKind(V2), V1, V2)
ACTIVE(U103(tt, V1, V2)) → ISNATILISTKIND(V2)
ACTIVE(U104(tt, V1, V2)) → MARK(U105(isNat(V1), V2))
ACTIVE(U104(tt, V1, V2)) → U105¹(isNat(V1), V2)
ACTIVE(U104(tt, V1, V2)) → ISNAT(V1)
ACTIVE(U105(tt, V2)) → MARK(U106(isNatIList(V2)))
ACTIVE(U105(tt, V2)) → U106¹(isNatIList(V2))
ACTIVE(U105(tt, V2)) → ISNATILIST(V2)
ACTIVE(U106(tt)) → MARK(tt)
ACTIVE(U11(tt, V1)) → MARK(U12(isNatIListKind(V1), V1))
ACTIVE(U11(tt, V1)) → U12¹(isNatIListKind(V1), V1)
ACTIVE(U11(tt, V1)) → ISNATILISTKIND(V1)
ACTIVE(U111(tt, L, N)) → MARK(U112(isNatIListKind(L), L, N))
ACTIVE(U111(tt, L, N)) → U112¹(isNatIListKind(L), L, N)
ACTIVE(U111(tt, L, N)) → ISNATILISTKIND(L)
ACTIVE(U112(tt, L, N)) → MARK(U113(isNat(N), L, N))
ACTIVE(U112(tt, L, N)) → U113¹(isNat(N), L, N)
ACTIVE(U112(tt, L, N)) → ISNAT(N)
ACTIVE(U113(tt, L, N)) → MARK(U114(isNatKind(N), L))
ACTIVE(U113(tt, L, N)) → U114¹(isNatKind(N), L)
ACTIVE(U113(tt, L, N)) → ISNATKIND(N)
ACTIVE(U114(tt, L)) → MARK(s(length(L)))
ACTIVE(U114(tt, L)) → S(length(L))
ACTIVE(U114(tt, L)) → LENGTH(L)
ACTIVE(U12(tt, V1)) → MARK(U13(isNatList(V1)))
ACTIVE(U12(tt, V1)) → U13¹(isNatList(V1))
ACTIVE(U12(tt, V1)) → ISNATLIST(V1)
ACTIVE(U121(tt, IL)) → MARK(U122(isNatIListKind(IL)))
ACTIVE(U121(tt, IL)) → U122¹(isNatIListKind(IL))
ACTIVE(U121(tt, IL)) → 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(U131(tt, IL, M, N)) → U132¹(isNatIListKind(IL), IL, M, N)
ACTIVE(U131(tt, IL, M, N)) → ISNATILISTKIND(IL)
ACTIVE(U132(tt, IL, M, N)) → MARK(U133(isNat(M), IL, M, N))
ACTIVE(U132(tt, IL, M, N)) → U133¹(isNat(M), IL, M, N)
ACTIVE(U132(tt, IL, M, N)) → ISNAT(M)
ACTIVE(U133(tt, IL, M, N)) → MARK(U134(isNatKind(M), IL, M, N))
ACTIVE(U133(tt, IL, M, N)) → U134¹(isNatKind(M), IL, M, N)
ACTIVE(U133(tt, IL, M, N)) → ISNATKIND(M)
ACTIVE(U134(tt, IL, M, N)) → MARK(U135(isNat(N), IL, M, N))
ACTIVE(U134(tt, IL, M, N)) → U135¹(isNat(N), IL, M, N)
ACTIVE(U134(tt, IL, M, N)) → ISNAT(N)
ACTIVE(U135(tt, IL, M, N)) → MARK(U136(isNatKind(N), IL, M, N))
ACTIVE(U135(tt, IL, M, N)) → U136¹(isNatKind(N), IL, M, N)
ACTIVE(U135(tt, IL, M, N)) → ISNATKIND(N)
ACTIVE(U136(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U136(tt, IL, M, N)) → CONS(N, take(M, IL))
ACTIVE(U136(tt, IL, M, N)) → TAKE(M, IL)
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U21(tt, V1)) → U22¹(isNatKind(V1), V1)
ACTIVE(U21(tt, V1)) → ISNATKIND(V1)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U22(tt, V1)) → U23¹(isNat(V1))
ACTIVE(U22(tt, V1)) → ISNAT(V1)
ACTIVE(U23(tt)) → MARK(tt)
ACTIVE(U31(tt, V)) → MARK(U32(isNatIListKind(V), V))
ACTIVE(U31(tt, V)) → U32¹(isNatIListKind(V), V)
ACTIVE(U31(tt, V)) → ISNATILISTKIND(V)
ACTIVE(U32(tt, V)) → MARK(U33(isNatList(V)))
ACTIVE(U32(tt, V)) → U33¹(isNatList(V))
ACTIVE(U32(tt, V)) → ISNATLIST(V)
ACTIVE(U33(tt)) → MARK(tt)
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isNatKind(V1), V1, V2))
ACTIVE(U41(tt, V1, V2)) → U42¹(isNatKind(V1), V1, V2)
ACTIVE(U41(tt, V1, V2)) → ISNATKIND(V1)
ACTIVE(U42(tt, V1, V2)) → MARK(U43(isNatIListKind(V2), V1, V2))
ACTIVE(U42(tt, V1, V2)) → U43¹(isNatIListKind(V2), V1, V2)
ACTIVE(U42(tt, V1, V2)) → ISNATILISTKIND(V2)
ACTIVE(U43(tt, V1, V2)) → MARK(U44(isNatIListKind(V2), V1, V2))
ACTIVE(U43(tt, V1, V2)) → U44¹(isNatIListKind(V2), V1, V2)
ACTIVE(U43(tt, V1, V2)) → ISNATILISTKIND(V2)
ACTIVE(U44(tt, V1, V2)) → MARK(U45(isNat(V1), V2))
ACTIVE(U44(tt, V1, V2)) → U45¹(isNat(V1), V2)
ACTIVE(U44(tt, V1, V2)) → ISNAT(V1)
ACTIVE(U45(tt, V2)) → MARK(U46(isNatIList(V2)))
ACTIVE(U45(tt, V2)) → U46¹(isNatIList(V2))
ACTIVE(U45(tt, V2)) → ISNATILIST(V2)
ACTIVE(U46(tt)) → MARK(tt)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatIListKind(V2)))
ACTIVE(U51(tt, V2)) → U52¹(isNatIListKind(V2))
ACTIVE(U51(tt, V2)) → ISNATILISTKIND(V2)
ACTIVE(U52(tt)) → MARK(tt)
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIListKind(V2)))
ACTIVE(U61(tt, V2)) → U62¹(isNatIListKind(V2))
ACTIVE(U61(tt, V2)) → 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(U91(tt, V1, V2)) → U92¹(isNatKind(V1), V1, V2)
ACTIVE(U91(tt, V1, V2)) → ISNATKIND(V1)
ACTIVE(U92(tt, V1, V2)) → MARK(U93(isNatIListKind(V2), V1, V2))
ACTIVE(U92(tt, V1, V2)) → U93¹(isNatIListKind(V2), V1, V2)
ACTIVE(U92(tt, V1, V2)) → ISNATILISTKIND(V2)
ACTIVE(U93(tt, V1, V2)) → MARK(U94(isNatIListKind(V2), V1, V2))
ACTIVE(U93(tt, V1, V2)) → U94¹(isNatIListKind(V2), V1, V2)
ACTIVE(U93(tt, V1, V2)) → ISNATILISTKIND(V2)
ACTIVE(U94(tt, V1, V2)) → MARK(U95(isNat(V1), V2))
ACTIVE(U94(tt, V1, V2)) → U95¹(isNat(V1), V2)
ACTIVE(U94(tt, V1, V2)) → ISNAT(V1)
ACTIVE(U95(tt, V2)) → MARK(U96(isNatList(V2)))
ACTIVE(U95(tt, V2)) → U96¹(isNatList(V2))
ACTIVE(U95(tt, V2)) → ISNATLIST(V2)
ACTIVE(U96(tt)) → MARK(tt)
ACTIVE(isNat(0)) → MARK(tt)
ACTIVE(isNat(length(V1))) → MARK(U11(isNatIListKind(V1), V1))
ACTIVE(isNat(length(V1))) → U11¹(isNatIListKind(V1), V1)
ACTIVE(isNat(length(V1))) → ISNATILISTKIND(V1)
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
ACTIVE(isNat(s(V1))) → U21¹(isNatKind(V1), V1)
ACTIVE(isNat(s(V1))) → ISNATKIND(V1)
ACTIVE(isNatIList(V)) → MARK(U31(isNatIListKind(V), V))
ACTIVE(isNatIList(V)) → U31¹(isNatIListKind(V), V)
ACTIVE(isNatIList(V)) → ISNATILISTKIND(V)
ACTIVE(isNatIList(zeros)) → MARK(tt)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNatKind(V1), V1, V2))
ACTIVE(isNatIList(cons(V1, V2))) → U41¹(isNatKind(V1), V1, V2)
ACTIVE(isNatIList(cons(V1, V2))) → ISNATKIND(V1)
ACTIVE(isNatIListKind(nil)) → MARK(tt)
ACTIVE(isNatIListKind(zeros)) → MARK(tt)
ACTIVE(isNatIListKind(cons(V1, V2))) → MARK(U51(isNatKind(V1), V2))
ACTIVE(isNatIListKind(cons(V1, V2))) → U51¹(isNatKind(V1), V2)
ACTIVE(isNatIListKind(cons(V1, V2))) → ISNATKIND(V1)
ACTIVE(isNatIListKind(take(V1, V2))) → MARK(U61(isNatKind(V1), V2))
ACTIVE(isNatIListKind(take(V1, V2))) → U61¹(isNatKind(V1), V2)
ACTIVE(isNatIListKind(take(V1, V2))) → ISNATKIND(V1)
ACTIVE(isNatKind(0)) → MARK(tt)
ACTIVE(isNatKind(length(V1))) → MARK(U71(isNatIListKind(V1)))
ACTIVE(isNatKind(length(V1))) → U71¹(isNatIListKind(V1))
ACTIVE(isNatKind(length(V1))) → ISNATILISTKIND(V1)
ACTIVE(isNatKind(s(V1))) → MARK(U81(isNatKind(V1)))
ACTIVE(isNatKind(s(V1))) → U81¹(isNatKind(V1))
ACTIVE(isNatKind(s(V1))) → ISNATKIND(V1)
ACTIVE(isNatList(nil)) → MARK(tt)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U91(isNatKind(V1), V1, V2))
ACTIVE(isNatList(cons(V1, V2))) → U91¹(isNatKind(V1), V1, V2)
ACTIVE(isNatList(cons(V1, V2))) → ISNATKIND(V1)
ACTIVE(isNatList(take(V1, V2))) → MARK(U101(isNatKind(V1), V1, V2))
ACTIVE(isNatList(take(V1, V2))) → U101¹(isNatKind(V1), V1, V2)
ACTIVE(isNatList(take(V1, V2))) → ISNATKIND(V1)
ACTIVE(length(nil)) → MARK(0)
ACTIVE(length(cons(N, L))) → MARK(U111(isNatList(L), L, N))
ACTIVE(length(cons(N, L))) → U111¹(isNatList(L), L, N)
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
ACTIVE(take(0, IL)) → MARK(U121(isNatIList(IL), IL))
ACTIVE(take(0, IL)) → U121¹(isNatIList(IL), IL)
ACTIVE(take(0, IL)) → ISNATILIST(IL)
ACTIVE(take(s(M), cons(N, IL))) → MARK(U131(isNatIList(IL), IL, M, N))
ACTIVE(take(s(M), cons(N, IL))) → U131¹(isNatIList(IL), IL, M, N)
ACTIVE(take(s(M), cons(N, IL))) → ISNATILIST(IL)
MARK(zeros) → ACTIVE(zeros)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(0) → ACTIVE(0)
MARK(U101(X1, X2, X3)) → ACTIVE(U101(mark(X1), X2, X3))
MARK(U101(X1, X2, X3)) → U101¹(mark(X1), X2, X3)
MARK(U101(X1, X2, X3)) → MARK(X1)
MARK(tt) → ACTIVE(tt)
MARK(U102(X1, X2, X3)) → ACTIVE(U102(mark(X1), X2, X3))
MARK(U102(X1, X2, X3)) → U102¹(mark(X1), X2, X3)
MARK(U102(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
MARK(U103(X1, X2, X3)) → ACTIVE(U103(mark(X1), X2, X3))
MARK(U103(X1, X2, X3)) → U103¹(mark(X1), X2, X3)
MARK(U103(X1, X2, X3)) → MARK(X1)
MARK(isNatIListKind(X)) → ACTIVE(isNatIListKind(X))
MARK(U104(X1, X2, X3)) → ACTIVE(U104(mark(X1), X2, X3))
MARK(U104(X1, X2, X3)) → U104¹(mark(X1), X2, X3)
MARK(U104(X1, X2, X3)) → MARK(X1)
MARK(U105(X1, X2)) → ACTIVE(U105(mark(X1), X2))
MARK(U105(X1, X2)) → U105¹(mark(X1), X2)
MARK(U105(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U106(X)) → ACTIVE(U106(mark(X)))
MARK(U106(X)) → U106¹(mark(X))
MARK(U106(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U11(X1, X2)) → U11¹(mark(X1), X2)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(U12(X1, X2)) → U12¹(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U111(X1, X2, X3)) → ACTIVE(U111(mark(X1), X2, X3))
MARK(U111(X1, X2, X3)) → U111¹(mark(X1), X2, X3)
MARK(U111(X1, X2, X3)) → MARK(X1)
MARK(U112(X1, X2, X3)) → ACTIVE(U112(mark(X1), X2, X3))
MARK(U112(X1, X2, X3)) → U112¹(mark(X1), X2, X3)
MARK(U112(X1, X2, X3)) → MARK(X1)
MARK(U113(X1, X2, X3)) → ACTIVE(U113(mark(X1), X2, X3))
MARK(U113(X1, X2, X3)) → U113¹(mark(X1), X2, X3)
MARK(U113(X1, X2, X3)) → MARK(X1)
MARK(U114(X1, X2)) → ACTIVE(U114(mark(X1), X2))
MARK(U114(X1, X2)) → U114¹(mark(X1), X2)
MARK(U114(X1, X2)) → MARK(X1)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → S(mark(X))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(length(X)) → LENGTH(mark(X))
MARK(length(X)) → MARK(X)
MARK(U13(X)) → ACTIVE(U13(mark(X)))
MARK(U13(X)) → U13¹(mark(X))
MARK(U13(X)) → MARK(X)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(U121(X1, X2)) → ACTIVE(U121(mark(X1), X2))
MARK(U121(X1, X2)) → U121¹(mark(X1), X2)
MARK(U121(X1, X2)) → MARK(X1)
MARK(U122(X)) → ACTIVE(U122(mark(X)))
MARK(U122(X)) → U122¹(mark(X))
MARK(U122(X)) → MARK(X)
MARK(nil) → ACTIVE(nil)
MARK(U131(X1, X2, X3, X4)) → ACTIVE(U131(mark(X1), X2, X3, X4))
MARK(U131(X1, X2, X3, X4)) → U131¹(mark(X1), X2, X3, X4)
MARK(U131(X1, X2, X3, X4)) → MARK(X1)
MARK(U132(X1, X2, X3, X4)) → ACTIVE(U132(mark(X1), X2, X3, X4))
MARK(U132(X1, X2, X3, X4)) → U132¹(mark(X1), X2, X3, X4)
MARK(U132(X1, X2, X3, X4)) → MARK(X1)
MARK(U133(X1, X2, X3, X4)) → ACTIVE(U133(mark(X1), X2, X3, X4))
MARK(U133(X1, X2, X3, X4)) → U133¹(mark(X1), X2, X3, X4)
MARK(U133(X1, X2, X3, X4)) → MARK(X1)
MARK(U134(X1, X2, X3, X4)) → ACTIVE(U134(mark(X1), X2, X3, X4))
MARK(U134(X1, X2, X3, X4)) → U134¹(mark(X1), X2, X3, X4)
MARK(U134(X1, X2, X3, X4)) → MARK(X1)
MARK(U135(X1, X2, X3, X4)) → ACTIVE(U135(mark(X1), X2, X3, X4))
MARK(U135(X1, X2, X3, X4)) → U135¹(mark(X1), X2, X3, X4)
MARK(U135(X1, X2, X3, X4)) → MARK(X1)
MARK(U136(X1, X2, X3, X4)) → ACTIVE(U136(mark(X1), X2, X3, X4))
MARK(U136(X1, X2, X3, X4)) → U136¹(mark(X1), X2, X3, X4)
MARK(U136(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(take(X1, X2)) → TAKE(mark(X1), mark(X2))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U21(X1, X2)) → U21¹(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U22(X1, X2)) → U22¹(mark(X1), X2)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → ACTIVE(U23(mark(X)))
MARK(U23(X)) → U23¹(mark(X))
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U31(X1, X2)) → U31¹(mark(X1), X2)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X1, X2)) → ACTIVE(U32(mark(X1), X2))
MARK(U32(X1, X2)) → U32¹(mark(X1), X2)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U33(X)) → ACTIVE(U33(mark(X)))
MARK(U33(X)) → U33¹(mark(X))
MARK(U33(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U41(X1, X2, X3)) → U41¹(mark(X1), X2, X3)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2, X3)) → ACTIVE(U42(mark(X1), X2, X3))
MARK(U42(X1, X2, X3)) → U42¹(mark(X1), X2, X3)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(U43(X1, X2, X3)) → ACTIVE(U43(mark(X1), X2, X3))
