(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(mark(X1), X2) → TAKE(X1, X2)
TAKE(active(X1), 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(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(x1)
U72(x1, x2)  =  U72
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length(x1)
U81(x1)  =  U81(x1)
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Recursive path order with status [RPO].
Precedence:
TAKE2 > U711
zeros > cons > isNatIList > U31 > tt > U72 > s > mark1 > length1 > active1 > U711
zeros > cons > isNatIList > U31 > tt > U72 > s > mark1 > U811 > active1 > U711
zeros > cons > isNatIList > U31 > tt > nil > 0 > mark1 > length1 > active1 > U711
zeros > cons > isNatIList > U31 > tt > nil > 0 > mark1 > U811 > active1 > U711
zeros > cons > isNatIList > U41 > U42 > tt > U72 > s > mark1 > length1 > active1 > U711
zeros > cons > isNatIList > U41 > U42 > tt > U72 > s > mark1 > U811 > active1 > U711
zeros > cons > isNatIList > U41 > U42 > tt > nil > 0 > mark1 > length1 > active1 > U711
zeros > cons > isNatIList > U41 > U42 > tt > nil > 0 > mark1 > U811 > active1 > U711
zeros > cons > isNatIList > isNat > U11 > tt > U72 > s > mark1 > length1 > active1 > U711
zeros > cons > isNatIList > isNat > U11 > tt > U72 > s > mark1 > U811 > active1 > U711
zeros > cons > isNatIList > isNat > U11 > tt > nil > 0 > mark1 > length1 > active1 > U711
zeros > cons > isNatIList > isNat > U11 > tt > nil > 0 > mark1 > U811 > active1 > U711
zeros > cons > isNatIList > isNat > U21 > tt > U72 > s > mark1 > length1 > active1 > U711
zeros > cons > isNatIList > isNat > U21 > tt > U72 > s > mark1 > U811 > active1 > U711
zeros > cons > isNatIList > isNat > U21 > tt > nil > 0 > mark1 > length1 > active1 > U711
zeros > cons > isNatIList > isNat > U21 > tt > nil > 0 > mark1 > U811 > active1 > U711
zeros > cons > isNatList > U51 > U52 > tt > U72 > s > mark1 > length1 > active1 > U711
zeros > cons > isNatList > U51 > U52 > tt > U72 > s > mark1 > U811 > active1 > U711
zeros > cons > isNatList > U51 > U52 > tt > nil > 0 > mark1 > length1 > active1 > U711
zeros > cons > isNatList > U51 > U52 > tt > nil > 0 > mark1 > U811 > active1 > U711
zeros > cons > isNatList > U61 > U62 > tt > U72 > s > mark1 > length1 > active1 > U711
zeros > cons > isNatList > U61 > U62 > tt > U72 > s > mark1 > U811 > active1 > U711
zeros > cons > isNatList > U61 > U62 > tt > nil > 0 > mark1 > length1 > active1 > U711
zeros > cons > isNatList > U61 > U62 > tt > nil > 0 > mark1 > U811 > active1 > U711
zeros > cons > isNatList > isNat > U11 > tt > U72 > s > mark1 > length1 > active1 > U711
zeros > cons > isNatList > isNat > U11 > tt > U72 > s > mark1 > U811 > active1 > U711
zeros > cons > isNatList > isNat > U11 > tt > nil > 0 > mark1 > length1 > active1 > U711
zeros > cons > isNatList > isNat > U11 > tt > nil > 0 > mark1 > U811 > active1 > U711
zeros > cons > isNatList > isNat > U21 > tt > U72 > s > mark1 > length1 > active1 > U711
zeros > cons > isNatList > isNat > U21 > tt > U72 > s > mark1 > U811 > active1 > U711
zeros > cons > isNatList > isNat > U21 > tt > nil > 0 > mark1 > length1 > active1 > U711
zeros > cons > isNatList > isNat > U21 > tt > nil > 0 > mark1 > U811 > active1 > U711
take > U91 > U92 > U93 > cons > isNatIList > U31 > tt > U72 > s > mark1 > length1 > active1 > U711
take > U91 > U92 > U93 > cons > isNatIList > U31 > tt > U72 > s > mark1 > U811 > active1 > U711
take > U91 > U92 > U93 > cons > isNatIList > U31 > tt > nil > 0 > mark1 > length1 > active1 > U711
take > U91 > U92 > U93 > cons > isNatIList > U31 > tt > nil > 0 > mark1 > U811 > active1 > U711
take > U91 > U92 > U93 > cons > isNatIList > U41 > U42 > tt > U72 > s > mark1 > length1 > active1 > U711
take > U91 > U92 > U93 > cons > isNatIList > U41 > U42 > tt > U72 > s > mark1 > U811 > active1 > U711
take > U91 > U92 > U93 > cons > isNatIList > U41 > U42 > tt > nil > 0 > mark1 > length1 > active1 > U711
take > U91 > U92 > U93 > cons > isNatIList > U41 > U42 > tt > nil > 0 > mark1 > U811 > active1 > U711
take > U91 > U92 > U93 > cons > isNatIList > isNat > U11 > tt > U72 > s > mark1 > length1 > active1 > U711
take > U91 > U92 > U93 > cons > isNatIList > isNat > U11 > tt > U72 > s > mark1 > U811 > active1 > U711
take > U91 > U92 > U93 > cons > isNatIList > isNat > U11 > tt > nil > 0 > mark1 > length1 > active1 > U711
take > U91 > U92 > U93 > cons > isNatIList > isNat > U11 > tt > nil > 0 > mark1 > U811 > active1 > U711
take > U91 > U92 > U93 > cons > isNatIList > isNat > U21 > tt > U72 > s > mark1 > length1 > active1 > U711
take > U91 > U92 > U93 > cons > isNatIList > isNat > U21 > tt > U72 > s > mark1 > U811 > active1 > U711
take > U91 > U92 > U93 > cons > isNatIList > isNat > U21 > tt > nil > 0 > mark1 > length1 > active1 > U711
take > U91 > U92 > U93 > cons > isNatIList > isNat > U21 > tt > nil > 0 > mark1 > U811 > active1 > U711
take > U91 > U92 > U93 > cons > isNatList > U51 > U52 > tt > U72 > s > mark1 > length1 > active1 > U711
take > U91 > U92 > U93 > cons > isNatList > U51 > U52 > tt > U72 > s > mark1 > U811 > active1 > U711
take > U91 > U92 > U93 > cons > isNatList > U51 > U52 > tt > nil > 0 > mark1 > length1 > active1 > U711
take > U91 > U92 > U93 > cons > isNatList > U51 > U52 > tt > nil > 0 > mark1 > U811 > active1 > U711
take > U91 > U92 > U93 > cons > isNatList > U61 > U62 > tt > U72 > s > mark1 > length1 > active1 > U711
take > U91 > U92 > U93 > cons > isNatList > U61 > U62 > tt > U72 > s > mark1 > U811 > active1 > U711
take > U91 > U92 > U93 > cons > isNatList > U61 > U62 > tt > nil > 0 > mark1 > length1 > active1 > U711
take > U91 > U92 > U93 > cons > isNatList > U61 > U62 > tt > nil > 0 > mark1 > U811 > active1 > U711
take > U91 > U92 > U93 > cons > isNatList > isNat > U11 > tt > U72 > s > mark1 > length1 > active1 > U711
take > U91 > U92 > U93 > cons > isNatList > isNat > U11 > tt > U72 > s > mark1 > U811 > active1 > U711
take > U91 > U92 > U93 > cons > isNatList > isNat > U11 > tt > nil > 0 > mark1 > length1 > active1 > U711
take > U91 > U92 > U93 > cons > isNatList > isNat > U11 > tt > nil > 0 > mark1 > U811 > active1 > U711
take > U91 > U92 > U93 > cons > isNatList > isNat > U21 > tt > U72 > s > mark1 > length1 > active1 > U711
take > U91 > U92 > U93 > cons > isNatList > isNat > U21 > tt > U72 > s > mark1 > U811 > active1 > U711
take > U91 > U92 > U93 > cons > isNatList > isNat > U21 > tt > nil > 0 > mark1 > length1 > active1 > U711
take > U91 > U92 > U93 > cons > isNatList > isNat > U21 > tt > nil > 0 > mark1 > U811 > active1 > U711

Status:
TAKE2: [1,2]
mark1: [1]
active1: multiset
zeros: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: multiset
U31: []
U41: []
U42: multiset
isNatIList: multiset
U51: []
U52: []
isNatList: multiset
U61: []
U62: []
U711: [1]
U72: []
isNat: multiset
s: []
length1: multiset
U811: multiset
nil: multiset
U91: multiset
U92: []
U93: []
take: []

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:
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.

