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(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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(U11(tt)) → MARK(tt)
ACTIVE(U21(tt)) → MARK(tt)
ACTIVE(U31(tt)) → MARK(tt)
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
ACTIVE(U41(tt, V2)) → U42¹(isNatIList(V2))
ACTIVE(U41(tt, V2)) → ISNATILIST(V2)
ACTIVE(U42(tt)) → MARK(tt)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
ACTIVE(U51(tt, V2)) → U52¹(isNatList(V2))
ACTIVE(U51(tt, V2)) → ISNATLIST(V2)
ACTIVE(U52(tt)) → MARK(tt)
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
ACTIVE(U61(tt, V2)) → U62¹(isNatIList(V2))
ACTIVE(U61(tt, V2)) → ISNATILIST(V2)
ACTIVE(U62(tt)) → MARK(tt)
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
ACTIVE(U71(tt, L, N)) → U72¹(isNat(N), L)
ACTIVE(U71(tt, L, N)) → ISNAT(N)
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
ACTIVE(U72(tt, L)) → S(length(L))
ACTIVE(U72(tt, L)) → LENGTH(L)
ACTIVE(U81(tt)) → MARK(nil)
ACTIVE(U91(tt, IL, M, N)) → MARK(U92(isNat(M), IL, M, N))
ACTIVE(U91(tt, IL, M, N)) → U92¹(isNat(M), IL, M, N)
ACTIVE(U91(tt, IL, M, N)) → ISNAT(M)
ACTIVE(U92(tt, IL, M, N)) → MARK(U93(isNat(N), IL, M, N))
ACTIVE(U92(tt, IL, M, N)) → U93¹(isNat(N), IL, M, N)
ACTIVE(U92(tt, IL, M, N)) → ISNAT(N)
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
ACTIVE(U93(tt, IL, M, N)) → CONS(N, take(M, IL))
ACTIVE(U93(tt, IL, M, N)) → TAKE(M, IL)
ACTIVE(isNat(0)) → MARK(tt)
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
ACTIVE(isNat(length(V1))) → U11¹(isNatList(V1))
ACTIVE(isNat(length(V1))) → ISNATLIST(V1)
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
ACTIVE(isNat(s(V1))) → U21¹(isNat(V1))
ACTIVE(isNat(s(V1))) → ISNAT(V1)
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
ACTIVE(isNatIList(V)) → U31¹(isNatList(V))
ACTIVE(isNatIList(V)) → ISNATLIST(V)
ACTIVE(isNatIList(zeros)) → MARK(tt)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
ACTIVE(isNatIList(cons(V1, V2))) → U41¹(isNat(V1), V2)
ACTIVE(isNatIList(cons(V1, V2))) → ISNAT(V1)
ACTIVE(isNatList(nil)) → MARK(tt)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
ACTIVE(isNatList(cons(V1, V2))) → U51¹(isNat(V1), V2)
ACTIVE(isNatList(cons(V1, V2))) → ISNAT(V1)
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
ACTIVE(isNatList(take(V1, V2))) → U61¹(isNat(V1), V2)
ACTIVE(isNatList(take(V1, V2))) → ISNAT(V1)
ACTIVE(length(nil)) → MARK(0)
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
ACTIVE(length(cons(N, L))) → U71¹(isNatList(L), L, N)
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
ACTIVE(take(0, IL)) → MARK(U81(isNatIList(IL)))
ACTIVE(take(0, IL)) → U81¹(isNatIList(IL))
ACTIVE(take(0, IL)) → ISNATILIST(IL)
ACTIVE(take(s(M), cons(N, IL))) → MARK(U91(isNatIList(IL), IL, M, N))
ACTIVE(take(s(M), cons(N, IL))) → U91¹(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(U11(X)) → ACTIVE(U11(mark(X)))
MARK(U11(X)) → U11¹(mark(X))
MARK(U11(X)) → MARK(X)
MARK(tt) → ACTIVE(tt)
MARK(U21(X)) → ACTIVE(U21(mark(X)))
MARK(U21(X)) → U21¹(mark(X))
MARK(U21(X)) → MARK(X)
MARK(U31(X)) → ACTIVE(U31(mark(X)))
MARK(U31(X)) → U31¹(mark(X))
MARK(U31(X)) → MARK(X)
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
MARK(U41(X1, X2)) → U41¹(mark(X1), X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → ACTIVE(U42(mark(X)))
MARK(U42(X)) → U42¹(mark(X))
MARK(U42(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(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(isNatList(X)) → ACTIVE(isNatList(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(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
MARK(U71(X1, X2, X3)) → U71¹(mark(X1), X2, X3)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(U72(X1, X2)) → U72¹(mark(X1), X2)
MARK(U72(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
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(U81(X)) → ACTIVE(U81(mark(X)))
MARK(U81(X)) → U81¹(mark(X))
MARK(U81(X)) → MARK(X)
MARK(nil) → ACTIVE(nil)
MARK(U91(X1, X2, X3, X4)) → ACTIVE(U91(mark(X1), X2, X3, X4))
MARK(U91(X1, X2, X3, X4)) → U91¹(mark(X1), X2, X3, X4)
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(U92(X1, X2, X3, X4)) → ACTIVE(U92(mark(X1), X2, X3, X4))
MARK(U92(X1, X2, X3, X4)) → U92¹(mark(X1), X2, X3, X4)
MARK(U92(X1, X2, X3, X4)) → MARK(X1)
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
MARK(U93(X1, X2, X3, X4)) → U93¹(mark(X1), X2, X3, X4)
MARK(U93(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)
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)
U11¹(mark(X)) → U11¹(X)
U11¹(active(X)) → U11¹(X)
U21¹(mark(X)) → U21¹(X)
U21¹(active(X)) → U21¹(X)
U31¹(mark(X)) → U31¹(X)
U31¹(active(X)) → U31¹(X)
U41¹(mark(X1), X2) → U41¹(X1, X2)
U41¹(X1, mark(X2)) → U41¹(X1, X2)
U41¹(active(X1), X2) → U41¹(X1, X2)
U41¹(X1, active(X2)) → U41¹(X1, X2)
U42¹(mark(X)) → U42¹(X)
U42¹(active(X)) → U42¹(X)
ISNATILIST(mark(X)) → ISNATILIST(X)
ISNATILIST(active(X)) → ISNATILIST(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)
ISNATLIST(mark(X)) → ISNATLIST(X)
ISNATLIST(active(X)) → ISNATLIST(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(X1), X2, X3) → U71¹(X1, X2, X3)
U71¹(X1, mark(X2), X3) → U71¹(X1, X2, X3)
U71¹(X1, X2, mark(X3)) → U71¹(X1, X2, X3)
U71¹(active(X1), X2, X3) → U71¹(X1, X2, X3)
U71¹(X1, active(X2), X3) → U71¹(X1, X2, X3)
U71¹(X1, X2, active(X3)) → U71¹(X1, X2, X3)
U72¹(mark(X1), X2) → U72¹(X1, X2)
U72¹(X1, mark(X2)) → U72¹(X1, X2)
U72¹(active(X1), X2) → U72¹(X1, X2)
U72¹(X1, active(X2)) → U72¹(X1, X2)
ISNAT(mark(X)) → ISNAT(X)
ISNAT(active(X)) → ISNAT(X)
S(mark(X)) → S(X)
S(active(X)) → S(X)
LENGTH(mark(X)) → LENGTH(X)
LENGTH(active(X)) → LENGTH(X)
U81¹(mark(X)) → U81¹(X)
U81¹(active(X)) → U81¹(X)
U91¹(mark(X1), X2, X3, X4) → U91¹(X1, X2, X3, X4)
U91¹(X1, mark(X2), X3, X4) → U91¹(X1, X2, X3, X4)
U91¹(X1, X2, mark(X3), X4) → U91¹(X1, X2, X3, X4)
U91¹(X1, X2, X3, mark(X4)) → U91¹(X1, X2, X3, X4)
U91¹(active(X1), X2, X3, X4) → U91¹(X1, X2, X3, X4)
U91¹(X1, active(X2), X3, X4) → U91¹(X1, X2, X3, X4)
U91¹(X1, X2, active(X3), X4) → U91¹(X1, X2, X3, X4)
U91¹(X1, X2, X3, active(X4)) → U91¹(X1, X2, X3, X4)
U92¹(mark(X1), X2, X3, X4) → U92¹(X1, X2, X3, X4)
U92¹(X1, mark(X2), X3, X4) → U92¹(X1, X2, X3, X4)
U92¹(X1, X2, mark(X3), X4) → U92¹(X1, X2, X3, X4)
U92¹(X1, X2, X3, mark(X4)) → U92¹(X1, X2, X3, X4)
U92¹(active(X1), X2, X3, X4) → U92¹(X1, X2, X3, X4)
U92¹(X1, active(X2), X3, X4) → U92¹(X1, X2, X3, X4)
U92¹(X1, X2, active(X3), X4) → U92¹(X1, X2, X3, X4)
U92¹(X1, X2, X3, active(X4)) → U92¹(X1, X2, X3, X4)
U93¹(mark(X1), X2, X3, X4) → U93¹(X1, X2, X3, X4)
U93¹(X1, mark(X2), X3, X4) → U93¹(X1, X2, X3, X4)
U93¹(X1, X2, mark(X3), X4) → U93¹(X1, X2, X3, X4)
U93¹(X1, X2, X3, mark(X4)) → U93¹(X1, X2, X3, X4)
U93¹(active(X1), X2, X3, X4) → U93¹(X1, X2, X3, X4)
U93¹(X1, active(X2), X3, X4) → U93¹(X1, X2, X3, X4)
U93¹(X1, X2, active(X3), X4) → U93¹(X1, X2, X3, X4)
U93¹(X1, X2, X3, active(X4)) → U93¹(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)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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 23 SCCs with 68 less nodes.

(4) Complex Obligation (AND)

(5) 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(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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.

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)  =  TAKE(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat(x1)
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > mark₁ > active₁ > TAKE₁
zeros > cons > mark₁ > active₁ > 0
zeros > cons > mark₁ > active₁ > s
zeros > tt > U72 > mark₁ > active₁ > TAKE₁
zeros > tt > U72 > mark₁ > active₁ > 0
zeros > tt > U72 > mark₁ > active₁ > s
isNatIList > U31 > tt > U72 > mark₁ > active₁ > TAKE₁
isNatIList > U31 > tt > U72 > mark₁ > active₁ > 0
isNatIList > U31 > tt > U72 > mark₁ > active₁ > s
isNatIList > U41 > U42 > tt > U72 > mark₁ > active₁ > TAKE₁
isNatIList > U41 > U42 > tt > U72 > mark₁ > active₁ > 0
isNatIList > U41 > U42 > tt > U72 > mark₁ > active₁ > s
isNatList > U51 > U52 > tt > U72 > mark₁ > active₁ > TAKE₁
isNatList > U51 > U52 > tt > U72 > mark₁ > active₁ > 0
isNatList > U51 > U52 > tt > U72 > mark₁ > active₁ > s
isNatList > U61 > U62 > tt > U72 > mark₁ > active₁ > TAKE₁
isNatList > U61 > U62 > tt > U72 > mark₁ > active₁ > 0
isNatList > U61 > U62 > tt > U72 > mark₁ > active₁ > s
isNat₁ > U11 > tt > U72 > mark₁ > active₁ > TAKE₁
isNat₁ > U11 > tt > U72 > mark₁ > active₁ > 0
isNat₁ > U11 > tt > U72 > mark₁ > active₁ > s
isNat₁ > U21 > tt > U72 > mark₁ > active₁ > TAKE₁
isNat₁ > U21 > tt > U72 > mark₁ > active₁ > 0
isNat₁ > U21 > tt > U72 > mark₁ > active₁ > s
length > U71 > U72 > mark₁ > active₁ > TAKE₁
length > U71 > U72 > mark₁ > active₁ > 0
length > U71 > U72 > mark₁ > active₁ > s
take > U61 > U62 > tt > U72 > mark₁ > active₁ > TAKE₁
take > U61 > U62 > tt > U72 > mark₁ > active₁ > 0
take > U61 > U62 > tt > U72 > mark₁ > active₁ > s
take > U81 > nil > mark₁ > active₁ > TAKE₁
take > U81 > nil > mark₁ > active₁ > 0
take > U81 > nil > mark₁ > active₁ > s
take > U91 > U92 > U93 > cons > mark₁ > active₁ > TAKE₁
take > U91 > U92 > U93 > cons > mark₁ > active₁ > 0
take > U91 > U92 > U93 > cons > mark₁ > active₁ > s

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(7) 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(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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.

TAKE(mark(X1), X2) → TAKE(X1, X2)
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)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  x1
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U41 > U42 > tt > mark₁ > nil > active₁
zeros > 0 > tt > mark₁ > nil > active₁
isNatIList > U31 > tt > mark₁ > nil > active₁
isNatIList > U41 > U42 > tt > mark₁ > nil > active₁
isNatList > U51 > U52 > tt > mark₁ > nil > active₁
isNatList > U61 > U62 > tt > mark₁ > nil > active₁
length > 0 > tt > mark₁ > nil > active₁
length > U11 > tt > mark₁ > nil > active₁
length > U71 > U72 > s > U21 > tt > mark₁ > nil > active₁
take > U61 > U62 > tt > mark₁ > nil > active₁
take > U81 > mark₁ > nil > active₁
take > U91 > U92 > U93 > cons > U41 > U42 > tt > mark₁ > nil > active₁

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(9) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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:

U93¹(X1, mark(X2), X3, X4) → U93¹(X1, X2, X3, X4)
U93¹(mark(X1), X2, X3, X4) → U93¹(X1, X2, X3, X4)
U93¹(X1, X2, mark(X3), X4) → U93¹(X1, X2, X3, X4)
U93¹(X1, X2, X3, mark(X4)) → U93¹(X1, X2, X3, X4)
U93¹(active(X1), X2, X3, X4) → U93¹(X1, X2, X3, X4)
U93¹(X1, active(X2), X3, X4) → U93¹(X1, X2, X3, X4)
U93¹(X1, X2, active(X3), X4) → U93¹(X1, X2, X3, X4)
U93¹(X1, X2, X3, active(X4)) → U93¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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.

U93¹(X1, mark(X2), X3, X4) → U93¹(X1, X2, X3, X4)
U93¹(mark(X1), X2, X3, X4) → U93¹(X1, X2, X3, X4)
U93¹(X1, X2, mark(X3), X4) → U93¹(X1, X2, X3, X4)
U93¹(X1, X2, X3, mark(X4)) → U93¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U93¹(x1, x2, x3, x4)  =  U93¹(x1, x2, x3, x4)
mark(x1)  =  mark(x1)
active(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71(x1)
U72(x1, x2)  =  U72
isNat(x1)  =  isNat(x1)
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81(x1)
nil  =  nil
U91(x1, x2, x3, x4)  =  x1
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > tt > U72 > s > mark₁ > U93^1₄ > U81₁
zeros > tt > U72 > s > mark₁ > cons > U81₁
zeros > tt > U72 > s > mark₁ > U71₁ > U81₁
zeros > tt > U72 > s > mark₁ > nil > 0 > U81₁
zeros > tt > U92 > U93 > mark₁ > U93^1₄ > U81₁
zeros > tt > U92 > U93 > mark₁ > cons > U81₁
zeros > tt > U92 > U93 > mark₁ > U71₁ > U81₁
zeros > tt > U92 > U93 > mark₁ > nil > 0 > U81₁
length > U11 > tt > U72 > s > mark₁ > U93^1₄ > U81₁
length > U11 > tt > U72 > s > mark₁ > cons > U81₁
length > U11 > tt > U72 > s > mark₁ > U71₁ > U81₁
length > U11 > tt > U72 > s > mark₁ > nil > 0 > U81₁
length > U11 > tt > U92 > U93 > mark₁ > U93^1₄ > U81₁
length > U11 > tt > U92 > U93 > mark₁ > cons > U81₁
length > U11 > tt > U92 > U93 > mark₁ > U71₁ > U81₁
length > U11 > tt > U92 > U93 > mark₁ > nil > 0 > U81₁
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > U93^1₄ > U81₁
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > cons > U81₁
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > U71₁ > U81₁
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > nil > 0 > U81₁
length > isNatList > U51 > U52 > tt > U92 > U93 > mark₁ > U93^1₄ > U81₁
length > isNatList > U51 > U52 > tt > U92 > U93 > mark₁ > cons > U81₁
length > isNatList > U51 > U52 > tt > U92 > U93 > mark₁ > U71₁ > U81₁
length > isNatList > U51 > U52 > tt > U92 > U93 > mark₁ > nil > 0 > U81₁
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > U93^1₄ > U81₁
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > cons > U81₁
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > U71₁ > U81₁
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > nil > 0 > U81₁
length > isNatList > U61 > U62 > tt > U92 > U93 > mark₁ > U93^1₄ > U81₁
length > isNatList > U61 > U62 > tt > U92 > U93 > mark₁ > cons > U81₁
length > isNatList > U61 > U62 > tt > U92 > U93 > mark₁ > U71₁ > U81₁
length > isNatList > U61 > U62 > tt > U92 > U93 > mark₁ > nil > 0 > U81₁
take > isNatIList > U31 > tt > U72 > s > mark₁ > U93^1₄ > U81₁
take > isNatIList > U31 > tt > U72 > s > mark₁ > cons > U81₁
take > isNatIList > U31 > tt > U72 > s > mark₁ > U71₁ > U81₁
take > isNatIList > U31 > tt > U72 > s > mark₁ > nil > 0 > U81₁
take > isNatIList > U31 > tt > U92 > U93 > mark₁ > U93^1₄ > U81₁
take > isNatIList > U31 > tt > U92 > U93 > mark₁ > cons > U81₁
take > isNatIList > U31 > tt > U92 > U93 > mark₁ > U71₁ > U81₁
take > isNatIList > U31 > tt > U92 > U93 > mark₁ > nil > 0 > U81₁
take > isNatIList > U41 > U42 > tt > U72 > s > mark₁ > U93^1₄ > U81₁
take > isNatIList > U41 > U42 > tt > U72 > s > mark₁ > cons > U81₁
take > isNatIList > U41 > U42 > tt > U72 > s > mark₁ > U71₁ > U81₁
take > isNatIList > U41 > U42 > tt > U72 > s > mark₁ > nil > 0 > U81₁
take > isNatIList > U41 > U42 > tt > U92 > U93 > mark₁ > U93^1₄ > U81₁
take > isNatIList > U41 > U42 > tt > U92 > U93 > mark₁ > cons > U81₁
take > isNatIList > U41 > U42 > tt > U92 > U93 > mark₁ > U71₁ > U81₁
take > isNatIList > U41 > U42 > tt > U92 > U93 > mark₁ > nil > 0 > U81₁
take > isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > U93^1₄ > U81₁
take > isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > cons > U81₁
take > isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > U71₁ > U81₁
take > isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > nil > 0 > U81₁
take > isNatIList > isNatList > U51 > U52 > tt > U92 > U93 > mark₁ > U93^1₄ > U81₁
take > isNatIList > isNatList > U51 > U52 > tt > U92 > U93 > mark₁ > cons > U81₁
take > isNatIList > isNatList > U51 > U52 > tt > U92 > U93 > mark₁ > U71₁ > U81₁
take > isNatIList > isNatList > U51 > U52 > tt > U92 > U93 > mark₁ > nil > 0 > U81₁
take > isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > U93^1₄ > U81₁
take > isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > cons > U81₁
take > isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > U71₁ > U81₁
take > isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > nil > 0 > U81₁
take > isNatIList > isNatList > U61 > U62 > tt > U92 > U93 > mark₁ > U93^1₄ > U81₁
take > isNatIList > isNatList > U61 > U62 > tt > U92 > U93 > mark₁ > cons > U81₁
take > isNatIList > isNatList > U61 > U62 > tt > U92 > U93 > mark₁ > U71₁ > U81₁
take > isNatIList > isNatList > U61 > U62 > tt > U92 > U93 > mark₁ > nil > 0 > U81₁
take > isNatIList > isNat₁ > U21 > tt > U72 > s > mark₁ > U93^1₄ > U81₁
take > isNatIList > isNat₁ > U21 > tt > U72 > s > mark₁ > cons > U81₁
take > isNatIList > isNat₁ > U21 > tt > U72 > s > mark₁ > U71₁ > U81₁
take > isNatIList > isNat₁ > U21 > tt > U72 > s > mark₁ > nil > 0 > U81₁
take > isNatIList > isNat₁ > U21 > tt > U92 > U93 > mark₁ > U93^1₄ > U81₁
take > isNatIList > isNat₁ > U21 > tt > U92 > U93 > mark₁ > cons > U81₁
take > isNatIList > isNat₁ > U21 > tt > U92 > U93 > mark₁ > U71₁ > U81₁
take > isNatIList > isNat₁ > U21 > tt > U92 > U93 > mark₁ > nil > 0 > U81₁

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(14) Obligation:

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

U93¹(active(X1), X2, X3, X4) → U93¹(X1, X2, X3, X4)
U93¹(X1, active(X2), X3, X4) → U93¹(X1, X2, X3, X4)
U93¹(X1, X2, active(X3), X4) → U93¹(X1, X2, X3, X4)
U93¹(X1, X2, X3, active(X4)) → U93¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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.

U93¹(X1, X2, active(X3), X4) → U93¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U93¹(x1, x2, x3, x4)  =  x3
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > tt > mark₁ > active₁ > 0
zeros > tt > mark₁ > active₁ > nil
zeros > tt > cons > active₁ > 0
zeros > tt > cons > active₁ > nil
zeros > tt > s > active₁ > 0
zeros > tt > s > active₁ > nil
isNatIList > U31 > tt > mark₁ > active₁ > 0
isNatIList > U31 > tt > mark₁ > active₁ > nil
isNatIList > U31 > tt > cons > active₁ > 0
isNatIList > U31 > tt > cons > active₁ > nil
isNatIList > U31 > tt > s > active₁ > 0
isNatIList > U31 > tt > s > active₁ > nil
isNatIList > U41 > U42 > tt > mark₁ > active₁ > 0
isNatIList > U41 > U42 > tt > mark₁ > active₁ > nil
isNatIList > U41 > U42 > tt > cons > active₁ > 0
isNatIList > U41 > U42 > tt > cons > active₁ > nil
isNatIList > U41 > U42 > tt > s > active₁ > 0
isNatIList > U41 > U42 > tt > s > active₁ > nil
isNatList > U51 > U52 > tt > mark₁ > active₁ > 0
isNatList > U51 > U52 > tt > mark₁ > active₁ > nil
isNatList > U51 > U52 > tt > cons > active₁ > 0
isNatList > U51 > U52 > tt > cons > active₁ > nil
isNatList > U51 > U52 > tt > s > active₁ > 0
isNatList > U51 > U52 > tt > s > active₁ > nil
isNatList > U61 > U62 > tt > mark₁ > active₁ > 0
isNatList > U61 > U62 > tt > mark₁ > active₁ > nil
isNatList > U61 > U62 > tt > cons > active₁ > 0
isNatList > U61 > U62 > tt > cons > active₁ > nil
isNatList > U61 > U62 > tt > s > active₁ > 0
isNatList > U61 > U62 > tt > s > active₁ > nil
isNat > U11 > tt > mark₁ > active₁ > 0
isNat > U11 > tt > mark₁ > active₁ > nil
isNat > U11 > tt > cons > active₁ > 0
isNat > U11 > tt > cons > active₁ > nil
isNat > U11 > tt > s > active₁ > 0
isNat > U11 > tt > s > active₁ > nil
isNat > U21 > tt > mark₁ > active₁ > 0
isNat > U21 > tt > mark₁ > active₁ > nil
isNat > U21 > tt > cons > active₁ > 0
isNat > U21 > tt > cons > active₁ > nil
isNat > U21 > tt > s > active₁ > 0
isNat > U21 > tt > s > active₁ > nil
length > U71 > U72 > mark₁ > active₁ > 0
length > U71 > U72 > mark₁ > active₁ > nil
length > U71 > U72 > s > active₁ > 0
length > U71 > U72 > s > active₁ > nil
take > U61 > U62 > tt > mark₁ > active₁ > 0
take > U61 > U62 > tt > mark₁ > active₁ > nil
take > U61 > U62 > tt > cons > active₁ > 0
take > U61 > U62 > tt > cons > active₁ > nil
take > U61 > U62 > tt > s > active₁ > 0
take > U61 > U62 > tt > s > active₁ > nil
take > U81 > mark₁ > active₁ > 0
take > U81 > mark₁ > active₁ > nil
take > U91 > U92 > U93 > mark₁ > active₁ > 0
take > U91 > U92 > U93 > mark₁ > active₁ > nil
take > U91 > U92 > U93 > cons > active₁ > 0
take > U91 > U92 > U93 > cons > active₁ > nil

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(16) Obligation:

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

U93¹(active(X1), X2, X3, X4) → U93¹(X1, X2, X3, X4)
U93¹(X1, active(X2), X3, X4) → U93¹(X1, X2, X3, X4)
U93¹(X1, X2, X3, active(X4)) → U93¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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.