MARK(U43(X1, X2, X3)) → U43¹(mark(X1), X2, X3)
MARK(U43(X1, X2, X3)) → MARK(X1)
MARK(U44(X1, X2, X3)) → ACTIVE(U44(mark(X1), X2, X3))
MARK(U44(X1, X2, X3)) → U44¹(mark(X1), X2, X3)
MARK(U44(X1, X2, X3)) → MARK(X1)
MARK(U45(X1, X2)) → ACTIVE(U45(mark(X1), X2))
MARK(U45(X1, X2)) → U45¹(mark(X1), X2)
MARK(U45(X1, X2)) → MARK(X1)
MARK(U46(X)) → ACTIVE(U46(mark(X)))
MARK(U46(X)) → U46¹(mark(X))
MARK(U46(X)) → MARK(X)
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U51(X1, X2)) → U51¹(mark(X1), X2)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → ACTIVE(U52(mark(X)))
MARK(U52(X)) → U52¹(mark(X))
MARK(U52(X)) → MARK(X)
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U61(X1, X2)) → U61¹(mark(X1), X2)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U62(X)) → ACTIVE(U62(mark(X)))
MARK(U62(X)) → U62¹(mark(X))
MARK(U62(X)) → MARK(X)
MARK(U71(X)) → ACTIVE(U71(mark(X)))
MARK(U71(X)) → U71¹(mark(X))
MARK(U71(X)) → MARK(X)
MARK(U81(X)) → ACTIVE(U81(mark(X)))
MARK(U81(X)) → U81¹(mark(X))
MARK(U81(X)) → MARK(X)
MARK(U91(X1, X2, X3)) → ACTIVE(U91(mark(X1), X2, X3))
MARK(U91(X1, X2, X3)) → U91¹(mark(X1), X2, X3)
MARK(U91(X1, X2, X3)) → MARK(X1)
MARK(U92(X1, X2, X3)) → ACTIVE(U92(mark(X1), X2, X3))
MARK(U92(X1, X2, X3)) → U92¹(mark(X1), X2, X3)
MARK(U92(X1, X2, X3)) → MARK(X1)
MARK(U93(X1, X2, X3)) → ACTIVE(U93(mark(X1), X2, X3))
MARK(U93(X1, X2, X3)) → U93¹(mark(X1), X2, X3)
MARK(U93(X1, X2, X3)) → MARK(X1)
MARK(U94(X1, X2, X3)) → ACTIVE(U94(mark(X1), X2, X3))
MARK(U94(X1, X2, X3)) → U94¹(mark(X1), X2, X3)
MARK(U94(X1, X2, X3)) → MARK(X1)
MARK(U95(X1, X2)) → ACTIVE(U95(mark(X1), X2))
MARK(U95(X1, X2)) → U95¹(mark(X1), X2)
MARK(U95(X1, X2)) → MARK(X1)
MARK(U96(X)) → ACTIVE(U96(mark(X)))
MARK(U96(X)) → U96¹(mark(X))
MARK(U96(X)) → 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)
U101¹(mark(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(X1, mark(X2), X3) → U101¹(X1, X2, X3)
U101¹(X1, X2, mark(X3)) → U101¹(X1, X2, X3)
U101¹(active(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(X1, active(X2), X3) → U101¹(X1, X2, X3)
U101¹(X1, X2, active(X3)) → U101¹(X1, X2, X3)
U102¹(mark(X1), X2, X3) → U102¹(X1, X2, X3)
U102¹(X1, mark(X2), X3) → U102¹(X1, X2, X3)
U102¹(X1, X2, mark(X3)) → U102¹(X1, X2, X3)
U102¹(active(X1), X2, X3) → U102¹(X1, X2, X3)
U102¹(X1, active(X2), X3) → U102¹(X1, X2, X3)
U102¹(X1, X2, active(X3)) → U102¹(X1, X2, X3)
ISNATKIND(mark(X)) → ISNATKIND(X)
ISNATKIND(active(X)) → ISNATKIND(X)
U103¹(mark(X1), X2, X3) → U103¹(X1, X2, X3)
U103¹(X1, mark(X2), X3) → U103¹(X1, X2, X3)
U103¹(X1, X2, mark(X3)) → U103¹(X1, X2, X3)
U103¹(active(X1), X2, X3) → U103¹(X1, X2, X3)
U103¹(X1, active(X2), X3) → U103¹(X1, X2, X3)
U103¹(X1, X2, active(X3)) → U103¹(X1, X2, X3)
ISNATILISTKIND(mark(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(active(X)) → ISNATILISTKIND(X)
U104¹(mark(X1), X2, X3) → U104¹(X1, X2, X3)
U104¹(X1, mark(X2), X3) → U104¹(X1, X2, X3)
U104¹(X1, X2, mark(X3)) → U104¹(X1, X2, X3)
U104¹(active(X1), X2, X3) → U104¹(X1, X2, X3)
U104¹(X1, active(X2), X3) → U104¹(X1, X2, X3)
U104¹(X1, X2, active(X3)) → U104¹(X1, X2, X3)
U105¹(mark(X1), X2) → U105¹(X1, X2)
U105¹(X1, mark(X2)) → U105¹(X1, X2)
U105¹(active(X1), X2) → U105¹(X1, X2)
U105¹(X1, active(X2)) → U105¹(X1, X2)
ISNAT(mark(X)) → ISNAT(X)
ISNAT(active(X)) → ISNAT(X)
U106¹(mark(X)) → U106¹(X)
U106¹(active(X)) → U106¹(X)
ISNATILIST(mark(X)) → ISNATILIST(X)
ISNATILIST(active(X)) → ISNATILIST(X)
U11¹(mark(X1), X2) → U11¹(X1, X2)
U11¹(X1, mark(X2)) → U11¹(X1, X2)
U11¹(active(X1), X2) → U11¹(X1, X2)
U11¹(X1, active(X2)) → U11¹(X1, X2)
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)
U111¹(mark(X1), X2, X3) → U111¹(X1, X2, X3)
U111¹(X1, mark(X2), X3) → U111¹(X1, X2, X3)
U111¹(X1, X2, mark(X3)) → U111¹(X1, X2, X3)
U111¹(active(X1), X2, X3) → U111¹(X1, X2, X3)
U111¹(X1, active(X2), X3) → U111¹(X1, X2, X3)
U111¹(X1, X2, active(X3)) → U111¹(X1, X2, X3)
U112¹(mark(X1), X2, X3) → U112¹(X1, X2, X3)
U112¹(X1, mark(X2), X3) → U112¹(X1, X2, X3)
U112¹(X1, X2, mark(X3)) → U112¹(X1, X2, X3)
U112¹(active(X1), X2, X3) → U112¹(X1, X2, X3)
U112¹(X1, active(X2), X3) → U112¹(X1, X2, X3)
U112¹(X1, X2, active(X3)) → U112¹(X1, X2, X3)
U113¹(mark(X1), X2, X3) → U113¹(X1, X2, X3)
U113¹(X1, mark(X2), X3) → U113¹(X1, X2, X3)
U113¹(X1, X2, mark(X3)) → U113¹(X1, X2, X3)
U113¹(active(X1), X2, X3) → U113¹(X1, X2, X3)
U113¹(X1, active(X2), X3) → U113¹(X1, X2, X3)
U113¹(X1, X2, active(X3)) → U113¹(X1, X2, X3)
U114¹(mark(X1), X2) → U114¹(X1, X2)
U114¹(X1, mark(X2)) → U114¹(X1, X2)
U114¹(active(X1), X2) → U114¹(X1, X2)
U114¹(X1, active(X2)) → U114¹(X1, X2)
S(mark(X)) → S(X)
S(active(X)) → S(X)
LENGTH(mark(X)) → LENGTH(X)
LENGTH(active(X)) → LENGTH(X)
U13¹(mark(X)) → U13¹(X)
U13¹(active(X)) → U13¹(X)
ISNATLIST(mark(X)) → ISNATLIST(X)
ISNATLIST(active(X)) → ISNATLIST(X)
U121¹(mark(X1), X2) → U121¹(X1, X2)
U121¹(X1, mark(X2)) → U121¹(X1, X2)
U121¹(active(X1), X2) → U121¹(X1, X2)
U121¹(X1, active(X2)) → U121¹(X1, X2)
U122¹(mark(X)) → U122¹(X)
U122¹(active(X)) → U122¹(X)
U131¹(mark(X1), X2, X3, X4) → U131¹(X1, X2, X3, X4)
U131¹(X1, mark(X2), X3, X4) → U131¹(X1, X2, X3, X4)
U131¹(X1, X2, mark(X3), X4) → U131¹(X1, X2, X3, X4)
U131¹(X1, X2, X3, mark(X4)) → U131¹(X1, X2, X3, X4)
U131¹(active(X1), X2, X3, X4) → U131¹(X1, X2, X3, X4)
U131¹(X1, active(X2), X3, X4) → U131¹(X1, X2, X3, X4)
U131¹(X1, X2, active(X3), X4) → U131¹(X1, X2, X3, X4)
U131¹(X1, X2, X3, active(X4)) → U131¹(X1, X2, X3, X4)
U132¹(mark(X1), X2, X3, X4) → U132¹(X1, X2, X3, X4)
U132¹(X1, mark(X2), X3, X4) → U132¹(X1, X2, X3, X4)
U132¹(X1, X2, mark(X3), X4) → U132¹(X1, X2, X3, X4)
U132¹(X1, X2, X3, mark(X4)) → U132¹(X1, X2, X3, X4)
U132¹(active(X1), X2, X3, X4) → U132¹(X1, X2, X3, X4)
U132¹(X1, active(X2), X3, X4) → U132¹(X1, X2, X3, X4)
U132¹(X1, X2, active(X3), X4) → U132¹(X1, X2, X3, X4)
U132¹(X1, X2, X3, active(X4)) → U132¹(X1, X2, X3, X4)
U133¹(mark(X1), X2, X3, X4) → U133¹(X1, X2, X3, X4)
U133¹(X1, mark(X2), X3, X4) → U133¹(X1, X2, X3, X4)
U133¹(X1, X2, mark(X3), X4) → U133¹(X1, X2, X3, X4)
U133¹(X1, X2, X3, mark(X4)) → U133¹(X1, X2, X3, X4)
U133¹(active(X1), X2, X3, X4) → U133¹(X1, X2, X3, X4)
U133¹(X1, active(X2), X3, X4) → U133¹(X1, X2, X3, X4)
U133¹(X1, X2, active(X3), X4) → U133¹(X1, X2, X3, X4)
U133¹(X1, X2, X3, active(X4)) → U133¹(X1, X2, X3, X4)
U134¹(mark(X1), X2, X3, X4) → U134¹(X1, X2, X3, X4)
U134¹(X1, mark(X2), X3, X4) → U134¹(X1, X2, X3, X4)
U134¹(X1, X2, mark(X3), X4) → U134¹(X1, X2, X3, X4)
U134¹(X1, X2, X3, mark(X4)) → U134¹(X1, X2, X3, X4)
U134¹(active(X1), X2, X3, X4) → U134¹(X1, X2, X3, X4)
U134¹(X1, active(X2), X3, X4) → U134¹(X1, X2, X3, X4)
U134¹(X1, X2, active(X3), X4) → U134¹(X1, X2, X3, X4)
U134¹(X1, X2, X3, active(X4)) → U134¹(X1, X2, X3, X4)
U135¹(mark(X1), X2, X3, X4) → U135¹(X1, X2, X3, X4)
U135¹(X1, mark(X2), X3, X4) → U135¹(X1, X2, X3, X4)
U135¹(X1, X2, mark(X3), X4) → U135¹(X1, X2, X3, X4)
U135¹(X1, X2, X3, mark(X4)) → U135¹(X1, X2, X3, X4)
U135¹(active(X1), X2, X3, X4) → U135¹(X1, X2, X3, X4)
U135¹(X1, active(X2), X3, X4) → U135¹(X1, X2, X3, X4)
U135¹(X1, X2, active(X3), X4) → U135¹(X1, X2, X3, X4)
U135¹(X1, X2, X3, active(X4)) → U135¹(X1, X2, X3, X4)
U136¹(mark(X1), X2, X3, X4) → U136¹(X1, X2, X3, X4)
U136¹(X1, mark(X2), X3, X4) → U136¹(X1, X2, X3, X4)
U136¹(X1, X2, mark(X3), X4) → U136¹(X1, X2, X3, X4)
U136¹(X1, X2, X3, mark(X4)) → U136¹(X1, X2, X3, X4)
U136¹(active(X1), X2, X3, X4) → U136¹(X1, X2, X3, X4)
U136¹(X1, active(X2), X3, X4) → U136¹(X1, X2, X3, X4)
U136¹(X1, X2, active(X3), X4) → U136¹(X1, X2, X3, X4)
U136¹(X1, X2, X3, active(X4)) → U136¹(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)
U21¹(mark(X1), X2) → U21¹(X1, X2)
U21¹(X1, mark(X2)) → U21¹(X1, X2)
U21¹(active(X1), X2) → U21¹(X1, X2)
U21¹(X1, active(X2)) → U21¹(X1, X2)
U22¹(mark(X1), X2) → U22¹(X1, X2)
U22¹(X1, mark(X2)) → U22¹(X1, X2)
U22¹(active(X1), X2) → U22¹(X1, X2)
U22¹(X1, active(X2)) → U22¹(X1, X2)
U23¹(mark(X)) → U23¹(X)
U23¹(active(X)) → U23¹(X)
U31¹(mark(X1), X2) → U31¹(X1, X2)
U31¹(X1, mark(X2)) → U31¹(X1, X2)
U31¹(active(X1), X2) → U31¹(X1, X2)
U31¹(X1, active(X2)) → U31¹(X1, X2)
U32¹(mark(X1), X2) → U32¹(X1, X2)
U32¹(X1, mark(X2)) → U32¹(X1, X2)
U32¹(active(X1), X2) → U32¹(X1, X2)
U32¹(X1, active(X2)) → U32¹(X1, X2)
U33¹(mark(X)) → U33¹(X)
U33¹(active(X)) → U33¹(X)
U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, mark(X2), X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, mark(X3)) → U41¹(X1, X2, X3)
U41¹(active(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, active(X2), X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, active(X3)) → U41¹(X1, X2, X3)
U42¹(mark(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(X1, mark(X2), X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, mark(X3)) → U42¹(X1, X2, X3)
U42¹(active(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(X1, active(X2), X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, active(X3)) → U42¹(X1, X2, X3)
U43¹(mark(X1), X2, X3) → U43¹(X1, X2, X3)
U43¹(X1, mark(X2), X3) → U43¹(X1, X2, X3)
U43¹(X1, X2, mark(X3)) → U43¹(X1, X2, X3)
U43¹(active(X1), X2, X3) → U43¹(X1, X2, X3)
U43¹(X1, active(X2), X3) → U43¹(X1, X2, X3)
U43¹(X1, X2, active(X3)) → U43¹(X1, X2, X3)
U44¹(mark(X1), X2, X3) → U44¹(X1, X2, X3)
U44¹(X1, mark(X2), X3) → U44¹(X1, X2, X3)
U44¹(X1, X2, mark(X3)) → U44¹(X1, X2, X3)
U44¹(active(X1), X2, X3) → U44¹(X1, X2, X3)
U44¹(X1, active(X2), X3) → U44¹(X1, X2, X3)
U44¹(X1, X2, active(X3)) → U44¹(X1, X2, X3)
U45¹(mark(X1), X2) → U45¹(X1, X2)
U45¹(X1, mark(X2)) → U45¹(X1, X2)
U45¹(active(X1), X2) → U45¹(X1, X2)
U45¹(X1, active(X2)) → U45¹(X1, X2)
U46¹(mark(X)) → U46¹(X)
U46¹(active(X)) → U46¹(X)
U51¹(mark(X1), X2) → U51¹(X1, X2)
U51¹(X1, mark(X2)) → U51¹(X1, X2)
U51¹(active(X1), X2) → U51¹(X1, X2)
U51¹(X1, active(X2)) → U51¹(X1, X2)
U52¹(mark(X)) → U52¹(X)
U52¹(active(X)) → U52¹(X)
U61¹(mark(X1), X2) → U61¹(X1, X2)
U61¹(X1, mark(X2)) → U61¹(X1, X2)
U61¹(active(X1), X2) → U61¹(X1, X2)
U61¹(X1, active(X2)) → U61¹(X1, X2)
U62¹(mark(X)) → U62¹(X)
U62¹(active(X)) → U62¹(X)
U71¹(mark(X)) → U71¹(X)
U71¹(active(X)) → U71¹(X)
U81¹(mark(X)) → U81¹(X)
U81¹(active(X)) → U81¹(X)
U91¹(mark(X1), X2, X3) → U91¹(X1, X2, X3)
U91¹(X1, mark(X2), X3) → U91¹(X1, X2, X3)
U91¹(X1, X2, mark(X3)) → U91¹(X1, X2, X3)
U91¹(active(X1), X2, X3) → U91¹(X1, X2, X3)
U91¹(X1, active(X2), X3) → U91¹(X1, X2, X3)
U91¹(X1, X2, active(X3)) → U91¹(X1, X2, X3)
U92¹(mark(X1), X2, X3) → U92¹(X1, X2, X3)
U92¹(X1, mark(X2), X3) → U92¹(X1, X2, X3)
U92¹(X1, X2, mark(X3)) → U92¹(X1, X2, X3)
U92¹(active(X1), X2, X3) → U92¹(X1, X2, X3)
U92¹(X1, active(X2), X3) → U92¹(X1, X2, X3)
U92¹(X1, X2, active(X3)) → U92¹(X1, X2, X3)
U93¹(mark(X1), X2, X3) → U93¹(X1, X2, X3)
U93¹(X1, mark(X2), X3) → U93¹(X1, X2, X3)
U93¹(X1, X2, mark(X3)) → U93¹(X1, X2, X3)
U93¹(active(X1), X2, X3) → U93¹(X1, X2, X3)
U93¹(X1, active(X2), X3) → U93¹(X1, X2, X3)
U93¹(X1, X2, active(X3)) → U93¹(X1, X2, X3)
U94¹(mark(X1), X2, X3) → U94¹(X1, X2, X3)
U94¹(X1, mark(X2), X3) → U94¹(X1, X2, X3)
U94¹(X1, X2, mark(X3)) → U94¹(X1, X2, X3)
U94¹(active(X1), X2, X3) → U94¹(X1, X2, X3)
U94¹(X1, active(X2), X3) → U94¹(X1, X2, X3)
U94¹(X1, X2, active(X3)) → U94¹(X1, X2, X3)
U95¹(mark(X1), X2) → U95¹(X1, X2)
U95¹(X1, mark(X2)) → U95¹(X1, X2)
U95¹(active(X1), X2) → U95¹(X1, X2)
U95¹(X1, active(X2)) → U95¹(X1, X2)
U96¹(mark(X)) → U96¹(X)
U96¹(active(X)) → U96¹(X)