(8) PisEmptyProof (EQUIVALENT transformation)

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

(9) TRUE

(10) 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.

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U931(X1, X2, X3, mark(X4)) → U931(X1, X2, X3, X4)
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)  =  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
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(x1)

Recursive path order with status [RPO].
Precedence:
zeros > 0 > isNatIList > U31 > mark1 > active1 > cons > take1
zeros > 0 > isNatIList > U31 > mark1 > active1 > nil > take1
zeros > 0 > isNatIList > U31 > tt > active1 > cons > take1
zeros > 0 > isNatIList > U31 > tt > active1 > nil > take1
zeros > 0 > isNatIList > U41 > U42 > mark1 > active1 > cons > take1
zeros > 0 > isNatIList > U41 > U42 > mark1 > active1 > nil > take1
zeros > 0 > isNatIList > U41 > U42 > tt > active1 > cons > take1
zeros > 0 > isNatIList > U41 > U42 > tt > active1 > nil > take1
zeros > 0 > U81 > mark1 > active1 > cons > take1
zeros > 0 > U81 > mark1 > active1 > nil > take1
isNat1 > U11 > mark1 > active1 > cons > take1
isNat1 > U11 > mark1 > active1 > nil > take1
isNat1 > U11 > tt > active1 > cons > take1
isNat1 > U11 > tt > active1 > nil > take1
isNat1 > U21 > mark1 > active1 > cons > take1
isNat1 > U21 > mark1 > active1 > nil > take1
isNat1 > U21 > tt > active1 > cons > take1
isNat1 > U21 > tt > active1 > nil > take1
isNat1 > isNatList > U51 > U52 > mark1 > active1 > cons > take1
isNat1 > isNatList > U51 > U52 > mark1 > active1 > nil > take1
isNat1 > isNatList > U51 > U52 > tt > active1 > cons > take1
isNat1 > isNatList > U51 > U52 > tt > active1 > nil > take1
isNat1 > isNatList > U61 > isNatIList > U31 > mark1 > active1 > cons > take1
isNat1 > isNatList > U61 > isNatIList > U31 > mark1 > active1 > nil > take1
isNat1 > isNatList > U61 > isNatIList > U31 > tt > active1 > cons > take1
isNat1 > isNatList > U61 > isNatIList > U31 > tt > active1 > nil > take1
isNat1 > isNatList > U61 > isNatIList > U41 > U42 > mark1 > active1 > cons > take1
isNat1 > isNatList > U61 > isNatIList > U41 > U42 > mark1 > active1 > nil > take1
isNat1 > isNatList > U61 > isNatIList > U41 > U42 > tt > active1 > cons > take1
isNat1 > isNatList > U61 > isNatIList > U41 > U42 > tt > active1 > nil > take1
isNat1 > isNatList > U61 > U62 > mark1 > active1 > cons > take1
isNat1 > isNatList > U61 > U62 > mark1 > active1 > nil > take1
isNat1 > isNatList > U61 > U62 > tt > active1 > cons > take1
isNat1 > isNatList > U61 > U62 > tt > active1 > nil > take1
length > 0 > isNatIList > U31 > mark1 > active1 > cons > take1
length > 0 > isNatIList > U31 > mark1 > active1 > nil > take1
length > 0 > isNatIList > U31 > tt > active1 > cons > take1
length > 0 > isNatIList > U31 > tt > active1 > nil > take1
length > 0 > isNatIList > U41 > U42 > mark1 > active1 > cons > take1
length > 0 > isNatIList > U41 > U42 > mark1 > active1 > nil > take1
length > 0 > isNatIList > U41 > U42 > tt > active1 > cons > take1
length > 0 > isNatIList > U41 > U42 > tt > active1 > nil > take1
length > 0 > U81 > mark1 > active1 > cons > take1
length > 0 > U81 > mark1 > active1 > nil > take1
length > U71 > U72 > s > U91 > U92 > U93 > mark1 > active1 > cons > take1
length > U71 > U72 > s > U91 > U92 > U93 > mark1 > active1 > nil > take1

Status:
mark1: [1]
active1: multiset
zeros: multiset
cons: multiset
0: multiset
U11: multiset
tt: multiset
U21: multiset
U31: multiset
U41: []
U42: []
isNatIList: multiset
U51: multiset
U52: multiset
isNatList: []
U61: []
U62: []
U71: []
U72: []
isNat1: multiset
s: []
length: multiset
U81: []
nil: multiset
U91: multiset
U92: []
U93: []
take1: [1]

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)

(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(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)

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(mark(X1), X2, X3, X4) → U931(X1, X2, X3, X4)
U931(X1, X2, mark(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, 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
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  U91
U92(x1, x2, x3, x4)  =  x1
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Recursive path order with status [RPO].
Precedence:
zeros > 0 > tt > U72 > mark1 > U711 > U93^13
zeros > 0 > tt > U72 > s > U93^13
zeros > 0 > tt > U93 > mark1 > U711 > U93^13
zeros > 0 > tt > U93 > cons > U711 > U93^13
isNatIList > U31 > tt > U72 > mark1 > U711 > U93^13
isNatIList > U31 > tt > U72 > s > U93^13
isNatIList > U31 > tt > U93 > mark1 > U711 > U93^13
isNatIList > U31 > tt > U93 > cons > U711 > U93^13
isNatIList > U41 > U42 > tt > U72 > mark1 > U711 > U93^13
isNatIList > U41 > U42 > tt > U72 > s > U93^13
isNatIList > U41 > U42 > tt > U93 > mark1 > U711 > U93^13
isNatIList > U41 > U42 > tt > U93 > cons > U711 > U93^13
isNatIList > isNatList > U51 > U52 > tt > U72 > mark1 > U711 > U93^13
isNatIList > isNatList > U51 > U52 > tt > U72 > s > U93^13
isNatIList > isNatList > U51 > U52 > tt > U93 > mark1 > U711 > U93^13
isNatIList > isNatList > U51 > U52 > tt > U93 > cons > U711 > U93^13
isNatIList > isNatList > U61 > U62 > tt > U72 > mark1 > U711 > U93^13
isNatIList > isNatList > U61 > U62 > tt > U72 > s > U93^13
isNatIList > isNatList > U61 > U62 > tt > U93 > mark1 > U711 > U93^13
isNatIList > isNatList > U61 > U62 > tt > U93 > cons > U711 > U93^13
isNatIList > isNat > U11 > tt > U72 > mark1 > U711 > U93^13
isNatIList > isNat > U11 > tt > U72 > s > U93^13
isNatIList > isNat > U11 > tt > U93 > mark1 > U711 > U93^13
isNatIList > isNat > U11 > tt > U93 > cons > U711 > U93^13
isNatIList > isNat > U21 > tt > U72 > mark1 > U711 > U93^13
isNatIList > isNat > U21 > tt > U72 > s > U93^13
isNatIList > isNat > U21 > tt > U93 > mark1 > U711 > U93^13
isNatIList > isNat > U21 > tt > U93 > cons > U711 > U93^13
length > 0 > tt > U72 > mark1 > U711 > U93^13
length > 0 > tt > U72 > s > U93^13
length > 0 > tt > U93 > mark1 > U711 > U93^13
length > 0 > tt > U93 > cons > U711 > U93^13
length > U11 > tt > U72 > mark1 > U711 > U93^13
length > U11 > tt > U72 > s > U93^13
length > U11 > tt > U93 > mark1 > U711 > U93^13
length > U11 > tt > U93 > cons > U711 > U93^13
length > isNatList > U51 > U52 > tt > U72 > mark1 > U711 > U93^13
length > isNatList > U51 > U52 > tt > U72 > s > U93^13
length > isNatList > U51 > U52 > tt > U93 > mark1 > U711 > U93^13
length > isNatList > U51 > U52 > tt > U93 > cons > U711 > U93^13
length > isNatList > U61 > U62 > tt > U72 > mark1 > U711 > U93^13
length > isNatList > U61 > U62 > tt > U72 > s > U93^13
length > isNatList > U61 > U62 > tt > U93 > mark1 > U711 > U93^13
length > isNatList > U61 > U62 > tt > U93 > cons > U711 > U93^13
take > U61 > U62 > tt > U72 > mark1 > U711 > U93^13
take > U61 > U62 > tt > U72 > s > U93^13
take > U61 > U62 > tt > U93 > mark1 > U711 > U93^13
take > U61 > U62 > tt > U93 > cons > U711 > U93^13
take > U81 > nil > 0 > tt > U72 > mark1 > U711 > U93^13
take > U81 > nil > 0 > tt > U72 > s > U93^13
take > U81 > nil > 0 > tt > U93 > mark1 > U711 > U93^13
take > U81 > nil > 0 > tt > U93 > cons > U711 > U93^13
take > U91 > isNat > U11 > tt > U72 > mark1 > U711 > U93^13
take > U91 > isNat > U11 > tt > U72 > s > U93^13
take > U91 > isNat > U11 > tt > U93 > mark1 > U711 > U93^13
take > U91 > isNat > U11 > tt > U93 > cons > U711 > U93^13
take > U91 > isNat > U21 > tt > U72 > mark1 > U711 > U93^13
take > U91 > isNat > U21 > tt > U72 > s > U93^13
take > U91 > isNat > U21 > tt > U93 > mark1 > U711 > U93^13
take > U91 > isNat > U21 > tt > U93 > cons > U711 > U93^13