U93¹(X1, active(X2), X3, X4) → U93¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U93¹(x1, x2, x3, x4)  =  U93¹(x2, x3)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92(x1)
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > cons > active₁ > U93^1₂
zeros > mark₁ > s > active₁ > U93^1₂
zeros > mark₁ > nil > active₁ > U93^1₂
zeros > mark₁ > U92₁ > active₁ > U93^1₂
isNatIList > U31 > tt > U93 > mark₁ > cons > active₁ > U93^1₂
isNatIList > U31 > tt > U93 > mark₁ > s > active₁ > U93^1₂
isNatIList > U31 > tt > U93 > mark₁ > nil > active₁ > U93^1₂
isNatIList > U31 > tt > U93 > mark₁ > U92₁ > active₁ > U93^1₂
isNatIList > U41 > U42 > tt > U93 > mark₁ > cons > active₁ > U93^1₂
isNatIList > U41 > U42 > tt > U93 > mark₁ > s > active₁ > U93^1₂
isNatIList > U41 > U42 > tt > U93 > mark₁ > nil > active₁ > U93^1₂
isNatIList > U41 > U42 > tt > U93 > mark₁ > U92₁ > active₁ > U93^1₂
isNatIList > isNat > U11 > tt > U93 > mark₁ > cons > active₁ > U93^1₂
isNatIList > isNat > U11 > tt > U93 > mark₁ > s > active₁ > U93^1₂
isNatIList > isNat > U11 > tt > U93 > mark₁ > nil > active₁ > U93^1₂
isNatIList > isNat > U11 > tt > U93 > mark₁ > U92₁ > active₁ > U93^1₂
isNatIList > isNat > U21 > tt > U93 > mark₁ > cons > active₁ > U93^1₂
isNatIList > isNat > U21 > tt > U93 > mark₁ > s > active₁ > U93^1₂
isNatIList > isNat > U21 > tt > U93 > mark₁ > nil > active₁ > U93^1₂
isNatIList > isNat > U21 > tt > U93 > mark₁ > U92₁ > active₁ > U93^1₂
isNatList > U51 > U52 > tt > U93 > mark₁ > cons > active₁ > U93^1₂
isNatList > U51 > U52 > tt > U93 > mark₁ > s > active₁ > U93^1₂
isNatList > U51 > U52 > tt > U93 > mark₁ > nil > active₁ > U93^1₂
isNatList > U51 > U52 > tt > U93 > mark₁ > U92₁ > active₁ > U93^1₂
isNatList > U61 > U62 > tt > U93 > mark₁ > cons > active₁ > U93^1₂
isNatList > U61 > U62 > tt > U93 > mark₁ > s > active₁ > U93^1₂
isNatList > U61 > U62 > tt > U93 > mark₁ > nil > active₁ > U93^1₂
isNatList > U61 > U62 > tt > U93 > mark₁ > U92₁ > active₁ > U93^1₂
isNatList > isNat > U11 > tt > U93 > mark₁ > cons > active₁ > U93^1₂
isNatList > isNat > U11 > tt > U93 > mark₁ > s > active₁ > U93^1₂
isNatList > isNat > U11 > tt > U93 > mark₁ > nil > active₁ > U93^1₂
isNatList > isNat > U11 > tt > U93 > mark₁ > U92₁ > active₁ > U93^1₂
isNatList > isNat > U21 > tt > U93 > mark₁ > cons > active₁ > U93^1₂
isNatList > isNat > U21 > tt > U93 > mark₁ > s > active₁ > U93^1₂
isNatList > isNat > U21 > tt > U93 > mark₁ > nil > active₁ > U93^1₂
isNatList > isNat > U21 > tt > U93 > mark₁ > U92₁ > active₁ > U93^1₂
length > 0 > tt > U93 > mark₁ > cons > active₁ > U93^1₂
length > 0 > tt > U93 > mark₁ > s > active₁ > U93^1₂
length > 0 > tt > U93 > mark₁ > nil > active₁ > U93^1₂
length > 0 > tt > U93 > mark₁ > U92₁ > active₁ > U93^1₂
length > 0 > U81 > mark₁ > cons > active₁ > U93^1₂
length > 0 > U81 > mark₁ > s > active₁ > U93^1₂
length > 0 > U81 > mark₁ > nil > active₁ > U93^1₂
length > 0 > U81 > mark₁ > U92₁ > active₁ > U93^1₂
length > U71 > U72 > mark₁ > cons > active₁ > U93^1₂
length > U71 > U72 > mark₁ > s > active₁ > U93^1₂
length > U71 > U72 > mark₁ > nil > active₁ > U93^1₂
length > U71 > U72 > mark₁ > U92₁ > active₁ > U93^1₂
length > U71 > isNat > U11 > tt > U93 > mark₁ > cons > active₁ > U93^1₂
length > U71 > isNat > U11 > tt > U93 > mark₁ > s > active₁ > U93^1₂
length > U71 > isNat > U11 > tt > U93 > mark₁ > nil > active₁ > U93^1₂
length > U71 > isNat > U11 > tt > U93 > mark₁ > U92₁ > active₁ > U93^1₂
length > U71 > isNat > U21 > tt > U93 > mark₁ > cons > active₁ > U93^1₂
length > U71 > isNat > U21 > tt > U93 > mark₁ > s > active₁ > U93^1₂
length > U71 > isNat > U21 > tt > U93 > mark₁ > nil > active₁ > U93^1₂
length > U71 > isNat > U21 > tt > U93 > mark₁ > U92₁ > active₁ > U93^1₂
take > U61 > U62 > tt > U93 > mark₁ > cons > active₁ > U93^1₂
take > U61 > U62 > tt > U93 > mark₁ > s > active₁ > U93^1₂
take > U61 > U62 > tt > U93 > mark₁ > nil > active₁ > U93^1₂
take > U61 > U62 > tt > U93 > mark₁ > U92₁ > active₁ > U93^1₂
take > U81 > mark₁ > cons > active₁ > U93^1₂
take > U81 > mark₁ > s > active₁ > U93^1₂
take > U81 > mark₁ > nil > active₁ > U93^1₂
take > U81 > mark₁ > U92₁ > active₁ > U93^1₂
take > U91 > isNat > U11 > tt > U93 > mark₁ > cons > active₁ > U93^1₂
take > U91 > isNat > U11 > tt > U93 > mark₁ > s > active₁ > U93^1₂
take > U91 > isNat > U11 > tt > U93 > mark₁ > nil > active₁ > U93^1₂
take > U91 > isNat > U11 > tt > U93 > mark₁ > U92₁ > active₁ > U93^1₂
take > U91 > isNat > U21 > tt > U93 > mark₁ > cons > active₁ > U93^1₂
take > U91 > isNat > U21 > tt > U93 > mark₁ > s > active₁ > U93^1₂
take > U91 > isNat > U21 > tt > U93 > mark₁ > nil > active₁ > U93^1₂
take > U91 > isNat > U21 > tt > U93 > mark₁ > U92₁ > active₁ > U93^1₂

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(18) Obligation:

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

U93¹(active(X1), X2, X3, X4) → U93¹(X1, X2, X3, X4)
U93¹(X1, X2, X3, active(X4)) → U93¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U93¹(X1, X2, X3, active(X4)) → U93¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U93¹(x1, x2, x3, x4)  =  U93¹(x2, x3, x4)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList(x1)
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length(x1)
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U71 > U72 > mark₁ > s > active₁ > U93^1₃
zeros > cons > U71 > U72 > mark₁ > length₁ > 0 > active₁ > U93^1₃
zeros > tt > s > active₁ > U93^1₃
zeros > tt > length₁ > 0 > active₁ > U93^1₃
isNatIList₁ > U31 > mark₁ > s > active₁ > U93^1₃
isNatIList₁ > U31 > mark₁ > length₁ > 0 > active₁ > U93^1₃
isNatIList₁ > U31 > tt > s > active₁ > U93^1₃
isNatIList₁ > U31 > tt > length₁ > 0 > active₁ > U93^1₃
isNatIList₁ > U41 > U42 > mark₁ > s > active₁ > U93^1₃
isNatIList₁ > U41 > U42 > mark₁ > length₁ > 0 > active₁ > U93^1₃
isNatIList₁ > U41 > U42 > tt > s > active₁ > U93^1₃
isNatIList₁ > U41 > U42 > tt > length₁ > 0 > active₁ > U93^1₃
isNatList₁ > U51 > U52 > mark₁ > s > active₁ > U93^1₃
isNatList₁ > U51 > U52 > mark₁ > length₁ > 0 > active₁ > U93^1₃
isNatList₁ > U51 > U52 > tt > s > active₁ > U93^1₃
isNatList₁ > U51 > U52 > tt > length₁ > 0 > active₁ > U93^1₃
isNatList₁ > U61 > U62 > mark₁ > s > active₁ > U93^1₃
isNatList₁ > U61 > U62 > mark₁ > length₁ > 0 > active₁ > U93^1₃
isNatList₁ > U61 > U62 > tt > s > active₁ > U93^1₃
isNatList₁ > U61 > U62 > tt > length₁ > 0 > active₁ > U93^1₃
isNat > U11 > mark₁ > s > active₁ > U93^1₃
isNat > U11 > mark₁ > length₁ > 0 > active₁ > U93^1₃
isNat > U11 > tt > s > active₁ > U93^1₃
isNat > U11 > tt > length₁ > 0 > active₁ > U93^1₃
isNat > U21 > mark₁ > s > active₁ > U93^1₃
isNat > U21 > mark₁ > length₁ > 0 > active₁ > U93^1₃
isNat > U21 > tt > s > active₁ > U93^1₃
isNat > U21 > tt > length₁ > 0 > active₁ > U93^1₃
take > U61 > U62 > mark₁ > s > active₁ > U93^1₃
take > U61 > U62 > mark₁ > length₁ > 0 > active₁ > U93^1₃
take > U61 > U62 > tt > s > active₁ > U93^1₃
take > U61 > U62 > tt > length₁ > 0 > active₁ > U93^1₃
take > U81 > nil > mark₁ > s > active₁ > U93^1₃
take > U81 > nil > mark₁ > length₁ > 0 > active₁ > U93^1₃
take > U81 > nil > tt > s > active₁ > U93^1₃
take > U81 > nil > tt > length₁ > 0 > active₁ > U93^1₃
take > U91 > U92 > U93 > cons > U71 > U72 > mark₁ > s > active₁ > U93^1₃
take > U91 > U92 > U93 > cons > U71 > U72 > mark₁ > length₁ > 0 > active₁ > U93^1₃

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(20) Obligation:

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

U93¹(active(X1), X2, X3, X4) → U93¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U93¹(active(X1), X2, X3, X4) → U93¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U93¹(x1, x2, x3, x4)  =  U93¹(x1, x2, x3, x4)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71(x1)
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81(x1)
nil  =  nil
U91(x1, x2, x3, x4)  =  U91(x1)
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > U71₁
zeros > mark₁ > U81₁ > active₁
zeros > mark₁ > U91₁ > active₁
zeros > cons > U71₁
zeros > cons > U91₁ > active₁
isNat > U11 > tt > U72 > s > mark₁ > U71₁
isNat > U11 > tt > U72 > s > mark₁ > U81₁ > active₁
isNat > U11 > tt > U72 > s > mark₁ > U91₁ > active₁
isNat > U11 > tt > nil > 0 > mark₁ > U71₁
isNat > U11 > tt > nil > 0 > mark₁ > U81₁ > active₁
isNat > U11 > tt > nil > 0 > mark₁ > U91₁ > active₁
isNat > U11 > tt > U92 > U93 > mark₁ > U71₁
isNat > U11 > tt > U92 > U93 > mark₁ > U81₁ > active₁
isNat > U11 > tt > U92 > U93 > mark₁ > U91₁ > active₁
isNat > U11 > tt > U92 > U93 > cons > U71₁
isNat > U11 > tt > U92 > U93 > cons > U91₁ > active₁
isNat > U21 > tt > U72 > s > mark₁ > U71₁
isNat > U21 > tt > U72 > s > mark₁ > U81₁ > active₁
isNat > U21 > tt > U72 > s > mark₁ > U91₁ > active₁
isNat > U21 > tt > nil > 0 > mark₁ > U71₁
isNat > U21 > tt > nil > 0 > mark₁ > U81₁ > active₁
isNat > U21 > tt > nil > 0 > mark₁ > U91₁ > active₁
isNat > U21 > tt > U92 > U93 > mark₁ > U71₁
isNat > U21 > tt > U92 > U93 > mark₁ > U81₁ > active₁
isNat > U21 > tt > U92 > U93 > mark₁ > U91₁ > active₁
isNat > U21 > tt > U92 > U93 > cons > U71₁
isNat > U21 > tt > U92 > U93 > cons > U91₁ > active₁
isNat > isNatList > U51 > U52 > tt > U72 > s > mark₁ > U71₁
isNat > isNatList > U51 > U52 > tt > U72 > s > mark₁ > U81₁ > active₁
isNat > isNatList > U51 > U52 > tt > U72 > s > mark₁ > U91₁ > active₁
isNat > isNatList > U51 > U52 > tt > nil > 0 > mark₁ > U71₁
isNat > isNatList > U51 > U52 > tt > nil > 0 > mark₁ > U81₁ > active₁
isNat > isNatList > U51 > U52 > tt > nil > 0 > mark₁ > U91₁ > active₁
isNat > isNatList > U51 > U52 > tt > U92 > U93 > mark₁ > U71₁
isNat > isNatList > U51 > U52 > tt > U92 > U93 > mark₁ > U81₁ > active₁
isNat > isNatList > U51 > U52 > tt > U92 > U93 > mark₁ > U91₁ > active₁
isNat > isNatList > U51 > U52 > tt > U92 > U93 > cons > U71₁
isNat > isNatList > U51 > U52 > tt > U92 > U93 > cons > U91₁ > active₁
isNat > isNatList > U61 > U62 > tt > U72 > s > mark₁ > U71₁
isNat > isNatList > U61 > U62 > tt > U72 > s > mark₁ > U81₁ > active₁
isNat > isNatList > U61 > U62 > tt > U72 > s > mark₁ > U91₁ > active₁
isNat > isNatList > U61 > U62 > tt > nil > 0 > mark₁ > U71₁
isNat > isNatList > U61 > U62 > tt > nil > 0 > mark₁ > U81₁ > active₁
isNat > isNatList > U61 > U62 > tt > nil > 0 > mark₁ > U91₁ > active₁
isNat > isNatList > U61 > U62 > tt > U92 > U93 > mark₁ > U71₁
isNat > isNatList > U61 > U62 > tt > U92 > U93 > mark₁ > U81₁ > active₁
isNat > isNatList > U61 > U62 > tt > U92 > U93 > mark₁ > U91₁ > active₁
isNat > isNatList > U61 > U62 > tt > U92 > U93 > cons > U71₁
isNat > isNatList > U61 > U62 > tt > U92 > U93 > cons > U91₁ > active₁
length > U11 > tt > U72 > s > mark₁ > U71₁
length > U11 > tt > U72 > s > mark₁ > U81₁ > active₁
length > U11 > tt > U72 > s > mark₁ > U91₁ > active₁
length > U11 > tt > nil > 0 > mark₁ > U71₁
length > U11 > tt > nil > 0 > mark₁ > U81₁ > active₁
length > U11 > tt > nil > 0 > mark₁ > U91₁ > active₁
length > U11 > tt > U92 > U93 > mark₁ > U71₁
length > U11 > tt > U92 > U93 > mark₁ > U81₁ > active₁
length > U11 > tt > U92 > U93 > mark₁ > U91₁ > active₁
length > U11 > tt > U92 > U93 > cons > U71₁
length > U11 > tt > U92 > U93 > cons > U91₁ > active₁
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > U71₁
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > U81₁ > active₁
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > U91₁ > active₁
length > isNatList > U51 > U52 > tt > nil > 0 > mark₁ > U71₁
length > isNatList > U51 > U52 > tt > nil > 0 > mark₁ > U81₁ > active₁
length > isNatList > U51 > U52 > tt > nil > 0 > mark₁ > U91₁ > active₁
length > isNatList > U51 > U52 > tt > U92 > U93 > mark₁ > U71₁
length > isNatList > U51 > U52 > tt > U92 > U93 > mark₁ > U81₁ > active₁
length > isNatList > U51 > U52 > tt > U92 > U93 > mark₁ > U91₁ > active₁
length > isNatList > U51 > U52 > tt > U92 > U93 > cons > U71₁
length > isNatList > U51 > U52 > tt > U92 > U93 > cons > U91₁ > active₁
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > U71₁
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > U81₁ > active₁
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > U91₁ > active₁
length > isNatList > U61 > U62 > tt > nil > 0 > mark₁ > U71₁
length > isNatList > U61 > U62 > tt > nil > 0 > mark₁ > U81₁ > active₁
length > isNatList > U61 > U62 > tt > nil > 0 > mark₁ > U91₁ > active₁
length > isNatList > U61 > U62 > tt > U92 > U93 > mark₁ > U71₁
length > isNatList > U61 > U62 > tt > U92 > U93 > mark₁ > U81₁ > active₁
length > isNatList > U61 > U62 > tt > U92 > U93 > mark₁ > U91₁ > active₁
length > isNatList > U61 > U62 > tt > U92 > U93 > cons > U71₁
length > isNatList > U61 > U62 > tt > U92 > U93 > cons > U91₁ > active₁
take > isNatIList > U31 > tt > U72 > s > mark₁ > U71₁
take > isNatIList > U31 > tt > U72 > s > mark₁ > U81₁ > active₁
take > isNatIList > U31 > tt > U72 > s > mark₁ > U91₁ > active₁
take > isNatIList > U31 > tt > nil > 0 > mark₁ > U71₁
take > isNatIList > U31 > tt > nil > 0 > mark₁ > U81₁ > active₁
take > isNatIList > U31 > tt > nil > 0 > mark₁ > U91₁ > active₁
take > isNatIList > U31 > tt > U92 > U93 > mark₁ > U71₁
take > isNatIList > U31 > tt > U92 > U93 > mark₁ > U81₁ > active₁
take > isNatIList > U31 > tt > U92 > U93 > mark₁ > U91₁ > active₁
take > isNatIList > U31 > tt > U92 > U93 > cons > U71₁
take > isNatIList > U31 > tt > U92 > U93 > cons > U91₁ > active₁
take > isNatIList > U41 > U42 > tt > U72 > s > mark₁ > U71₁
take > isNatIList > U41 > U42 > tt > U72 > s > mark₁ > U81₁ > active₁
take > isNatIList > U41 > U42 > tt > U72 > s > mark₁ > U91₁ > active₁
take > isNatIList > U41 > U42 > tt > nil > 0 > mark₁ > U71₁
take > isNatIList > U41 > U42 > tt > nil > 0 > mark₁ > U81₁ > active₁
take > isNatIList > U41 > U42 > tt > nil > 0 > mark₁ > U91₁ > active₁
take > isNatIList > U41 > U42 > tt > U92 > U93 > mark₁ > U71₁
take > isNatIList > U41 > U42 > tt > U92 > U93 > mark₁ > U81₁ > active₁
take > isNatIList > U41 > U42 > tt > U92 > U93 > mark₁ > U91₁ > active₁
take > isNatIList > U41 > U42 > tt > U92 > U93 > cons > U71₁
take > isNatIList > U41 > U42 > tt > U92 > U93 > cons > U91₁ > active₁
take > U61 > U62 > tt > U72 > s > mark₁ > U71₁
take > U61 > U62 > tt > U72 > s > mark₁ > U81₁ > active₁
take > U61 > U62 > tt > U72 > s > mark₁ > U91₁ > active₁
take > U61 > U62 > tt > nil > 0 > mark₁ > U71₁
take > U61 > U62 > tt > nil > 0 > mark₁ > U81₁ > active₁
take > U61 > U62 > tt > nil > 0 > mark₁ > U91₁ > active₁
take > U61 > U62 > tt > U92 > U93 > mark₁ > U71₁
take > U61 > U62 > tt > U92 > U93 > mark₁ > U81₁ > active₁
take > U61 > U62 > tt > U92 > U93 > mark₁ > U91₁ > active₁
take > U61 > U62 > tt > U92 > U93 > cons > U71₁
take > U61 > U62 > tt > U92 > U93 > cons > U91₁ > active₁

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(22) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(23) PisEmptyProof (EQUIVALENT transformation)

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

(24) TRUE

(25) Obligation:

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

U92¹(X1, mark(X2), X3, X4) → U92¹(X1, X2, X3, X4)
U92¹(mark(X1), X2, X3, X4) → U92¹(X1, X2, X3, X4)
U92¹(X1, X2, mark(X3), X4) → U92¹(X1, X2, X3, X4)
U92¹(X1, X2, X3, mark(X4)) → U92¹(X1, X2, X3, X4)
U92¹(active(X1), X2, X3, X4) → U92¹(X1, X2, X3, X4)
U92¹(X1, active(X2), X3, X4) → U92¹(X1, X2, X3, X4)
U92¹(X1, X2, active(X3), X4) → U92¹(X1, X2, X3, X4)
U92¹(X1, X2, X3, active(X4)) → U92¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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.