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 55 SCCs with 165 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

U96¹(active(X)) → U96¹(X)
U96¹(mark(X)) → U96¹(X)

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U96¹(active(X)) → U96¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U96¹(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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

U96¹(mark(X)) → U96¹(X)

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

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U96¹(mark(X)) → U96¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:

mark₁ > U96^1₁

The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

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

(10) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(11) TRUE

(12) Obligation:

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

U95¹(X1, mark(X2)) → U95¹(X1, X2)
U95¹(mark(X1), X2) → U95¹(X1, X2)
U95¹(active(X1), X2) → U95¹(X1, X2)
U95¹(X1, active(X2)) → U95¹(X1, X2)

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

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U95¹(X1, mark(X2)) → U95¹(X1, X2)
U95¹(X1, active(X2)) → U95¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U95¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(14) Obligation:

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

U95¹(mark(X1), X2) → U95¹(X1, X2)
U95¹(active(X1), X2) → U95¹(X1, X2)

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U95¹(mark(X1), X2) → U95¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U95¹(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(16) Obligation:

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

U95¹(active(X1), X2) → U95¹(X1, X2)

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

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U95¹(active(X1), X2) → U95¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U95¹(x1, x2)  =  U95¹(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

active₁ > U95^1₁

The following usable rules [FROCOS05] were oriented: none

(18) Obligation:

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

(19) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(20) TRUE

(21) Obligation:

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

U94¹(X1, mark(X2), X3) → U94¹(X1, X2, X3)
U94¹(mark(X1), X2, X3) → U94¹(X1, X2, X3)
U94¹(X1, X2, mark(X3)) → U94¹(X1, X2, X3)
U94¹(active(X1), X2, X3) → U94¹(X1, X2, X3)
U94¹(X1, active(X2), X3) → U94¹(X1, X2, X3)
U94¹(X1, X2, active(X3)) → U94¹(X1, X2, X3)

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

(22) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U94¹(X1, X2, mark(X3)) → U94¹(X1, X2, X3)
U94¹(X1, X2, active(X3)) → U94¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U94¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(23) Obligation:

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

U94¹(X1, mark(X2), X3) → U94¹(X1, X2, X3)
U94¹(mark(X1), X2, X3) → U94¹(X1, X2, X3)
U94¹(active(X1), X2, X3) → U94¹(X1, X2, X3)
U94¹(X1, active(X2), X3) → U94¹(X1, X2, X3)

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

(24) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U94¹(mark(X1), X2, X3) → U94¹(X1, X2, X3)
U94¹(active(X1), X2, X3) → U94¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U94¹(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(25) Obligation:

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

U94¹(X1, mark(X2), X3) → U94¹(X1, X2, X3)
U94¹(X1, active(X2), X3) → U94¹(X1, X2, X3)