Status:
U93^13: [1,3,2]
mark1: [1]
zeros: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: multiset
U31: []
U41: multiset
U42: multiset
isNatIList: multiset
U51: multiset
U52: multiset
isNatList: multiset
U61: multiset
U62: []
U711: multiset
U72: multiset
isNat: multiset
s: []
length: []
U81: []
nil: multiset
U91: []
U93: []
take: []

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(X1, mark(X2), X3, 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)

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

Recursive path order with status [RPO].
Precedence:
zeros > cons > mark1 > 0 > active1 > tt > s
zeros > cons > mark1 > 0 > active1 > nil
isNatIList > U31 > mark1 > 0 > active1 > tt > s
isNatIList > U31 > mark1 > 0 > active1 > nil
isNatIList > U41 > U42 > mark1 > 0 > active1 > tt > s
isNatIList > U41 > U42 > mark1 > 0 > active1 > nil
isNatIList > isNatList > U51 > U52 > mark1 > 0 > active1 > tt > s
isNatIList > isNatList > U51 > U52 > mark1 > 0 > active1 > nil
isNatIList > isNatList > U61 > U62 > mark1 > 0 > active1 > tt > s
isNatIList > isNatList > U61 > U62 > mark1 > 0 > active1 > nil
isNatIList > isNatList > isNat1 > U11 > mark1 > 0 > active1 > tt > s
isNatIList > isNatList > isNat1 > U11 > mark1 > 0 > active1 > nil
isNatIList > isNatList > isNat1 > U21 > mark1 > 0 > active1 > tt > s
isNatIList > isNatList > isNat1 > U21 > mark1 > 0 > active1 > nil
length > isNatList > U51 > U52 > mark1 > 0 > active1 > tt > s
length > isNatList > U51 > U52 > mark1 > 0 > active1 > nil
length > isNatList > U61 > U62 > mark1 > 0 > active1 > tt > s
length > isNatList > U61 > U62 > mark1 > 0 > active1 > nil
length > isNatList > isNat1 > U11 > mark1 > 0 > active1 > tt > s
length > isNatList > isNat1 > U11 > mark1 > 0 > active1 > nil
length > isNatList > isNat1 > U21 > mark1 > 0 > active1 > tt > s
length > isNatList > isNat1 > U21 > mark1 > 0 > active1 > nil
length > U71 > U72 > mark1 > 0 > active1 > tt > s
length > U71 > U72 > mark1 > 0 > active1 > nil
take > U81 > mark1 > 0 > active1 > tt > s
take > U81 > mark1 > 0 > active1 > nil
take > U91 > U92 > U93 > cons > mark1 > 0 > active1 > tt > s
take > U91 > U92 > U93 > cons > mark1 > 0 > active1 > nil

Status:
U93^11: [1]
mark1: multiset
active1: multiset
zeros: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: []
U31: []
U41: []
U42: multiset
isNatIList: multiset
U51: []
U52: []
isNatList: multiset
U61: multiset
U62: multiset
U71: multiset
U72: []
isNat1: [1]
s: []
length: []
U81: []
nil: multiset
U91: []
U92: []
U93: multiset
take: []

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, X2, active(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.

(17) 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)
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)  =  U931(x1, 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(x1)
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Recursive path order with status [RPO].
Precedence:
U93^12 > active1
zeros > 0 > tt > nil > mark1 > s > active1
zeros > 0 > tt > nil > mark1 > U911 > active1
zeros > 0 > tt > U92 > U93 > cons > mark1 > s > active1
zeros > 0 > tt > U92 > U93 > cons > mark1 > U911 > active1
zeros > 0 > U81 > nil > mark1 > s > active1
zeros > 0 > U81 > nil > mark1 > U911 > active1
length > 0 > tt > nil > mark1 > s > active1
length > 0 > tt > nil > mark1 > U911 > active1
length > 0 > tt > U92 > U93 > cons > mark1 > s > active1
length > 0 > tt > U92 > U93 > cons > mark1 > U911 > active1
length > 0 > U81 > nil > mark1 > s > active1
length > 0 > U81 > nil > mark1 > U911 > active1
length > U11 > tt > nil > mark1 > s > active1
length > U11 > tt > nil > mark1 > U911 > active1
length > U11 > tt > U92 > U93 > cons > mark1 > s > active1
length > U11 > tt > U92 > U93 > cons > mark1 > U911 > active1
length > isNatList > U51 > U52 > tt > nil > mark1 > s > active1
length > isNatList > U51 > U52 > tt > nil > mark1 > U911 > active1
length > isNatList > U51 > U52 > tt > U92 > U93 > cons > mark1 > s > active1
length > isNatList > U51 > U52 > tt > U92 > U93 > cons > mark1 > U911 > active1
length > isNatList > U61 > U62 > tt > nil > mark1 > s > active1
length > isNatList > U61 > U62 > tt > nil > mark1 > U911 > active1
length > isNatList > U61 > U62 > tt > U92 > U93 > cons > mark1 > s > active1
length > isNatList > U61 > U62 > tt > U92 > U93 > cons > mark1 > U911 > active1
length > U71 > U72 > mark1 > s > active1
length > U71 > U72 > mark1 > U911 > active1
take > isNatIList > U31 > tt > nil > mark1 > s > active1
take > isNatIList > U31 > tt > nil > mark1 > U911 > active1
take > isNatIList > U31 > tt > U92 > U93 > cons > mark1 > s > active1
take > isNatIList > U31 > tt > U92 > U93 > cons > mark1 > U911 > active1
take > isNatIList > U41 > U42 > tt > nil > mark1 > s > active1
take > isNatIList > U41 > U42 > tt > nil > mark1 > U911 > active1
take > isNatIList > U41 > U42 > tt > U92 > U93 > cons > mark1 > s > active1
take > isNatIList > U41 > U42 > tt > U92 > U93 > cons > mark1 > U911 > active1
take > U61 > U62 > tt > nil > mark1 > s > active1
take > U61 > U62 > tt > nil > mark1 > U911 > active1
take > U61 > U62 > tt > U92 > U93 > cons > mark1 > s > active1
take > U61 > U62 > tt > U92 > U93 > cons > mark1 > U911 > active1
take > isNat > U11 > tt > nil > mark1 > s > active1
take > isNat > U11 > tt > nil > mark1 > U911 > active1
take > isNat > U11 > tt > U92 > U93 > cons > mark1 > s > active1
take > isNat > U11 > tt > U92 > U93 > cons > mark1 > U911 > active1
take > isNat > U21 > tt > nil > mark1 > s > active1
take > isNat > U21 > tt > nil > mark1 > U911 > active1
take > isNat > U21 > tt > U92 > U93 > cons > mark1 > s > active1
take > isNat > U21 > tt > U92 > U93 > cons > mark1 > U911 > active1
take > U81 > nil > mark1 > s > active1
take > U81 > nil > mark1 > U911 > active1

Status:
U93^12: multiset
active1: multiset
zeros: multiset
mark1: [1]
cons: []
0: multiset
U11: multiset
tt: multiset
U21: multiset
U31: multiset
U41: multiset
U42: []
isNatIList: multiset
U51: []
U52: []
isNatList: multiset
U61: []
U62: multiset
U71: []
U72: multiset
isNat: multiset
s: multiset
length: []
U81: multiset
nil: multiset
U911: multiset
U92: []
U93: multiset
take: []

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:
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.