U92¹(X1, X2, mark(X3), X4) → U92¹(X1, X2, X3, X4)
U92¹(X1, X2, active(X3), X4) → U92¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U92¹(x1, x2, x3, x4)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > s > active₁ > nil
zeros > cons > active₁ > nil
zeros > 0 > active₁ > nil
zeros > tt > s > active₁ > nil
length > 0 > active₁ > nil
length > U11 > mark₁ > s > active₁ > nil
length > U11 > tt > s > active₁ > nil
length > isNatList > U51 > U52 > mark₁ > s > active₁ > nil
length > isNatList > U51 > U52 > tt > s > active₁ > nil
length > isNatList > U61 > U62 > mark₁ > s > active₁ > nil
length > isNatList > U61 > U62 > tt > s > active₁ > nil
length > U71 > U72 > mark₁ > s > active₁ > nil
take > isNatIList > U31 > mark₁ > s > active₁ > nil
take > isNatIList > U31 > tt > s > active₁ > nil
take > isNatIList > U41 > U42 > mark₁ > s > active₁ > nil
take > isNatIList > U41 > U42 > tt > s > active₁ > nil
take > U61 > U62 > mark₁ > s > active₁ > nil
take > U61 > U62 > tt > s > active₁ > nil
take > isNat > U11 > mark₁ > s > active₁ > nil
take > isNat > U11 > tt > s > active₁ > nil
take > isNat > U21 > mark₁ > s > active₁ > nil
take > isNat > U21 > tt > s > active₁ > nil
take > U81 > mark₁ > s > active₁ > nil
take > U91 > U92 > U93 > mark₁ > s > active₁ > nil
take > U91 > U92 > U93 > cons > active₁ > nil

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(27) Obligation:

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

U92¹(X1, mark(X2), X3, X4) → U92¹(X1, X2, X3, X4)
U92¹(mark(X1), X2, X3, X4) → U92¹(X1, X2, X3, X4)
U92¹(X1, X2, X3, mark(X4)) → U92¹(X1, X2, X3, X4)
U92¹(active(X1), X2, X3, X4) → U92¹(X1, X2, X3, X4)
U92¹(X1, active(X2), X3, X4) → U92¹(X1, X2, X3, X4)
U92¹(X1, X2, X3, active(X4)) → U92¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(28) 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, X4) → U92¹(X1, X2, X3, X4)
U92¹(mark(X1), X2, X3, X4) → U92¹(X1, X2, X3, X4)
U92¹(active(X1), X2, X3, X4) → U92¹(X1, X2, X3, X4)
U92¹(X1, active(X2), X3, X4) → U92¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U92¹(x1, x2, x3, x4)  =  U92¹(x1, x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  x1
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > U92^1₃
zeros > mark₁ > active₁ > tt
zeros > mark₁ > cons
length > 0 > isNatIList₁ > U31 > mark₁ > U92^1₃
length > 0 > isNatIList₁ > U31 > mark₁ > active₁ > tt
length > 0 > isNatIList₁ > U31 > mark₁ > cons
length > 0 > isNatIList₁ > U41 > U42 > mark₁ > U92^1₃
length > 0 > isNatIList₁ > U41 > U42 > mark₁ > active₁ > tt
length > 0 > isNatIList₁ > U41 > U42 > mark₁ > cons
length > 0 > isNatIList₁ > isNatList > U51 > U52 > mark₁ > U92^1₃
length > 0 > isNatIList₁ > isNatList > U51 > U52 > mark₁ > active₁ > tt
length > 0 > isNatIList₁ > isNatList > U51 > U52 > mark₁ > cons
length > 0 > isNatIList₁ > isNatList > U61 > U62 > mark₁ > U92^1₃
length > 0 > isNatIList₁ > isNatList > U61 > U62 > mark₁ > active₁ > tt
length > 0 > isNatIList₁ > isNatList > U61 > U62 > mark₁ > cons
length > 0 > U81 > nil > mark₁ > U92^1₃
length > 0 > U81 > nil > mark₁ > active₁ > tt
length > 0 > U81 > nil > mark₁ > cons
length > U11 > mark₁ > U92^1₃
length > U11 > mark₁ > active₁ > tt
length > U11 > mark₁ > cons
length > U71 > U72 > s > U21 > mark₁ > U92^1₃
length > U71 > U72 > s > U21 > mark₁ > active₁ > tt
length > U71 > U72 > s > U21 > mark₁ > cons
take > U81 > nil > mark₁ > U92^1₃
take > U81 > nil > mark₁ > active₁ > tt
take > U81 > nil > mark₁ > cons
take > U91 > U92 > U93 > mark₁ > U92^1₃
take > U91 > U92 > U93 > mark₁ > active₁ > tt
take > U91 > U92 > U93 > mark₁ > cons

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(29) Obligation:

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

U92¹(X1, X2, X3, mark(X4)) → U92¹(X1, X2, X3, X4)
U92¹(X1, X2, X3, active(X4)) → U92¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(30) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U92¹(X1, X2, X3, mark(X4)) → U92¹(X1, X2, X3, X4)
U92¹(X1, X2, X3, active(X4)) → U92¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U92¹(x1, x2, x3, x4)  =  x4
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList(x1)
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat(x1)
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > 0 > mark₁ > active₁
zeros > tt > U93 > mark₁ > active₁
zeros > tt > U93 > cons > active₁
isNatIList > U31 > tt > U93 > mark₁ > active₁
isNatIList > U31 > tt > U93 > cons > active₁
isNatIList > U41 > U42 > tt > U93 > mark₁ > active₁
isNatIList > U41 > U42 > tt > U93 > cons > active₁
isNatIList > isNatList₁ > U51 > U52 > tt > U93 > mark₁ > active₁
isNatIList > isNatList₁ > U51 > U52 > tt > U93 > cons > active₁
isNatIList > isNatList₁ > U61 > U62 > tt > U93 > mark₁ > active₁
isNatIList > isNatList₁ > U61 > U62 > tt > U93 > cons > active₁
isNat₁ > U11 > tt > U93 > mark₁ > active₁
isNat₁ > U11 > tt > U93 > cons > active₁
isNat₁ > U21 > tt > U93 > mark₁ > active₁
isNat₁ > U21 > tt > U93 > cons > active₁
isNat₁ > isNatList₁ > U51 > U52 > tt > U93 > mark₁ > active₁
isNat₁ > isNatList₁ > U51 > U52 > tt > U93 > cons > active₁
isNat₁ > isNatList₁ > U61 > U62 > tt > U93 > mark₁ > active₁
isNat₁ > isNatList₁ > U61 > U62 > tt > U93 > cons > active₁
length > 0 > mark₁ > active₁
length > U11 > tt > U93 > mark₁ > active₁
length > U11 > tt > U93 > cons > active₁
length > isNatList₁ > U51 > U52 > tt > U93 > mark₁ > active₁
length > isNatList₁ > U51 > U52 > tt > U93 > cons > active₁
length > isNatList₁ > U61 > U62 > tt > U93 > mark₁ > active₁
length > isNatList₁ > U61 > U62 > tt > U93 > cons > active₁
length > U71 > U72 > s > U91 > U92 > U93 > mark₁ > active₁
length > U71 > U72 > s > U91 > U92 > U93 > cons > active₁
take > U61 > U62 > tt > U93 > mark₁ > active₁
take > U61 > U62 > tt > U93 > cons > active₁
take > U81 > nil > 0 > mark₁ > active₁
take > U81 > nil > tt > U93 > mark₁ > active₁
take > U81 > nil > tt > U93 > cons > active₁
take > U91 > U92 > U93 > mark₁ > active₁
take > U91 > U92 > U93 > cons > active₁

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(31) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(32) PisEmptyProof (EQUIVALENT transformation)

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

(33) TRUE

(34) Obligation:

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

U91¹(X1, mark(X2), X3, X4) → U91¹(X1, X2, X3, X4)
U91¹(mark(X1), X2, X3, X4) → U91¹(X1, X2, X3, X4)
U91¹(X1, X2, mark(X3), X4) → U91¹(X1, X2, X3, X4)
U91¹(X1, X2, X3, mark(X4)) → U91¹(X1, X2, X3, X4)
U91¹(active(X1), X2, X3, X4) → U91¹(X1, X2, X3, X4)
U91¹(X1, active(X2), X3, X4) → U91¹(X1, X2, X3, X4)
U91¹(X1, X2, active(X3), X4) → U91¹(X1, X2, X3, X4)
U91¹(X1, X2, X3, active(X4)) → U91¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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.

U91¹(X1, X2, X3, mark(X4)) → U91¹(X1, X2, X3, X4)
U91¹(X1, X2, X3, active(X4)) → U91¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U91¹(x1, x2, x3, x4)  =  x4
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList(x1)
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81(x1)
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93(x1)
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > active₁ > s
zeros > mark₁ > active₁ > U81₁ > nil
zeros > mark₁ > active₁ > U93₁ > cons
zeros > 0 > active₁ > s
zeros > 0 > active₁ > U81₁ > nil
zeros > 0 > active₁ > U93₁ > cons
isNatList₁ > U51 > U52 > tt > mark₁ > active₁ > s
isNatList₁ > U51 > U52 > tt > mark₁ > active₁ > U81₁ > nil
isNatList₁ > U51 > U52 > tt > mark₁ > active₁ > U93₁ > cons
isNatList₁ > U61 > isNatIList > U31 > tt > mark₁ > active₁ > s
isNatList₁ > U61 > isNatIList > U31 > tt > mark₁ > active₁ > U81₁ > nil
isNatList₁ > U61 > isNatIList > U31 > tt > mark₁ > active₁ > U93₁ > cons
isNatList₁ > U61 > isNatIList > U41 > U42 > tt > mark₁ > active₁ > s
isNatList₁ > U61 > isNatIList > U41 > U42 > tt > mark₁ > active₁ > U81₁ > nil
isNatList₁ > U61 > isNatIList > U41 > U42 > tt > mark₁ > active₁ > U93₁ > cons
isNatList₁ > U61 > isNatIList > isNat > U11 > tt > mark₁ > active₁ > s
isNatList₁ > U61 > isNatIList > isNat > U11 > tt > mark₁ > active₁ > U81₁ > nil
isNatList₁ > U61 > isNatIList > isNat > U11 > tt > mark₁ > active₁ > U93₁ > cons
isNatList₁ > U61 > isNatIList > isNat > U21 > tt > mark₁ > active₁ > s
isNatList₁ > U61 > isNatIList > isNat > U21 > tt > mark₁ > active₁ > U81₁ > nil
isNatList₁ > U61 > isNatIList > isNat > U21 > tt > mark₁ > active₁ > U93₁ > cons
isNatList₁ > U61 > U62 > tt > mark₁ > active₁ > s
isNatList₁ > U61 > U62 > tt > mark₁ > active₁ > U81₁ > nil
isNatList₁ > U61 > U62 > tt > mark₁ > active₁ > U93₁ > cons
length > 0 > active₁ > s
length > 0 > active₁ > U81₁ > nil
length > 0 > active₁ > U93₁ > cons
length > U71 > U72 > mark₁ > active₁ > s
length > U71 > U72 > mark₁ > active₁ > U81₁ > nil
length > U71 > U72 > mark₁ > active₁ > U93₁ > cons
length > U71 > isNat > U11 > tt > mark₁ > active₁ > s
length > U71 > isNat > U11 > tt > mark₁ > active₁ > U81₁ > nil
length > U71 > isNat > U11 > tt > mark₁ > active₁ > U93₁ > cons
length > U71 > isNat > U21 > tt > mark₁ > active₁ > s
length > U71 > isNat > U21 > tt > mark₁ > active₁ > U81₁ > nil
length > U71 > isNat > U21 > tt > mark₁ > active₁ > U93₁ > cons
take > isNatIList > U31 > tt > mark₁ > active₁ > s
take > isNatIList > U31 > tt > mark₁ > active₁ > U81₁ > nil
take > isNatIList > U31 > tt > mark₁ > active₁ > U93₁ > cons
take > isNatIList > U41 > U42 > tt > mark₁ > active₁ > s
take > isNatIList > U41 > U42 > tt > mark₁ > active₁ > U81₁ > nil
take > isNatIList > U41 > U42 > tt > mark₁ > active₁ > U93₁ > cons
take > isNatIList > isNat > U11 > tt > mark₁ > active₁ > s
take > isNatIList > isNat > U11 > tt > mark₁ > active₁ > U81₁ > nil
take > isNatIList > isNat > U11 > tt > mark₁ > active₁ > U93₁ > cons
take > isNatIList > isNat > U21 > tt > mark₁ > active₁ > s
take > isNatIList > isNat > U21 > tt > mark₁ > active₁ > U81₁ > nil
take > isNatIList > isNat > U21 > tt > mark₁ > active₁ > U93₁ > cons
take > U91 > U92 > isNat > U11 > tt > mark₁ > active₁ > s
take > U91 > U92 > isNat > U11 > tt > mark₁ > active₁ > U81₁ > nil
take > U91 > U92 > isNat > U11 > tt > mark₁ > active₁ > U93₁ > cons
take > U91 > U92 > isNat > U21 > tt > mark₁ > active₁ > s
take > U91 > U92 > isNat > U21 > tt > mark₁ > active₁ > U81₁ > nil
take > U91 > U92 > isNat > U21 > tt > mark₁ > active₁ > U93₁ > cons

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(36) Obligation:

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

U91¹(X1, mark(X2), X3, X4) → U91¹(X1, X2, X3, X4)
U91¹(mark(X1), X2, X3, X4) → U91¹(X1, X2, X3, X4)
U91¹(X1, X2, mark(X3), X4) → U91¹(X1, X2, X3, X4)
U91¹(active(X1), X2, X3, X4) → U91¹(X1, X2, X3, X4)
U91¹(X1, active(X2), X3, X4) → U91¹(X1, X2, X3, X4)
U91¹(X1, X2, active(X3), X4) → U91¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(37) 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, X4) → U91¹(X1, X2, X3, X4)
U91¹(mark(X1), X2, X3, X4) → U91¹(X1, X2, X3, X4)
U91¹(X1, X2, mark(X3), X4) → U91¹(X1, X2, X3, X4)
U91¹(active(X1), X2, X3, X4) → U91¹(X1, X2, X3, X4)
U91¹(X1, active(X2), X3, X4) → U91¹(X1, X2, X3, X4)
U91¹(X1, X2, active(X3), X4) → U91¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U91¹(x1, x2, x3, x4)  =  U91¹(x1, x2, x3, x4)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71(x1)
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U51 > U52 > tt > U72 > mark₁ > active₁ > U91^1₄ > U71₁
zeros > cons > U51 > U52 > tt > U72 > mark₁ > s > U71₁
zeros > cons > isNat > U11 > tt > U72 > mark₁ > active₁ > U91^1₄ > U71₁
zeros > cons > isNat > U11 > tt > U72 > mark₁ > s > U71₁
zeros > cons > isNat > U21 > tt > U72 > mark₁ > active₁ > U91^1₄ > U71₁
zeros > cons > isNat > U21 > tt > U72 > mark₁ > s > U71₁
zeros > 0 > mark₁ > active₁ > U91^1₄ > U71₁
zeros > 0 > mark₁ > s > U71₁
isNatIList > U31 > tt > U72 > mark₁ > active₁ > U91^1₄ > U71₁
isNatIList > U31 > tt > U72 > mark₁ > s > U71₁
isNatIList > U41 > U42 > tt > U72 > mark₁ > active₁ > U91^1₄ > U71₁
isNatIList > U41 > U42 > tt > U72 > mark₁ > s > U71₁
isNatIList > isNatList > U51 > U52 > tt > U72 > mark₁ > active₁ > U91^1₄ > U71₁
isNatIList > isNatList > U51 > U52 > tt > U72 > mark₁ > s > U71₁
isNatIList > isNatList > U61 > U62 > tt > U72 > mark₁ > active₁ > U91^1₄ > U71₁
isNatIList > isNatList > U61 > U62 > tt > U72 > mark₁ > s > U71₁
isNatIList > isNatList > isNat > U11 > tt > U72 > mark₁ > active₁ > U91^1₄ > U71₁
isNatIList > isNatList > isNat > U11 > tt > U72 > mark₁ > s > U71₁
isNatIList > isNatList > isNat > U21 > tt > U72 > mark₁ > active₁ > U91^1₄ > U71₁
isNatIList > isNatList > isNat > U21 > tt > U72 > mark₁ > s > U71₁
length > 0 > mark₁ > active₁ > U91^1₄ > U71₁
length > 0 > mark₁ > s > U71₁
length > isNatList > U51 > U52 > tt > U72 > mark₁ > active₁ > U91^1₄ > U71₁
length > isNatList > U51 > U52 > tt > U72 > mark₁ > s > U71₁
length > isNatList > U61 > U62 > tt > U72 > mark₁ > active₁ > U91^1₄ > U71₁
length > isNatList > U61 > U62 > tt > U72 > mark₁ > s > U71₁
length > isNatList > isNat > U11 > tt > U72 > mark₁ > active₁ > U91^1₄ > U71₁
length > isNatList > isNat > U11 > tt > U72 > mark₁ > s > U71₁
length > isNatList > isNat > U21 > tt > U72 > mark₁ > active₁ > U91^1₄ > U71₁
length > isNatList > isNat > U21 > tt > U72 > mark₁ > s > U71₁
take > U61 > U62 > tt > U72 > mark₁ > active₁ > U91^1₄ > U71₁
take > U61 > U62 > tt > U72 > mark₁ > s > U71₁
take > U81 > nil > 0 > mark₁ > active₁ > U91^1₄ > U71₁
take > U81 > nil > 0 > mark₁ > s > U71₁
take > U81 > nil > tt > U72 > mark₁ > active₁ > U91^1₄ > U71₁
take > U81 > nil > tt > U72 > mark₁ > s > U71₁
take > U91 > U92 > U93 > cons > U51 > U52 > tt > U72 > mark₁ > active₁ > U91^1₄ > U71₁
take > U91 > U92 > U93 > cons > U51 > U52 > tt > U72 > mark₁ > s > U71₁
take > U91 > U92 > U93 > cons > isNat > U11 > tt > U72 > mark₁ > active₁ > U91^1₄ > U71₁
take > U91 > U92 > U93 > cons > isNat > U11 > tt > U72 > mark₁ > s > U71₁
take > U91 > U92 > U93 > cons > isNat > U21 > tt > U72 > mark₁ > active₁ > U91^1₄ > U71₁
take > U91 > U92 > U93 > cons > isNat > U21 > tt > U72 > mark₁ > s > U71₁

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(38) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(39) PisEmptyProof (EQUIVALENT transformation)

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

(40) TRUE

(41) 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(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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.

U81¹(mark(X)) → U81¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U81¹(x1)  =  U81¹(x1)
active(x1)  =  x1
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > cons > U81^1₁
zeros > mark₁ > tt > U81^1₁
isNatList > U51 > U52 > mark₁ > cons > U81^1₁
isNatList > U51 > U52 > mark₁ > tt > U81^1₁
isNatList > U61 > U62 > mark₁ > cons > U81^1₁
isNatList > U61 > U62 > mark₁ > tt > U81^1₁
isNat > U11 > mark₁ > cons > U81^1₁
isNat > U11 > mark₁ > tt > U81^1₁
isNat > U21 > mark₁ > cons > U81^1₁
isNat > U21 > mark₁ > tt > U81^1₁
length > 0 > isNatIList₁ > U31 > mark₁ > cons > U81^1₁
length > 0 > isNatIList₁ > U31 > mark₁ > tt > U81^1₁
length > 0 > isNatIList₁ > U41 > U42 > mark₁ > cons > U81^1₁
length > 0 > isNatIList₁ > U41 > U42 > mark₁ > tt > U81^1₁
length > U11 > mark₁ > cons > U81^1₁
length > U11 > mark₁ > tt > U81^1₁
length > U71 > U72 > s > mark₁ > cons > U81^1₁
length > U71 > U72 > s > mark₁ > tt > U81^1₁
take > isNatIList₁ > U31 > mark₁ > cons > U81^1₁
take > isNatIList₁ > U31 > mark₁ > tt > U81^1₁
take > isNatIList₁ > U41 > U42 > mark₁ > cons > U81^1₁
take > isNatIList₁ > U41 > U42 > mark₁ > tt > U81^1₁
take > U61 > U62 > mark₁ > cons > U81^1₁
take > U61 > U62 > mark₁ > tt > U81^1₁
take > U81 > mark₁ > cons > U81^1₁
take > U81 > mark₁ > tt > U81^1₁
take > U81 > nil > tt > U81^1₁
take > U91 > U92 > U93 > mark₁ > cons > U81^1₁
take > U91 > U92 > U93 > mark₁ > tt > U81^1₁

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(43) Obligation:

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

U81¹(active(X)) → U81¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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.

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)  =  U81¹(x1)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > tt > cons > mark₁ > active₁
zeros > tt > cons > mark₁ > nil
zeros > tt > U72 > s > mark₁ > active₁
zeros > tt > U72 > s > mark₁ > nil
isNatIList > U31 > tt > cons > mark₁ > active₁
isNatIList > U31 > tt > cons > mark₁ > nil
isNatIList > U31 > tt > U72 > s > mark₁ > active₁
isNatIList > U31 > tt > U72 > s > mark₁ > nil
isNatIList > U41 > U42 > tt > cons > mark₁ > active₁
isNatIList > U41 > U42 > tt > cons > mark₁ > nil
isNatIList > U41 > U42 > tt > U72 > s > mark₁ > active₁
isNatIList > U41 > U42 > tt > U72 > s > mark₁ > nil
isNatIList > isNatList > U51 > U52 > tt > cons > mark₁ > active₁
isNatIList > isNatList > U51 > U52 > tt > cons > mark₁ > nil
isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > active₁
isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > nil
isNatIList > isNatList > U61 > U62 > tt > cons > mark₁ > active₁
isNatIList > isNatList > U61 > U62 > tt > cons > mark₁ > nil
isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > active₁
isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > nil
isNatIList > isNat > U11 > tt > cons > mark₁ > active₁
isNatIList > isNat > U11 > tt > cons > mark₁ > nil
isNatIList > isNat > U11 > tt > U72 > s > mark₁ > active₁
isNatIList > isNat > U11 > tt > U72 > s > mark₁ > nil
isNatIList > isNat > U21 > tt > cons > mark₁ > active₁
isNatIList > isNat > U21 > tt > cons > mark₁ > nil
isNatIList > isNat > U21 > tt > U72 > s > mark₁ > active₁
isNatIList > isNat > U21 > tt > U72 > s > mark₁ > nil
length > 0 > tt > cons > mark₁ > active₁
length > 0 > tt > cons > mark₁ > nil
length > 0 > tt > U72 > s > mark₁ > active₁
length > 0 > tt > U72 > s > mark₁ > nil
length > U11 > tt > cons > mark₁ > active₁
length > U11 > tt > cons > mark₁ > nil
length > U11 > tt > U72 > s > mark₁ > active₁
length > U11 > tt > U72 > s > mark₁ > nil
length > isNatList > U51 > U52 > tt > cons > mark₁ > active₁
length > isNatList > U51 > U52 > tt > cons > mark₁ > nil
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > active₁
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > nil
length > isNatList > U61 > U62 > tt > cons > mark₁ > active₁
length > isNatList > U61 > U62 > tt > cons > mark₁ > nil
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > active₁
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > nil
length > U71 > U72 > s > mark₁ > active₁
length > U71 > U72 > s > mark₁ > nil
take > U61 > U62 > tt > cons > mark₁ > active₁
take > U61 > U62 > tt > cons > mark₁ > nil
take > U61 > U62 > tt > U72 > s > mark₁ > active₁
take > U61 > U62 > tt > U72 > s > mark₁ > nil
take > isNat > U11 > tt > cons > mark₁ > active₁
take > isNat > U11 > tt > cons > mark₁ > nil
take > isNat > U11 > tt > U72 > s > mark₁ > active₁
take > isNat > U11 > tt > U72 > s > mark₁ > nil
take > isNat > U21 > tt > cons > mark₁ > active₁
take > isNat > U21 > tt > cons > mark₁ > nil
take > isNat > U21 > tt > U72 > s > mark₁ > active₁
take > isNat > U21 > tt > U72 > s > mark₁ > nil
take > U81 > mark₁ > active₁
take > U81 > mark₁ > nil
take > U91 > U92 > U93 > cons > mark₁ > active₁
take > U91 > U92 > U93 > cons > mark₁ > nil

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(45) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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:

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(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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.

LENGTH(mark(X)) → LENGTH(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
LENGTH(x1)  =  LENGTH(x1)
active(x1)  =  x1
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > cons > LENGTH₁
zeros > mark₁ > tt > LENGTH₁
isNatList > U51 > U52 > mark₁ > cons > LENGTH₁
isNatList > U51 > U52 > mark₁ > tt > LENGTH₁
isNatList > U61 > U62 > mark₁ > cons > LENGTH₁
isNatList > U61 > U62 > mark₁ > tt > LENGTH₁
isNat > U11 > mark₁ > cons > LENGTH₁
isNat > U11 > mark₁ > tt > LENGTH₁
isNat > U21 > mark₁ > cons > LENGTH₁
isNat > U21 > mark₁ > tt > LENGTH₁
length > 0 > isNatIList₁ > U31 > mark₁ > cons > LENGTH₁
length > 0 > isNatIList₁ > U31 > mark₁ > tt > LENGTH₁
length > 0 > isNatIList₁ > U41 > U42 > mark₁ > cons > LENGTH₁
length > 0 > isNatIList₁ > U41 > U42 > mark₁ > tt > LENGTH₁
length > U11 > mark₁ > cons > LENGTH₁
length > U11 > mark₁ > tt > LENGTH₁
length > U71 > U72 > s > mark₁ > cons > LENGTH₁
length > U71 > U72 > s > mark₁ > tt > LENGTH₁
take > isNatIList₁ > U31 > mark₁ > cons > LENGTH₁
take > isNatIList₁ > U31 > mark₁ > tt > LENGTH₁
take > isNatIList₁ > U41 > U42 > mark₁ > cons > LENGTH₁
take > isNatIList₁ > U41 > U42 > mark₁ > tt > LENGTH₁
take > U61 > U62 > mark₁ > cons > LENGTH₁
take > U61 > U62 > mark₁ > tt > LENGTH₁
take > U81 > mark₁ > cons > LENGTH₁
take > U81 > mark₁ > tt > LENGTH₁
take > U81 > nil > tt > LENGTH₁
take > U91 > U92 > U93 > mark₁ > cons > LENGTH₁
take > U91 > U92 > U93 > mark₁ > tt > LENGTH₁

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(50) Obligation:

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

LENGTH(active(X)) → LENGTH(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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.