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

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U94¹(X1, mark(X2), X3) → U94¹(X1, X2, X3)
U94¹(X1, active(X2), X3) → U94¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U94¹(x1, x2, x3)  =  U94¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

mark₁ > U94^1₁
active₁ > U94^1₁

The following usable rules [FROCOS05] were oriented: none

(27) Obligation:

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

(28) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(29) TRUE

(30) Obligation:

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

U93¹(X1, mark(X2), X3) → U93¹(X1, X2, X3)
U93¹(mark(X1), X2, X3) → U93¹(X1, X2, X3)
U93¹(X1, X2, mark(X3)) → U93¹(X1, X2, X3)
U93¹(active(X1), X2, X3) → U93¹(X1, X2, X3)
U93¹(X1, active(X2), X3) → U93¹(X1, X2, X3)
U93¹(X1, X2, active(X3)) → U93¹(X1, X2, X3)

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

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U93¹(X1, X2, mark(X3)) → U93¹(X1, X2, X3)
U93¹(X1, X2, active(X3)) → U93¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U93¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(32) Obligation:

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

U93¹(X1, mark(X2), X3) → U93¹(X1, X2, X3)
U93¹(mark(X1), X2, X3) → U93¹(X1, X2, X3)
U93¹(active(X1), X2, X3) → U93¹(X1, X2, X3)
U93¹(X1, active(X2), X3) → U93¹(X1, X2, X3)

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

(33) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U93¹(mark(X1), X2, X3) → U93¹(X1, X2, X3)
U93¹(active(X1), X2, X3) → U93¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U93¹(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(34) Obligation:

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

U93¹(X1, mark(X2), X3) → U93¹(X1, X2, X3)
U93¹(X1, active(X2), X3) → U93¹(X1, X2, X3)

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

(35) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U93¹(X1, mark(X2), X3) → U93¹(X1, X2, X3)
U93¹(X1, active(X2), X3) → U93¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U93¹(x1, x2, x3)  =  U93¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

mark₁ > U93^1₁
active₁ > U93^1₁

The following usable rules [FROCOS05] were oriented: none

(36) Obligation:

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

(37) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(38) TRUE

(39) Obligation:

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

U92¹(X1, mark(X2), X3) → U92¹(X1, X2, X3)
U92¹(mark(X1), X2, X3) → U92¹(X1, X2, X3)
U92¹(X1, X2, mark(X3)) → U92¹(X1, X2, X3)
U92¹(active(X1), X2, X3) → U92¹(X1, X2, X3)
U92¹(X1, active(X2), X3) → U92¹(X1, X2, X3)
U92¹(X1, X2, active(X3)) → U92¹(X1, X2, X3)

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

(40) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U92¹(X1, X2, mark(X3)) → U92¹(X1, X2, X3)
U92¹(X1, X2, active(X3)) → U92¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U92¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(41) Obligation:

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

U92¹(X1, mark(X2), X3) → U92¹(X1, X2, X3)
U92¹(mark(X1), X2, X3) → U92¹(X1, X2, X3)
U92¹(active(X1), X2, X3) → U92¹(X1, X2, X3)
U92¹(X1, active(X2), X3) → U92¹(X1, X2, X3)

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

(42) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U92¹(mark(X1), X2, X3) → U92¹(X1, X2, X3)
U92¹(active(X1), X2, X3) → U92¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U92¹(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(43) Obligation:

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

U92¹(X1, mark(X2), X3) → U92¹(X1, X2, X3)
U92¹(X1, active(X2), X3) → U92¹(X1, X2, X3)

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

(44) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U92¹(X1, mark(X2), X3) → U92¹(X1, X2, X3)
U92¹(X1, active(X2), X3) → U92¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U92¹(x1, x2, x3)  =  U92¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

mark₁ > U92^1₁
active₁ > U92^1₁

The following usable rules [FROCOS05] were oriented: none

(45) Obligation:

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

(46) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(47) TRUE

(48) Obligation:

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

U91¹(X1, mark(X2), X3) → U91¹(X1, X2, X3)
U91¹(mark(X1), X2, X3) → U91¹(X1, X2, X3)
U91¹(X1, X2, mark(X3)) → U91¹(X1, X2, X3)
U91¹(active(X1), X2, X3) → U91¹(X1, X2, X3)
U91¹(X1, active(X2), X3) → U91¹(X1, X2, X3)
U91¹(X1, X2, active(X3)) → U91¹(X1, X2, X3)

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

(49) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U91¹(X1, X2, mark(X3)) → U91¹(X1, X2, X3)
U91¹(X1, X2, active(X3)) → U91¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U91¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(50) Obligation:

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

U91¹(X1, mark(X2), X3) → U91¹(X1, X2, X3)
U91¹(mark(X1), X2, X3) → U91¹(X1, X2, X3)
U91¹(active(X1), X2, X3) → U91¹(X1, X2, X3)
U91¹(X1, active(X2), X3) → U91¹(X1, X2, X3)

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

(51) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U91¹(mark(X1), X2, X3) → U91¹(X1, X2, X3)
U91¹(active(X1), X2, X3) → U91¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U91¹(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(52) Obligation:

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

U91¹(X1, mark(X2), X3) → U91¹(X1, X2, X3)
U91¹(X1, active(X2), X3) → U91¹(X1, X2, X3)

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

(53) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U91¹(X1, mark(X2), X3) → U91¹(X1, X2, X3)
U91¹(X1, active(X2), X3) → U91¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U91¹(x1, x2, x3)  =  U91¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

mark₁ > U91^1₁
active₁ > U91^1₁

The following usable rules [FROCOS05] were oriented: none

(54) Obligation:

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

(55) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(56) TRUE

(57) Obligation:

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

U81¹(active(X)) → U81¹(X)
U81¹(mark(X)) → U81¹(X)

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

(58) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(active(X)) → U81¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U81¹(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(59) Obligation:

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

U81¹(mark(X)) → U81¹(X)

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

(60) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(mark(X)) → U81¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:

mark₁ > U81^1₁

The following usable rules [FROCOS05] were oriented: none

(61) Obligation:

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

(62) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(63) TRUE

(64) Obligation:

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

U71¹(active(X)) → U71¹(X)
U71¹(mark(X)) → U71¹(X)

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

(65) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(active(X)) → U71¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U71¹(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(66) Obligation:

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

U71¹(mark(X)) → U71¹(X)

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

(67) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(mark(X)) → U71¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:

mark₁ > U71^1₁

The following usable rules [FROCOS05] were oriented: none

(68) Obligation:

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

(69) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(70) TRUE

(71) Obligation:

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

U62¹(active(X)) → U62¹(X)
U62¹(mark(X)) → U62¹(X)

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

(72) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U62¹(active(X)) → U62¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U62¹(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(73) Obligation:

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

U62¹(mark(X)) → U62¹(X)

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

(74) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U62¹(mark(X)) → U62¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:

mark₁ > U62^1₁

The following usable rules [FROCOS05] were oriented: none

(75) Obligation:

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

(76) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(77) TRUE

(78) Obligation:

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

U61¹(X1, mark(X2)) → U61¹(X1, X2)
U61¹(mark(X1), X2) → U61¹(X1, X2)
U61¹(active(X1), X2) → U61¹(X1, X2)
U61¹(X1, active(X2)) → U61¹(X1, X2)

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

(79) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(X1, mark(X2)) → U61¹(X1, X2)
U61¹(X1, active(X2)) → U61¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(80) Obligation:

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

U61¹(mark(X1), X2) → U61¹(X1, X2)
U61¹(active(X1), X2) → U61¹(X1, X2)

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

(81) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(mark(X1), X2) → U61¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(82) Obligation:

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

U61¹(active(X1), X2) → U61¹(X1, X2)

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

(83) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(active(X1), X2) → U61¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2)  =  U61¹(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

active₁ > U61^1₁

The following usable rules [FROCOS05] were oriented: none

(84) Obligation:

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

(85) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(86) TRUE

(87) Obligation:

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

U52¹(active(X)) → U52¹(X)
U52¹(mark(X)) → U52¹(X)

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

(88) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U52¹(active(X)) → U52¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U52¹(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(89) Obligation:

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

U52¹(mark(X)) → U52¹(X)

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

(90) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U52¹(mark(X)) → U52¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:

mark₁ > U52^1₁

The following usable rules [FROCOS05] were oriented: none

(91) Obligation:

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

(92) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(93) TRUE

(94) Obligation:

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

U51¹(X1, mark(X2)) → U51¹(X1, X2)
U51¹(mark(X1), X2) → U51¹(X1, X2)
U51¹(active(X1), X2) → U51¹(X1, X2)
U51¹(X1, active(X2)) → U51¹(X1, X2)

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

(95) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(X1, mark(X2)) → U51¹(X1, X2)
U51¹(X1, active(X2)) → U51¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(96) Obligation:

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

U51¹(mark(X1), X2) → U51¹(X1, X2)
U51¹(active(X1), X2) → U51¹(X1, X2)

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

(97) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(mark(X1), X2) → U51¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(98) Obligation:

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

U51¹(active(X1), X2) → U51¹(X1, X2)

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

(99) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(active(X1), X2) → U51¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2)  =  U51¹(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

active₁ > U51^1₁

The following usable rules [FROCOS05] were oriented: none

(100) Obligation:

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

(101) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(102) TRUE

(103) Obligation:

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

U46¹(active(X)) → U46¹(X)
U46¹(mark(X)) → U46¹(X)

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

(104) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U46¹(active(X)) → U46¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U46¹(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(105) Obligation:

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

U46¹(mark(X)) → U46¹(X)

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

(106) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U46¹(mark(X)) → U46¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:

mark₁ > U46^1₁

The following usable rules [FROCOS05] were oriented: none

(107) Obligation:

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

(108) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(109) TRUE

(110) Obligation:

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

U45¹(X1, mark(X2)) → U45¹(X1, X2)
U45¹(mark(X1), X2) → U45¹(X1, X2)
U45¹(active(X1), X2) → U45¹(X1, X2)
U45¹(X1, active(X2)) → U45¹(X1, X2)

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

(111) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U45¹(X1, mark(X2)) → U45¹(X1, X2)
U45¹(X1, active(X2)) → U45¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U45¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(112) Obligation:

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

U45¹(mark(X1), X2) → U45¹(X1, X2)
U45¹(active(X1), X2) → U45¹(X1, X2)

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

(113) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U45¹(mark(X1), X2) → U45¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U45¹(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(114) Obligation:

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

U45¹(active(X1), X2) → U45¹(X1, X2)

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

(115) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U45¹(active(X1), X2) → U45¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U45¹(x1, x2)  =  U45¹(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

active₁ > U45^1₁

The following usable rules [FROCOS05] were oriented: none

(116) Obligation:

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

(117) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(118) TRUE

(119) Obligation:

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

U44¹(X1, mark(X2), X3) → U44¹(X1, X2, X3)
U44¹(mark(X1), X2, X3) → U44¹(X1, X2, X3)
U44¹(X1, X2, mark(X3)) → U44¹(X1, X2, X3)
U44¹(active(X1), X2, X3) → U44¹(X1, X2, X3)
U44¹(X1, active(X2), X3) → U44¹(X1, X2, X3)
U44¹(X1, X2, active(X3)) → U44¹(X1, X2, X3)

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

(120) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U44¹(X1, X2, mark(X3)) → U44¹(X1, X2, X3)
U44¹(X1, X2, active(X3)) → U44¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U44¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(121) Obligation:

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

U44¹(X1, mark(X2), X3) → U44¹(X1, X2, X3)
U44¹(mark(X1), X2, X3) → U44¹(X1, X2, X3)
U44¹(active(X1), X2, X3) → U44¹(X1, X2, X3)
U44¹(X1, active(X2), X3) → U44¹(X1, X2, X3)

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

(122) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U44¹(mark(X1), X2, X3) → U44¹(X1, X2, X3)
U44¹(active(X1), X2, X3) → U44¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U44¹(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(123) Obligation:

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

U44¹(X1, mark(X2), X3) → U44¹(X1, X2, X3)
U44¹(X1, active(X2), X3) → U44¹(X1, X2, X3)

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

(124) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U44¹(X1, mark(X2), X3) → U44¹(X1, X2, X3)
U44¹(X1, active(X2), X3) → U44¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U44¹(x1, x2, x3)  =  U44¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

mark₁ > U44^1₁
active₁ > U44^1₁

The following usable rules [FROCOS05] were oriented: none

(125) Obligation:

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

(126) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(127) TRUE

(128) Obligation:

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

U43¹(X1, mark(X2), X3) → U43¹(X1, X2, X3)
U43¹(mark(X1), X2, X3) → U43¹(X1, X2, X3)
U43¹(X1, X2, mark(X3)) → U43¹(X1, X2, X3)
U43¹(active(X1), X2, X3) → U43¹(X1, X2, X3)
U43¹(X1, active(X2), X3) → U43¹(X1, X2, X3)
U43¹(X1, X2, active(X3)) → U43¹(X1, X2, X3)

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

(129) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U43¹(X1, X2, mark(X3)) → U43¹(X1, X2, X3)
U43¹(X1, X2, active(X3)) → U43¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U43¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(130) Obligation:

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

U43¹(X1, mark(X2), X3) → U43¹(X1, X2, X3)
U43¹(mark(X1), X2, X3) → U43¹(X1, X2, X3)
U43¹(active(X1), X2, X3) → U43¹(X1, X2, X3)
U43¹(X1, active(X2), X3) → U43¹(X1, X2, X3)

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

(131) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U43¹(mark(X1), X2, X3) → U43¹(X1, X2, X3)
U43¹(active(X1), X2, X3) → U43¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U43¹(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(132) Obligation:

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

U43¹(X1, mark(X2), X3) → U43¹(X1, X2, X3)
U43¹(X1, active(X2), X3) → U43¹(X1, X2, X3)

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

(133) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U43¹(X1, mark(X2), X3) → U43¹(X1, X2, X3)
U43¹(X1, active(X2), X3) → U43¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U43¹(x1, x2, x3)  =  U43¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

mark₁ > U43^1₁
active₁ > U43^1₁

The following usable rules [FROCOS05] were oriented: none

(134) Obligation:

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

(135) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(136) TRUE

(137) Obligation:

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

U42¹(X1, mark(X2), X3) → U42¹(X1, X2, X3)
U42¹(mark(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, mark(X3)) → U42¹(X1, X2, X3)
U42¹(active(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(X1, active(X2), X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, active(X3)) → U42¹(X1, X2, X3)

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

(138) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(X1, X2, mark(X3)) → U42¹(X1, X2, X3)
U42¹(X1, X2, active(X3)) → U42¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U42¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(139) Obligation:

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

U42¹(X1, mark(X2), X3) → U42¹(X1, X2, X3)
U42¹(mark(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(active(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(X1, active(X2), X3) → U42¹(X1, X2, X3)

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

(140) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(mark(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(active(X1), X2, X3) → U42¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U42¹(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(141) Obligation:

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

U42¹(X1, mark(X2), X3) → U42¹(X1, X2, X3)
U42¹(X1, active(X2), X3) → U42¹(X1, X2, X3)

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

(142) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(X1, mark(X2), X3) → U42¹(X1, X2, X3)
U42¹(X1, active(X2), X3) → U42¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U42¹(x1, x2, x3)  =  U42¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

mark₁ > U42^1₁
active₁ > U42^1₁

The following usable rules [FROCOS05] were oriented: none

(143) Obligation:

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

(144) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(145) TRUE

(146) Obligation:

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

U41¹(X1, mark(X2), X3) → U41¹(X1, X2, X3)
U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, mark(X3)) → U41¹(X1, X2, X3)
U41¹(active(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, active(X2), X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, active(X3)) → U41¹(X1, X2, X3)

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

(147) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(X1, X2, mark(X3)) → U41¹(X1, X2, X3)
U41¹(X1, X2, active(X3)) → U41¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(148) Obligation:

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

U41¹(X1, mark(X2), X3) → U41¹(X1, X2, X3)
U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(active(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, active(X2), X3) → U41¹(X1, X2, X3)

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

(149) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(active(X1), X2, X3) → U41¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(150) Obligation:

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

U41¹(X1, mark(X2), X3) → U41¹(X1, X2, X3)
U41¹(X1, active(X2), X3) → U41¹(X1, X2, X3)

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

(151) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(X1, mark(X2), X3) → U41¹(X1, X2, X3)
U41¹(X1, active(X2), X3) → U41¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2, x3)  =  U41¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

mark₁ > U41^1₁
active₁ > U41^1₁

The following usable rules [FROCOS05] were oriented: none

(152) Obligation:

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

(153) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(154) TRUE

(155) Obligation:

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

U33¹(active(X)) → U33¹(X)
U33¹(mark(X)) → U33¹(X)

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

(156) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U33¹(active(X)) → U33¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U33¹(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(157) Obligation:

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

U33¹(mark(X)) → U33¹(X)

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

(158) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U33¹(mark(X)) → U33¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:

mark₁ > U33^1₁

The following usable rules [FROCOS05] were oriented: none

(159) Obligation:

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

(160) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(161) TRUE

(162) Obligation:

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

U32¹(X1, mark(X2)) → U32¹(X1, X2)
U32¹(mark(X1), X2) → U32¹(X1, X2)
U32¹(active(X1), X2) → U32¹(X1, X2)
U32¹(X1, active(X2)) → U32¹(X1, X2)

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

(163) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U32¹(X1, mark(X2)) → U32¹(X1, X2)
U32¹(X1, active(X2)) → U32¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U32¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(164) Obligation:

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

U32¹(mark(X1), X2) → U32¹(X1, X2)
U32¹(active(X1), X2) → U32¹(X1, X2)

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

(165) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U32¹(mark(X1), X2) → U32¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U32¹(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(166) Obligation:

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

U32¹(active(X1), X2) → U32¹(X1, X2)

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

(167) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U32¹(active(X1), X2) → U32¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U32¹(x1, x2)  =  U32¹(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

active₁ > U32^1₁

The following usable rules [FROCOS05] were oriented: none

(168) Obligation:

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

(169) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(170) TRUE

(171) Obligation:

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

U31¹(X1, mark(X2)) → U31¹(X1, X2)
U31¹(mark(X1), X2) → U31¹(X1, X2)
U31¹(active(X1), X2) → U31¹(X1, X2)
U31¹(X1, active(X2)) → U31¹(X1, X2)

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

(172) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(X1, mark(X2)) → U31¹(X1, X2)
U31¹(X1, active(X2)) → U31¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(173) Obligation:

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

U31¹(mark(X1), X2) → U31¹(X1, X2)
U31¹(active(X1), X2) → U31¹(X1, X2)

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

(174) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(mark(X1), X2) → U31¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(175) Obligation:

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

U31¹(active(X1), X2) → U31¹(X1, X2)

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

(176) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(active(X1), X2) → U31¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1, x2)  =  U31¹(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

active₁ > U31^1₁

The following usable rules [FROCOS05] were oriented: none

(177) Obligation:

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

(178) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(179) TRUE

(180) Obligation:

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

U23¹(active(X)) → U23¹(X)
U23¹(mark(X)) → U23¹(X)

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

(181) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U23¹(active(X)) → U23¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U23¹(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(182) Obligation:

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

U23¹(mark(X)) → U23¹(X)

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

(183) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U23¹(mark(X)) → U23¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:

mark₁ > U23^1₁

The following usable rules [FROCOS05] were oriented: none

(184) Obligation:

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

(185) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(186) TRUE

(187) Obligation:

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

U22¹(X1, mark(X2)) → U22¹(X1, X2)
U22¹(mark(X1), X2) → U22¹(X1, X2)
U22¹(active(X1), X2) → U22¹(X1, X2)
U22¹(X1, active(X2)) → U22¹(X1, X2)

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

(188) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(X1, mark(X2)) → U22¹(X1, X2)
U22¹(X1, active(X2)) → U22¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U22¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(189) Obligation:

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

U22¹(mark(X1), X2) → U22¹(X1, X2)
U22¹(active(X1), X2) → U22¹(X1, X2)

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

(190) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(mark(X1), X2) → U22¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U22¹(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(191) Obligation:

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

U22¹(active(X1), X2) → U22¹(X1, X2)

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

(192) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(active(X1), X2) → U22¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U22¹(x1, x2)  =  U22¹(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

active₁ > U22^1₁

The following usable rules [FROCOS05] were oriented: none

(193) Obligation:

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

(194) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(195) TRUE

(196) Obligation:

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

U21¹(X1, mark(X2)) → U21¹(X1, X2)
U21¹(mark(X1), X2) → U21¹(X1, X2)
U21¹(active(X1), X2) → U21¹(X1, X2)
U21¹(X1, active(X2)) → U21¹(X1, X2)

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

(197) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(X1, mark(X2)) → U21¹(X1, X2)
U21¹(X1, active(X2)) → U21¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(198) Obligation:

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

U21¹(mark(X1), X2) → U21¹(X1, X2)
U21¹(active(X1), X2) → U21¹(X1, X2)

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

(199) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(mark(X1), X2) → U21¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(200) Obligation:

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

U21¹(active(X1), X2) → U21¹(X1, X2)

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

(201) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(active(X1), X2) → U21¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1, x2)  =  U21¹(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

active₁ > U21^1₁

The following usable rules [FROCOS05] were oriented: none

(202) Obligation:

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

(203) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(204) TRUE

(205) Obligation:

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

TAKE(X1, mark(X2)) → TAKE(X1, X2)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)

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

(206) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(X1, mark(X2)) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAKE(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(207) Obligation:

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

TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(active(X1), X2) → TAKE(X1, X2)

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

(208) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(mark(X1), X2) → TAKE(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAKE(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(209) Obligation:

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

TAKE(active(X1), X2) → TAKE(X1, X2)

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

(210) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(active(X1), X2) → TAKE(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAKE(x1, x2)  =  TAKE(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

active₁ > TAKE₁

The following usable rules [FROCOS05] were oriented: none

(211) Obligation:

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

(212) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(213) TRUE

(214) Obligation:

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

U136¹(X1, mark(X2), X3, X4) → U136¹(X1, X2, X3, X4)
U136¹(mark(X1), X2, X3, X4) → U136¹(X1, X2, X3, X4)
U136¹(X1, X2, mark(X3), X4) → U136¹(X1, X2, X3, X4)
U136¹(X1, X2, X3, mark(X4)) → U136¹(X1, X2, X3, X4)
U136¹(active(X1), X2, X3, X4) → U136¹(X1, X2, X3, X4)
U136¹(X1, active(X2), X3, X4) → U136¹(X1, X2, X3, X4)
U136¹(X1, X2, active(X3), X4) → U136¹(X1, X2, X3, X4)
U136¹(X1, X2, X3, active(X4)) → U136¹(X1, X2, X3, X4)

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

(215) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U136¹(X1, X2, mark(X3), X4) → U136¹(X1, X2, X3, X4)
U136¹(X1, X2, active(X3), X4) → U136¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U136¹(x1, x2, x3, x4)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(216) Obligation:

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

U136¹(X1, mark(X2), X3, X4) → U136¹(X1, X2, X3, X4)
U136¹(mark(X1), X2, X3, X4) → U136¹(X1, X2, X3, X4)
U136¹(X1, X2, X3, mark(X4)) → U136¹(X1, X2, X3, X4)
U136¹(active(X1), X2, X3, X4) → U136¹(X1, X2, X3, X4)
U136¹(X1, active(X2), X3, X4) → U136¹(X1, X2, X3, X4)
U136¹(X1, X2, X3, active(X4)) → U136¹(X1, X2, X3, X4)

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

(217) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U136¹(X1, X2, X3, mark(X4)) → U136¹(X1, X2, X3, X4)
U136¹(X1, X2, X3, active(X4)) → U136¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U136¹(x1, x2, x3, x4)  =  x4
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(218) Obligation:

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

U136¹(X1, mark(X2), X3, X4) → U136¹(X1, X2, X3, X4)
U136¹(mark(X1), X2, X3, X4) → U136¹(X1, X2, X3, X4)
U136¹(active(X1), X2, X3, X4) → U136¹(X1, X2, X3, X4)
U136¹(X1, active(X2), X3, X4) → U136¹(X1, X2, X3, X4)

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

(219) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U136¹(X1, active(X2), X3, X4) → U136¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U136¹(x1, x2, x3, x4)  =  U136¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(220) Obligation:

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

U136¹(X1, mark(X2), X3, X4) → U136¹(X1, X2, X3, X4)
U136¹(mark(X1), X2, X3, X4) → U136¹(X1, X2, X3, X4)
U136¹(active(X1), X2, X3, X4) → U136¹(X1, X2, X3, X4)

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

(221) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U136¹(mark(X1), X2, X3, X4) → U136¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U136¹(x1, x2, x3, x4)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(222) Obligation:

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

U136¹(X1, mark(X2), X3, X4) → U136¹(X1, X2, X3, X4)
U136¹(active(X1), X2, X3, X4) → U136¹(X1, X2, X3, X4)

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

(223) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U136¹(X1, mark(X2), X3, X4) → U136¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U136¹(x1, x2, x3, x4)  =  U136¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

mark₁ > U136^1₁
active₁ > U136^1₁

The following usable rules [FROCOS05] were oriented: none

(224) Obligation:

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

U136¹(active(X1), X2, X3, X4) → U136¹(X1, X2, X3, X4)

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

(225) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U136¹(active(X1), X2, X3, X4) → U136¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U136¹(x1, x2, x3, x4)  =  U136¹(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(226) Obligation:

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

(227) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(228) TRUE

(229) Obligation:

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

U135¹(X1, mark(X2), X3, X4) → U135¹(X1, X2, X3, X4)
U135¹(mark(X1), X2, X3, X4) → U135¹(X1, X2, X3, X4)
U135¹(X1, X2, mark(X3), X4) → U135¹(X1, X2, X3, X4)
U135¹(X1, X2, X3, mark(X4)) → U135¹(X1, X2, X3, X4)
U135¹(active(X1), X2, X3, X4) → U135¹(X1, X2, X3, X4)
U135¹(X1, active(X2), X3, X4) → U135¹(X1, X2, X3, X4)
U135¹(X1, X2, active(X3), X4) → U135¹(X1, X2, X3, X4)
U135¹(X1, X2, X3, active(X4)) → U135¹(X1, X2, X3, X4)