(19) PisEmptyProof (EQUIVALENT transformation)

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

(20) TRUE

(21) Obligation:

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

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.

(22) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U921(mark(X1), X2, X3, X4) → U921(X1, X2, X3, X4)
U921(active(X1), 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)
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(x1, x2)

Recursive path order with status [RPO].
Precedence:
zeros > cons > active1 > U92^11
zeros > cons > active1 > nil
zeros > cons > active1 > take2
zeros > 0 > tt > active1 > U92^11
zeros > 0 > tt > active1 > nil
zeros > 0 > tt > active1 > take2
zeros > 0 > U81 > mark1 > active1 > U92^11
zeros > 0 > U81 > mark1 > active1 > nil
zeros > 0 > U81 > mark1 > active1 > take2
isNatList > U51 > U52 > mark1 > active1 > U92^11
isNatList > U51 > U52 > mark1 > active1 > nil
isNatList > U51 > U52 > mark1 > active1 > take2
isNatList > U51 > U52 > tt > active1 > U92^11
isNatList > U51 > U52 > tt > active1 > nil
isNatList > U51 > U52 > tt > active1 > take2
isNatList > U61 > U62 > mark1 > active1 > U92^11
isNatList > U61 > U62 > mark1 > active1 > nil
isNatList > U61 > U62 > mark1 > active1 > take2
isNatList > U61 > U62 > tt > active1 > U92^11
isNatList > U61 > U62 > tt > active1 > nil
isNatList > U61 > U62 > tt > active1 > take2
isNatList > isNat > U11 > mark1 > active1 > U92^11
isNatList > isNat > U11 > mark1 > active1 > nil
isNatList > isNat > U11 > mark1 > active1 > take2
isNatList > isNat > U11 > tt > active1 > U92^11
isNatList > isNat > U11 > tt > active1 > nil
isNatList > isNat > U11 > tt > active1 > take2
isNatList > isNat > U21 > mark1 > active1 > U92^11
isNatList > isNat > U21 > mark1 > active1 > nil
isNatList > isNat > U21 > mark1 > active1 > take2
isNatList > isNat > U21 > tt > active1 > U92^11
isNatList > isNat > U21 > tt > active1 > nil
isNatList > isNat > U21 > tt > active1 > take2
length > 0 > tt > active1 > U92^11
length > 0 > tt > active1 > nil
length > 0 > tt > active1 > take2
length > 0 > U81 > mark1 > active1 > U92^11
length > 0 > U81 > mark1 > active1 > nil
length > 0 > U81 > mark1 > active1 > take2
length > U71 > U72 > s > U21 > mark1 > active1 > U92^11
length > U71 > U72 > s > U21 > mark1 > active1 > nil
length > U71 > U72 > s > U21 > mark1 > active1 > take2
length > U71 > U72 > s > U21 > tt > active1 > U92^11
length > U71 > U72 > s > U21 > tt > active1 > nil
length > U71 > U72 > s > U21 > tt > active1 > take2
length > U71 > U72 > s > isNatIList > U31 > mark1 > active1 > U92^11
length > U71 > U72 > s > isNatIList > U31 > mark1 > active1 > nil
length > U71 > U72 > s > isNatIList > U31 > mark1 > active1 > take2
length > U71 > U72 > s > isNatIList > U31 > tt > active1 > U92^11
length > U71 > U72 > s > isNatIList > U31 > tt > active1 > nil
length > U71 > U72 > s > isNatIList > U31 > tt > active1 > take2
length > U71 > U72 > s > isNatIList > U41 > U42 > mark1 > active1 > U92^11
length > U71 > U72 > s > isNatIList > U41 > U42 > mark1 > active1 > nil
length > U71 > U72 > s > isNatIList > U41 > U42 > mark1 > active1 > take2
length > U71 > U72 > s > isNatIList > U41 > U42 > tt > active1 > U92^11
length > U71 > U72 > s > isNatIList > U41 > U42 > tt > active1 > nil
length > U71 > U72 > s > isNatIList > U41 > U42 > tt > active1 > take2
length > U71 > U72 > s > U91 > U92 > U93 > mark1 > active1 > U92^11
length > U71 > U72 > s > U91 > U92 > U93 > mark1 > active1 > nil
length > U71 > U72 > s > U91 > U92 > U93 > mark1 > active1 > take2
length > U71 > U72 > s > U91 > U92 > U93 > cons > active1 > U92^11
length > U71 > U72 > s > U91 > U92 > U93 > cons > active1 > nil
length > U71 > U72 > s > U91 > U92 > U93 > cons > active1 > take2
length > U71 > isNat > U11 > mark1 > active1 > U92^11
length > U71 > isNat > U11 > mark1 > active1 > nil
length > U71 > isNat > U11 > mark1 > active1 > take2
length > U71 > isNat > U11 > tt > active1 > U92^11
length > U71 > isNat > U11 > tt > active1 > nil
length > U71 > isNat > U11 > tt > active1 > take2
length > U71 > isNat > U21 > mark1 > active1 > U92^11
length > U71 > isNat > U21 > mark1 > active1 > nil
length > U71 > isNat > U21 > mark1 > active1 > take2
length > U71 > isNat > U21 > tt > active1 > U92^11
length > U71 > isNat > U21 > tt > active1 > nil
length > U71 > isNat > U21 > tt > active1 > take2

Status:
U92^11: multiset
mark1: [1]
active1: multiset
zeros: multiset
cons: multiset
0: multiset
U11: multiset
tt: multiset
U21: []
U31: multiset
U41: multiset
U42: []
isNatIList: []
U51: []
U52: multiset
isNatList: multiset
U61: []
U62: multiset
U71: []
U72: []
isNat: multiset
s: []
length: []
U81: multiset
nil: multiset
U91: []
U92: []
U93: []
take2: [2,1]

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)

(23) Obligation:

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

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(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.

(24) 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)
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(x3)
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(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(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

Recursive path order with status [RPO].
Precedence:
zeros > mark1 > cons
zeros > mark1 > 0
zeros > mark1 > tt
zeros > mark1 > s
isNatList1 > U51 > U52 > mark1 > cons
isNatList1 > U51 > U52 > mark1 > 0
isNatList1 > U51 > U52 > mark1 > tt
isNatList1 > U51 > U52 > mark1 > s
isNatList1 > U61 > isNatIList1 > U31 > mark1 > cons
isNatList1 > U61 > isNatIList1 > U31 > mark1 > 0
isNatList1 > U61 > isNatIList1 > U31 > mark1 > tt
isNatList1 > U61 > isNatIList1 > U31 > mark1 > s
isNatList1 > U61 > isNatIList1 > U41 > U42 > mark1 > cons
isNatList1 > U61 > isNatIList1 > U41 > U42 > mark1 > 0
isNatList1 > U61 > isNatIList1 > U41 > U42 > mark1 > tt
isNatList1 > U61 > isNatIList1 > U41 > U42 > mark1 > s
isNatList1 > U61 > isNatIList1 > isNat1 > U21 > mark1 > cons
isNatList1 > U61 > isNatIList1 > isNat1 > U21 > mark1 > 0
isNatList1 > U61 > isNatIList1 > isNat1 > U21 > mark1 > tt
isNatList1 > U61 > isNatIList1 > isNat1 > U21 > mark1 > s
isNatList1 > U61 > U62 > mark1 > cons
isNatList1 > U61 > U62 > mark1 > 0
isNatList1 > U61 > U62 > mark1 > tt
isNatList1 > U61 > U62 > mark1 > s
length > U11 > mark1 > cons
length > U11 > mark1 > 0
length > U11 > mark1 > tt
length > U11 > mark1 > s
length > U71 > U72 > mark1 > cons
length > U71 > U72 > mark1 > 0
length > U71 > U72 > mark1 > tt
length > U71 > U72 > mark1 > s
length > U71 > isNat1 > U21 > mark1 > cons
length > U71 > isNat1 > U21 > mark1 > 0
length > U71 > isNat1 > U21 > mark1 > tt
length > U71 > isNat1 > U21 > mark1 > s
take > U81 > nil > mark1 > cons
take > U81 > nil > mark1 > 0
take > U81 > nil > mark1 > tt
take > U81 > nil > mark1 > s
take > U91 > U92 > isNat1 > U21 > mark1 > cons
take > U91 > U92 > isNat1 > U21 > mark1 > 0
take > U91 > U92 > isNat1 > U21 > mark1 > tt
take > U91 > U92 > isNat1 > U21 > mark1 > s
take > U91 > U92 > U93 > mark1 > cons
take > U91 > U92 > U93 > mark1 > 0
take > U91 > U92 > U93 > mark1 > tt
take > U91 > U92 > U93 > mark1 > s