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)  =  LENGTH(x1)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > tt > cons > mark₁ > active₁
zeros > tt > cons > mark₁ > nil
zeros > tt > U72 > s > mark₁ > active₁
zeros > tt > U72 > s > mark₁ > nil
isNatIList > U31 > tt > cons > mark₁ > active₁
isNatIList > U31 > tt > cons > mark₁ > nil
isNatIList > U31 > tt > U72 > s > mark₁ > active₁
isNatIList > U31 > tt > U72 > s > mark₁ > nil
isNatIList > U41 > U42 > tt > cons > mark₁ > active₁
isNatIList > U41 > U42 > tt > cons > mark₁ > nil
isNatIList > U41 > U42 > tt > U72 > s > mark₁ > active₁
isNatIList > U41 > U42 > tt > U72 > s > mark₁ > nil
isNatIList > isNatList > U51 > U52 > tt > cons > mark₁ > active₁
isNatIList > isNatList > U51 > U52 > tt > cons > mark₁ > nil
isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > active₁
isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > nil
isNatIList > isNatList > U61 > U62 > tt > cons > mark₁ > active₁
isNatIList > isNatList > U61 > U62 > tt > cons > mark₁ > nil
isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > active₁
isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > nil
isNatIList > isNat > U11 > tt > cons > mark₁ > active₁
isNatIList > isNat > U11 > tt > cons > mark₁ > nil
isNatIList > isNat > U11 > tt > U72 > s > mark₁ > active₁
isNatIList > isNat > U11 > tt > U72 > s > mark₁ > nil
isNatIList > isNat > U21 > tt > cons > mark₁ > active₁
isNatIList > isNat > U21 > tt > cons > mark₁ > nil
isNatIList > isNat > U21 > tt > U72 > s > mark₁ > active₁
isNatIList > isNat > U21 > tt > U72 > s > mark₁ > nil
length > 0 > tt > cons > mark₁ > active₁
length > 0 > tt > cons > mark₁ > nil
length > 0 > tt > U72 > s > mark₁ > active₁
length > 0 > tt > U72 > s > mark₁ > nil
length > U11 > tt > cons > mark₁ > active₁
length > U11 > tt > cons > mark₁ > nil
length > U11 > tt > U72 > s > mark₁ > active₁
length > U11 > tt > U72 > s > mark₁ > nil
length > isNatList > U51 > U52 > tt > cons > mark₁ > active₁
length > isNatList > U51 > U52 > tt > cons > mark₁ > nil
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > active₁
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > nil
length > isNatList > U61 > U62 > tt > cons > mark₁ > active₁
length > isNatList > U61 > U62 > tt > cons > mark₁ > nil
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > active₁
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > nil
length > U71 > U72 > s > mark₁ > active₁
length > U71 > U72 > s > mark₁ > nil
take > U61 > U62 > tt > cons > mark₁ > active₁
take > U61 > U62 > tt > cons > mark₁ > nil
take > U61 > U62 > tt > U72 > s > mark₁ > active₁
take > U61 > U62 > tt > U72 > s > mark₁ > nil
take > isNat > U11 > tt > cons > mark₁ > active₁
take > isNat > U11 > tt > cons > mark₁ > nil
take > isNat > U11 > tt > U72 > s > mark₁ > active₁
take > isNat > U11 > tt > U72 > s > mark₁ > nil
take > isNat > U21 > tt > cons > mark₁ > active₁
take > isNat > U21 > tt > cons > mark₁ > nil
take > isNat > U21 > tt > U72 > s > mark₁ > active₁
take > isNat > U21 > tt > U72 > s > mark₁ > nil
take > U81 > mark₁ > active₁
take > U81 > mark₁ > nil
take > U91 > U92 > U93 > cons > mark₁ > active₁
take > U91 > U92 > U93 > cons > mark₁ > nil

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(52) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(53) PisEmptyProof (EQUIVALENT transformation)

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

(54) TRUE

(55) 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(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(56) 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: Combined order from the following AFS and order.
S(x1)  =  S(x1)
active(x1)  =  x1
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > cons > S₁
zeros > mark₁ > tt > S₁
isNatList > U51 > U52 > mark₁ > cons > S₁
isNatList > U51 > U52 > mark₁ > tt > S₁
isNatList > U61 > U62 > mark₁ > cons > S₁
isNatList > U61 > U62 > mark₁ > tt > S₁
isNat > U11 > mark₁ > cons > S₁
isNat > U11 > mark₁ > tt > S₁
isNat > U21 > mark₁ > cons > S₁
isNat > U21 > mark₁ > tt > S₁
length > 0 > isNatIList₁ > U31 > mark₁ > cons > S₁
length > 0 > isNatIList₁ > U31 > mark₁ > tt > S₁
length > 0 > isNatIList₁ > U41 > U42 > mark₁ > cons > S₁
length > 0 > isNatIList₁ > U41 > U42 > mark₁ > tt > S₁
length > U11 > mark₁ > cons > S₁
length > U11 > mark₁ > tt > S₁
length > U71 > U72 > s > mark₁ > cons > S₁
length > U71 > U72 > s > mark₁ > tt > S₁
take > isNatIList₁ > U31 > mark₁ > cons > S₁
take > isNatIList₁ > U31 > mark₁ > tt > S₁
take > isNatIList₁ > U41 > U42 > mark₁ > cons > S₁
take > isNatIList₁ > U41 > U42 > mark₁ > tt > S₁
take > U61 > U62 > mark₁ > cons > S₁
take > U61 > U62 > mark₁ > tt > S₁
take > U81 > mark₁ > cons > S₁
take > U81 > mark₁ > tt > S₁
take > U81 > nil > tt > S₁
take > U91 > U92 > U93 > mark₁ > cons > S₁
take > U91 > U92 > U93 > mark₁ > tt > S₁

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(57) Obligation:

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

S(active(X)) → S(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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.

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)  =  S(x1)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > tt > cons > mark₁ > active₁
zeros > tt > cons > mark₁ > nil
zeros > tt > U72 > s > mark₁ > active₁
zeros > tt > U72 > s > mark₁ > nil
isNatIList > U31 > tt > cons > mark₁ > active₁
isNatIList > U31 > tt > cons > mark₁ > nil
isNatIList > U31 > tt > U72 > s > mark₁ > active₁
isNatIList > U31 > tt > U72 > s > mark₁ > nil
isNatIList > U41 > U42 > tt > cons > mark₁ > active₁
isNatIList > U41 > U42 > tt > cons > mark₁ > nil
isNatIList > U41 > U42 > tt > U72 > s > mark₁ > active₁
isNatIList > U41 > U42 > tt > U72 > s > mark₁ > nil
isNatIList > isNatList > U51 > U52 > tt > cons > mark₁ > active₁
isNatIList > isNatList > U51 > U52 > tt > cons > mark₁ > nil
isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > active₁
isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > nil
isNatIList > isNatList > U61 > U62 > tt > cons > mark₁ > active₁
isNatIList > isNatList > U61 > U62 > tt > cons > mark₁ > nil
isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > active₁
isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > nil
isNatIList > isNat > U11 > tt > cons > mark₁ > active₁
isNatIList > isNat > U11 > tt > cons > mark₁ > nil
isNatIList > isNat > U11 > tt > U72 > s > mark₁ > active₁
isNatIList > isNat > U11 > tt > U72 > s > mark₁ > nil
isNatIList > isNat > U21 > tt > cons > mark₁ > active₁
isNatIList > isNat > U21 > tt > cons > mark₁ > nil
isNatIList > isNat > U21 > tt > U72 > s > mark₁ > active₁
isNatIList > isNat > U21 > tt > U72 > s > mark₁ > nil
length > 0 > tt > cons > mark₁ > active₁
length > 0 > tt > cons > mark₁ > nil
length > 0 > tt > U72 > s > mark₁ > active₁
length > 0 > tt > U72 > s > mark₁ > nil
length > U11 > tt > cons > mark₁ > active₁
length > U11 > tt > cons > mark₁ > nil
length > U11 > tt > U72 > s > mark₁ > active₁
length > U11 > tt > U72 > s > mark₁ > nil
length > isNatList > U51 > U52 > tt > cons > mark₁ > active₁
length > isNatList > U51 > U52 > tt > cons > mark₁ > nil
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > active₁
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > nil
length > isNatList > U61 > U62 > tt > cons > mark₁ > active₁
length > isNatList > U61 > U62 > tt > cons > mark₁ > nil
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > active₁
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > nil
length > U71 > U72 > s > mark₁ > active₁
length > U71 > U72 > s > mark₁ > nil
take > U61 > U62 > tt > cons > mark₁ > active₁
take > U61 > U62 > tt > cons > mark₁ > nil
take > U61 > U62 > tt > U72 > s > mark₁ > active₁
take > U61 > U62 > tt > U72 > s > mark₁ > nil
take > isNat > U11 > tt > cons > mark₁ > active₁
take > isNat > U11 > tt > cons > mark₁ > nil
take > isNat > U11 > tt > U72 > s > mark₁ > active₁
take > isNat > U11 > tt > U72 > s > mark₁ > nil
take > isNat > U21 > tt > cons > mark₁ > active₁
take > isNat > U21 > tt > cons > mark₁ > nil
take > isNat > U21 > tt > U72 > s > mark₁ > active₁
take > isNat > U21 > tt > U72 > s > mark₁ > nil
take > U81 > mark₁ > active₁
take > U81 > mark₁ > nil
take > U91 > U92 > U93 > cons > mark₁ > active₁
take > U91 > U92 > U93 > cons > mark₁ > nil

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(59) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(60) PisEmptyProof (EQUIVALENT transformation)

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

(61) TRUE

(62) 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(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(63) 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: Combined order from the following AFS and order.
ISNAT(x1)  =  ISNAT(x1)
active(x1)  =  x1
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > cons > ISNAT₁
zeros > mark₁ > tt > ISNAT₁
isNatList > U51 > U52 > mark₁ > cons > ISNAT₁
isNatList > U51 > U52 > mark₁ > tt > ISNAT₁
isNatList > U61 > U62 > mark₁ > cons > ISNAT₁
isNatList > U61 > U62 > mark₁ > tt > ISNAT₁
isNat > U11 > mark₁ > cons > ISNAT₁
isNat > U11 > mark₁ > tt > ISNAT₁
isNat > U21 > mark₁ > cons > ISNAT₁
isNat > U21 > mark₁ > tt > ISNAT₁
length > 0 > isNatIList₁ > U31 > mark₁ > cons > ISNAT₁
length > 0 > isNatIList₁ > U31 > mark₁ > tt > ISNAT₁
length > 0 > isNatIList₁ > U41 > U42 > mark₁ > cons > ISNAT₁
length > 0 > isNatIList₁ > U41 > U42 > mark₁ > tt > ISNAT₁
length > U11 > mark₁ > cons > ISNAT₁
length > U11 > mark₁ > tt > ISNAT₁
length > U71 > U72 > s > mark₁ > cons > ISNAT₁
length > U71 > U72 > s > mark₁ > tt > ISNAT₁
take > isNatIList₁ > U31 > mark₁ > cons > ISNAT₁
take > isNatIList₁ > U31 > mark₁ > tt > ISNAT₁
take > isNatIList₁ > U41 > U42 > mark₁ > cons > ISNAT₁
take > isNatIList₁ > U41 > U42 > mark₁ > tt > ISNAT₁
take > U61 > U62 > mark₁ > cons > ISNAT₁
take > U61 > U62 > mark₁ > tt > ISNAT₁
take > U81 > mark₁ > cons > ISNAT₁
take > U81 > mark₁ > tt > ISNAT₁
take > U81 > nil > tt > ISNAT₁
take > U91 > U92 > U93 > mark₁ > cons > ISNAT₁
take > U91 > U92 > U93 > mark₁ > tt > ISNAT₁

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(64) Obligation:

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

ISNAT(active(X)) → ISNAT(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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.

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)  =  ISNAT(x1)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > tt > cons > mark₁ > active₁
zeros > tt > cons > mark₁ > nil
zeros > tt > U72 > s > mark₁ > active₁
zeros > tt > U72 > s > mark₁ > nil
isNatIList > U31 > tt > cons > mark₁ > active₁
isNatIList > U31 > tt > cons > mark₁ > nil
isNatIList > U31 > tt > U72 > s > mark₁ > active₁
isNatIList > U31 > tt > U72 > s > mark₁ > nil
isNatIList > U41 > U42 > tt > cons > mark₁ > active₁
isNatIList > U41 > U42 > tt > cons > mark₁ > nil
isNatIList > U41 > U42 > tt > U72 > s > mark₁ > active₁
isNatIList > U41 > U42 > tt > U72 > s > mark₁ > nil
isNatIList > isNatList > U51 > U52 > tt > cons > mark₁ > active₁
isNatIList > isNatList > U51 > U52 > tt > cons > mark₁ > nil
isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > active₁
isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > nil
isNatIList > isNatList > U61 > U62 > tt > cons > mark₁ > active₁
isNatIList > isNatList > U61 > U62 > tt > cons > mark₁ > nil
isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > active₁
isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > nil
isNatIList > isNat > U11 > tt > cons > mark₁ > active₁
isNatIList > isNat > U11 > tt > cons > mark₁ > nil
isNatIList > isNat > U11 > tt > U72 > s > mark₁ > active₁
isNatIList > isNat > U11 > tt > U72 > s > mark₁ > nil
isNatIList > isNat > U21 > tt > cons > mark₁ > active₁
isNatIList > isNat > U21 > tt > cons > mark₁ > nil
isNatIList > isNat > U21 > tt > U72 > s > mark₁ > active₁
isNatIList > isNat > U21 > tt > U72 > s > mark₁ > nil
length > 0 > tt > cons > mark₁ > active₁
length > 0 > tt > cons > mark₁ > nil
length > 0 > tt > U72 > s > mark₁ > active₁
length > 0 > tt > U72 > s > mark₁ > nil
length > U11 > tt > cons > mark₁ > active₁
length > U11 > tt > cons > mark₁ > nil
length > U11 > tt > U72 > s > mark₁ > active₁
length > U11 > tt > U72 > s > mark₁ > nil
length > isNatList > U51 > U52 > tt > cons > mark₁ > active₁
length > isNatList > U51 > U52 > tt > cons > mark₁ > nil
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > active₁
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > nil
length > isNatList > U61 > U62 > tt > cons > mark₁ > active₁
length > isNatList > U61 > U62 > tt > cons > mark₁ > nil
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > active₁
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > nil
length > U71 > U72 > s > mark₁ > active₁
length > U71 > U72 > s > mark₁ > nil
take > U61 > U62 > tt > cons > mark₁ > active₁
take > U61 > U62 > tt > cons > mark₁ > nil
take > U61 > U62 > tt > U72 > s > mark₁ > active₁
take > U61 > U62 > tt > U72 > s > mark₁ > nil
take > isNat > U11 > tt > cons > mark₁ > active₁
take > isNat > U11 > tt > cons > mark₁ > nil
take > isNat > U11 > tt > U72 > s > mark₁ > active₁
take > isNat > U11 > tt > U72 > s > mark₁ > nil
take > isNat > U21 > tt > cons > mark₁ > active₁
take > isNat > U21 > tt > cons > mark₁ > nil
take > isNat > U21 > tt > U72 > s > mark₁ > active₁
take > isNat > U21 > tt > U72 > s > mark₁ > nil
take > U81 > mark₁ > active₁
take > U81 > mark₁ > nil
take > U91 > U92 > U93 > cons > mark₁ > active₁
take > U91 > U92 > U93 > cons > mark₁ > nil

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(66) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(67) PisEmptyProof (EQUIVALENT transformation)

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

(68) TRUE

(69) Obligation:

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

U72¹(X1, mark(X2)) → U72¹(X1, X2)
U72¹(mark(X1), X2) → U72¹(X1, X2)
U72¹(active(X1), X2) → U72¹(X1, X2)
U72¹(X1, active(X2)) → U72¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(70) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U72¹(X1, mark(X2)) → U72¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U72¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat(x1)
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁
zeros > cons
zeros > 0
zeros > tt
isNatIList > U31 > mark₁
isNatIList > U31 > tt
isNatIList > U41 > U42 > mark₁
isNatIList > U41 > U42 > tt
isNatList > U51 > U52 > mark₁
isNatList > U51 > U52 > tt
isNatList > U61 > U62 > mark₁
isNatList > U61 > U62 > tt
isNat₁ > U11 > mark₁
isNat₁ > U11 > tt
isNat₁ > U21 > mark₁
isNat₁ > U21 > tt
length > 0
length > U11 > mark₁
length > U11 > tt
length > U71 > U72 > s > mark₁
take > U61 > U62 > mark₁
take > U61 > U62 > tt
take > U81 > mark₁
take > U81 > nil
take > U91 > U92 > U93 > mark₁
take > U91 > U92 > U93 > cons

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(71) Obligation:

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

U72¹(mark(X1), X2) → U72¹(X1, X2)
U72¹(active(X1), X2) → U72¹(X1, X2)
U72¹(X1, active(X2)) → U72¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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.

U72¹(X1, active(X2)) → U72¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U72¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72(x1)
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92(x1)
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > 0 > tt > s > mark₁ > active₁ > U72₁
zeros > 0 > tt > s > mark₁ > cons > U72₁
zeros > 0 > tt > s > mark₁ > nil > U72₁
zeros > 0 > tt > s > mark₁ > U92₁ > U72₁
zeros > 0 > tt > U93 > mark₁ > active₁ > U72₁
zeros > 0 > tt > U93 > mark₁ > cons > U72₁
zeros > 0 > tt > U93 > mark₁ > nil > U72₁
zeros > 0 > tt > U93 > mark₁ > U92₁ > U72₁
isNatIList > U31 > tt > s > mark₁ > active₁ > U72₁
isNatIList > U31 > tt > s > mark₁ > cons > U72₁
isNatIList > U31 > tt > s > mark₁ > nil > U72₁
isNatIList > U31 > tt > s > mark₁ > U92₁ > U72₁
isNatIList > U31 > tt > U93 > mark₁ > active₁ > U72₁
isNatIList > U31 > tt > U93 > mark₁ > cons > U72₁
isNatIList > U31 > tt > U93 > mark₁ > nil > U72₁
isNatIList > U31 > tt > U93 > mark₁ > U92₁ > U72₁
isNatIList > U41 > U42 > tt > s > mark₁ > active₁ > U72₁
isNatIList > U41 > U42 > tt > s > mark₁ > cons > U72₁
isNatIList > U41 > U42 > tt > s > mark₁ > nil > U72₁
isNatIList > U41 > U42 > tt > s > mark₁ > U92₁ > U72₁
isNatIList > U41 > U42 > tt > U93 > mark₁ > active₁ > U72₁
isNatIList > U41 > U42 > tt > U93 > mark₁ > cons > U72₁
isNatIList > U41 > U42 > tt > U93 > mark₁ > nil > U72₁
isNatIList > U41 > U42 > tt > U93 > mark₁ > U92₁ > U72₁
isNatIList > isNat > U11 > tt > s > mark₁ > active₁ > U72₁
isNatIList > isNat > U11 > tt > s > mark₁ > cons > U72₁
isNatIList > isNat > U11 > tt > s > mark₁ > nil > U72₁
isNatIList > isNat > U11 > tt > s > mark₁ > U92₁ > U72₁
isNatIList > isNat > U11 > tt > U93 > mark₁ > active₁ > U72₁
isNatIList > isNat > U11 > tt > U93 > mark₁ > cons > U72₁
isNatIList > isNat > U11 > tt > U93 > mark₁ > nil > U72₁
isNatIList > isNat > U11 > tt > U93 > mark₁ > U92₁ > U72₁
isNatIList > isNat > U21 > tt > s > mark₁ > active₁ > U72₁
isNatIList > isNat > U21 > tt > s > mark₁ > cons > U72₁
isNatIList > isNat > U21 > tt > s > mark₁ > nil > U72₁
isNatIList > isNat > U21 > tt > s > mark₁ > U92₁ > U72₁
isNatIList > isNat > U21 > tt > U93 > mark₁ > active₁ > U72₁
isNatIList > isNat > U21 > tt > U93 > mark₁ > cons > U72₁
isNatIList > isNat > U21 > tt > U93 > mark₁ > nil > U72₁
isNatIList > isNat > U21 > tt > U93 > mark₁ > U92₁ > U72₁
isNatList > U51 > U52 > tt > s > mark₁ > active₁ > U72₁
isNatList > U51 > U52 > tt > s > mark₁ > cons > U72₁
isNatList > U51 > U52 > tt > s > mark₁ > nil > U72₁
isNatList > U51 > U52 > tt > s > mark₁ > U92₁ > U72₁
isNatList > U51 > U52 > tt > U93 > mark₁ > active₁ > U72₁
isNatList > U51 > U52 > tt > U93 > mark₁ > cons > U72₁
isNatList > U51 > U52 > tt > U93 > mark₁ > nil > U72₁
isNatList > U51 > U52 > tt > U93 > mark₁ > U92₁ > U72₁
isNatList > U61 > U62 > tt > s > mark₁ > active₁ > U72₁
isNatList > U61 > U62 > tt > s > mark₁ > cons > U72₁
isNatList > U61 > U62 > tt > s > mark₁ > nil > U72₁
isNatList > U61 > U62 > tt > s > mark₁ > U92₁ > U72₁
isNatList > U61 > U62 > tt > U93 > mark₁ > active₁ > U72₁
isNatList > U61 > U62 > tt > U93 > mark₁ > cons > U72₁
isNatList > U61 > U62 > tt > U93 > mark₁ > nil > U72₁
isNatList > U61 > U62 > tt > U93 > mark₁ > U92₁ > U72₁
isNatList > isNat > U11 > tt > s > mark₁ > active₁ > U72₁
isNatList > isNat > U11 > tt > s > mark₁ > cons > U72₁
isNatList > isNat > U11 > tt > s > mark₁ > nil > U72₁
isNatList > isNat > U11 > tt > s > mark₁ > U92₁ > U72₁
isNatList > isNat > U11 > tt > U93 > mark₁ > active₁ > U72₁
isNatList > isNat > U11 > tt > U93 > mark₁ > cons > U72₁
isNatList > isNat > U11 > tt > U93 > mark₁ > nil > U72₁
isNatList > isNat > U11 > tt > U93 > mark₁ > U92₁ > U72₁
isNatList > isNat > U21 > tt > s > mark₁ > active₁ > U72₁
isNatList > isNat > U21 > tt > s > mark₁ > cons > U72₁
isNatList > isNat > U21 > tt > s > mark₁ > nil > U72₁
isNatList > isNat > U21 > tt > s > mark₁ > U92₁ > U72₁
isNatList > isNat > U21 > tt > U93 > mark₁ > active₁ > U72₁
isNatList > isNat > U21 > tt > U93 > mark₁ > cons > U72₁
isNatList > isNat > U21 > tt > U93 > mark₁ > nil > U72₁
isNatList > isNat > U21 > tt > U93 > mark₁ > U92₁ > U72₁
length > 0 > tt > s > mark₁ > active₁ > U72₁
length > 0 > tt > s > mark₁ > cons > U72₁
length > 0 > tt > s > mark₁ > nil > U72₁
length > 0 > tt > s > mark₁ > U92₁ > U72₁
length > 0 > tt > U93 > mark₁ > active₁ > U72₁
length > 0 > tt > U93 > mark₁ > cons > U72₁
length > 0 > tt > U93 > mark₁ > nil > U72₁
length > 0 > tt > U93 > mark₁ > U92₁ > U72₁
length > U71 > isNat > U11 > tt > s > mark₁ > active₁ > U72₁
length > U71 > isNat > U11 > tt > s > mark₁ > cons > U72₁
length > U71 > isNat > U11 > tt > s > mark₁ > nil > U72₁
length > U71 > isNat > U11 > tt > s > mark₁ > U92₁ > U72₁
length > U71 > isNat > U11 > tt > U93 > mark₁ > active₁ > U72₁
length > U71 > isNat > U11 > tt > U93 > mark₁ > cons > U72₁
length > U71 > isNat > U11 > tt > U93 > mark₁ > nil > U72₁
length > U71 > isNat > U11 > tt > U93 > mark₁ > U92₁ > U72₁
length > U71 > isNat > U21 > tt > s > mark₁ > active₁ > U72₁
length > U71 > isNat > U21 > tt > s > mark₁ > cons > U72₁
length > U71 > isNat > U21 > tt > s > mark₁ > nil > U72₁
length > U71 > isNat > U21 > tt > s > mark₁ > U92₁ > U72₁
length > U71 > isNat > U21 > tt > U93 > mark₁ > active₁ > U72₁
length > U71 > isNat > U21 > tt > U93 > mark₁ > cons > U72₁
length > U71 > isNat > U21 > tt > U93 > mark₁ > nil > U72₁
length > U71 > isNat > U21 > tt > U93 > mark₁ > U92₁ > U72₁
take > U81 > mark₁ > active₁ > U72₁
take > U81 > mark₁ > cons > U72₁
take > U81 > mark₁ > nil > U72₁
take > U81 > mark₁ > U92₁ > U72₁
take > U91 > isNat > U11 > tt > s > mark₁ > active₁ > U72₁
take > U91 > isNat > U11 > tt > s > mark₁ > cons > U72₁
take > U91 > isNat > U11 > tt > s > mark₁ > nil > U72₁
take > U91 > isNat > U11 > tt > s > mark₁ > U92₁ > U72₁
take > U91 > isNat > U11 > tt > U93 > mark₁ > active₁ > U72₁
take > U91 > isNat > U11 > tt > U93 > mark₁ > cons > U72₁
take > U91 > isNat > U11 > tt > U93 > mark₁ > nil > U72₁
take > U91 > isNat > U11 > tt > U93 > mark₁ > U92₁ > U72₁
take > U91 > isNat > U21 > tt > s > mark₁ > active₁ > U72₁
take > U91 > isNat > U21 > tt > s > mark₁ > cons > U72₁
take > U91 > isNat > U21 > tt > s > mark₁ > nil > U72₁
take > U91 > isNat > U21 > tt > s > mark₁ > U92₁ > U72₁
take > U91 > isNat > U21 > tt > U93 > mark₁ > active₁ > U72₁
take > U91 > isNat > U21 > tt > U93 > mark₁ > cons > U72₁
take > U91 > isNat > U21 > tt > U93 > mark₁ > nil > U72₁
take > U91 > isNat > U21 > tt > U93 > mark₁ > U92₁ > U72₁

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(73) Obligation:

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

U72¹(mark(X1), X2) → U72¹(X1, X2)
U72¹(active(X1), X2) → U72¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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.