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

(230) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U135¹(X1, X2, mark(X3), X4) → U135¹(X1, X2, X3, X4)
U135¹(X1, X2, active(X3), X4) → U135¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U135¹(x1, x2, x3, x4)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(231) Obligation:

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

U135¹(X1, mark(X2), X3, X4) → U135¹(X1, X2, X3, X4)
U135¹(mark(X1), X2, X3, X4) → U135¹(X1, X2, X3, X4)
U135¹(X1, X2, X3, mark(X4)) → U135¹(X1, X2, X3, X4)
U135¹(active(X1), X2, X3, X4) → U135¹(X1, X2, X3, X4)
U135¹(X1, active(X2), X3, X4) → U135¹(X1, X2, X3, X4)
U135¹(X1, X2, X3, active(X4)) → U135¹(X1, X2, X3, X4)

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

(232) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U135¹(X1, X2, X3, mark(X4)) → U135¹(X1, X2, X3, X4)
U135¹(X1, X2, X3, active(X4)) → U135¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U135¹(x1, x2, x3, x4)  =  x4
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(233) Obligation:

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

U135¹(X1, mark(X2), X3, X4) → U135¹(X1, X2, X3, X4)
U135¹(mark(X1), X2, X3, X4) → U135¹(X1, X2, X3, X4)
U135¹(active(X1), X2, X3, X4) → U135¹(X1, X2, X3, X4)
U135¹(X1, active(X2), X3, X4) → U135¹(X1, X2, X3, X4)

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

(234) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U135¹(X1, active(X2), X3, X4) → U135¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U135¹(x1, x2, x3, x4)  =  U135¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(235) Obligation:

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

U135¹(X1, mark(X2), X3, X4) → U135¹(X1, X2, X3, X4)
U135¹(mark(X1), X2, X3, X4) → U135¹(X1, X2, X3, X4)
U135¹(active(X1), X2, X3, X4) → U135¹(X1, X2, X3, X4)

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

(236) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U135¹(mark(X1), X2, X3, X4) → U135¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U135¹(x1, x2, x3, x4)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(237) Obligation:

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

U135¹(X1, mark(X2), X3, X4) → U135¹(X1, X2, X3, X4)
U135¹(active(X1), X2, X3, X4) → U135¹(X1, X2, X3, X4)

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

(238) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U135¹(X1, mark(X2), X3, X4) → U135¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U135¹(x1, x2, x3, x4)  =  U135¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

mark₁ > U135^1₁
active₁ > U135^1₁

The following usable rules [FROCOS05] were oriented: none

(239) Obligation:

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

U135¹(active(X1), X2, X3, X4) → U135¹(X1, X2, X3, X4)

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

(240) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U135¹(active(X1), X2, X3, X4) → U135¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U135¹(x1, x2, x3, x4)  =  U135¹(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(241) Obligation:

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

(242) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(243) TRUE

(244) Obligation:

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

U134¹(X1, mark(X2), X3, X4) → U134¹(X1, X2, X3, X4)
U134¹(mark(X1), X2, X3, X4) → U134¹(X1, X2, X3, X4)
U134¹(X1, X2, mark(X3), X4) → U134¹(X1, X2, X3, X4)
U134¹(X1, X2, X3, mark(X4)) → U134¹(X1, X2, X3, X4)
U134¹(active(X1), X2, X3, X4) → U134¹(X1, X2, X3, X4)
U134¹(X1, active(X2), X3, X4) → U134¹(X1, X2, X3, X4)
U134¹(X1, X2, active(X3), X4) → U134¹(X1, X2, X3, X4)
U134¹(X1, X2, X3, active(X4)) → U134¹(X1, X2, X3, X4)

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

(245) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U134¹(X1, X2, mark(X3), X4) → U134¹(X1, X2, X3, X4)
U134¹(X1, X2, active(X3), X4) → U134¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U134¹(x1, x2, x3, x4)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(246) Obligation:

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

U134¹(X1, mark(X2), X3, X4) → U134¹(X1, X2, X3, X4)
U134¹(mark(X1), X2, X3, X4) → U134¹(X1, X2, X3, X4)
U134¹(X1, X2, X3, mark(X4)) → U134¹(X1, X2, X3, X4)
U134¹(active(X1), X2, X3, X4) → U134¹(X1, X2, X3, X4)
U134¹(X1, active(X2), X3, X4) → U134¹(X1, X2, X3, X4)
U134¹(X1, X2, X3, active(X4)) → U134¹(X1, X2, X3, X4)

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

(247) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U134¹(X1, X2, X3, mark(X4)) → U134¹(X1, X2, X3, X4)
U134¹(X1, X2, X3, active(X4)) → U134¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U134¹(x1, x2, x3, x4)  =  x4
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(248) Obligation:

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

U134¹(X1, mark(X2), X3, X4) → U134¹(X1, X2, X3, X4)
U134¹(mark(X1), X2, X3, X4) → U134¹(X1, X2, X3, X4)
U134¹(active(X1), X2, X3, X4) → U134¹(X1, X2, X3, X4)
U134¹(X1, active(X2), X3, X4) → U134¹(X1, X2, X3, X4)

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

(249) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U134¹(X1, active(X2), X3, X4) → U134¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U134¹(x1, x2, x3, x4)  =  U134¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(250) Obligation:

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

U134¹(X1, mark(X2), X3, X4) → U134¹(X1, X2, X3, X4)
U134¹(mark(X1), X2, X3, X4) → U134¹(X1, X2, X3, X4)
U134¹(active(X1), X2, X3, X4) → U134¹(X1, X2, X3, X4)

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

(251) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U134¹(mark(X1), X2, X3, X4) → U134¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U134¹(x1, x2, x3, x4)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(252) Obligation:

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

U134¹(X1, mark(X2), X3, X4) → U134¹(X1, X2, X3, X4)
U134¹(active(X1), X2, X3, X4) → U134¹(X1, X2, X3, X4)

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

(253) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U134¹(X1, mark(X2), X3, X4) → U134¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U134¹(x1, x2, x3, x4)  =  U134¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

mark₁ > U134^1₁
active₁ > U134^1₁

The following usable rules [FROCOS05] were oriented: none

(254) Obligation:

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

U134¹(active(X1), X2, X3, X4) → U134¹(X1, X2, X3, X4)

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

(255) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U134¹(active(X1), X2, X3, X4) → U134¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U134¹(x1, x2, x3, x4)  =  U134¹(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(256) Obligation:

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

(257) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(258) TRUE

(259) Obligation:

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

U133¹(X1, mark(X2), X3, X4) → U133¹(X1, X2, X3, X4)
U133¹(mark(X1), X2, X3, X4) → U133¹(X1, X2, X3, X4)
U133¹(X1, X2, mark(X3), X4) → U133¹(X1, X2, X3, X4)
U133¹(X1, X2, X3, mark(X4)) → U133¹(X1, X2, X3, X4)
U133¹(active(X1), X2, X3, X4) → U133¹(X1, X2, X3, X4)
U133¹(X1, active(X2), X3, X4) → U133¹(X1, X2, X3, X4)
U133¹(X1, X2, active(X3), X4) → U133¹(X1, X2, X3, X4)
U133¹(X1, X2, X3, active(X4)) → U133¹(X1, X2, X3, X4)

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

(260) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U133¹(X1, X2, mark(X3), X4) → U133¹(X1, X2, X3, X4)
U133¹(X1, X2, active(X3), X4) → U133¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U133¹(x1, x2, x3, x4)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(261) Obligation:

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

U133¹(X1, mark(X2), X3, X4) → U133¹(X1, X2, X3, X4)
U133¹(mark(X1), X2, X3, X4) → U133¹(X1, X2, X3, X4)
U133¹(X1, X2, X3, mark(X4)) → U133¹(X1, X2, X3, X4)
U133¹(active(X1), X2, X3, X4) → U133¹(X1, X2, X3, X4)
U133¹(X1, active(X2), X3, X4) → U133¹(X1, X2, X3, X4)
U133¹(X1, X2, X3, active(X4)) → U133¹(X1, X2, X3, X4)

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

(262) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U133¹(X1, X2, X3, mark(X4)) → U133¹(X1, X2, X3, X4)
U133¹(X1, X2, X3, active(X4)) → U133¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U133¹(x1, x2, x3, x4)  =  x4
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(263) Obligation:

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

U133¹(X1, mark(X2), X3, X4) → U133¹(X1, X2, X3, X4)
U133¹(mark(X1), X2, X3, X4) → U133¹(X1, X2, X3, X4)
U133¹(active(X1), X2, X3, X4) → U133¹(X1, X2, X3, X4)
U133¹(X1, active(X2), X3, X4) → U133¹(X1, X2, X3, X4)

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

(264) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U133¹(X1, active(X2), X3, X4) → U133¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U133¹(x1, x2, x3, x4)  =  U133¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(265) Obligation:

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

U133¹(X1, mark(X2), X3, X4) → U133¹(X1, X2, X3, X4)
U133¹(mark(X1), X2, X3, X4) → U133¹(X1, X2, X3, X4)
U133¹(active(X1), X2, X3, X4) → U133¹(X1, X2, X3, X4)

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

(266) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U133¹(mark(X1), X2, X3, X4) → U133¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U133¹(x1, x2, x3, x4)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(267) Obligation:

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

U133¹(X1, mark(X2), X3, X4) → U133¹(X1, X2, X3, X4)
U133¹(active(X1), X2, X3, X4) → U133¹(X1, X2, X3, X4)

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

(268) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U133¹(X1, mark(X2), X3, X4) → U133¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U133¹(x1, x2, x3, x4)  =  U133¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

mark₁ > U133^1₁
active₁ > U133^1₁

The following usable rules [FROCOS05] were oriented: none

(269) Obligation:

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

U133¹(active(X1), X2, X3, X4) → U133¹(X1, X2, X3, X4)

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

(270) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U133¹(active(X1), X2, X3, X4) → U133¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U133¹(x1, x2, x3, x4)  =  U133¹(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(271) Obligation:

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

(272) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(273) TRUE

(274) Obligation:

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

U132¹(X1, mark(X2), X3, X4) → U132¹(X1, X2, X3, X4)
U132¹(mark(X1), X2, X3, X4) → U132¹(X1, X2, X3, X4)
U132¹(X1, X2, mark(X3), X4) → U132¹(X1, X2, X3, X4)
U132¹(X1, X2, X3, mark(X4)) → U132¹(X1, X2, X3, X4)
U132¹(active(X1), X2, X3, X4) → U132¹(X1, X2, X3, X4)
U132¹(X1, active(X2), X3, X4) → U132¹(X1, X2, X3, X4)
U132¹(X1, X2, active(X3), X4) → U132¹(X1, X2, X3, X4)
U132¹(X1, X2, X3, active(X4)) → U132¹(X1, X2, X3, X4)

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

(275) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U132¹(X1, X2, mark(X3), X4) → U132¹(X1, X2, X3, X4)
U132¹(X1, X2, active(X3), X4) → U132¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U132¹(x1, x2, x3, x4)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(276) Obligation:

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

U132¹(X1, mark(X2), X3, X4) → U132¹(X1, X2, X3, X4)
U132¹(mark(X1), X2, X3, X4) → U132¹(X1, X2, X3, X4)
U132¹(X1, X2, X3, mark(X4)) → U132¹(X1, X2, X3, X4)
U132¹(active(X1), X2, X3, X4) → U132¹(X1, X2, X3, X4)
U132¹(X1, active(X2), X3, X4) → U132¹(X1, X2, X3, X4)
U132¹(X1, X2, X3, active(X4)) → U132¹(X1, X2, X3, X4)

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

(277) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U132¹(X1, X2, X3, mark(X4)) → U132¹(X1, X2, X3, X4)
U132¹(X1, X2, X3, active(X4)) → U132¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U132¹(x1, x2, x3, x4)  =  x4
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(278) Obligation:

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

U132¹(X1, mark(X2), X3, X4) → U132¹(X1, X2, X3, X4)
U132¹(mark(X1), X2, X3, X4) → U132¹(X1, X2, X3, X4)
U132¹(active(X1), X2, X3, X4) → U132¹(X1, X2, X3, X4)
U132¹(X1, active(X2), X3, X4) → U132¹(X1, X2, X3, X4)

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

(279) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U132¹(X1, active(X2), X3, X4) → U132¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U132¹(x1, x2, x3, x4)  =  U132¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(280) Obligation:

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

U132¹(X1, mark(X2), X3, X4) → U132¹(X1, X2, X3, X4)
U132¹(mark(X1), X2, X3, X4) → U132¹(X1, X2, X3, X4)
U132¹(active(X1), X2, X3, X4) → U132¹(X1, X2, X3, X4)

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

(281) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U132¹(mark(X1), X2, X3, X4) → U132¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U132¹(x1, x2, x3, x4)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(282) Obligation:

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

U132¹(X1, mark(X2), X3, X4) → U132¹(X1, X2, X3, X4)
U132¹(active(X1), X2, X3, X4) → U132¹(X1, X2, X3, X4)