Status:
U92^11: [1]
mark1: multiset
zeros: multiset
cons: multiset
0: multiset
U11: []
tt: multiset
U21: []
U31: []
U41: multiset
U42: multiset
isNatIList1: multiset
U51: multiset
U52: multiset
isNatList1: multiset
U61: []
U62: multiset
U71: multiset
U72: []
isNat1: [1]
s: multiset
length: []
U81: []
nil: multiset
U91: []
U92: multiset
U93: []
take: multiset

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)

(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(X1, X2, X3, mark(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, mark(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(x2, x3)
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(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatList(x1)  =  isNatList
U61(x1, x2)  =  U61
U62(x1)  =  U62
U71(x1, x2, x3)  =  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

Recursive path order with status [RPO].
Precedence:
zeros > mark1 > cons > U92^12
zeros > mark1 > 0 > U92^12
zeros > mark1 > s > U92^12
zeros > mark1 > nil > U92^12
isNatIList1 > U31 > tt > U72 > mark1 > cons > U92^12
isNatIList1 > U31 > tt > U72 > mark1 > 0 > U92^12
isNatIList1 > U31 > tt > U72 > mark1 > s > U92^12
isNatIList1 > U31 > tt > U72 > mark1 > nil > U92^12
isNatIList1 > U41 > U42 > tt > U72 > mark1 > cons > U92^12
isNatIList1 > U41 > U42 > tt > U72 > mark1 > 0 > U92^12
isNatIList1 > U41 > U42 > tt > U72 > mark1 > s > U92^12
isNatIList1 > U41 > U42 > tt > U72 > mark1 > nil > U92^12
isNatIList1 > isNatList > U51 > U52 > tt > U72 > mark1 > cons > U92^12
isNatIList1 > isNatList > U51 > U52 > tt > U72 > mark1 > 0 > U92^12
isNatIList1 > isNatList > U51 > U52 > tt > U72 > mark1 > s > U92^12
isNatIList1 > isNatList > U51 > U52 > tt > U72 > mark1 > nil > U92^12
isNatIList1 > isNatList > U61 > U62 > tt > U72 > mark1 > cons > U92^12
isNatIList1 > isNatList > U61 > U62 > tt > U72 > mark1 > 0 > U92^12
isNatIList1 > isNatList > U61 > U62 > tt > U72 > mark1 > s > U92^12
isNatIList1 > isNatList > U61 > U62 > tt > U72 > mark1 > nil > U92^12
isNat > U11 > tt > U72 > mark1 > cons > U92^12
isNat > U11 > tt > U72 > mark1 > 0 > U92^12
isNat > U11 > tt > U72 > mark1 > s > U92^12
isNat > U11 > tt > U72 > mark1 > nil > U92^12
isNat > U21 > tt > U72 > mark1 > cons > U92^12
isNat > U21 > tt > U72 > mark1 > 0 > U92^12
isNat > U21 > tt > U72 > mark1 > s > U92^12
isNat > U21 > tt > U72 > mark1 > nil > U92^12
isNat > isNatList > U51 > U52 > tt > U72 > mark1 > cons > U92^12
isNat > isNatList > U51 > U52 > tt > U72 > mark1 > 0 > U92^12
isNat > isNatList > U51 > U52 > tt > U72 > mark1 > s > U92^12
isNat > isNatList > U51 > U52 > tt > U72 > mark1 > nil > U92^12
isNat > isNatList > U61 > U62 > tt > U72 > mark1 > cons > U92^12
isNat > isNatList > U61 > U62 > tt > U72 > mark1 > 0 > U92^12
isNat > isNatList > U61 > U62 > tt > U72 > mark1 > s > U92^12
isNat > isNatList > U61 > U62 > tt > U72 > mark1 > nil > U92^12
length > isNatList > U51 > U52 > tt > U72 > mark1 > cons > U92^12
length > isNatList > U51 > U52 > tt > U72 > mark1 > 0 > U92^12
length > isNatList > U51 > U52 > tt > U72 > mark1 > s > U92^12
length > isNatList > U51 > U52 > tt > U72 > mark1 > nil > U92^12
length > isNatList > U61 > U62 > tt > U72 > mark1 > cons > U92^12
length > isNatList > U61 > U62 > tt > U72 > mark1 > 0 > U92^12
length > isNatList > U61 > U62 > tt > U72 > mark1 > s > U92^12
length > isNatList > U61 > U62 > tt > U72 > mark1 > nil > U92^12
take > U61 > U62 > tt > U72 > mark1 > cons > U92^12
take > U61 > U62 > tt > U72 > mark1 > 0 > U92^12
take > U61 > U62 > tt > U72 > mark1 > s > U92^12
take > U61 > U62 > tt > U72 > mark1 > nil > U92^12
take > U81 > mark1 > cons > U92^12
take > U81 > mark1 > 0 > U92^12
take > U81 > mark1 > s > U92^12
take > U81 > mark1 > nil > U92^12
take > U91 > U92 > U93 > mark1 > cons > U92^12
take > U91 > U92 > U93 > mark1 > 0 > U92^12
take > U91 > U92 > U93 > mark1 > s > U92^12
take > U91 > U92 > U93 > mark1 > nil > U92^12

Status:
U92^12: [2,1]
mark1: [1]
zeros: multiset
cons: []
0: multiset
U11: multiset
tt: multiset
U21: multiset
U31: []
U41: []
U42: multiset
isNatIList1: multiset
U51: []
U52: []
isNatList: multiset
U61: []
U62: []
U72: multiset
isNat: []
s: multiset
length: []
U81: multiset
nil: multiset
U91: []
U92: []
U93: []
take: []

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, X2, X3, mark(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.

(28) 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)  =  U921(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
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(x1)
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  take