U72¹(mark(X1), X2) → U72¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U72¹(x1, x2)  =  U72¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList(x1)
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  x1
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  x1
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > s > U72^1₂
zeros > cons > U72^1₂
isNatIList > U31 > tt > mark₁ > s > U72^1₂
isNatIList > U31 > tt > cons > U72^1₂
isNatIList > U41 > U42 > tt > mark₁ > s > U72^1₂
isNatIList > U41 > U42 > tt > cons > U72^1₂
isNatIList > isNatList₁ > U51 > U52 > tt > mark₁ > s > U72^1₂
isNatIList > isNatList₁ > U51 > U52 > tt > cons > U72^1₂
isNatIList > isNatList₁ > U61 > U62 > tt > mark₁ > s > U72^1₂
isNatIList > isNatList₁ > U61 > U62 > tt > cons > U72^1₂
isNatIList > isNatList₁ > isNat > U11 > tt > mark₁ > s > U72^1₂
isNatIList > isNatList₁ > isNat > U11 > tt > cons > U72^1₂
isNatIList > isNatList₁ > isNat > U21 > tt > mark₁ > s > U72^1₂
isNatIList > isNatList₁ > isNat > U21 > tt > cons > U72^1₂
length > 0 > tt > mark₁ > s > U72^1₂
length > 0 > tt > cons > U72^1₂
length > isNatList₁ > U51 > U52 > tt > mark₁ > s > U72^1₂
length > isNatList₁ > U51 > U52 > tt > cons > U72^1₂
length > isNatList₁ > U61 > U62 > tt > mark₁ > s > U72^1₂
length > isNatList₁ > U61 > U62 > tt > cons > U72^1₂
length > isNatList₁ > isNat > U11 > tt > mark₁ > s > U72^1₂
length > isNatList₁ > isNat > U11 > tt > cons > U72^1₂
length > isNatList₁ > isNat > U21 > tt > mark₁ > s > U72^1₂
length > isNatList₁ > isNat > U21 > tt > cons > U72^1₂
length > U71 > isNat > U11 > tt > mark₁ > s > U72^1₂
length > U71 > isNat > U11 > tt > cons > U72^1₂
length > U71 > isNat > U21 > tt > mark₁ > s > U72^1₂
length > U71 > isNat > U21 > tt > cons > U72^1₂
take > U81 > nil > tt > mark₁ > s > U72^1₂
take > U81 > nil > tt > cons > U72^1₂
take > U91 > U92 > isNat > U11 > tt > mark₁ > s > U72^1₂
take > U91 > U92 > isNat > U11 > tt > cons > U72^1₂
take > U91 > U92 > isNat > U21 > tt > mark₁ > s > U72^1₂
take > U91 > U92 > isNat > U21 > tt > cons > U72^1₂

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(75) Obligation:

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

U72¹(active(X1), X2) → U72¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(76) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U72¹(active(X1), X2) → U72¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U72¹(x1, x2)  =  U72¹(x1)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > cons > active₁ > U72^1₁
zeros > mark₁ > s > active₁ > U72^1₁
zeros > mark₁ > nil > active₁ > U72^1₁
zeros > 0 > active₁ > U72^1₁
zeros > tt > cons > active₁ > U72^1₁
zeros > tt > s > active₁ > U72^1₁
zeros > tt > nil > active₁ > U72^1₁
isNatIList > U31 > mark₁ > cons > active₁ > U72^1₁
isNatIList > U31 > mark₁ > s > active₁ > U72^1₁
isNatIList > U31 > mark₁ > nil > active₁ > U72^1₁
isNatIList > U31 > tt > cons > active₁ > U72^1₁
isNatIList > U31 > tt > s > active₁ > U72^1₁
isNatIList > U31 > tt > nil > active₁ > U72^1₁
isNatIList > U41 > U42 > mark₁ > cons > active₁ > U72^1₁
isNatIList > U41 > U42 > mark₁ > s > active₁ > U72^1₁
isNatIList > U41 > U42 > mark₁ > nil > active₁ > U72^1₁
isNatIList > U41 > U42 > tt > cons > active₁ > U72^1₁
isNatIList > U41 > U42 > tt > s > active₁ > U72^1₁
isNatIList > U41 > U42 > tt > nil > active₁ > U72^1₁
isNatList > U51 > U52 > mark₁ > cons > active₁ > U72^1₁
isNatList > U51 > U52 > mark₁ > s > active₁ > U72^1₁
isNatList > U51 > U52 > mark₁ > nil > active₁ > U72^1₁
isNatList > U51 > U52 > tt > cons > active₁ > U72^1₁
isNatList > U51 > U52 > tt > s > active₁ > U72^1₁
isNatList > U51 > U52 > tt > nil > active₁ > U72^1₁
isNatList > U61 > U62 > mark₁ > cons > active₁ > U72^1₁
isNatList > U61 > U62 > mark₁ > s > active₁ > U72^1₁
isNatList > U61 > U62 > mark₁ > nil > active₁ > U72^1₁
isNatList > U61 > U62 > tt > cons > active₁ > U72^1₁
isNatList > U61 > U62 > tt > s > active₁ > U72^1₁
isNatList > U61 > U62 > tt > nil > active₁ > U72^1₁
isNat > U11 > mark₁ > cons > active₁ > U72^1₁
isNat > U11 > mark₁ > s > active₁ > U72^1₁
isNat > U11 > mark₁ > nil > active₁ > U72^1₁
isNat > U11 > tt > cons > active₁ > U72^1₁
isNat > U11 > tt > s > active₁ > U72^1₁
isNat > U11 > tt > nil > active₁ > U72^1₁
isNat > U21 > mark₁ > cons > active₁ > U72^1₁
isNat > U21 > mark₁ > s > active₁ > U72^1₁
isNat > U21 > mark₁ > nil > active₁ > U72^1₁
isNat > U21 > tt > cons > active₁ > U72^1₁
isNat > U21 > tt > s > active₁ > U72^1₁
isNat > U21 > tt > nil > active₁ > U72^1₁
length > 0 > active₁ > U72^1₁
length > U11 > mark₁ > cons > active₁ > U72^1₁
length > U11 > mark₁ > s > active₁ > U72^1₁
length > U11 > mark₁ > nil > active₁ > U72^1₁
length > U11 > tt > cons > active₁ > U72^1₁
length > U11 > tt > s > active₁ > U72^1₁
length > U11 > tt > nil > active₁ > U72^1₁
length > U71 > U72 > mark₁ > cons > active₁ > U72^1₁
length > U71 > U72 > mark₁ > s > active₁ > U72^1₁
length > U71 > U72 > mark₁ > nil > active₁ > U72^1₁
take > U81 > mark₁ > cons > active₁ > U72^1₁
take > U81 > mark₁ > s > active₁ > U72^1₁
take > U81 > mark₁ > nil > active₁ > U72^1₁
take > U91 > U92 > U93 > mark₁ > cons > active₁ > U72^1₁
take > U91 > U92 > U93 > mark₁ > s > active₁ > U72^1₁
take > U91 > U92 > U93 > mark₁ > nil > active₁ > U72^1₁

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(77) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(78) PisEmptyProof (EQUIVALENT transformation)

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

(79) TRUE

(80) Obligation:

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

U71¹(X1, mark(X2), X3) → U71¹(X1, X2, X3)
U71¹(mark(X1), X2, X3) → U71¹(X1, X2, X3)
U71¹(X1, X2, mark(X3)) → U71¹(X1, X2, X3)
U71¹(active(X1), X2, X3) → U71¹(X1, X2, X3)
U71¹(X1, active(X2), X3) → U71¹(X1, X2, X3)
U71¹(X1, X2, active(X3)) → U71¹(X1, X2, X3)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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.

U71¹(X1, mark(X2), X3) → U71¹(X1, X2, X3)
U71¹(X1, active(X2), X3) → U71¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U71¹(x1, x2, x3)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > cons > active₁ > 0
zeros > mark₁ > cons > active₁ > tt
zeros > mark₁ > nil > active₁ > 0
zeros > mark₁ > nil > active₁ > tt
isNatIList₁ > U31 > mark₁ > cons > active₁ > 0
isNatIList₁ > U31 > mark₁ > cons > active₁ > tt
isNatIList₁ > U31 > mark₁ > nil > active₁ > 0
isNatIList₁ > U31 > mark₁ > nil > active₁ > tt
isNatIList₁ > U41 > U42 > mark₁ > cons > active₁ > 0
isNatIList₁ > U41 > U42 > mark₁ > cons > active₁ > tt
isNatIList₁ > U41 > U42 > mark₁ > nil > active₁ > 0
isNatIList₁ > U41 > U42 > mark₁ > nil > active₁ > tt
isNatList > U51 > U52 > mark₁ > cons > active₁ > 0
isNatList > U51 > U52 > mark₁ > cons > active₁ > tt
isNatList > U51 > U52 > mark₁ > nil > active₁ > 0
isNatList > U51 > U52 > mark₁ > nil > active₁ > tt
isNatList > U61 > U62 > mark₁ > cons > active₁ > 0
isNatList > U61 > U62 > mark₁ > cons > active₁ > tt
isNatList > U61 > U62 > mark₁ > nil > active₁ > 0
isNatList > U61 > U62 > mark₁ > nil > active₁ > tt
isNat > U11 > mark₁ > cons > active₁ > 0
isNat > U11 > mark₁ > cons > active₁ > tt
isNat > U11 > mark₁ > nil > active₁ > 0
isNat > U11 > mark₁ > nil > active₁ > tt
isNat > U21 > mark₁ > cons > active₁ > 0
isNat > U21 > mark₁ > cons > active₁ > tt
isNat > U21 > mark₁ > nil > active₁ > 0
isNat > U21 > mark₁ > nil > active₁ > tt
length > U11 > mark₁ > cons > active₁ > 0
length > U11 > mark₁ > cons > active₁ > tt
length > U11 > mark₁ > nil > active₁ > 0
length > U11 > mark₁ > nil > active₁ > tt
length > U71 > U72 > mark₁ > cons > active₁ > 0
length > U71 > U72 > mark₁ > cons > active₁ > tt
length > U71 > U72 > mark₁ > nil > active₁ > 0
length > U71 > U72 > mark₁ > nil > active₁ > tt
length > U71 > U72 > s > active₁ > 0
length > U71 > U72 > s > active₁ > tt
take > U61 > U62 > mark₁ > cons > active₁ > 0
take > U61 > U62 > mark₁ > cons > active₁ > tt
take > U61 > U62 > mark₁ > nil > active₁ > 0
take > U61 > U62 > mark₁ > nil > active₁ > tt
take > U81 > mark₁ > cons > active₁ > 0
take > U81 > mark₁ > cons > active₁ > tt
take > U81 > mark₁ > nil > active₁ > 0
take > U81 > mark₁ > nil > active₁ > tt
take > U91 > U92 > U93 > mark₁ > cons > active₁ > 0
take > U91 > U92 > U93 > mark₁ > cons > active₁ > tt
take > U91 > U92 > U93 > mark₁ > nil > active₁ > 0
take > U91 > U92 > U93 > mark₁ > nil > active₁ > tt

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(82) Obligation:

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

U71¹(mark(X1), X2, X3) → U71¹(X1, X2, X3)
U71¹(X1, X2, mark(X3)) → U71¹(X1, X2, X3)
U71¹(active(X1), X2, X3) → U71¹(X1, X2, X3)
U71¹(X1, X2, active(X3)) → U71¹(X1, X2, X3)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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.

U71¹(X1, X2, mark(X3)) → U71¹(X1, X2, X3)
U71¹(X1, X2, active(X3)) → U71¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U71¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList(x1)
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > mark₁ > active₁
isNatIList₁ > U31 > tt > cons > mark₁ > active₁
isNatIList₁ > U31 > tt > nil > 0 > mark₁ > active₁
isNatIList₁ > U41 > U42 > tt > cons > mark₁ > active₁
isNatIList₁ > U41 > U42 > tt > nil > 0 > mark₁ > active₁
isNat > U11 > tt > cons > mark₁ > active₁
isNat > U11 > tt > nil > 0 > mark₁ > active₁
isNat > U21 > tt > cons > mark₁ > active₁
isNat > U21 > tt > nil > 0 > mark₁ > active₁
isNat > isNatList₁ > U51 > U52 > tt > cons > mark₁ > active₁
isNat > isNatList₁ > U51 > U52 > tt > nil > 0 > mark₁ > active₁
isNat > isNatList₁ > U61 > U62 > tt > cons > mark₁ > active₁
isNat > isNatList₁ > U61 > U62 > tt > nil > 0 > mark₁ > active₁
length > U11 > tt > cons > mark₁ > active₁
length > U11 > tt > nil > 0 > mark₁ > active₁
length > isNatList₁ > U51 > U52 > tt > cons > mark₁ > active₁
length > isNatList₁ > U51 > U52 > tt > nil > 0 > mark₁ > active₁
length > isNatList₁ > U61 > U62 > tt > cons > mark₁ > active₁
length > isNatList₁ > U61 > U62 > tt > nil > 0 > mark₁ > active₁
length > U71 > U72 > s > U21 > tt > cons > mark₁ > active₁
length > U71 > U72 > s > U21 > tt > nil > 0 > mark₁ > active₁
length > U71 > U72 > s > U91 > U92 > U93 > cons > mark₁ > active₁
take > U81 > nil > 0 > mark₁ > active₁
take > U91 > U92 > U93 > cons > mark₁ > active₁

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(84) Obligation:

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

U71¹(mark(X1), X2, X3) → U71¹(X1, X2, X3)
U71¹(active(X1), X2, X3) → U71¹(X1, X2, X3)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(85) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(mark(X1), X2, X3) → U71¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U71¹(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList(x1)
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  x1
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > tt > cons > U71 > U72 > mark₁ > 0
zeros > tt > cons > U71 > U72 > mark₁ > s
isNatIList > U31 > tt > cons > U71 > U72 > mark₁ > 0
isNatIList > U31 > tt > cons > U71 > U72 > mark₁ > s
isNatIList > U41 > U42 > tt > cons > U71 > U72 > mark₁ > 0
isNatIList > U41 > U42 > tt > cons > U71 > U72 > mark₁ > s
isNatList₁ > U51 > U52 > tt > cons > U71 > U72 > mark₁ > 0
isNatList₁ > U51 > U52 > tt > cons > U71 > U72 > mark₁ > s
isNatList₁ > U61 > U62 > tt > cons > U71 > U72 > mark₁ > 0
isNatList₁ > U61 > U62 > tt > cons > U71 > U72 > mark₁ > s
isNat > U11 > tt > cons > U71 > U72 > mark₁ > 0
isNat > U11 > tt > cons > U71 > U72 > mark₁ > s
isNat > U21 > tt > cons > U71 > U72 > mark₁ > 0
isNat > U21 > tt > cons > U71 > U72 > mark₁ > s
take > U61 > U62 > tt > cons > U71 > U72 > mark₁ > 0
take > U61 > U62 > tt > cons > U71 > U72 > mark₁ > s
take > U81 > nil > mark₁ > 0
take > U81 > nil > mark₁ > s
take > U91 > U92 > U93 > cons > U71 > U72 > mark₁ > 0
take > U91 > U92 > U93 > cons > U71 > U72 > mark₁ > s

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(86) Obligation:

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

U71¹(active(X1), X2, X3) → U71¹(X1, X2, X3)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(87) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(active(X1), X2, X3) → U71¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U71¹(x1, x2, x3)  =  x1
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U41 > U42 > tt > mark₁ > active₁
zeros > cons > U51 > U52 > tt > mark₁ > active₁
zeros > 0 > tt > mark₁ > active₁
isNatList > U51 > U52 > tt > mark₁ > active₁
isNatList > U61 > U62 > tt > mark₁ > active₁
isNat > U11 > tt > mark₁ > active₁
isNat > U21 > tt > mark₁ > active₁
length > 0 > tt > mark₁ > active₁
length > U11 > tt > mark₁ > active₁
length > U71 > U72 > s > mark₁ > active₁
take > isNatIList > U31 > tt > mark₁ > active₁
take > isNatIList > U41 > U42 > tt > mark₁ > active₁
take > U61 > U62 > tt > mark₁ > active₁
take > U81 > nil > tt > mark₁ > active₁
take > U91 > U92 > U93 > cons > U41 > U42 > tt > mark₁ > active₁
take > U91 > U92 > U93 > cons > U51 > U52 > tt > mark₁ > active₁

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(88) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(89) PisEmptyProof (EQUIVALENT transformation)

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

(90) TRUE

(91) 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(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(92) 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: Combined order from the following AFS and order.
U62¹(x1)  =  U62¹(x1)
active(x1)  =  x1
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > cons > U62^1₁
zeros > mark₁ > tt > U62^1₁
isNatList > U51 > U52 > mark₁ > cons > U62^1₁
isNatList > U51 > U52 > mark₁ > tt > U62^1₁
isNatList > U61 > U62 > mark₁ > cons > U62^1₁
isNatList > U61 > U62 > mark₁ > tt > U62^1₁
isNat > U11 > mark₁ > cons > U62^1₁
isNat > U11 > mark₁ > tt > U62^1₁
isNat > U21 > mark₁ > cons > U62^1₁
isNat > U21 > mark₁ > tt > U62^1₁
length > 0 > isNatIList₁ > U31 > mark₁ > cons > U62^1₁
length > 0 > isNatIList₁ > U31 > mark₁ > tt > U62^1₁
length > 0 > isNatIList₁ > U41 > U42 > mark₁ > cons > U62^1₁
length > 0 > isNatIList₁ > U41 > U42 > mark₁ > tt > U62^1₁
length > U11 > mark₁ > cons > U62^1₁
length > U11 > mark₁ > tt > U62^1₁
length > U71 > U72 > s > mark₁ > cons > U62^1₁
length > U71 > U72 > s > mark₁ > tt > U62^1₁
take > isNatIList₁ > U31 > mark₁ > cons > U62^1₁
take > isNatIList₁ > U31 > mark₁ > tt > U62^1₁
take > isNatIList₁ > U41 > U42 > mark₁ > cons > U62^1₁
take > isNatIList₁ > U41 > U42 > mark₁ > tt > U62^1₁
take > U61 > U62 > mark₁ > cons > U62^1₁
take > U61 > U62 > mark₁ > tt > U62^1₁
take > U81 > mark₁ > cons > U62^1₁
take > U81 > mark₁ > tt > U62^1₁
take > U81 > nil > tt > U62^1₁
take > U91 > U92 > U93 > mark₁ > cons > U62^1₁
take > U91 > U92 > U93 > mark₁ > tt > U62^1₁

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(93) Obligation:

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

U62¹(active(X)) → U62¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(94) 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)  =  U62¹(x1)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > tt > cons > mark₁ > active₁
zeros > tt > cons > mark₁ > nil
zeros > tt > U72 > s > mark₁ > active₁
zeros > tt > U72 > s > mark₁ > nil
isNatIList > U31 > tt > cons > mark₁ > active₁
isNatIList > U31 > tt > cons > mark₁ > nil
isNatIList > U31 > tt > U72 > s > mark₁ > active₁
isNatIList > U31 > tt > U72 > s > mark₁ > nil
isNatIList > U41 > U42 > tt > cons > mark₁ > active₁
isNatIList > U41 > U42 > tt > cons > mark₁ > nil
isNatIList > U41 > U42 > tt > U72 > s > mark₁ > active₁
isNatIList > U41 > U42 > tt > U72 > s > mark₁ > nil
isNatIList > isNatList > U51 > U52 > tt > cons > mark₁ > active₁
isNatIList > isNatList > U51 > U52 > tt > cons > mark₁ > nil
isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > active₁
isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > nil
isNatIList > isNatList > U61 > U62 > tt > cons > mark₁ > active₁
isNatIList > isNatList > U61 > U62 > tt > cons > mark₁ > nil
isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > active₁
isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > nil
isNatIList > isNat > U11 > tt > cons > mark₁ > active₁
isNatIList > isNat > U11 > tt > cons > mark₁ > nil
isNatIList > isNat > U11 > tt > U72 > s > mark₁ > active₁
isNatIList > isNat > U11 > tt > U72 > s > mark₁ > nil
isNatIList > isNat > U21 > tt > cons > mark₁ > active₁
isNatIList > isNat > U21 > tt > cons > mark₁ > nil
isNatIList > isNat > U21 > tt > U72 > s > mark₁ > active₁
isNatIList > isNat > U21 > tt > U72 > s > mark₁ > nil
length > 0 > tt > cons > mark₁ > active₁
length > 0 > tt > cons > mark₁ > nil
length > 0 > tt > U72 > s > mark₁ > active₁
length > 0 > tt > U72 > s > mark₁ > nil
length > U11 > tt > cons > mark₁ > active₁
length > U11 > tt > cons > mark₁ > nil
length > U11 > tt > U72 > s > mark₁ > active₁
length > U11 > tt > U72 > s > mark₁ > nil
length > isNatList > U51 > U52 > tt > cons > mark₁ > active₁
length > isNatList > U51 > U52 > tt > cons > mark₁ > nil
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > active₁
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > nil
length > isNatList > U61 > U62 > tt > cons > mark₁ > active₁
length > isNatList > U61 > U62 > tt > cons > mark₁ > nil
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > active₁
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > nil
length > U71 > U72 > s > mark₁ > active₁
length > U71 > U72 > s > mark₁ > nil
take > U61 > U62 > tt > cons > mark₁ > active₁
take > U61 > U62 > tt > cons > mark₁ > nil
take > U61 > U62 > tt > U72 > s > mark₁ > active₁
take > U61 > U62 > tt > U72 > s > mark₁ > nil
take > isNat > U11 > tt > cons > mark₁ > active₁
take > isNat > U11 > tt > cons > mark₁ > nil
take > isNat > U11 > tt > U72 > s > mark₁ > active₁
take > isNat > U11 > tt > U72 > s > mark₁ > nil
take > isNat > U21 > tt > cons > mark₁ > active₁
take > isNat > U21 > tt > cons > mark₁ > nil
take > isNat > U21 > tt > U72 > s > mark₁ > active₁
take > isNat > U21 > tt > U72 > s > mark₁ > nil
take > U81 > mark₁ > active₁
take > U81 > mark₁ > nil
take > U91 > U92 > U93 > cons > mark₁ > active₁
take > U91 > U92 > U93 > cons > mark₁ > nil

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(95) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(96) PisEmptyProof (EQUIVALENT transformation)

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

(97) TRUE

(98) 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(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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.