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

(283) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U132¹(X1, mark(X2), X3, X4) → U132¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U132¹(x1, x2, x3, x4)  =  U132¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

mark₁ > U132^1₁
active₁ > U132^1₁

The following usable rules [FROCOS05] were oriented: none

(284) Obligation:

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

U132¹(active(X1), X2, X3, X4) → U132¹(X1, X2, X3, X4)

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

(285) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U132¹(active(X1), X2, X3, X4) → U132¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U132¹(x1, x2, x3, x4)  =  U132¹(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(286) Obligation:

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

(287) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(288) TRUE

(289) Obligation:

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

U131¹(X1, mark(X2), X3, X4) → U131¹(X1, X2, X3, X4)
U131¹(mark(X1), X2, X3, X4) → U131¹(X1, X2, X3, X4)
U131¹(X1, X2, mark(X3), X4) → U131¹(X1, X2, X3, X4)
U131¹(X1, X2, X3, mark(X4)) → U131¹(X1, X2, X3, X4)
U131¹(active(X1), X2, X3, X4) → U131¹(X1, X2, X3, X4)
U131¹(X1, active(X2), X3, X4) → U131¹(X1, X2, X3, X4)
U131¹(X1, X2, active(X3), X4) → U131¹(X1, X2, X3, X4)
U131¹(X1, X2, X3, active(X4)) → U131¹(X1, X2, X3, X4)

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

(290) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U131¹(X1, X2, mark(X3), X4) → U131¹(X1, X2, X3, X4)
U131¹(X1, X2, active(X3), X4) → U131¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U131¹(x1, x2, x3, x4)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(291) Obligation:

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

U131¹(X1, mark(X2), X3, X4) → U131¹(X1, X2, X3, X4)
U131¹(mark(X1), X2, X3, X4) → U131¹(X1, X2, X3, X4)
U131¹(X1, X2, X3, mark(X4)) → U131¹(X1, X2, X3, X4)
U131¹(active(X1), X2, X3, X4) → U131¹(X1, X2, X3, X4)
U131¹(X1, active(X2), X3, X4) → U131¹(X1, X2, X3, X4)
U131¹(X1, X2, X3, active(X4)) → U131¹(X1, X2, X3, X4)

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

(292) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U131¹(X1, X2, X3, mark(X4)) → U131¹(X1, X2, X3, X4)
U131¹(X1, X2, X3, active(X4)) → U131¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U131¹(x1, x2, x3, x4)  =  x4
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(293) Obligation:

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

U131¹(X1, mark(X2), X3, X4) → U131¹(X1, X2, X3, X4)
U131¹(mark(X1), X2, X3, X4) → U131¹(X1, X2, X3, X4)
U131¹(active(X1), X2, X3, X4) → U131¹(X1, X2, X3, X4)
U131¹(X1, active(X2), X3, X4) → U131¹(X1, X2, X3, X4)

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

(294) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U131¹(X1, active(X2), X3, X4) → U131¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U131¹(x1, x2, x3, x4)  =  U131¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(295) Obligation:

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

U131¹(X1, mark(X2), X3, X4) → U131¹(X1, X2, X3, X4)
U131¹(mark(X1), X2, X3, X4) → U131¹(X1, X2, X3, X4)
U131¹(active(X1), X2, X3, X4) → U131¹(X1, X2, X3, X4)

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

(296) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U131¹(mark(X1), X2, X3, X4) → U131¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U131¹(x1, x2, x3, x4)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(297) Obligation:

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

U131¹(X1, mark(X2), X3, X4) → U131¹(X1, X2, X3, X4)
U131¹(active(X1), X2, X3, X4) → U131¹(X1, X2, X3, X4)

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

(298) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U131¹(X1, mark(X2), X3, X4) → U131¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U131¹(x1, x2, x3, x4)  =  U131¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

mark₁ > U131^1₁
active₁ > U131^1₁

The following usable rules [FROCOS05] were oriented: none

(299) Obligation:

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

U131¹(active(X1), X2, X3, X4) → U131¹(X1, X2, X3, X4)

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

(300) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U131¹(active(X1), X2, X3, X4) → U131¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U131¹(x1, x2, x3, x4)  =  U131¹(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(301) Obligation:

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

(302) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(303) TRUE

(304) Obligation:

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

U122¹(active(X)) → U122¹(X)
U122¹(mark(X)) → U122¹(X)

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

(305) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U122¹(active(X)) → U122¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U122¹(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(306) Obligation:

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

U122¹(mark(X)) → U122¹(X)

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

(307) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U122¹(mark(X)) → U122¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:

mark₁ > U122^1₁

The following usable rules [FROCOS05] were oriented: none

(308) Obligation:

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

(309) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(310) TRUE

(311) Obligation:

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

U121¹(X1, mark(X2)) → U121¹(X1, X2)
U121¹(mark(X1), X2) → U121¹(X1, X2)
U121¹(active(X1), X2) → U121¹(X1, X2)
U121¹(X1, active(X2)) → U121¹(X1, X2)

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

(312) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U121¹(X1, mark(X2)) → U121¹(X1, X2)
U121¹(X1, active(X2)) → U121¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U121¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(313) Obligation:

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

U121¹(mark(X1), X2) → U121¹(X1, X2)
U121¹(active(X1), X2) → U121¹(X1, X2)

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

(314) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U121¹(mark(X1), X2) → U121¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U121¹(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(315) Obligation:

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

U121¹(active(X1), X2) → U121¹(X1, X2)

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

(316) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U121¹(active(X1), X2) → U121¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U121¹(x1, x2)  =  U121¹(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

active₁ > U121^1₁

The following usable rules [FROCOS05] were oriented: none

(317) Obligation:

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

(318) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(319) TRUE

(320) Obligation:

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

ISNATLIST(active(X)) → ISNATLIST(X)
ISNATLIST(mark(X)) → ISNATLIST(X)

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

(321) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATLIST(active(X)) → ISNATLIST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATLIST(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(322) Obligation:

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

ISNATLIST(mark(X)) → ISNATLIST(X)

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

(323) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATLIST(mark(X)) → ISNATLIST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:

mark₁ > ISNATLIST₁

The following usable rules [FROCOS05] were oriented: none

(324) Obligation:

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

(325) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(326) TRUE

(327) Obligation:

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

U13¹(active(X)) → U13¹(X)
U13¹(mark(X)) → U13¹(X)

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

(328) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U13¹(active(X)) → U13¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U13¹(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(329) Obligation:

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

U13¹(mark(X)) → U13¹(X)

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

(330) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U13¹(mark(X)) → U13¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:

mark₁ > U13^1₁

The following usable rules [FROCOS05] were oriented: none

(331) Obligation:

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

(332) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(333) TRUE

(334) Obligation:

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

LENGTH(active(X)) → LENGTH(X)
LENGTH(mark(X)) → LENGTH(X)

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

(335) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

LENGTH(active(X)) → LENGTH(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
LENGTH(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(336) Obligation:

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

LENGTH(mark(X)) → LENGTH(X)

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

(337) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

LENGTH(mark(X)) → LENGTH(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:

mark₁ > LENGTH₁

The following usable rules [FROCOS05] were oriented: none

(338) Obligation:

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

(339) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(340) TRUE

(341) Obligation:

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

S(active(X)) → S(X)
S(mark(X)) → S(X)

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

(342) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(active(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(343) Obligation:

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

S(mark(X)) → S(X)

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

(344) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(mark(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:

mark₁ > S₁

The following usable rules [FROCOS05] were oriented: none

(345) Obligation:

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

(346) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(347) TRUE

(348) Obligation:

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

U114¹(X1, mark(X2)) → U114¹(X1, X2)
U114¹(mark(X1), X2) → U114¹(X1, X2)
U114¹(active(X1), X2) → U114¹(X1, X2)
U114¹(X1, active(X2)) → U114¹(X1, X2)

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

(349) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U114¹(X1, mark(X2)) → U114¹(X1, X2)
U114¹(X1, active(X2)) → U114¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U114¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(350) Obligation:

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

U114¹(mark(X1), X2) → U114¹(X1, X2)
U114¹(active(X1), X2) → U114¹(X1, X2)

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

(351) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U114¹(mark(X1), X2) → U114¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U114¹(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(352) Obligation:

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

U114¹(active(X1), X2) → U114¹(X1, X2)

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

(353) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U114¹(active(X1), X2) → U114¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U114¹(x1, x2)  =  U114¹(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

active₁ > U114^1₁

The following usable rules [FROCOS05] were oriented: none

(354) Obligation:

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

(355) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(356) TRUE

(357) Obligation:

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

U113¹(X1, mark(X2), X3) → U113¹(X1, X2, X3)
U113¹(mark(X1), X2, X3) → U113¹(X1, X2, X3)
U113¹(X1, X2, mark(X3)) → U113¹(X1, X2, X3)
U113¹(active(X1), X2, X3) → U113¹(X1, X2, X3)
U113¹(X1, active(X2), X3) → U113¹(X1, X2, X3)
U113¹(X1, X2, active(X3)) → U113¹(X1, X2, X3)

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

(358) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U113¹(X1, X2, mark(X3)) → U113¹(X1, X2, X3)
U113¹(X1, X2, active(X3)) → U113¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U113¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(359) Obligation:

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

U113¹(X1, mark(X2), X3) → U113¹(X1, X2, X3)
U113¹(mark(X1), X2, X3) → U113¹(X1, X2, X3)
U113¹(active(X1), X2, X3) → U113¹(X1, X2, X3)
U113¹(X1, active(X2), X3) → U113¹(X1, X2, X3)

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

(360) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U113¹(mark(X1), X2, X3) → U113¹(X1, X2, X3)
U113¹(active(X1), X2, X3) → U113¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U113¹(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(361) Obligation:

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

U113¹(X1, mark(X2), X3) → U113¹(X1, X2, X3)
U113¹(X1, active(X2), X3) → U113¹(X1, X2, X3)

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

(362) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U113¹(X1, mark(X2), X3) → U113¹(X1, X2, X3)
U113¹(X1, active(X2), X3) → U113¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U113¹(x1, x2, x3)  =  U113¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

mark₁ > U113^1₁
active₁ > U113^1₁

The following usable rules [FROCOS05] were oriented: none

(363) Obligation:

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

(364) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(365) TRUE

(366) Obligation:

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

U112¹(X1, mark(X2), X3) → U112¹(X1, X2, X3)
U112¹(mark(X1), X2, X3) → U112¹(X1, X2, X3)
U112¹(X1, X2, mark(X3)) → U112¹(X1, X2, X3)
U112¹(active(X1), X2, X3) → U112¹(X1, X2, X3)
U112¹(X1, active(X2), X3) → U112¹(X1, X2, X3)
U112¹(X1, X2, active(X3)) → U112¹(X1, X2, X3)

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

(367) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U112¹(X1, X2, mark(X3)) → U112¹(X1, X2, X3)
U112¹(X1, X2, active(X3)) → U112¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U112¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(368) Obligation:

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

U112¹(X1, mark(X2), X3) → U112¹(X1, X2, X3)
U112¹(mark(X1), X2, X3) → U112¹(X1, X2, X3)
U112¹(active(X1), X2, X3) → U112¹(X1, X2, X3)
U112¹(X1, active(X2), X3) → U112¹(X1, X2, X3)

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

(369) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U112¹(mark(X1), X2, X3) → U112¹(X1, X2, X3)
U112¹(active(X1), X2, X3) → U112¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U112¹(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(370) Obligation:

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

U112¹(X1, mark(X2), X3) → U112¹(X1, X2, X3)
U112¹(X1, active(X2), X3) → U112¹(X1, X2, X3)

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

(371) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U112¹(X1, mark(X2), X3) → U112¹(X1, X2, X3)
U112¹(X1, active(X2), X3) → U112¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U112¹(x1, x2, x3)  =  U112¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

mark₁ > U112^1₁
active₁ > U112^1₁

The following usable rules [FROCOS05] were oriented: none

(372) Obligation:

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

(373) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(374) TRUE

(375) Obligation:

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

U111¹(X1, mark(X2), X3) → U111¹(X1, X2, X3)
U111¹(mark(X1), X2, X3) → U111¹(X1, X2, X3)
U111¹(X1, X2, mark(X3)) → U111¹(X1, X2, X3)
U111¹(active(X1), X2, X3) → U111¹(X1, X2, X3)
U111¹(X1, active(X2), X3) → U111¹(X1, X2, X3)
U111¹(X1, X2, active(X3)) → U111¹(X1, X2, X3)

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

(376) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U111¹(X1, X2, mark(X3)) → U111¹(X1, X2, X3)
U111¹(X1, X2, active(X3)) → U111¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U111¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(377) Obligation:

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

U111¹(X1, mark(X2), X3) → U111¹(X1, X2, X3)
U111¹(mark(X1), X2, X3) → U111¹(X1, X2, X3)
U111¹(active(X1), X2, X3) → U111¹(X1, X2, X3)
U111¹(X1, active(X2), X3) → U111¹(X1, X2, X3)

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

(378) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U111¹(mark(X1), X2, X3) → U111¹(X1, X2, X3)
U111¹(active(X1), X2, X3) → U111¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U111¹(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(379) Obligation:

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

U111¹(X1, mark(X2), X3) → U111¹(X1, X2, X3)
U111¹(X1, active(X2), X3) → U111¹(X1, X2, X3)

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

(380) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U111¹(X1, mark(X2), X3) → U111¹(X1, X2, X3)
U111¹(X1, active(X2), X3) → U111¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U111¹(x1, x2, x3)  =  U111¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

mark₁ > U111^1₁
active₁ > U111^1₁

The following usable rules [FROCOS05] were oriented: none

(381) Obligation:

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

(382) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(383) TRUE

(384) Obligation:

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

U12¹(X1, mark(X2)) → U12¹(X1, X2)
U12¹(mark(X1), X2) → U12¹(X1, X2)
U12¹(active(X1), X2) → U12¹(X1, X2)
U12¹(X1, active(X2)) → U12¹(X1, X2)

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

(385) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(X1, mark(X2)) → U12¹(X1, X2)
U12¹(X1, active(X2)) → U12¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(386) Obligation:

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

U12¹(mark(X1), X2) → U12¹(X1, X2)
U12¹(active(X1), X2) → U12¹(X1, X2)

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

(387) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(mark(X1), X2) → U12¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(388) Obligation:

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

U12¹(active(X1), X2) → U12¹(X1, X2)

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

(389) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(active(X1), X2) → U12¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2)  =  U12¹(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

active₁ > U12^1₁

The following usable rules [FROCOS05] were oriented: none

(390) Obligation:

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

(391) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(392) TRUE

(393) Obligation:

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

U11¹(X1, mark(X2)) → U11¹(X1, X2)
U11¹(mark(X1), X2) → U11¹(X1, X2)
U11¹(active(X1), X2) → U11¹(X1, X2)
U11¹(X1, active(X2)) → U11¹(X1, X2)

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

(394) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(X1, mark(X2)) → U11¹(X1, X2)
U11¹(X1, active(X2)) → U11¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(395) Obligation:

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

U11¹(mark(X1), X2) → U11¹(X1, X2)
U11¹(active(X1), X2) → U11¹(X1, X2)

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

(396) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(mark(X1), X2) → U11¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(397) Obligation:

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

U11¹(active(X1), X2) → U11¹(X1, X2)

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

(398) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(active(X1), X2) → U11¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2)  =  U11¹(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

active₁ > U11^1₁

The following usable rules [FROCOS05] were oriented: none

(399) Obligation:

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

(400) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(401) TRUE

(402) Obligation:

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

ISNATILIST(active(X)) → ISNATILIST(X)
ISNATILIST(mark(X)) → ISNATILIST(X)

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

(403) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATILIST(active(X)) → ISNATILIST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATILIST(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(404) Obligation:

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

ISNATILIST(mark(X)) → ISNATILIST(X)

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

(405) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATILIST(mark(X)) → ISNATILIST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:

mark₁ > ISNATILIST₁

The following usable rules [FROCOS05] were oriented: none

(406) Obligation:

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

(407) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(408) TRUE

(409) Obligation:

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

U106¹(active(X)) → U106¹(X)
U106¹(mark(X)) → U106¹(X)

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

(410) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U106¹(active(X)) → U106¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U106¹(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(411) Obligation:

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

U106¹(mark(X)) → U106¹(X)

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

(412) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U106¹(mark(X)) → U106¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:

mark₁ > U106^1₁

The following usable rules [FROCOS05] were oriented: none

(413) Obligation:

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

(414) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(415) TRUE

(416) Obligation:

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

ISNAT(active(X)) → ISNAT(X)
ISNAT(mark(X)) → ISNAT(X)

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

(417) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNAT(active(X)) → ISNAT(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNAT(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(418) Obligation:

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

ISNAT(mark(X)) → ISNAT(X)

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

(419) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNAT(mark(X)) → ISNAT(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:

mark₁ > ISNAT₁

The following usable rules [FROCOS05] were oriented: none

(420) Obligation:

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

(421) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(422) TRUE

(423) Obligation:

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

U105¹(X1, mark(X2)) → U105¹(X1, X2)
U105¹(mark(X1), X2) → U105¹(X1, X2)
U105¹(active(X1), X2) → U105¹(X1, X2)
U105¹(X1, active(X2)) → U105¹(X1, X2)

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

(424) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U105¹(X1, mark(X2)) → U105¹(X1, X2)
U105¹(X1, active(X2)) → U105¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U105¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(425) Obligation:

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

U105¹(mark(X1), X2) → U105¹(X1, X2)
U105¹(active(X1), X2) → U105¹(X1, X2)

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

(426) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U105¹(mark(X1), X2) → U105¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U105¹(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(427) Obligation:

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

U105¹(active(X1), X2) → U105¹(X1, X2)

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

(428) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U105¹(active(X1), X2) → U105¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U105¹(x1, x2)  =  U105¹(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

active₁ > U105^1₁

The following usable rules [FROCOS05] were oriented: none

(429) Obligation:

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

(430) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(431) TRUE

(432) Obligation:

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

U104¹(X1, mark(X2), X3) → U104¹(X1, X2, X3)
U104¹(mark(X1), X2, X3) → U104¹(X1, X2, X3)
U104¹(X1, X2, mark(X3)) → U104¹(X1, X2, X3)
U104¹(active(X1), X2, X3) → U104¹(X1, X2, X3)
U104¹(X1, active(X2), X3) → U104¹(X1, X2, X3)
U104¹(X1, X2, active(X3)) → U104¹(X1, X2, X3)

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

(433) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U104¹(X1, X2, mark(X3)) → U104¹(X1, X2, X3)
U104¹(X1, X2, active(X3)) → U104¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U104¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(434) Obligation:

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

U104¹(X1, mark(X2), X3) → U104¹(X1, X2, X3)
U104¹(mark(X1), X2, X3) → U104¹(X1, X2, X3)
U104¹(active(X1), X2, X3) → U104¹(X1, X2, X3)
U104¹(X1, active(X2), X3) → U104¹(X1, X2, X3)

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

(435) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U104¹(mark(X1), X2, X3) → U104¹(X1, X2, X3)
U104¹(active(X1), X2, X3) → U104¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U104¹(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(436) Obligation:

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

U104¹(X1, mark(X2), X3) → U104¹(X1, X2, X3)
U104¹(X1, active(X2), X3) → U104¹(X1, X2, X3)

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

(437) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U104¹(X1, mark(X2), X3) → U104¹(X1, X2, X3)
U104¹(X1, active(X2), X3) → U104¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U104¹(x1, x2, x3)  =  U104¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

mark₁ > U104^1₁
active₁ > U104^1₁

The following usable rules [FROCOS05] were oriented: none

(438) Obligation:

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

(439) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(440) TRUE

(441) Obligation:

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

ISNATILISTKIND(active(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(mark(X)) → ISNATILISTKIND(X)

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

(442) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATILISTKIND(active(X)) → ISNATILISTKIND(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATILISTKIND(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(443) Obligation:

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

ISNATILISTKIND(mark(X)) → ISNATILISTKIND(X)

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

(444) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATILISTKIND(mark(X)) → ISNATILISTKIND(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:

mark₁ > ISNATILISTKIND₁

The following usable rules [FROCOS05] were oriented: none

(445) Obligation:

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

(446) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(447) TRUE

(448) Obligation:

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

U103¹(X1, mark(X2), X3) → U103¹(X1, X2, X3)
U103¹(mark(X1), X2, X3) → U103¹(X1, X2, X3)
U103¹(X1, X2, mark(X3)) → U103¹(X1, X2, X3)
U103¹(active(X1), X2, X3) → U103¹(X1, X2, X3)
U103¹(X1, active(X2), X3) → U103¹(X1, X2, X3)
U103¹(X1, X2, active(X3)) → U103¹(X1, X2, X3)

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

(449) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U103¹(X1, X2, mark(X3)) → U103¹(X1, X2, X3)
U103¹(X1, X2, active(X3)) → U103¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U103¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(450) Obligation:

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

U103¹(X1, mark(X2), X3) → U103¹(X1, X2, X3)
U103¹(mark(X1), X2, X3) → U103¹(X1, X2, X3)
U103¹(active(X1), X2, X3) → U103¹(X1, X2, X3)
U103¹(X1, active(X2), X3) → U103¹(X1, X2, X3)

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

(451) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U103¹(mark(X1), X2, X3) → U103¹(X1, X2, X3)
U103¹(active(X1), X2, X3) → U103¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U103¹(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(452) Obligation:

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

U103¹(X1, mark(X2), X3) → U103¹(X1, X2, X3)
U103¹(X1, active(X2), X3) → U103¹(X1, X2, X3)

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

(453) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U103¹(X1, mark(X2), X3) → U103¹(X1, X2, X3)
U103¹(X1, active(X2), X3) → U103¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U103¹(x1, x2, x3)  =  U103¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

mark₁ > U103^1₁
active₁ > U103^1₁

The following usable rules [FROCOS05] were oriented: none

(454) Obligation:

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

(455) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(456) TRUE

(457) Obligation:

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

ISNATKIND(active(X)) → ISNATKIND(X)
ISNATKIND(mark(X)) → ISNATKIND(X)

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

(458) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATKIND(active(X)) → ISNATKIND(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATKIND(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(459) Obligation:

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

ISNATKIND(mark(X)) → ISNATKIND(X)

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

(460) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATKIND(mark(X)) → ISNATKIND(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:

mark₁ > ISNATKIND₁

The following usable rules [FROCOS05] were oriented: none

(461) Obligation:

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

(462) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(463) TRUE

(464) Obligation:

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

U102¹(X1, mark(X2), X3) → U102¹(X1, X2, X3)
U102¹(mark(X1), X2, X3) → U102¹(X1, X2, X3)
U102¹(X1, X2, mark(X3)) → U102¹(X1, X2, X3)
U102¹(active(X1), X2, X3) → U102¹(X1, X2, X3)
U102¹(X1, active(X2), X3) → U102¹(X1, X2, X3)
U102¹(X1, X2, active(X3)) → U102¹(X1, X2, X3)

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

(465) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U102¹(X1, X2, mark(X3)) → U102¹(X1, X2, X3)
U102¹(X1, X2, active(X3)) → U102¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U102¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(466) Obligation:

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

U102¹(X1, mark(X2), X3) → U102¹(X1, X2, X3)
U102¹(mark(X1), X2, X3) → U102¹(X1, X2, X3)
U102¹(active(X1), X2, X3) → U102¹(X1, X2, X3)
U102¹(X1, active(X2), X3) → U102¹(X1, X2, X3)

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

(467) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U102¹(mark(X1), X2, X3) → U102¹(X1, X2, X3)
U102¹(active(X1), X2, X3) → U102¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U102¹(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(468) Obligation:

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

U102¹(X1, mark(X2), X3) → U102¹(X1, X2, X3)
U102¹(X1, active(X2), X3) → U102¹(X1, X2, X3)

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

(469) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U102¹(X1, mark(X2), X3) → U102¹(X1, X2, X3)
U102¹(X1, active(X2), X3) → U102¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U102¹(x1, x2, x3)  =  U102¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

mark₁ > U102^1₁
active₁ > U102^1₁

The following usable rules [FROCOS05] were oriented: none

(470) Obligation:

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

(471) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(472) TRUE

(473) Obligation:

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

U101¹(X1, mark(X2), X3) → U101¹(X1, X2, X3)
U101¹(mark(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(X1, X2, mark(X3)) → U101¹(X1, X2, X3)
U101¹(active(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(X1, active(X2), X3) → U101¹(X1, X2, X3)
U101¹(X1, X2, active(X3)) → U101¹(X1, X2, X3)

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

(474) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U101¹(X1, X2, mark(X3)) → U101¹(X1, X2, X3)
U101¹(X1, X2, active(X3)) → U101¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U101¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(475) Obligation:

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

U101¹(X1, mark(X2), X3) → U101¹(X1, X2, X3)
U101¹(mark(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(active(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(X1, active(X2), X3) → U101¹(X1, X2, X3)

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

(476) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U101¹(mark(X1), X2, X3) → U101¹(X1, X2, X3)
U101¹(active(X1), X2, X3) → U101¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U101¹(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(477) Obligation:

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

U101¹(X1, mark(X2), X3) → U101¹(X1, X2, X3)
U101¹(X1, active(X2), X3) → U101¹(X1, X2, X3)

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

(478) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U101¹(X1, mark(X2), X3) → U101¹(X1, X2, X3)
U101¹(X1, active(X2), X3) → U101¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U101¹(x1, x2, x3)  =  U101¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

mark₁ > U101^1₁
active₁ > U101^1₁

The following usable rules [FROCOS05] were oriented: none

(479) Obligation:

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

(480) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(481) TRUE

(482) Obligation:

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

CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)

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

(483) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(484) Obligation:

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

CONS(mark(X1), X2) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)

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

(485) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(mark(X1), X2) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(486) Obligation:

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

CONS(active(X1), X2) → CONS(X1, X2)

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

(487) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(active(X1), X2) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  CONS(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

active₁ > CONS₁

The following usable rules [FROCOS05] were oriented: none

(488) Obligation:

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

(489) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(490) TRUE

(491) Obligation:

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

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

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