Recursive path order with status [RPO].
Precedence:
zeros > 0 > active1
zeros > tt > U92 > U93 > mark1 > U92^11 > active1
zeros > tt > U92 > U93 > mark1 > cons > active1
zeros > tt > U92 > U93 > mark1 > s > active1
zeros > tt > U92 > U93 > mark1 > U911 > active1
isNatList > U51 > U52 > tt > U92 > U93 > mark1 > U92^11 > active1
isNatList > U51 > U52 > tt > U92 > U93 > mark1 > cons > active1
isNatList > U51 > U52 > tt > U92 > U93 > mark1 > s > active1
isNatList > U51 > U52 > tt > U92 > U93 > mark1 > U911 > active1
isNatList > U61 > isNatIList > U31 > tt > U92 > U93 > mark1 > U92^11 > active1
isNatList > U61 > isNatIList > U31 > tt > U92 > U93 > mark1 > cons > active1
isNatList > U61 > isNatIList > U31 > tt > U92 > U93 > mark1 > s > active1
isNatList > U61 > isNatIList > U31 > tt > U92 > U93 > mark1 > U911 > active1
isNatList > U61 > isNatIList > U41 > U42 > tt > U92 > U93 > mark1 > U92^11 > active1
isNatList > U61 > isNatIList > U41 > U42 > tt > U92 > U93 > mark1 > cons > active1
isNatList > U61 > isNatIList > U41 > U42 > tt > U92 > U93 > mark1 > s > active1
isNatList > U61 > isNatIList > U41 > U42 > tt > U92 > U93 > mark1 > U911 > active1
isNatList > U61 > U62 > tt > U92 > U93 > mark1 > U92^11 > active1
isNatList > U61 > U62 > tt > U92 > U93 > mark1 > cons > active1
isNatList > U61 > U62 > tt > U92 > U93 > mark1 > s > active1
isNatList > U61 > U62 > tt > U92 > U93 > mark1 > U911 > active1
isNat1 > U11 > tt > U92 > U93 > mark1 > U92^11 > active1
isNat1 > U11 > tt > U92 > U93 > mark1 > cons > active1
isNat1 > U11 > tt > U92 > U93 > mark1 > s > active1
isNat1 > U11 > tt > U92 > U93 > mark1 > U911 > active1
isNat1 > U21 > tt > U92 > U93 > mark1 > U92^11 > active1
isNat1 > U21 > tt > U92 > U93 > mark1 > cons > active1
isNat1 > U21 > tt > U92 > U93 > mark1 > s > active1
isNat1 > U21 > tt > U92 > U93 > mark1 > U911 > active1
length > 0 > active1
length > U11 > tt > U92 > U93 > mark1 > U92^11 > active1
length > U11 > tt > U92 > U93 > mark1 > cons > active1
length > U11 > tt > U92 > U93 > mark1 > s > active1
length > U11 > tt > U92 > U93 > mark1 > U911 > active1
length > U71 > U72 > mark1 > U92^11 > active1
length > U71 > U72 > mark1 > cons > active1
length > U71 > U72 > mark1 > s > active1
length > U71 > U72 > mark1 > U911 > active1
take > isNatIList > U31 > tt > U92 > U93 > mark1 > U92^11 > active1
take > isNatIList > U31 > tt > U92 > U93 > mark1 > cons > active1
take > isNatIList > U31 > tt > U92 > U93 > mark1 > s > active1
take > isNatIList > U31 > tt > U92 > U93 > mark1 > U911 > active1
take > isNatIList > U41 > U42 > tt > U92 > U93 > mark1 > U92^11 > active1
take > isNatIList > U41 > U42 > tt > U92 > U93 > mark1 > cons > active1
take > isNatIList > U41 > U42 > tt > U92 > U93 > mark1 > s > active1
take > isNatIList > U41 > U42 > tt > U92 > U93 > mark1 > U911 > active1
take > U81 > nil > 0 > active1
take > U81 > nil > tt > U92 > U93 > mark1 > U92^11 > active1
take > U81 > nil > tt > U92 > U93 > mark1 > cons > active1
take > U81 > nil > tt > U92 > U93 > mark1 > s > active1
take > U81 > nil > tt > U92 > U93 > mark1 > U911 > active1

Status:
U92^11: multiset
mark1: multiset
active1: multiset
zeros: multiset
cons: multiset
0: multiset
U11: multiset
tt: multiset
U21: multiset
U31: multiset
U41: multiset
U42: []
isNatIList: multiset
U51: multiset
U52: multiset
isNatList: multiset
U61: []
U62: []
U71: []
U72: multiset
isNat1: [1]
s: []
length: multiset
U81: multiset
nil: multiset
U911: [1]
U92: []
U93: []
take: multiset

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, active(X2), X3, X4) → U921(X1, X2, X3, X4)
U921(X1, X2, active(X3), 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, 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)  =  U921(x1, 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

Recursive path order with status [RPO].
Precedence:
zeros > cons > mark1 > s > active1 > tt > U92^12
zeros > cons > mark1 > nil > 0 > active1 > tt > U92^12
isNatIList > U31 > mark1 > s > active1 > tt > U92^12
isNatIList > U31 > mark1 > nil > 0 > active1 > tt > U92^12
isNatIList > U41 > U42 > mark1 > s > active1 > tt > U92^12
isNatIList > U41 > U42 > mark1 > nil > 0 > active1 > tt > U92^12
isNatIList > isNat > U11 > mark1 > s > active1 > tt > U92^12
isNatIList > isNat > U11 > mark1 > nil > 0 > active1 > tt > U92^12
isNatIList > isNat > U21 > mark1 > s > active1 > tt > U92^12
isNatIList > isNat > U21 > mark1 > nil > 0 > active1 > tt > U92^12
isNatList > U51 > U52 > mark1 > s > active1 > tt > U92^12
isNatList > U51 > U52 > mark1 > nil > 0 > active1 > tt > U92^12
isNatList > U61 > U62 > mark1 > s > active1 > tt > U92^12
isNatList > U61 > U62 > mark1 > nil > 0 > active1 > tt > U92^12
isNatList > isNat > U11 > mark1 > s > active1 > tt > U92^12
isNatList > isNat > U11 > mark1 > nil > 0 > active1 > tt > U92^12
isNatList > isNat > U21 > mark1 > s > active1 > tt > U92^12
isNatList > isNat > U21 > mark1 > nil > 0 > active1 > tt > U92^12
length > U71 > U72 > mark1 > s > active1 > tt > U92^12
length > U71 > U72 > mark1 > nil > 0 > active1 > tt > U92^12
length > U71 > isNat > U11 > mark1 > s > active1 > tt > U92^12
length > U71 > isNat > U11 > mark1 > nil > 0 > active1 > tt > U92^12
length > U71 > isNat > U21 > mark1 > s > active1 > tt > U92^12
length > U71 > isNat > U21 > mark1 > nil > 0 > active1 > tt > U92^12
take > U61 > U62 > mark1 > s > active1 > tt > U92^12
take > U61 > U62 > mark1 > nil > 0 > active1 > tt > U92^12
take > U81 > mark1 > s > active1 > tt > U92^12
take > U81 > mark1 > nil > 0 > active1 > tt > U92^12
take > U91 > U92 > isNat > U11 > mark1 > s > active1 > tt > U92^12
take > U91 > U92 > isNat > U11 > mark1 > nil > 0 > active1 > tt > U92^12
take > U91 > U92 > isNat > U21 > mark1 > s > active1 > tt > U92^12
take > U91 > U92 > isNat > U21 > mark1 > nil > 0 > active1 > tt > U92^12
take > U91 > U92 > U93 > cons > mark1 > s > active1 > tt > U92^12
take > U91 > U92 > U93 > cons > mark1 > nil > 0 > active1 > tt > U92^12

Status:
U92^12: multiset
active1: multiset
zeros: multiset
mark1: [1]
cons: []
0: multiset
U11: multiset
tt: multiset
U21: multiset
U31: multiset
U41: []
U42: []
isNatIList: []
U51: []
U52: multiset
isNatList: multiset
U61: []
U62: []
U71: []
U72: []
isNat: multiset
s: multiset
length: []
U81: []
nil: multiset
U91: []
U92: []
U93: multiset
take: []

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:
The TRS P consists of the following rules:

U921(X1, active(X2), X3, 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.

(32) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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)  =  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
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

Recursive path order with status [RPO].
Precedence:
zeros > cons > U71 > U72 > mark1 > active1
zeros > cons > U71 > U72 > s > active1
zeros > 0 > mark1 > active1
zeros > tt > U72 > mark1 > active1
zeros > tt > U72 > s > active1
zeros > tt > nil > active1
isNatList1 > U51 > U52 > tt > U72 > mark1 > active1
isNatList1 > U51 > U52 > tt > U72 > s > active1
isNatList1 > U51 > U52 > tt > nil > active1
isNatList1 > U61 > isNatIList > U31 > tt > U72 > mark1 > active1
isNatList1 > U61 > isNatIList > U31 > tt > U72 > s > active1
isNatList1 > U61 > isNatIList > U31 > tt > nil > active1
isNatList1 > U61 > isNatIList > U41 > U42 > tt > U72 > mark1 > active1
isNatList1 > U61 > isNatIList > U41 > U42 > tt > U72 > s > active1
isNatList1 > U61 > isNatIList > U41 > U42 > tt > nil > active1
isNatList1 > U61 > U62 > tt > U72 > mark1 > active1
isNatList1 > U61 > U62 > tt > U72 > s > active1
isNatList1 > U61 > U62 > tt > nil > active1
isNatList1 > isNat > U11 > tt > U72 > mark1 > active1
isNatList1 > isNat > U11 > tt > U72 > s > active1
isNatList1 > isNat > U11 > tt > nil > active1
isNatList1 > isNat > U21 > tt > U72 > mark1 > active1
isNatList1 > isNat > U21 > tt > U72 > s > active1
isNatList1 > isNat > U21 > tt > nil > active1
length > 0 > mark1 > active1
length > U11 > tt > U72 > mark1 > active1
length > U11 > tt > U72 > s > active1
length > U11 > tt > nil > active1
length > U71 > U72 > mark1 > active1
length > U71 > U72 > s > active1
take > isNatIList > U31 > tt > U72 > mark1 > active1
take > isNatIList > U31 > tt > U72 > s > active1
take > isNatIList > U31 > tt > nil > active1
take > isNatIList > U41 > U42 > tt > U72 > mark1 > active1
take > isNatIList > U41 > U42 > tt > U72 > s > active1
take > isNatIList > U41 > U42 > tt > nil > active1
take > U81 > mark1 > active1
take > U81 > nil > active1
take > U91 > isNat > U11 > tt > U72 > mark1 > active1
take > U91 > isNat > U11 > tt > U72 > s > active1
take > U91 > isNat > U11 > tt > nil > active1
take > U91 > isNat > U21 > tt > U72 > mark1 > active1
take > U91 > isNat > U21 > tt > U72 > s > active1
take > U91 > isNat > U21 > tt > nil > active1
take > U91 > U92 > U93 > cons > U71 > U72 > mark1 > active1
take > U91 > U92 > U93 > cons > U71 > U72 > s > active1