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)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81(x1)
nil  =  nil
U91(x1, x2, x3, x4)  =  U91(x1)
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > 0 > mark₁ > active₁ > U81₁
zeros > 0 > mark₁ > active₁ > nil
zeros > 0 > mark₁ > U91₁
zeros > tt > s > mark₁ > active₁ > U81₁
zeros > tt > s > mark₁ > active₁ > nil
zeros > tt > s > mark₁ > U91₁
zeros > tt > U92 > U93 > cons > mark₁ > active₁ > U81₁
zeros > tt > U92 > U93 > cons > mark₁ > active₁ > nil
zeros > tt > U92 > U93 > cons > mark₁ > U91₁
isNatList > U51 > U52 > tt > s > mark₁ > active₁ > U81₁
isNatList > U51 > U52 > tt > s > mark₁ > active₁ > nil
isNatList > U51 > U52 > tt > s > mark₁ > U91₁
isNatList > U51 > U52 > tt > U92 > U93 > cons > mark₁ > active₁ > U81₁
isNatList > U51 > U52 > tt > U92 > U93 > cons > mark₁ > active₁ > nil
isNatList > U51 > U52 > tt > U92 > U93 > cons > mark₁ > U91₁
isNatList > U61 > isNatIList > U31 > tt > s > mark₁ > active₁ > U81₁
isNatList > U61 > isNatIList > U31 > tt > s > mark₁ > active₁ > nil
isNatList > U61 > isNatIList > U31 > tt > s > mark₁ > U91₁
isNatList > U61 > isNatIList > U31 > tt > U92 > U93 > cons > mark₁ > active₁ > U81₁
isNatList > U61 > isNatIList > U31 > tt > U92 > U93 > cons > mark₁ > active₁ > nil
isNatList > U61 > isNatIList > U31 > tt > U92 > U93 > cons > mark₁ > U91₁
isNatList > U61 > isNatIList > U41 > U42 > tt > s > mark₁ > active₁ > U81₁
isNatList > U61 > isNatIList > U41 > U42 > tt > s > mark₁ > active₁ > nil
isNatList > U61 > isNatIList > U41 > U42 > tt > s > mark₁ > U91₁
isNatList > U61 > isNatIList > U41 > U42 > tt > U92 > U93 > cons > mark₁ > active₁ > U81₁
isNatList > U61 > isNatIList > U41 > U42 > tt > U92 > U93 > cons > mark₁ > active₁ > nil
isNatList > U61 > isNatIList > U41 > U42 > tt > U92 > U93 > cons > mark₁ > U91₁
isNatList > U61 > U62 > tt > s > mark₁ > active₁ > U81₁
isNatList > U61 > U62 > tt > s > mark₁ > active₁ > nil
isNatList > U61 > U62 > tt > s > mark₁ > U91₁
isNatList > U61 > U62 > tt > U92 > U93 > cons > mark₁ > active₁ > U81₁
isNatList > U61 > U62 > tt > U92 > U93 > cons > mark₁ > active₁ > nil
isNatList > U61 > U62 > tt > U92 > U93 > cons > mark₁ > U91₁
isNat > U11 > tt > s > mark₁ > active₁ > U81₁
isNat > U11 > tt > s > mark₁ > active₁ > nil
isNat > U11 > tt > s > mark₁ > U91₁
isNat > U11 > tt > U92 > U93 > cons > mark₁ > active₁ > U81₁
isNat > U11 > tt > U92 > U93 > cons > mark₁ > active₁ > nil
isNat > U11 > tt > U92 > U93 > cons > mark₁ > U91₁
isNat > U21 > tt > s > mark₁ > active₁ > U81₁
isNat > U21 > tt > s > mark₁ > active₁ > nil
isNat > U21 > tt > s > mark₁ > U91₁
isNat > U21 > tt > U92 > U93 > cons > mark₁ > active₁ > U81₁
isNat > U21 > tt > U92 > U93 > cons > mark₁ > active₁ > nil
isNat > U21 > tt > U92 > U93 > cons > mark₁ > U91₁
length > 0 > mark₁ > active₁ > U81₁
length > 0 > mark₁ > active₁ > nil
length > 0 > mark₁ > U91₁
length > U11 > tt > s > mark₁ > active₁ > U81₁
length > U11 > tt > s > mark₁ > active₁ > nil
length > U11 > tt > s > mark₁ > U91₁
length > U11 > tt > U92 > U93 > cons > mark₁ > active₁ > U81₁
length > U11 > tt > U92 > U93 > cons > mark₁ > active₁ > nil
length > U11 > tt > U92 > U93 > cons > mark₁ > U91₁
length > U71 > U72 > s > mark₁ > active₁ > U81₁
length > U71 > U72 > s > mark₁ > active₁ > nil
length > U71 > U72 > s > mark₁ > U91₁
take > isNatIList > U31 > tt > s > mark₁ > active₁ > U81₁
take > isNatIList > U31 > tt > s > mark₁ > active₁ > nil
take > isNatIList > U31 > tt > s > mark₁ > U91₁
take > isNatIList > U31 > tt > U92 > U93 > cons > mark₁ > active₁ > U81₁
take > isNatIList > U31 > tt > U92 > U93 > cons > mark₁ > active₁ > nil
take > isNatIList > U31 > tt > U92 > U93 > cons > mark₁ > U91₁
take > isNatIList > U41 > U42 > tt > s > mark₁ > active₁ > U81₁
take > isNatIList > U41 > U42 > tt > s > mark₁ > active₁ > nil
take > isNatIList > U41 > U42 > tt > s > mark₁ > U91₁
take > isNatIList > U41 > U42 > tt > U92 > U93 > cons > mark₁ > active₁ > U81₁
take > isNatIList > U41 > U42 > tt > U92 > U93 > cons > mark₁ > active₁ > nil
take > isNatIList > U41 > U42 > tt > U92 > U93 > cons > mark₁ > U91₁

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(100) 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(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(101) 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)
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)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList(x1)
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > cons > active₁ > nil > 0
zeros > tt > cons > active₁ > nil > 0
isNatIList > U31 > mark₁ > cons > active₁ > nil > 0
isNatIList > U31 > tt > cons > active₁ > nil > 0
isNatIList > U41 > U42 > mark₁ > cons > active₁ > nil > 0
isNatIList > U41 > U42 > tt > cons > active₁ > nil > 0
isNatList₁ > U51 > U52 > mark₁ > cons > active₁ > nil > 0
isNatList₁ > U51 > U52 > tt > cons > active₁ > nil > 0
isNatList₁ > U61 > U62 > mark₁ > cons > active₁ > nil > 0
isNatList₁ > U61 > U62 > tt > cons > active₁ > nil > 0
isNat > U11 > mark₁ > cons > active₁ > nil > 0
isNat > U11 > tt > cons > active₁ > nil > 0
isNat > U21 > mark₁ > cons > active₁ > nil > 0
isNat > U21 > tt > cons > active₁ > nil > 0
length > U11 > mark₁ > cons > active₁ > nil > 0
length > U11 > tt > cons > active₁ > nil > 0
length > U71 > U72 > mark₁ > cons > active₁ > nil > 0
length > U71 > U72 > s > active₁ > nil > 0
take > U61 > U62 > mark₁ > cons > active₁ > nil > 0
take > U61 > U62 > tt > cons > active₁ > nil > 0
take > U81 > mark₁ > cons > active₁ > nil > 0
take > U91 > U92 > U93 > mark₁ > cons > active₁ > nil > 0

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(102) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(103) PisEmptyProof (EQUIVALENT transformation)

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

(104) TRUE

(105) 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(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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.

ISNATLIST(mark(X)) → ISNATLIST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATLIST(x1)  =  ISNATLIST(x1)
active(x1)  =  x1
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > cons > ISNATLIST₁
zeros > mark₁ > tt > ISNATLIST₁
isNatList > U51 > U52 > mark₁ > cons > ISNATLIST₁
isNatList > U51 > U52 > mark₁ > tt > ISNATLIST₁
isNatList > U61 > U62 > mark₁ > cons > ISNATLIST₁
isNatList > U61 > U62 > mark₁ > tt > ISNATLIST₁
isNat > U11 > mark₁ > cons > ISNATLIST₁
isNat > U11 > mark₁ > tt > ISNATLIST₁
isNat > U21 > mark₁ > cons > ISNATLIST₁
isNat > U21 > mark₁ > tt > ISNATLIST₁
length > 0 > isNatIList₁ > U31 > mark₁ > cons > ISNATLIST₁
length > 0 > isNatIList₁ > U31 > mark₁ > tt > ISNATLIST₁
length > 0 > isNatIList₁ > U41 > U42 > mark₁ > cons > ISNATLIST₁
length > 0 > isNatIList₁ > U41 > U42 > mark₁ > tt > ISNATLIST₁
length > U11 > mark₁ > cons > ISNATLIST₁
length > U11 > mark₁ > tt > ISNATLIST₁
length > U71 > U72 > s > mark₁ > cons > ISNATLIST₁
length > U71 > U72 > s > mark₁ > tt > ISNATLIST₁
take > isNatIList₁ > U31 > mark₁ > cons > ISNATLIST₁
take > isNatIList₁ > U31 > mark₁ > tt > ISNATLIST₁
take > isNatIList₁ > U41 > U42 > mark₁ > cons > ISNATLIST₁
take > isNatIList₁ > U41 > U42 > mark₁ > tt > ISNATLIST₁
take > U61 > U62 > mark₁ > cons > ISNATLIST₁
take > U61 > U62 > mark₁ > tt > ISNATLIST₁
take > U81 > mark₁ > cons > ISNATLIST₁
take > U81 > mark₁ > tt > ISNATLIST₁
take > U81 > nil > tt > ISNATLIST₁
take > U91 > U92 > U93 > mark₁ > cons > ISNATLIST₁
take > U91 > U92 > U93 > mark₁ > tt > ISNATLIST₁

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(107) Obligation:

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

ISNATLIST(active(X)) → ISNATLIST(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(108) 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)  =  ISNATLIST(x1)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > tt > cons > mark₁ > active₁
zeros > tt > cons > mark₁ > nil
zeros > tt > U72 > s > mark₁ > active₁
zeros > tt > U72 > s > mark₁ > nil
isNatIList > U31 > tt > cons > mark₁ > active₁
isNatIList > U31 > tt > cons > mark₁ > nil
isNatIList > U31 > tt > U72 > s > mark₁ > active₁
isNatIList > U31 > tt > U72 > s > mark₁ > nil
isNatIList > U41 > U42 > tt > cons > mark₁ > active₁
isNatIList > U41 > U42 > tt > cons > mark₁ > nil
isNatIList > U41 > U42 > tt > U72 > s > mark₁ > active₁
isNatIList > U41 > U42 > tt > U72 > s > mark₁ > nil
isNatIList > isNatList > U51 > U52 > tt > cons > mark₁ > active₁
isNatIList > isNatList > U51 > U52 > tt > cons > mark₁ > nil
isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > active₁
isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > nil
isNatIList > isNatList > U61 > U62 > tt > cons > mark₁ > active₁
isNatIList > isNatList > U61 > U62 > tt > cons > mark₁ > nil
isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > active₁
isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > nil
isNatIList > isNat > U11 > tt > cons > mark₁ > active₁
isNatIList > isNat > U11 > tt > cons > mark₁ > nil
isNatIList > isNat > U11 > tt > U72 > s > mark₁ > active₁
isNatIList > isNat > U11 > tt > U72 > s > mark₁ > nil
isNatIList > isNat > U21 > tt > cons > mark₁ > active₁
isNatIList > isNat > U21 > tt > cons > mark₁ > nil
isNatIList > isNat > U21 > tt > U72 > s > mark₁ > active₁
isNatIList > isNat > U21 > tt > U72 > s > mark₁ > nil
length > 0 > tt > cons > mark₁ > active₁
length > 0 > tt > cons > mark₁ > nil
length > 0 > tt > U72 > s > mark₁ > active₁
length > 0 > tt > U72 > s > mark₁ > nil
length > U11 > tt > cons > mark₁ > active₁
length > U11 > tt > cons > mark₁ > nil
length > U11 > tt > U72 > s > mark₁ > active₁
length > U11 > tt > U72 > s > mark₁ > nil
length > isNatList > U51 > U52 > tt > cons > mark₁ > active₁
length > isNatList > U51 > U52 > tt > cons > mark₁ > nil
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > active₁
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > nil
length > isNatList > U61 > U62 > tt > cons > mark₁ > active₁
length > isNatList > U61 > U62 > tt > cons > mark₁ > nil
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > active₁
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > nil
length > U71 > U72 > s > mark₁ > active₁
length > U71 > U72 > s > mark₁ > nil
take > U61 > U62 > tt > cons > mark₁ > active₁
take > U61 > U62 > tt > cons > mark₁ > nil
take > U61 > U62 > tt > U72 > s > mark₁ > active₁
take > U61 > U62 > tt > U72 > s > mark₁ > nil
take > isNat > U11 > tt > cons > mark₁ > active₁
take > isNat > U11 > tt > cons > mark₁ > nil
take > isNat > U11 > tt > U72 > s > mark₁ > active₁
take > isNat > U11 > tt > U72 > s > mark₁ > nil
take > isNat > U21 > tt > cons > mark₁ > active₁
take > isNat > U21 > tt > cons > mark₁ > nil
take > isNat > U21 > tt > U72 > s > mark₁ > active₁
take > isNat > U21 > tt > U72 > s > mark₁ > nil
take > U81 > mark₁ > active₁
take > U81 > mark₁ > nil
take > U91 > U92 > U93 > cons > mark₁ > active₁
take > U91 > U92 > U93 > cons > mark₁ > nil

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(109) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(110) PisEmptyProof (EQUIVALENT transformation)

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

(111) TRUE

(112) 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(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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.

U52¹(mark(X)) → U52¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U52¹(x1)  =  U52¹(x1)
active(x1)  =  x1
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > cons > U52^1₁
zeros > mark₁ > tt > U52^1₁
isNatList > U51 > U52 > mark₁ > cons > U52^1₁
isNatList > U51 > U52 > mark₁ > tt > U52^1₁
isNatList > U61 > U62 > mark₁ > cons > U52^1₁
isNatList > U61 > U62 > mark₁ > tt > U52^1₁
isNat > U11 > mark₁ > cons > U52^1₁
isNat > U11 > mark₁ > tt > U52^1₁
isNat > U21 > mark₁ > cons > U52^1₁
isNat > U21 > mark₁ > tt > U52^1₁
length > 0 > isNatIList₁ > U31 > mark₁ > cons > U52^1₁
length > 0 > isNatIList₁ > U31 > mark₁ > tt > U52^1₁
length > 0 > isNatIList₁ > U41 > U42 > mark₁ > cons > U52^1₁
length > 0 > isNatIList₁ > U41 > U42 > mark₁ > tt > U52^1₁
length > U11 > mark₁ > cons > U52^1₁
length > U11 > mark₁ > tt > U52^1₁
length > U71 > U72 > s > mark₁ > cons > U52^1₁
length > U71 > U72 > s > mark₁ > tt > U52^1₁
take > isNatIList₁ > U31 > mark₁ > cons > U52^1₁
take > isNatIList₁ > U31 > mark₁ > tt > U52^1₁
take > isNatIList₁ > U41 > U42 > mark₁ > cons > U52^1₁
take > isNatIList₁ > U41 > U42 > mark₁ > tt > U52^1₁
take > U61 > U62 > mark₁ > cons > U52^1₁
take > U61 > U62 > mark₁ > tt > U52^1₁
take > U81 > mark₁ > cons > U52^1₁
take > U81 > mark₁ > tt > U52^1₁
take > U81 > nil > tt > U52^1₁
take > U91 > U92 > U93 > mark₁ > cons > U52^1₁
take > U91 > U92 > U93 > mark₁ > tt > U52^1₁

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(114) Obligation:

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

U52¹(active(X)) → U52¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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.

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)  =  U52¹(x1)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > tt > cons > mark₁ > active₁
zeros > tt > cons > mark₁ > nil
zeros > tt > U72 > s > mark₁ > active₁
zeros > tt > U72 > s > mark₁ > nil
isNatIList > U31 > tt > cons > mark₁ > active₁
isNatIList > U31 > tt > cons > mark₁ > nil
isNatIList > U31 > tt > U72 > s > mark₁ > active₁
isNatIList > U31 > tt > U72 > s > mark₁ > nil
isNatIList > U41 > U42 > tt > cons > mark₁ > active₁
isNatIList > U41 > U42 > tt > cons > mark₁ > nil
isNatIList > U41 > U42 > tt > U72 > s > mark₁ > active₁
isNatIList > U41 > U42 > tt > U72 > s > mark₁ > nil
isNatIList > isNatList > U51 > U52 > tt > cons > mark₁ > active₁
isNatIList > isNatList > U51 > U52 > tt > cons > mark₁ > nil
isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > active₁
isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > nil
isNatIList > isNatList > U61 > U62 > tt > cons > mark₁ > active₁
isNatIList > isNatList > U61 > U62 > tt > cons > mark₁ > nil
isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > active₁
isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > nil
isNatIList > isNat > U11 > tt > cons > mark₁ > active₁
isNatIList > isNat > U11 > tt > cons > mark₁ > nil
isNatIList > isNat > U11 > tt > U72 > s > mark₁ > active₁
isNatIList > isNat > U11 > tt > U72 > s > mark₁ > nil
isNatIList > isNat > U21 > tt > cons > mark₁ > active₁
isNatIList > isNat > U21 > tt > cons > mark₁ > nil
isNatIList > isNat > U21 > tt > U72 > s > mark₁ > active₁
isNatIList > isNat > U21 > tt > U72 > s > mark₁ > nil
length > 0 > tt > cons > mark₁ > active₁
length > 0 > tt > cons > mark₁ > nil
length > 0 > tt > U72 > s > mark₁ > active₁
length > 0 > tt > U72 > s > mark₁ > nil
length > U11 > tt > cons > mark₁ > active₁
length > U11 > tt > cons > mark₁ > nil
length > U11 > tt > U72 > s > mark₁ > active₁
length > U11 > tt > U72 > s > mark₁ > nil
length > isNatList > U51 > U52 > tt > cons > mark₁ > active₁
length > isNatList > U51 > U52 > tt > cons > mark₁ > nil
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > active₁
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > nil
length > isNatList > U61 > U62 > tt > cons > mark₁ > active₁
length > isNatList > U61 > U62 > tt > cons > mark₁ > nil
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > active₁
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > nil
length > U71 > U72 > s > mark₁ > active₁
length > U71 > U72 > s > mark₁ > nil
take > U61 > U62 > tt > cons > mark₁ > active₁
take > U61 > U62 > tt > cons > mark₁ > nil
take > U61 > U62 > tt > U72 > s > mark₁ > active₁
take > U61 > U62 > tt > U72 > s > mark₁ > nil
take > isNat > U11 > tt > cons > mark₁ > active₁
take > isNat > U11 > tt > cons > mark₁ > nil
take > isNat > U11 > tt > U72 > s > mark₁ > active₁
take > isNat > U11 > tt > U72 > s > mark₁ > nil
take > isNat > U21 > tt > cons > mark₁ > active₁
take > isNat > U21 > tt > cons > mark₁ > nil
take > isNat > U21 > tt > U72 > s > mark₁ > active₁
take > isNat > U21 > tt > U72 > s > mark₁ > nil
take > U81 > mark₁ > active₁
take > U81 > mark₁ > nil
take > U91 > U92 > U93 > cons > mark₁ > active₁
take > U91 > U92 > U93 > cons > mark₁ > nil

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(116) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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:

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(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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.

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 remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2)  =  U51¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > active₁ > cons
zeros > mark₁ > active₁ > nil
zeros > tt > active₁ > cons
zeros > tt > active₁ > nil
isNatIList > U31 > mark₁ > active₁ > cons
isNatIList > U31 > mark₁ > active₁ > nil
isNatIList > U31 > tt > active₁ > cons
isNatIList > U31 > tt > active₁ > nil
isNatIList > U41 > U42 > mark₁ > active₁ > cons
isNatIList > U41 > U42 > mark₁ > active₁ > nil
isNatIList > U41 > U42 > tt > active₁ > cons
isNatIList > U41 > U42 > tt > active₁ > nil
isNatIList > isNat > U11 > mark₁ > active₁ > cons
isNatIList > isNat > U11 > mark₁ > active₁ > nil
isNatIList > isNat > U11 > tt > active₁ > cons
isNatIList > isNat > U11 > tt > active₁ > nil
isNatIList > isNat > U21 > mark₁ > active₁ > cons
isNatIList > isNat > U21 > mark₁ > active₁ > nil
isNatIList > isNat > U21 > tt > active₁ > cons
isNatIList > isNat > U21 > tt > active₁ > nil
isNatList > U51 > U52 > mark₁ > active₁ > cons
isNatList > U51 > U52 > mark₁ > active₁ > nil
isNatList > U51 > U52 > tt > active₁ > cons
isNatList > U51 > U52 > tt > active₁ > nil
isNatList > U61 > U62 > mark₁ > active₁ > cons
isNatList > U61 > U62 > mark₁ > active₁ > nil
isNatList > U61 > U62 > tt > active₁ > cons
isNatList > U61 > U62 > tt > active₁ > nil
isNatList > isNat > U11 > mark₁ > active₁ > cons
isNatList > isNat > U11 > mark₁ > active₁ > nil
isNatList > isNat > U11 > tt > active₁ > cons
isNatList > isNat > U11 > tt > active₁ > nil
isNatList > isNat > U21 > mark₁ > active₁ > cons
isNatList > isNat > U21 > mark₁ > active₁ > nil
isNatList > isNat > U21 > tt > active₁ > cons
isNatList > isNat > U21 > tt > active₁ > nil
length > 0 > tt > active₁ > cons
length > 0 > tt > active₁ > nil
length > 0 > U81 > mark₁ > active₁ > cons
length > 0 > U81 > mark₁ > active₁ > nil
length > U71 > U72 > s > U21 > mark₁ > active₁ > cons
length > U71 > U72 > s > U21 > mark₁ > active₁ > nil
length > U71 > U72 > s > U21 > tt > active₁ > cons
length > U71 > U72 > s > U21 > tt > active₁ > nil
length > U71 > isNat > U11 > mark₁ > active₁ > cons
length > U71 > isNat > U11 > mark₁ > active₁ > nil
length > U71 > isNat > U11 > tt > active₁ > cons
length > U71 > isNat > U11 > tt > active₁ > nil
length > U71 > isNat > U21 > mark₁ > active₁ > cons
length > U71 > isNat > U21 > mark₁ > active₁ > nil
length > U71 > isNat > U21 > tt > active₁ > cons
length > U71 > isNat > U21 > tt > active₁ > nil
take > U61 > U62 > mark₁ > active₁ > cons
take > U61 > U62 > mark₁ > active₁ > nil
take > U61 > U62 > tt > active₁ > cons
take > U61 > U62 > tt > active₁ > nil
take > U81 > mark₁ > active₁ > cons
take > U81 > mark₁ > active₁ > nil
take > U91 > isNat > U11 > mark₁ > active₁ > cons
take > U91 > isNat > U11 > mark₁ > active₁ > nil
take > U91 > isNat > U11 > tt > active₁ > cons
take > U91 > isNat > U11 > tt > active₁ > nil
take > U91 > isNat > U21 > mark₁ > active₁ > cons
take > U91 > isNat > U21 > mark₁ > active₁ > nil
take > U91 > isNat > U21 > tt > active₁ > cons
take > U91 > isNat > U21 > tt > active₁ > nil
take > U91 > U92 > U93 > mark₁ > active₁ > cons
take > U91 > U92 > U93 > mark₁ > active₁ > nil

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(121) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(122) PisEmptyProof (EQUIVALENT transformation)

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

(123) TRUE

(124) 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(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(125) 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: Combined order from the following AFS and order.
ISNATILIST(x1)  =  ISNATILIST(x1)
active(x1)  =  x1
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > cons > ISNATILIST₁
zeros > mark₁ > tt > ISNATILIST₁
isNatList > U51 > U52 > mark₁ > cons > ISNATILIST₁
isNatList > U51 > U52 > mark₁ > tt > ISNATILIST₁
isNatList > U61 > U62 > mark₁ > cons > ISNATILIST₁
isNatList > U61 > U62 > mark₁ > tt > ISNATILIST₁
isNat > U11 > mark₁ > cons > ISNATILIST₁
isNat > U11 > mark₁ > tt > ISNATILIST₁
isNat > U21 > mark₁ > cons > ISNATILIST₁
isNat > U21 > mark₁ > tt > ISNATILIST₁
length > 0 > isNatIList₁ > U31 > mark₁ > cons > ISNATILIST₁
length > 0 > isNatIList₁ > U31 > mark₁ > tt > ISNATILIST₁
length > 0 > isNatIList₁ > U41 > U42 > mark₁ > cons > ISNATILIST₁
length > 0 > isNatIList₁ > U41 > U42 > mark₁ > tt > ISNATILIST₁
length > U11 > mark₁ > cons > ISNATILIST₁
length > U11 > mark₁ > tt > ISNATILIST₁
length > U71 > U72 > s > mark₁ > cons > ISNATILIST₁
length > U71 > U72 > s > mark₁ > tt > ISNATILIST₁
take > isNatIList₁ > U31 > mark₁ > cons > ISNATILIST₁
take > isNatIList₁ > U31 > mark₁ > tt > ISNATILIST₁
take > isNatIList₁ > U41 > U42 > mark₁ > cons > ISNATILIST₁
take > isNatIList₁ > U41 > U42 > mark₁ > tt > ISNATILIST₁
take > U61 > U62 > mark₁ > cons > ISNATILIST₁
take > U61 > U62 > mark₁ > tt > ISNATILIST₁
take > U81 > mark₁ > cons > ISNATILIST₁
take > U81 > mark₁ > tt > ISNATILIST₁
take > U81 > nil > tt > ISNATILIST₁
take > U91 > U92 > U93 > mark₁ > cons > ISNATILIST₁
take > U91 > U92 > U93 > mark₁ > tt > ISNATILIST₁

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(126) Obligation:

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

ISNATILIST(active(X)) → ISNATILIST(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(127) 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)  =  ISNATILIST(x1)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > tt > cons > mark₁ > active₁
zeros > tt > cons > mark₁ > nil
zeros > tt > U72 > s > mark₁ > active₁
zeros > tt > U72 > s > mark₁ > nil
isNatIList > U31 > tt > cons > mark₁ > active₁
isNatIList > U31 > tt > cons > mark₁ > nil
isNatIList > U31 > tt > U72 > s > mark₁ > active₁
isNatIList > U31 > tt > U72 > s > mark₁ > nil
isNatIList > U41 > U42 > tt > cons > mark₁ > active₁
isNatIList > U41 > U42 > tt > cons > mark₁ > nil
isNatIList > U41 > U42 > tt > U72 > s > mark₁ > active₁
isNatIList > U41 > U42 > tt > U72 > s > mark₁ > nil
isNatIList > isNatList > U51 > U52 > tt > cons > mark₁ > active₁
isNatIList > isNatList > U51 > U52 > tt > cons > mark₁ > nil
isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > active₁
isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > nil
isNatIList > isNatList > U61 > U62 > tt > cons > mark₁ > active₁
isNatIList > isNatList > U61 > U62 > tt > cons > mark₁ > nil
isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > active₁
isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > nil
isNatIList > isNat > U11 > tt > cons > mark₁ > active₁
isNatIList > isNat > U11 > tt > cons > mark₁ > nil
isNatIList > isNat > U11 > tt > U72 > s > mark₁ > active₁
isNatIList > isNat > U11 > tt > U72 > s > mark₁ > nil
isNatIList > isNat > U21 > tt > cons > mark₁ > active₁
isNatIList > isNat > U21 > tt > cons > mark₁ > nil
isNatIList > isNat > U21 > tt > U72 > s > mark₁ > active₁
isNatIList > isNat > U21 > tt > U72 > s > mark₁ > nil
length > 0 > tt > cons > mark₁ > active₁
length > 0 > tt > cons > mark₁ > nil
length > 0 > tt > U72 > s > mark₁ > active₁
length > 0 > tt > U72 > s > mark₁ > nil
length > U11 > tt > cons > mark₁ > active₁
length > U11 > tt > cons > mark₁ > nil
length > U11 > tt > U72 > s > mark₁ > active₁
length > U11 > tt > U72 > s > mark₁ > nil
length > isNatList > U51 > U52 > tt > cons > mark₁ > active₁
length > isNatList > U51 > U52 > tt > cons > mark₁ > nil
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > active₁
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > nil
length > isNatList > U61 > U62 > tt > cons > mark₁ > active₁
length > isNatList > U61 > U62 > tt > cons > mark₁ > nil
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > active₁
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > nil
length > U71 > U72 > s > mark₁ > active₁
length > U71 > U72 > s > mark₁ > nil
take > U61 > U62 > tt > cons > mark₁ > active₁
take > U61 > U62 > tt > cons > mark₁ > nil
take > U61 > U62 > tt > U72 > s > mark₁ > active₁
take > U61 > U62 > tt > U72 > s > mark₁ > nil
take > isNat > U11 > tt > cons > mark₁ > active₁
take > isNat > U11 > tt > cons > mark₁ > nil
take > isNat > U11 > tt > U72 > s > mark₁ > active₁
take > isNat > U11 > tt > U72 > s > mark₁ > nil
take > isNat > U21 > tt > cons > mark₁ > active₁
take > isNat > U21 > tt > cons > mark₁ > nil
take > isNat > U21 > tt > U72 > s > mark₁ > active₁
take > isNat > U21 > tt > U72 > s > mark₁ > nil
take > U81 > mark₁ > active₁
take > U81 > mark₁ > nil
take > U91 > U92 > U93 > cons > mark₁ > active₁
take > U91 > U92 > U93 > cons > mark₁ > nil

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(128) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(129) PisEmptyProof (EQUIVALENT transformation)

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

(130) TRUE

(131) Obligation:

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

U42¹(active(X)) → U42¹(X)
U42¹(mark(X)) → U42¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(132) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(mark(X)) → U42¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U42¹(x1)  =  U42¹(x1)
active(x1)  =  x1
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > cons > U42^1₁
zeros > mark₁ > tt > U42^1₁
isNatList > U51 > U52 > mark₁ > cons > U42^1₁
isNatList > U51 > U52 > mark₁ > tt > U42^1₁
isNatList > U61 > U62 > mark₁ > cons > U42^1₁
isNatList > U61 > U62 > mark₁ > tt > U42^1₁
isNat > U11 > mark₁ > cons > U42^1₁
isNat > U11 > mark₁ > tt > U42^1₁
isNat > U21 > mark₁ > cons > U42^1₁
isNat > U21 > mark₁ > tt > U42^1₁
length > 0 > isNatIList₁ > U31 > mark₁ > cons > U42^1₁
length > 0 > isNatIList₁ > U31 > mark₁ > tt > U42^1₁
length > 0 > isNatIList₁ > U41 > U42 > mark₁ > cons > U42^1₁
length > 0 > isNatIList₁ > U41 > U42 > mark₁ > tt > U42^1₁
length > U11 > mark₁ > cons > U42^1₁
length > U11 > mark₁ > tt > U42^1₁
length > U71 > U72 > s > mark₁ > cons > U42^1₁
length > U71 > U72 > s > mark₁ > tt > U42^1₁
take > isNatIList₁ > U31 > mark₁ > cons > U42^1₁
take > isNatIList₁ > U31 > mark₁ > tt > U42^1₁
take > isNatIList₁ > U41 > U42 > mark₁ > cons > U42^1₁
take > isNatIList₁ > U41 > U42 > mark₁ > tt > U42^1₁
take > U61 > U62 > mark₁ > cons > U42^1₁
take > U61 > U62 > mark₁ > tt > U42^1₁
take > U81 > mark₁ > cons > U42^1₁
take > U81 > mark₁ > tt > U42^1₁
take > U81 > nil > tt > U42^1₁
take > U91 > U92 > U93 > mark₁ > cons > U42^1₁
take > U91 > U92 > U93 > mark₁ > tt > U42^1₁

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(133) Obligation:

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

U42¹(active(X)) → U42¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(134) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(active(X)) → U42¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U42¹(x1)  =  U42¹(x1)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > tt > cons > mark₁ > active₁
zeros > tt > cons > mark₁ > nil
zeros > tt > U72 > s > mark₁ > active₁
zeros > tt > U72 > s > mark₁ > nil
isNatIList > U31 > tt > cons > mark₁ > active₁
isNatIList > U31 > tt > cons > mark₁ > nil
isNatIList > U31 > tt > U72 > s > mark₁ > active₁
isNatIList > U31 > tt > U72 > s > mark₁ > nil
isNatIList > U41 > U42 > tt > cons > mark₁ > active₁
isNatIList > U41 > U42 > tt > cons > mark₁ > nil
isNatIList > U41 > U42 > tt > U72 > s > mark₁ > active₁
isNatIList > U41 > U42 > tt > U72 > s > mark₁ > nil
isNatIList > isNatList > U51 > U52 > tt > cons > mark₁ > active₁
isNatIList > isNatList > U51 > U52 > tt > cons > mark₁ > nil
isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > active₁
isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > nil
isNatIList > isNatList > U61 > U62 > tt > cons > mark₁ > active₁
isNatIList > isNatList > U61 > U62 > tt > cons > mark₁ > nil
isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > active₁
isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > nil
isNatIList > isNat > U11 > tt > cons > mark₁ > active₁
isNatIList > isNat > U11 > tt > cons > mark₁ > nil
isNatIList > isNat > U11 > tt > U72 > s > mark₁ > active₁
isNatIList > isNat > U11 > tt > U72 > s > mark₁ > nil
isNatIList > isNat > U21 > tt > cons > mark₁ > active₁
isNatIList > isNat > U21 > tt > cons > mark₁ > nil
isNatIList > isNat > U21 > tt > U72 > s > mark₁ > active₁
isNatIList > isNat > U21 > tt > U72 > s > mark₁ > nil
length > 0 > tt > cons > mark₁ > active₁
length > 0 > tt > cons > mark₁ > nil
length > 0 > tt > U72 > s > mark₁ > active₁
length > 0 > tt > U72 > s > mark₁ > nil
length > U11 > tt > cons > mark₁ > active₁
length > U11 > tt > cons > mark₁ > nil
length > U11 > tt > U72 > s > mark₁ > active₁
length > U11 > tt > U72 > s > mark₁ > nil
length > isNatList > U51 > U52 > tt > cons > mark₁ > active₁
length > isNatList > U51 > U52 > tt > cons > mark₁ > nil
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > active₁
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > nil
length > isNatList > U61 > U62 > tt > cons > mark₁ > active₁
length > isNatList > U61 > U62 > tt > cons > mark₁ > nil
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > active₁
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > nil
length > U71 > U72 > s > mark₁ > active₁
length > U71 > U72 > s > mark₁ > nil
take > U61 > U62 > tt > cons > mark₁ > active₁
take > U61 > U62 > tt > cons > mark₁ > nil
take > U61 > U62 > tt > U72 > s > mark₁ > active₁
take > U61 > U62 > tt > U72 > s > mark₁ > nil
take > isNat > U11 > tt > cons > mark₁ > active₁
take > isNat > U11 > tt > cons > mark₁ > nil
take > isNat > U11 > tt > U72 > s > mark₁ > active₁
take > isNat > U11 > tt > U72 > s > mark₁ > nil
take > isNat > U21 > tt > cons > mark₁ > active₁
take > isNat > U21 > tt > cons > mark₁ > nil
take > isNat > U21 > tt > U72 > s > mark₁ > active₁
take > isNat > U21 > tt > U72 > s > mark₁ > nil
take > U81 > mark₁ > active₁
take > U81 > mark₁ > nil
take > U91 > U92 > U93 > cons > mark₁ > active₁
take > U91 > U92 > U93 > cons > mark₁ > nil

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(135) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(136) PisEmptyProof (EQUIVALENT transformation)

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

(137) TRUE

(138) Obligation:

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(139) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2)  =  U41¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList(x1)
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U51 > U52 > mark₁ > tt > active₁ > U41^1₂
zeros > cons > U51 > U52 > mark₁ > tt > active₁ > s
zeros > cons > U51 > U52 > mark₁ > tt > active₁ > nil
zeros > cons > U71 > U72 > mark₁ > tt > active₁ > U41^1₂
zeros > cons > U71 > U72 > mark₁ > tt > active₁ > s
zeros > cons > U71 > U72 > mark₁ > tt > active₁ > nil
zeros > cons > U71 > isNat > U11 > mark₁ > tt > active₁ > U41^1₂
zeros > cons > U71 > isNat > U11 > mark₁ > tt > active₁ > s
zeros > cons > U71 > isNat > U11 > mark₁ > tt > active₁ > nil
zeros > cons > U71 > isNat > U21 > mark₁ > tt > active₁ > U41^1₂
zeros > cons > U71 > isNat > U21 > mark₁ > tt > active₁ > s
zeros > cons > U71 > isNat > U21 > mark₁ > tt > active₁ > nil
zeros > 0 > tt > active₁ > U41^1₂
zeros > 0 > tt > active₁ > s
zeros > 0 > tt > active₁ > nil
isNatIList > U31 > mark₁ > tt > active₁ > U41^1₂
isNatIList > U31 > mark₁ > tt > active₁ > s
isNatIList > U31 > mark₁ > tt > active₁ > nil
isNatIList > U41 > U42 > mark₁ > tt > active₁ > U41^1₂
isNatIList > U41 > U42 > mark₁ > tt > active₁ > s
isNatIList > U41 > U42 > mark₁ > tt > active₁ > nil
isNatIList > isNatList₁ > U51 > U52 > mark₁ > tt > active₁ > U41^1₂
isNatIList > isNatList₁ > U51 > U52 > mark₁ > tt > active₁ > s
isNatIList > isNatList₁ > U51 > U52 > mark₁ > tt > active₁ > nil
isNatIList > isNatList₁ > isNat > U11 > mark₁ > tt > active₁ > U41^1₂
isNatIList > isNatList₁ > isNat > U11 > mark₁ > tt > active₁ > s
isNatIList > isNatList₁ > isNat > U11 > mark₁ > tt > active₁ > nil
isNatIList > isNatList₁ > isNat > U21 > mark₁ > tt > active₁ > U41^1₂
isNatIList > isNatList₁ > isNat > U21 > mark₁ > tt > active₁ > s
isNatIList > isNatList₁ > isNat > U21 > mark₁ > tt > active₁ > nil
length > 0 > tt > active₁ > U41^1₂
length > 0 > tt > active₁ > s
length > 0 > tt > active₁ > nil
length > isNatList₁ > U51 > U52 > mark₁ > tt > active₁ > U41^1₂
length > isNatList₁ > U51 > U52 > mark₁ > tt > active₁ > s
length > isNatList₁ > U51 > U52 > mark₁ > tt > active₁ > nil
length > isNatList₁ > isNat > U11 > mark₁ > tt > active₁ > U41^1₂
length > isNatList₁ > isNat > U11 > mark₁ > tt > active₁ > s
length > isNatList₁ > isNat > U11 > mark₁ > tt > active₁ > nil
length > isNatList₁ > isNat > U21 > mark₁ > tt > active₁ > U41^1₂
length > isNatList₁ > isNat > U21 > mark₁ > tt > active₁ > s
length > isNatList₁ > isNat > U21 > mark₁ > tt > active₁ > nil
length > U71 > U72 > mark₁ > tt > active₁ > U41^1₂
length > U71 > U72 > mark₁ > tt > active₁ > s
length > U71 > U72 > mark₁ > tt > active₁ > nil
length > U71 > isNat > U11 > mark₁ > tt > active₁ > U41^1₂
length > U71 > isNat > U11 > mark₁ > tt > active₁ > s
length > U71 > isNat > U11 > mark₁ > tt > active₁ > nil
length > U71 > isNat > U21 > mark₁ > tt > active₁ > U41^1₂
length > U71 > isNat > U21 > mark₁ > tt > active₁ > s
length > U71 > isNat > U21 > mark₁ > tt > active₁ > nil
take > U61 > U62 > mark₁ > tt > active₁ > U41^1₂
take > U61 > U62 > mark₁ > tt > active₁ > s
take > U61 > U62 > mark₁ > tt > active₁ > nil
take > U81 > mark₁ > tt > active₁ > U41^1₂
take > U81 > mark₁ > tt > active₁ > s
take > U81 > mark₁ > tt > active₁ > nil
take > U91 > U92 > U93 > cons > U51 > U52 > mark₁ > tt > active₁ > U41^1₂
take > U91 > U92 > U93 > cons > U51 > U52 > mark₁ > tt > active₁ > s
take > U91 > U92 > U93 > cons > U51 > U52 > mark₁ > tt > active₁ > nil
take > U91 > U92 > U93 > cons > U71 > U72 > mark₁ > tt > active₁ > U41^1₂
take > U91 > U92 > U93 > cons > U71 > U72 > mark₁ > tt > active₁ > s
take > U91 > U92 > U93 > cons > U71 > U72 > mark₁ > tt > active₁ > nil
take > U91 > U92 > U93 > cons > U71 > isNat > U11 > mark₁ > tt > active₁ > U41^1₂
take > U91 > U92 > U93 > cons > U71 > isNat > U11 > mark₁ > tt > active₁ > s
take > U91 > U92 > U93 > cons > U71 > isNat > U11 > mark₁ > tt > active₁ > nil
take > U91 > U92 > U93 > cons > U71 > isNat > U21 > mark₁ > tt > active₁ > U41^1₂
take > U91 > U92 > U93 > cons > U71 > isNat > U21 > mark₁ > tt > active₁ > s
take > U91 > U92 > U93 > cons > U71 > isNat > U21 > mark₁ > tt > active₁ > nil

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(140) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(141) PisEmptyProof (EQUIVALENT transformation)

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

(142) TRUE

(143) Obligation:

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

U31¹(active(X)) → U31¹(X)
U31¹(mark(X)) → U31¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(144) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(mark(X)) → U31¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1)  =  U31¹(x1)
active(x1)  =  x1
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > cons > U31^1₁
zeros > mark₁ > tt > U31^1₁
isNatList > U51 > U52 > mark₁ > cons > U31^1₁
isNatList > U51 > U52 > mark₁ > tt > U31^1₁
isNatList > U61 > U62 > mark₁ > cons > U31^1₁
isNatList > U61 > U62 > mark₁ > tt > U31^1₁
isNat > U11 > mark₁ > cons > U31^1₁
isNat > U11 > mark₁ > tt > U31^1₁
isNat > U21 > mark₁ > cons > U31^1₁
isNat > U21 > mark₁ > tt > U31^1₁
length > 0 > isNatIList₁ > U31 > mark₁ > cons > U31^1₁
length > 0 > isNatIList₁ > U31 > mark₁ > tt > U31^1₁
length > 0 > isNatIList₁ > U41 > U42 > mark₁ > cons > U31^1₁
length > 0 > isNatIList₁ > U41 > U42 > mark₁ > tt > U31^1₁
length > U11 > mark₁ > cons > U31^1₁
length > U11 > mark₁ > tt > U31^1₁
length > U71 > U72 > s > mark₁ > cons > U31^1₁
length > U71 > U72 > s > mark₁ > tt > U31^1₁
take > isNatIList₁ > U31 > mark₁ > cons > U31^1₁
take > isNatIList₁ > U31 > mark₁ > tt > U31^1₁
take > isNatIList₁ > U41 > U42 > mark₁ > cons > U31^1₁
take > isNatIList₁ > U41 > U42 > mark₁ > tt > U31^1₁
take > U61 > U62 > mark₁ > cons > U31^1₁
take > U61 > U62 > mark₁ > tt > U31^1₁
take > U81 > mark₁ > cons > U31^1₁
take > U81 > mark₁ > tt > U31^1₁
take > U81 > nil > tt > U31^1₁
take > U91 > U92 > U93 > mark₁ > cons > U31^1₁
take > U91 > U92 > U93 > mark₁ > tt > U31^1₁

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(145) Obligation:

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

U31¹(active(X)) → U31¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(146) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(active(X)) → U31¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1)  =  U31¹(x1)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > tt > cons > mark₁ > active₁
zeros > tt > cons > mark₁ > nil
zeros > tt > U72 > s > mark₁ > active₁
zeros > tt > U72 > s > mark₁ > nil
isNatIList > U31 > tt > cons > mark₁ > active₁
isNatIList > U31 > tt > cons > mark₁ > nil
isNatIList > U31 > tt > U72 > s > mark₁ > active₁
isNatIList > U31 > tt > U72 > s > mark₁ > nil
isNatIList > U41 > U42 > tt > cons > mark₁ > active₁
isNatIList > U41 > U42 > tt > cons > mark₁ > nil
isNatIList > U41 > U42 > tt > U72 > s > mark₁ > active₁
isNatIList > U41 > U42 > tt > U72 > s > mark₁ > nil
isNatIList > isNatList > U51 > U52 > tt > cons > mark₁ > active₁
isNatIList > isNatList > U51 > U52 > tt > cons > mark₁ > nil
isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > active₁
isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > nil
isNatIList > isNatList > U61 > U62 > tt > cons > mark₁ > active₁
isNatIList > isNatList > U61 > U62 > tt > cons > mark₁ > nil
isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > active₁
isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > nil
isNatIList > isNat > U11 > tt > cons > mark₁ > active₁
isNatIList > isNat > U11 > tt > cons > mark₁ > nil
isNatIList > isNat > U11 > tt > U72 > s > mark₁ > active₁
isNatIList > isNat > U11 > tt > U72 > s > mark₁ > nil
isNatIList > isNat > U21 > tt > cons > mark₁ > active₁
isNatIList > isNat > U21 > tt > cons > mark₁ > nil
isNatIList > isNat > U21 > tt > U72 > s > mark₁ > active₁
isNatIList > isNat > U21 > tt > U72 > s > mark₁ > nil
length > 0 > tt > cons > mark₁ > active₁
length > 0 > tt > cons > mark₁ > nil
length > 0 > tt > U72 > s > mark₁ > active₁
length > 0 > tt > U72 > s > mark₁ > nil
length > U11 > tt > cons > mark₁ > active₁
length > U11 > tt > cons > mark₁ > nil
length > U11 > tt > U72 > s > mark₁ > active₁
length > U11 > tt > U72 > s > mark₁ > nil
length > isNatList > U51 > U52 > tt > cons > mark₁ > active₁
length > isNatList > U51 > U52 > tt > cons > mark₁ > nil
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > active₁
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > nil
length > isNatList > U61 > U62 > tt > cons > mark₁ > active₁
length > isNatList > U61 > U62 > tt > cons > mark₁ > nil
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > active₁
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > nil
length > U71 > U72 > s > mark₁ > active₁
length > U71 > U72 > s > mark₁ > nil
take > U61 > U62 > tt > cons > mark₁ > active₁
take > U61 > U62 > tt > cons > mark₁ > nil
take > U61 > U62 > tt > U72 > s > mark₁ > active₁
take > U61 > U62 > tt > U72 > s > mark₁ > nil
take > isNat > U11 > tt > cons > mark₁ > active₁
take > isNat > U11 > tt > cons > mark₁ > nil
take > isNat > U11 > tt > U72 > s > mark₁ > active₁
take > isNat > U11 > tt > U72 > s > mark₁ > nil
take > isNat > U21 > tt > cons > mark₁ > active₁
take > isNat > U21 > tt > cons > mark₁ > nil
take > isNat > U21 > tt > U72 > s > mark₁ > active₁
take > isNat > U21 > tt > U72 > s > mark₁ > nil
take > U81 > mark₁ > active₁
take > U81 > mark₁ > nil
take > U91 > U92 > U93 > cons > mark₁ > active₁
take > U91 > U92 > U93 > cons > mark₁ > nil

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(147) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(148) PisEmptyProof (EQUIVALENT transformation)

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

(149) TRUE

(150) Obligation:

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

U21¹(active(X)) → U21¹(X)
U21¹(mark(X)) → U21¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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.

U21¹(mark(X)) → U21¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1)  =  U21¹(x1)
active(x1)  =  x1
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > cons > U21^1₁
zeros > mark₁ > tt > U21^1₁
isNatList > U51 > U52 > mark₁ > cons > U21^1₁
isNatList > U51 > U52 > mark₁ > tt > U21^1₁
isNatList > U61 > U62 > mark₁ > cons > U21^1₁
isNatList > U61 > U62 > mark₁ > tt > U21^1₁
isNat > U11 > mark₁ > cons > U21^1₁
isNat > U11 > mark₁ > tt > U21^1₁
isNat > U21 > mark₁ > cons > U21^1₁
isNat > U21 > mark₁ > tt > U21^1₁
length > 0 > isNatIList₁ > U31 > mark₁ > cons > U21^1₁
length > 0 > isNatIList₁ > U31 > mark₁ > tt > U21^1₁
length > 0 > isNatIList₁ > U41 > U42 > mark₁ > cons > U21^1₁
length > 0 > isNatIList₁ > U41 > U42 > mark₁ > tt > U21^1₁
length > U11 > mark₁ > cons > U21^1₁
length > U11 > mark₁ > tt > U21^1₁
length > U71 > U72 > s > mark₁ > cons > U21^1₁
length > U71 > U72 > s > mark₁ > tt > U21^1₁
take > isNatIList₁ > U31 > mark₁ > cons > U21^1₁
take > isNatIList₁ > U31 > mark₁ > tt > U21^1₁
take > isNatIList₁ > U41 > U42 > mark₁ > cons > U21^1₁
take > isNatIList₁ > U41 > U42 > mark₁ > tt > U21^1₁
take > U61 > U62 > mark₁ > cons > U21^1₁
take > U61 > U62 > mark₁ > tt > U21^1₁
take > U81 > mark₁ > cons > U21^1₁
take > U81 > mark₁ > tt > U21^1₁
take > U81 > nil > tt > U21^1₁
take > U91 > U92 > U93 > mark₁ > cons > U21^1₁
take > U91 > U92 > U93 > mark₁ > tt > U21^1₁

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(152) Obligation:

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

U21¹(active(X)) → U21¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(153) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(active(X)) → U21¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1)  =  U21¹(x1)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > tt > cons > mark₁ > active₁
zeros > tt > cons > mark₁ > nil
zeros > tt > U72 > s > mark₁ > active₁
zeros > tt > U72 > s > mark₁ > nil
isNatIList > U31 > tt > cons > mark₁ > active₁
isNatIList > U31 > tt > cons > mark₁ > nil
isNatIList > U31 > tt > U72 > s > mark₁ > active₁
isNatIList > U31 > tt > U72 > s > mark₁ > nil
isNatIList > U41 > U42 > tt > cons > mark₁ > active₁
isNatIList > U41 > U42 > tt > cons > mark₁ > nil
isNatIList > U41 > U42 > tt > U72 > s > mark₁ > active₁
isNatIList > U41 > U42 > tt > U72 > s > mark₁ > nil
isNatIList > isNatList > U51 > U52 > tt > cons > mark₁ > active₁
isNatIList > isNatList > U51 > U52 > tt > cons > mark₁ > nil
isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > active₁
isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > nil
isNatIList > isNatList > U61 > U62 > tt > cons > mark₁ > active₁
isNatIList > isNatList > U61 > U62 > tt > cons > mark₁ > nil
isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > active₁
isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > nil
isNatIList > isNat > U11 > tt > cons > mark₁ > active₁
isNatIList > isNat > U11 > tt > cons > mark₁ > nil
isNatIList > isNat > U11 > tt > U72 > s > mark₁ > active₁
isNatIList > isNat > U11 > tt > U72 > s > mark₁ > nil
isNatIList > isNat > U21 > tt > cons > mark₁ > active₁
isNatIList > isNat > U21 > tt > cons > mark₁ > nil
isNatIList > isNat > U21 > tt > U72 > s > mark₁ > active₁
isNatIList > isNat > U21 > tt > U72 > s > mark₁ > nil
length > 0 > tt > cons > mark₁ > active₁
length > 0 > tt > cons > mark₁ > nil
length > 0 > tt > U72 > s > mark₁ > active₁
length > 0 > tt > U72 > s > mark₁ > nil
length > U11 > tt > cons > mark₁ > active₁
length > U11 > tt > cons > mark₁ > nil
length > U11 > tt > U72 > s > mark₁ > active₁
length > U11 > tt > U72 > s > mark₁ > nil
length > isNatList > U51 > U52 > tt > cons > mark₁ > active₁
length > isNatList > U51 > U52 > tt > cons > mark₁ > nil
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > active₁
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > nil
length > isNatList > U61 > U62 > tt > cons > mark₁ > active₁
length > isNatList > U61 > U62 > tt > cons > mark₁ > nil
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > active₁
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > nil
length > U71 > U72 > s > mark₁ > active₁
length > U71 > U72 > s > mark₁ > nil
take > U61 > U62 > tt > cons > mark₁ > active₁
take > U61 > U62 > tt > cons > mark₁ > nil
take > U61 > U62 > tt > U72 > s > mark₁ > active₁
take > U61 > U62 > tt > U72 > s > mark₁ > nil
take > isNat > U11 > tt > cons > mark₁ > active₁
take > isNat > U11 > tt > cons > mark₁ > nil
take > isNat > U11 > tt > U72 > s > mark₁ > active₁
take > isNat > U11 > tt > U72 > s > mark₁ > nil
take > isNat > U21 > tt > cons > mark₁ > active₁
take > isNat > U21 > tt > cons > mark₁ > nil
take > isNat > U21 > tt > U72 > s > mark₁ > active₁
take > isNat > U21 > tt > U72 > s > mark₁ > nil
take > U81 > mark₁ > active₁
take > U81 > mark₁ > nil
take > U91 > U92 > U93 > cons > mark₁ > active₁
take > U91 > U92 > U93 > cons > mark₁ > nil

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(154) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(155) PisEmptyProof (EQUIVALENT transformation)

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

(156) TRUE

(157) Obligation:

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

U11¹(active(X)) → U11¹(X)
U11¹(mark(X)) → U11¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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.