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

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)

(33) 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.

(34) PisEmptyProof (EQUIVALENT transformation)

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

(35) TRUE

(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(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.

(37) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Recursive path order with status [RPO].
Precedence:
zeros > 0 > tt > cons
zeros > 0 > tt > s
zeros > 0 > tt > nil
zeros > 0 > U81 > mark1 > cons
zeros > 0 > U81 > mark1 > s
zeros > 0 > U81 > mark1 > nil
isNatIList > U31 > mark1 > cons
isNatIList > U31 > mark1 > s
isNatIList > U31 > mark1 > nil
isNatIList > U31 > tt > cons
isNatIList > U31 > tt > s
isNatIList > U31 > tt > nil
isNatIList > U41 > U42 > mark1 > cons
isNatIList > U41 > U42 > mark1 > s
isNatIList > U41 > U42 > mark1 > nil
isNatIList > U41 > U42 > tt > cons
isNatIList > U41 > U42 > tt > s
isNatIList > U41 > U42 > tt > nil
isNatIList > isNatList1 > U51 > U52 > mark1 > cons
isNatIList > isNatList1 > U51 > U52 > mark1 > s
isNatIList > isNatList1 > U51 > U52 > mark1 > nil
isNatIList > isNatList1 > U51 > U52 > tt > cons
isNatIList > isNatList1 > U51 > U52 > tt > s
isNatIList > isNatList1 > U51 > U52 > tt > nil
isNat > U11 > mark1 > cons
isNat > U11 > mark1 > s
isNat > U11 > mark1 > nil
isNat > U11 > tt > cons
isNat > U11 > tt > s
isNat > U11 > tt > nil
isNat > U21 > mark1 > cons
isNat > U21 > mark1 > s
isNat > U21 > mark1 > nil
isNat > U21 > tt > cons
isNat > U21 > tt > s
isNat > U21 > tt > nil
isNat > isNatList1 > U51 > U52 > mark1 > cons
isNat > isNatList1 > U51 > U52 > mark1 > s
isNat > isNatList1 > U51 > U52 > mark1 > nil
isNat > isNatList1 > U51 > U52 > tt > cons
isNat > isNatList1 > U51 > U52 > tt > s
isNat > isNatList1 > U51 > U52 > tt > nil
length > 0 > tt > cons
length > 0 > tt > s
length > 0 > tt > nil
length > 0 > U81 > mark1 > cons
length > 0 > U81 > mark1 > s
length > 0 > U81 > mark1 > nil
length > U11 > mark1 > cons
length > U11 > mark1 > s
length > U11 > mark1 > nil
length > U11 > tt > cons
length > U11 > tt > s
length > U11 > tt > nil
length > isNatList1 > U51 > U52 > mark1 > cons
length > isNatList1 > U51 > U52 > mark1 > s
length > isNatList1 > U51 > U52 > mark1 > nil
length > isNatList1 > U51 > U52 > tt > cons
length > isNatList1 > U51 > U52 > tt > s
length > isNatList1 > U51 > U52 > tt > nil
length > U71 > U72 > mark1 > cons
length > U71 > U72 > mark1 > s
length > U71 > U72 > mark1 > nil
take > U61 > U62 > mark1 > cons
take > U61 > U62 > mark1 > s
take > U61 > U62 > mark1 > nil
take > U61 > U62 > tt > cons
take > U61 > U62 > tt > s
take > U61 > U62 > tt > nil
take > U81 > mark1 > cons
take > U81 > mark1 > s
take > U81 > mark1 > nil
take > U91 > U92 > U93 > mark1 > cons
take > U91 > U92 > U93 > mark1 > s
take > U91 > U92 > U93 > mark1 > nil

Status:
U91^12: [2,1]
mark1: multiset
zeros: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: multiset
U31: multiset
U41: multiset
U42: []
isNatIList: multiset
U51: multiset
U52: multiset
isNatList1: multiset
U61: []
U62: []
U71: []
U72: multiset
isNat: multiset
s: multiset
length: multiset
U81: multiset
nil: multiset
U91: []
U92: []
U93: []
take: multiset

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:
The TRS P consists of the following rules:

U911(X1, mark(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)
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.

(39) 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(X1, X2, mark(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)  =  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)  =  x1
U72(x1, x2)  =  U72
isNat(x1)  =  isNat(x1)
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

Recursive path order with status [RPO].
Precedence:
zeros > cons > U41 > U42 > tt > U72 > mark1 > length1
zeros > cons > U41 > U42 > tt > U72 > s
zeros > cons > isNatList > U51 > U52 > tt > U72 > mark1 > length1
zeros > cons > isNatList > U51 > U52 > tt > U72 > s
zeros > cons > isNatList > U61 > U62 > tt > U72 > mark1 > length1
zeros > cons > isNatList > U61 > U62 > tt > U72 > s
zeros > 0 > tt > U72 > mark1 > length1
zeros > 0 > tt > U72 > s
take > isNatIList1 > U31 > tt > U72 > mark1 > length1
take > isNatIList1 > U31 > tt > U72 > s
take > isNatIList1 > U41 > U42 > tt > U72 > mark1 > length1
take > isNatIList1 > U41 > U42 > tt > U72 > s
take > isNatIList1 > isNat1 > U11 > tt > U72 > mark1 > length1
take > isNatIList1 > isNat1 > U11 > tt > U72 > s
take > isNatIList1 > isNat1 > U21 > tt > U72 > mark1 > length1
take > isNatIList1 > isNat1 > U21 > tt > U72 > s
take > isNatIList1 > isNat1 > isNatList > U51 > U52 > tt > U72 > mark1 > length1
take > isNatIList1 > isNat1 > isNatList > U51 > U52 > tt > U72 > s
take > isNatIList1 > isNat1 > isNatList > U61 > U62 > tt > U72 > mark1 > length1
take > isNatIList1 > isNat1 > isNatList > U61 > U62 > tt > U72 > s
take > U81 > nil > 0 > tt > U72 > mark1 > length1
take > U81 > nil > 0 > tt > U72 > s
take > U91 > U92 > isNat1 > U11 > tt > U72 > mark1 > length1
take > U91 > U92 > isNat1 > U11 > tt > U72 > s
take > U91 > U92 > isNat1 > U21 > tt > U72 > mark1 > length1
take > U91 > U92 > isNat1 > U21 > tt > U72 > s
take > U91 > U92 > isNat1 > isNatList > U51 > U52 > tt > U72 > mark1 > length1
take > U91 > U92 > isNat1 > isNatList > U51 > U52 > tt > U72 > s
take > U91 > U92 > isNat1 > isNatList > U61 > U62 > tt > U72 > mark1 > length1
take > U91 > U92 > isNat1 > isNatList > U61 > U62 > tt > U72 > s
take > U91 > U92 > U93 > cons > U41 > U42 > tt > U72 > mark1 > length1
take > U91 > U92 > U93 > cons > U41 > U42 > tt > U72 > s
take > U91 > U92 > U93 > cons > isNatList > U51 > U52 > tt > U72 > mark1 > length1
take > U91 > U92 > U93 > cons > isNatList > U51 > U52 > tt > U72 > s
take > U91 > U92 > U93 > cons > isNatList > U61 > U62 > tt > U72 > mark1 > length1
take > U91 > U92 > U93 > cons > isNatList > U61 > U62 > tt > U72 > s

Status:
U91^14: [2,1,4,3]
mark1: multiset
zeros: multiset
cons: []
0: multiset
U11: multiset
tt: multiset
U21: multiset
U31: []
U41: []
U42: []
isNatIList1: multiset
U51: multiset
U52: multiset
isNatList: multiset
U61: multiset
U62: multiset
U72: []
isNat1: multiset
s: multiset
length1: multiset
U81: []
nil: multiset
U91: []
U92: []
U93: multiset
take: []

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)

(40) Obligation:

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

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.