U11¹(mark(X)) → U11¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1)  =  U11¹(x1)
active(x1)  =  x1
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > cons > U11^1₁
zeros > mark₁ > tt > U11^1₁
isNatList > U51 > U52 > mark₁ > cons > U11^1₁
isNatList > U51 > U52 > mark₁ > tt > U11^1₁
isNatList > U61 > U62 > mark₁ > cons > U11^1₁
isNatList > U61 > U62 > mark₁ > tt > U11^1₁
isNat > U11 > mark₁ > cons > U11^1₁
isNat > U11 > mark₁ > tt > U11^1₁
isNat > U21 > mark₁ > cons > U11^1₁
isNat > U21 > mark₁ > tt > U11^1₁
length > 0 > isNatIList₁ > U31 > mark₁ > cons > U11^1₁
length > 0 > isNatIList₁ > U31 > mark₁ > tt > U11^1₁
length > 0 > isNatIList₁ > U41 > U42 > mark₁ > cons > U11^1₁
length > 0 > isNatIList₁ > U41 > U42 > mark₁ > tt > U11^1₁
length > U11 > mark₁ > cons > U11^1₁
length > U11 > mark₁ > tt > U11^1₁
length > U71 > U72 > s > mark₁ > cons > U11^1₁
length > U71 > U72 > s > mark₁ > tt > U11^1₁
take > isNatIList₁ > U31 > mark₁ > cons > U11^1₁
take > isNatIList₁ > U31 > mark₁ > tt > U11^1₁
take > isNatIList₁ > U41 > U42 > mark₁ > cons > U11^1₁
take > isNatIList₁ > U41 > U42 > mark₁ > tt > U11^1₁
take > U61 > U62 > mark₁ > cons > U11^1₁
take > U61 > U62 > mark₁ > tt > U11^1₁
take > U81 > mark₁ > cons > U11^1₁
take > U81 > mark₁ > tt > U11^1₁
take > U81 > nil > tt > U11^1₁
take > U91 > U92 > U93 > mark₁ > cons > U11^1₁
take > U91 > U92 > U93 > mark₁ > tt > U11^1₁

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(159) Obligation:

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

U11¹(active(X)) → U11¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(160) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(active(X)) → U11¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1)  =  U11¹(x1)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > tt > cons > mark₁ > active₁
zeros > tt > cons > mark₁ > nil
zeros > tt > U72 > s > mark₁ > active₁
zeros > tt > U72 > s > mark₁ > nil
isNatIList > U31 > tt > cons > mark₁ > active₁
isNatIList > U31 > tt > cons > mark₁ > nil
isNatIList > U31 > tt > U72 > s > mark₁ > active₁
isNatIList > U31 > tt > U72 > s > mark₁ > nil
isNatIList > U41 > U42 > tt > cons > mark₁ > active₁
isNatIList > U41 > U42 > tt > cons > mark₁ > nil
isNatIList > U41 > U42 > tt > U72 > s > mark₁ > active₁
isNatIList > U41 > U42 > tt > U72 > s > mark₁ > nil
isNatIList > isNatList > U51 > U52 > tt > cons > mark₁ > active₁
isNatIList > isNatList > U51 > U52 > tt > cons > mark₁ > nil
isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > active₁
isNatIList > isNatList > U51 > U52 > tt > U72 > s > mark₁ > nil
isNatIList > isNatList > U61 > U62 > tt > cons > mark₁ > active₁
isNatIList > isNatList > U61 > U62 > tt > cons > mark₁ > nil
isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > active₁
isNatIList > isNatList > U61 > U62 > tt > U72 > s > mark₁ > nil
isNatIList > isNat > U11 > tt > cons > mark₁ > active₁
isNatIList > isNat > U11 > tt > cons > mark₁ > nil
isNatIList > isNat > U11 > tt > U72 > s > mark₁ > active₁
isNatIList > isNat > U11 > tt > U72 > s > mark₁ > nil
isNatIList > isNat > U21 > tt > cons > mark₁ > active₁
isNatIList > isNat > U21 > tt > cons > mark₁ > nil
isNatIList > isNat > U21 > tt > U72 > s > mark₁ > active₁
isNatIList > isNat > U21 > tt > U72 > s > mark₁ > nil
length > 0 > tt > cons > mark₁ > active₁
length > 0 > tt > cons > mark₁ > nil
length > 0 > tt > U72 > s > mark₁ > active₁
length > 0 > tt > U72 > s > mark₁ > nil
length > U11 > tt > cons > mark₁ > active₁
length > U11 > tt > cons > mark₁ > nil
length > U11 > tt > U72 > s > mark₁ > active₁
length > U11 > tt > U72 > s > mark₁ > nil
length > isNatList > U51 > U52 > tt > cons > mark₁ > active₁
length > isNatList > U51 > U52 > tt > cons > mark₁ > nil
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > active₁
length > isNatList > U51 > U52 > tt > U72 > s > mark₁ > nil
length > isNatList > U61 > U62 > tt > cons > mark₁ > active₁
length > isNatList > U61 > U62 > tt > cons > mark₁ > nil
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > active₁
length > isNatList > U61 > U62 > tt > U72 > s > mark₁ > nil
length > U71 > U72 > s > mark₁ > active₁
length > U71 > U72 > s > mark₁ > nil
take > U61 > U62 > tt > cons > mark₁ > active₁
take > U61 > U62 > tt > cons > mark₁ > nil
take > U61 > U62 > tt > U72 > s > mark₁ > active₁
take > U61 > U62 > tt > U72 > s > mark₁ > nil
take > isNat > U11 > tt > cons > mark₁ > active₁
take > isNat > U11 > tt > cons > mark₁ > nil
take > isNat > U11 > tt > U72 > s > mark₁ > active₁
take > isNat > U11 > tt > U72 > s > mark₁ > nil
take > isNat > U21 > tt > cons > mark₁ > active₁
take > isNat > U21 > tt > cons > mark₁ > nil
take > isNat > U21 > tt > U72 > s > mark₁ > active₁
take > isNat > U21 > tt > U72 > s > mark₁ > nil
take > U81 > mark₁ > active₁
take > U81 > mark₁ > nil
take > U91 > U92 > U93 > cons > mark₁ > active₁
take > U91 > U92 > U93 > cons > mark₁ > nil

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(161) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(162) PisEmptyProof (EQUIVALENT transformation)

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

(163) TRUE

(164) 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(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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.

CONS(X1, mark(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)  =  x1
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > cons
zeros > mark₁ > 0
zeros > mark₁ > nil
isNatIList > U31 > tt > mark₁ > cons
isNatIList > U31 > tt > mark₁ > 0
isNatIList > U31 > tt > mark₁ > nil
isNatIList > U41 > U42 > tt > mark₁ > cons
isNatIList > U41 > U42 > tt > mark₁ > 0
isNatIList > U41 > U42 > tt > mark₁ > nil
isNatList > U51 > U52 > tt > mark₁ > cons
isNatList > U51 > U52 > tt > mark₁ > 0
isNatList > U51 > U52 > tt > mark₁ > nil
isNatList > U61 > U62 > tt > mark₁ > cons
isNatList > U61 > U62 > tt > mark₁ > 0
isNatList > U61 > U62 > tt > mark₁ > nil
isNat > U11 > tt > mark₁ > cons
isNat > U11 > tt > mark₁ > 0
isNat > U11 > tt > mark₁ > nil
isNat > U21 > tt > mark₁ > cons
isNat > U21 > tt > mark₁ > 0
isNat > U21 > tt > mark₁ > nil
length > U11 > tt > mark₁ > cons
length > U11 > tt > mark₁ > 0
length > U11 > tt > mark₁ > nil
length > U71 > U72 > s > mark₁ > cons
length > U71 > U72 > s > mark₁ > 0
length > U71 > U72 > s > mark₁ > nil
take > U61 > U62 > tt > mark₁ > cons
take > U61 > U62 > tt > mark₁ > 0
take > U61 > U62 > tt > mark₁ > nil
take > U81 > mark₁ > cons
take > U81 > mark₁ > 0
take > U81 > mark₁ > nil
take > U91 > U92 > U93 > mark₁ > cons
take > U91 > U92 > U93 > mark₁ > 0
take > U91 > U92 > U93 > mark₁ > nil

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(166) 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)
CONS(X1, active(X2)) → CONS(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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.

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)  =  CONS(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  x1
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length(x1)
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U51 > U52 > mark₁ > tt
zeros > cons > U51 > U52 > mark₁ > length₁ > 0
zeros > cons > U71 > U72 > mark₁ > tt
zeros > cons > U71 > U72 > mark₁ > length₁ > 0
zeros > cons > U71 > U72 > s
isNatIList > U31 > mark₁ > tt
isNatIList > U31 > mark₁ > length₁ > 0
isNatIList > U41 > U42 > mark₁ > tt
isNatIList > U41 > U42 > mark₁ > length₁ > 0
isNatIList > isNat > U11 > mark₁ > tt
isNatIList > isNat > U11 > mark₁ > length₁ > 0
isNatIList > isNat > U21 > mark₁ > tt
isNatIList > isNat > U21 > mark₁ > length₁ > 0
take > U61 > U62 > mark₁ > tt
take > U61 > U62 > mark₁ > length₁ > 0
take > U81 > nil > mark₁ > tt
take > U81 > nil > mark₁ > length₁ > 0
take > U91 > isNat > U11 > mark₁ > tt
take > U91 > isNat > U11 > mark₁ > length₁ > 0
take > U91 > isNat > U21 > mark₁ > tt
take > U91 > isNat > U21 > mark₁ > length₁ > 0
take > U91 > U92 > U93 > cons > U51 > U52 > mark₁ > tt
take > U91 > U92 > U93 > cons > U51 > U52 > mark₁ > length₁ > 0
take > U91 > U92 > U93 > cons > U71 > U72 > mark₁ > tt
take > U91 > U92 > U93 > cons > U71 > U72 > mark₁ > length₁ > 0
take > U91 > U92 > U93 > cons > U71 > U72 > s

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(168) Obligation:

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

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(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(169) 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)  =  x1
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72
isNat(x1)  =  isNat(x1)
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > 0 > mark₁ > active₁ > cons
zeros > tt > mark₁ > active₁ > cons
zeros > tt > s > active₁ > cons
isNatIList > U31 > tt > mark₁ > active₁ > cons
isNatIList > U31 > tt > s > active₁ > cons
isNatIList > U41 > U42 > tt > mark₁ > active₁ > cons
isNatIList > U41 > U42 > tt > s > active₁ > cons
isNatIList > isNatList > U51 > U52 > tt > mark₁ > active₁ > cons
isNatIList > isNatList > U51 > U52 > tt > s > active₁ > cons
isNatIList > isNatList > U61 > U62 > tt > mark₁ > active₁ > cons
isNatIList > isNatList > U61 > U62 > tt > s > active₁ > cons
isNatIList > isNatList > isNat₁ > U11 > tt > mark₁ > active₁ > cons
isNatIList > isNatList > isNat₁ > U11 > tt > s > active₁ > cons
isNatIList > isNatList > isNat₁ > U21 > tt > mark₁ > active₁ > cons
isNatIList > isNatList > isNat₁ > U21 > tt > s > active₁ > cons
length > 0 > mark₁ > active₁ > cons
length > U71 > U72 > mark₁ > active₁ > cons
length > U71 > U72 > s > active₁ > cons
take > U61 > U62 > tt > mark₁ > active₁ > cons
take > U61 > U62 > tt > s > active₁ > cons
take > isNat₁ > U11 > tt > mark₁ > active₁ > cons
take > isNat₁ > U11 > tt > s > active₁ > cons
take > isNat₁ > U21 > tt > mark₁ > active₁ > cons
take > isNat₁ > U21 > tt > s > active₁ > cons
take > U81 > nil > mark₁ > active₁ > cons
take > U91 > U92 > U93 > mark₁ > active₁ > cons

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(170) Obligation:

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

CONS(X1, active(X2)) → CONS(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(171) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
U41(x1, x2)  =  U41
U42(x1)  =  U42
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  U71
U72(x1, x2)  =  U72(x1)
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92(x1)
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > mark₁ > active₁ > U72₁
zeros > cons > mark₁ > U92₁ > U72₁
isNatIList₁ > U31 > tt > s > mark₁ > active₁ > U72₁
isNatIList₁ > U31 > tt > s > mark₁ > U92₁ > U72₁
isNatIList₁ > U31 > tt > nil > active₁ > U72₁
isNatIList₁ > U31 > tt > U93 > cons > mark₁ > active₁ > U72₁
isNatIList₁ > U31 > tt > U93 > cons > mark₁ > U92₁ > U72₁
isNatIList₁ > U41 > U42 > tt > s > mark₁ > active₁ > U72₁
isNatIList₁ > U41 > U42 > tt > s > mark₁ > U92₁ > U72₁
isNatIList₁ > U41 > U42 > tt > nil > active₁ > U72₁
isNatIList₁ > U41 > U42 > tt > U93 > cons > mark₁ > active₁ > U72₁
isNatIList₁ > U41 > U42 > tt > U93 > cons > mark₁ > U92₁ > U72₁
isNatIList₁ > isNat > U11 > tt > s > mark₁ > active₁ > U72₁
isNatIList₁ > isNat > U11 > tt > s > mark₁ > U92₁ > U72₁
isNatIList₁ > isNat > U11 > tt > nil > active₁ > U72₁
isNatIList₁ > isNat > U11 > tt > U93 > cons > mark₁ > active₁ > U72₁
isNatIList₁ > isNat > U11 > tt > U93 > cons > mark₁ > U92₁ > U72₁
isNatIList₁ > isNat > U21 > tt > s > mark₁ > active₁ > U72₁
isNatIList₁ > isNat > U21 > tt > s > mark₁ > U92₁ > U72₁
isNatIList₁ > isNat > U21 > tt > nil > active₁ > U72₁
isNatIList₁ > isNat > U21 > tt > U93 > cons > mark₁ > active₁ > U72₁
isNatIList₁ > isNat > U21 > tt > U93 > cons > mark₁ > U92₁ > U72₁
isNatList > U51 > U52 > tt > s > mark₁ > active₁ > U72₁
isNatList > U51 > U52 > tt > s > mark₁ > U92₁ > U72₁
isNatList > U51 > U52 > tt > nil > active₁ > U72₁
isNatList > U51 > U52 > tt > U93 > cons > mark₁ > active₁ > U72₁
isNatList > U51 > U52 > tt > U93 > cons > mark₁ > U92₁ > U72₁
isNatList > U61 > U62 > tt > s > mark₁ > active₁ > U72₁
isNatList > U61 > U62 > tt > s > mark₁ > U92₁ > U72₁
isNatList > U61 > U62 > tt > nil > active₁ > U72₁
isNatList > U61 > U62 > tt > U93 > cons > mark₁ > active₁ > U72₁
isNatList > U61 > U62 > tt > U93 > cons > mark₁ > U92₁ > U72₁
isNatList > isNat > U11 > tt > s > mark₁ > active₁ > U72₁
isNatList > isNat > U11 > tt > s > mark₁ > U92₁ > U72₁
isNatList > isNat > U11 > tt > nil > active₁ > U72₁
isNatList > isNat > U11 > tt > U93 > cons > mark₁ > active₁ > U72₁
isNatList > isNat > U11 > tt > U93 > cons > mark₁ > U92₁ > U72₁
isNatList > isNat > U21 > tt > s > mark₁ > active₁ > U72₁
isNatList > isNat > U21 > tt > s > mark₁ > U92₁ > U72₁
isNatList > isNat > U21 > tt > nil > active₁ > U72₁
isNatList > isNat > U21 > tt > U93 > cons > mark₁ > active₁ > U72₁
isNatList > isNat > U21 > tt > U93 > cons > mark₁ > U92₁ > U72₁
length > 0 > mark₁ > active₁ > U72₁
length > 0 > mark₁ > U92₁ > U72₁
length > U71 > isNat > U11 > tt > s > mark₁ > active₁ > U72₁
length > U71 > isNat > U11 > tt > s > mark₁ > U92₁ > U72₁
length > U71 > isNat > U11 > tt > nil > active₁ > U72₁
length > U71 > isNat > U11 > tt > U93 > cons > mark₁ > active₁ > U72₁
length > U71 > isNat > U11 > tt > U93 > cons > mark₁ > U92₁ > U72₁
length > U71 > isNat > U21 > tt > s > mark₁ > active₁ > U72₁
length > U71 > isNat > U21 > tt > s > mark₁ > U92₁ > U72₁
length > U71 > isNat > U21 > tt > nil > active₁ > U72₁
length > U71 > isNat > U21 > tt > U93 > cons > mark₁ > active₁ > U72₁
length > U71 > isNat > U21 > tt > U93 > cons > mark₁ > U92₁ > U72₁
take > U81 > mark₁ > active₁ > U72₁
take > U81 > mark₁ > U92₁ > U72₁
take > U81 > nil > active₁ > U72₁
take > U91 > isNat > U11 > tt > s > mark₁ > active₁ > U72₁
take > U91 > isNat > U11 > tt > s > mark₁ > U92₁ > U72₁
take > U91 > isNat > U11 > tt > nil > active₁ > U72₁
take > U91 > isNat > U11 > tt > U93 > cons > mark₁ > active₁ > U72₁
take > U91 > isNat > U11 > tt > U93 > cons > mark₁ > U92₁ > U72₁
take > U91 > isNat > U21 > tt > s > mark₁ > active₁ > U72₁
take > U91 > isNat > U21 > tt > s > mark₁ > U92₁ > U72₁
take > U91 > isNat > U21 > tt > nil > active₁ > U72₁
take > U91 > isNat > U21 > tt > U93 > cons > mark₁ > active₁ > U72₁
take > U91 > isNat > U21 > tt > U93 > cons > mark₁ > U92₁ > U72₁

The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

(172) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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

(173) PisEmptyProof (EQUIVALENT transformation)

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

(174) TRUE

(175) Obligation:

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

MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(U41(tt, V2)) → MARK(U42(isNatIList(V2)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(U11(X)) → ACTIVE(U11(mark(X)))
ACTIVE(U51(tt, V2)) → MARK(U52(isNatList(V2)))
MARK(U11(X)) → MARK(X)
MARK(U21(X)) → ACTIVE(U21(mark(X)))
ACTIVE(U61(tt, V2)) → MARK(U62(isNatIList(V2)))
MARK(U21(X)) → MARK(X)
MARK(U31(X)) → ACTIVE(U31(mark(X)))
ACTIVE(U71(tt, L, N)) → MARK(U72(isNat(N), L))
MARK(U31(X)) → MARK(X)
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
ACTIVE(U72(tt, L)) → MARK(s(length(L)))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → ACTIVE(U42(mark(X)))
ACTIVE(U91(tt, IL, M, N)) → MARK(U92(isNat(M), IL, M, N))
MARK(U42(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(U92(tt, IL, M, N)) → MARK(U93(isNat(N), IL, M, N))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
ACTIVE(U93(tt, IL, M, N)) → MARK(cons(N, take(M, IL)))
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → ACTIVE(U52(mark(X)))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatList(V1)))
MARK(U52(X)) → MARK(X)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNat(s(V1))) → MARK(U21(isNat(V1)))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
ACTIVE(isNatIList(V)) → MARK(U31(isNatList(V)))
MARK(U61(X1, X2)) → MARK(X1)
MARK(U62(X)) → ACTIVE(U62(mark(X)))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNat(V1), V2))
MARK(U62(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → ACTIVE(U71(mark(X1), X2, X3))
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(isNat(V1), V2))
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(isNatList(take(V1, V2))) → MARK(U61(isNat(V1), V2))
MARK(U72(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(length(cons(N, L))) → MARK(U71(isNatList(L), L, N))
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(take(0, IL)) → MARK(U81(isNatIList(IL)))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(take(s(M), cons(N, IL))) → MARK(U91(isNatIList(IL), IL, M, N))
MARK(length(X)) → MARK(X)
MARK(U81(X)) → ACTIVE(U81(mark(X)))
MARK(U81(X)) → MARK(X)
MARK(U91(X1, X2, X3, X4)) → ACTIVE(U91(mark(X1), X2, X3, X4))
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(U92(X1, X2, X3, X4)) → ACTIVE(U92(mark(X1), X2, X3, X4))
MARK(U92(X1, X2, X3, X4)) → MARK(X1)
MARK(U93(X1, X2, X3, X4)) → ACTIVE(U93(mark(X1), X2, X3, X4))
MARK(U93(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X)) → active(U31(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(U62(X)) → active(U62(mark(X)))
mark(U71(X1, X2, X3)) → active(U71(mark(X1), X2, X3))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(nil) → active(nil)
mark(U91(X1, X2, X3, X4)) → active(U91(mark(X1), X2, X3, X4))
mark(U92(X1, X2, X3, X4)) → active(U92(mark(X1), X2, X3, X4))
mark(U93(X1, X2, X3, X4)) → active(U93(mark(X1), X2, X3, X4))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
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)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(X)
U31(mark(X)) → U31(X)
U31(active(X)) → U31(X)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(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(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, mark(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, mark(X3)) → U71(X1, X2, X3)
U71(active(X1), X2, X3) → U71(X1, X2, X3)
U71(X1, active(X2), X3) → U71(X1, X2, X3)
U71(X1, X2, active(X3)) → U71(X1, X2, X3)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, mark(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, mark(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, mark(X4)) → U91(X1, X2, X3, X4)
U91(active(X1), X2, X3, X4) → U91(X1, X2, X3, X4)
U91(X1, active(X2), X3, X4) → U91(X1, X2, X3, X4)
U91(X1, X2, active(X3), X4) → U91(X1, X2, X3, X4)
U91(X1, X2, X3, active(X4)) → U91(X1, X2, X3, X4)
U92(mark(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, mark(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, mark(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, mark(X4)) → U92(X1, X2, X3, X4)
U92(active(X1), X2, X3, X4) → U92(X1, X2, X3, X4)
U92(X1, active(X2), X3, X4) → U92(X1, X2, X3, X4)
U92(X1, X2, active(X3), X4) → U92(X1, X2, X3, X4)
U92(X1, X2, X3, active(X4)) → U92(X1, X2, X3, X4)
U93(mark(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, mark(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, mark(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, mark(X4)) → U93(X1, X2, X3, X4)
U93(active(X1), X2, X3, X4) → U93(X1, X2, X3, X4)
U93(X1, active(X2), X3, X4) → U93(X1, X2, X3, X4)
U93(X1, X2, active(X3), X4) → U93(X1, X2, X3, X4)
U93(X1, X2, X3, active(X4)) → U93(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)

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