(41) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U911(active(X1), X2, 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)
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
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

Recursive path order with status [RPO].
Precedence:
zeros > 0 > U81 > mark1 > active1 > tt
zeros > 0 > U81 > mark1 > active1 > nil
zeros > 0 > U81 > mark1 > cons
isNatIList1 > U31 > mark1 > active1 > tt
isNatIList1 > U31 > mark1 > active1 > nil
isNatIList1 > U31 > mark1 > cons
isNatIList1 > U41 > U42 > mark1 > active1 > tt
isNatIList1 > U41 > U42 > mark1 > active1 > nil
isNatIList1 > U41 > U42 > mark1 > cons
isNatIList1 > isNat > U11 > mark1 > active1 > tt
isNatIList1 > isNat > U11 > mark1 > active1 > nil
isNatIList1 > isNat > U11 > mark1 > cons
isNatIList1 > isNat > U21 > mark1 > active1 > tt
isNatIList1 > isNat > U21 > mark1 > active1 > nil
isNatIList1 > isNat > U21 > mark1 > cons
isNatList > U51 > U52 > mark1 > active1 > tt
isNatList > U51 > U52 > mark1 > active1 > nil
isNatList > U51 > U52 > mark1 > cons
isNatList > U61 > U62 > mark1 > active1 > tt
isNatList > U61 > U62 > mark1 > active1 > nil
isNatList > U61 > U62 > mark1 > cons
isNatList > isNat > U11 > mark1 > active1 > tt
isNatList > isNat > U11 > mark1 > active1 > nil
isNatList > isNat > U11 > mark1 > cons
isNatList > isNat > U21 > mark1 > active1 > tt
isNatList > isNat > U21 > mark1 > active1 > nil
isNatList > isNat > U21 > mark1 > cons
length > 0 > U81 > mark1 > active1 > tt
length > 0 > U81 > mark1 > active1 > nil
length > 0 > U81 > mark1 > cons
length > U71 > U72 > s > isNat > U11 > mark1 > active1 > tt
length > U71 > U72 > s > isNat > U11 > mark1 > active1 > nil
length > U71 > U72 > s > isNat > U11 > mark1 > cons
length > U71 > U72 > s > isNat > U21 > mark1 > active1 > tt
length > U71 > U72 > s > isNat > U21 > mark1 > active1 > nil
length > U71 > U72 > s > isNat > U21 > mark1 > cons
take > U81 > mark1 > active1 > tt
take > U81 > mark1 > active1 > nil
take > U81 > mark1 > cons
take > U91 > U92 > isNat > U11 > mark1 > active1 > tt
take > U91 > U92 > isNat > U11 > mark1 > active1 > nil
take > U91 > U92 > isNat > U11 > mark1 > cons
take > U91 > U92 > isNat > U21 > mark1 > active1 > tt
take > U91 > U92 > isNat > U21 > mark1 > active1 > nil
take > U91 > U92 > isNat > U21 > mark1 > cons
take > U91 > U92 > U93 > mark1 > active1 > tt
take > U91 > U92 > U93 > mark1 > active1 > nil
take > U91 > U92 > U93 > mark1 > cons

Status:
U91^11: [1]
active1: multiset
zeros: multiset
mark1: multiset
cons: []
0: multiset
U11: multiset
tt: multiset
U21: multiset
U31: multiset
U41: []
U42: multiset
isNatIList1: [1]
U51: []
U52: []
isNatList: multiset
U61: multiset
U62: multiset
U71: []
U72: []
isNat: multiset
s: multiset
length: multiset
U81: []
nil: multiset
U91: multiset
U92: []
U93: []
take: []

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)

(42) Obligation:

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

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.

(43) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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(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(x1)
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

Recursive path order with status [RPO].
Precedence:
zeros > mark1 > cons > active1
zeros > mark1 > U711 > active1
zeros > mark1 > s > active1
isNat > U11 > tt > U72 > mark1 > cons > active1
isNat > U11 > tt > U72 > mark1 > U711 > active1
isNat > U11 > tt > U72 > mark1 > s > active1
isNat > U11 > tt > nil > mark1 > cons > active1
isNat > U11 > tt > nil > mark1 > U711 > active1
isNat > U11 > tt > nil > mark1 > s > active1
isNat > U11 > tt > U93 > mark1 > cons > active1
isNat > U11 > tt > U93 > mark1 > U711 > active1
isNat > U11 > tt > U93 > mark1 > s > active1
isNat > U21 > tt > U72 > mark1 > cons > active1
isNat > U21 > tt > U72 > mark1 > U711 > active1
isNat > U21 > tt > U72 > mark1 > s > active1
isNat > U21 > tt > nil > mark1 > cons > active1
isNat > U21 > tt > nil > mark1 > U711 > active1
isNat > U21 > tt > nil > mark1 > s > active1
isNat > U21 > tt > U93 > mark1 > cons > active1
isNat > U21 > tt > U93 > mark1 > U711 > active1
isNat > U21 > tt > U93 > mark1 > s > active1
length > 0 > isNatIList1 > U31 > tt > U72 > mark1 > cons > active1
length > 0 > isNatIList1 > U31 > tt > U72 > mark1 > U711 > active1
length > 0 > isNatIList1 > U31 > tt > U72 > mark1 > s > active1
length > 0 > isNatIList1 > U31 > tt > nil > mark1 > cons > active1
length > 0 > isNatIList1 > U31 > tt > nil > mark1 > U711 > active1
length > 0 > isNatIList1 > U31 > tt > nil > mark1 > s > active1
length > 0 > isNatIList1 > U31 > tt > U93 > mark1 > cons > active1
length > 0 > isNatIList1 > U31 > tt > U93 > mark1 > U711 > active1
length > 0 > isNatIList1 > U31 > tt > U93 > mark1 > s > active1
length > 0 > isNatIList1 > U41 > U42 > tt > U72 > mark1 > cons > active1
length > 0 > isNatIList1 > U41 > U42 > tt > U72 > mark1 > U711 > active1
length > 0 > isNatIList1 > U41 > U42 > tt > U72 > mark1 > s > active1
length > 0 > isNatIList1 > U41 > U42 > tt > nil > mark1 > cons > active1
length > 0 > isNatIList1 > U41 > U42 > tt > nil > mark1 > U711 > active1
length > 0 > isNatIList1 > U41 > U42 > tt > nil > mark1 > s > active1
length > 0 > isNatIList1 > U41 > U42 > tt > U93 > mark1 > cons > active1
length > 0 > isNatIList1 > U41 > U42 > tt > U93 > mark1 > U711 > active1
length > 0 > isNatIList1 > U41 > U42 > tt > U93 > mark1 > s > active1
length > 0 > U81 > nil > mark1 > cons > active1
length > 0 > U81 > nil > mark1 > U711 > active1
length > 0 > U81 > nil > mark1 > s > active1
length > U11 > tt > U72 > mark1 > cons > active1
length > U11 > tt > U72 > mark1 > U711 > active1
length > U11 > tt > U72 > mark1 > s > active1
length > U11 > tt > nil > mark1 > cons > active1
length > U11 > tt > nil > mark1 > U711 > active1
length > U11 > tt > nil > mark1 > s > active1
length > U11 > tt > U93 > mark1 > cons > active1
length > U11 > tt > U93 > mark1 > U711 > active1
length > U11 > tt > U93 > mark1 > s > active1
length > isNatList > U51 > U52 > tt > U72 > mark1 > cons > active1
length > isNatList > U51 > U52 > tt > U72 > mark1 > U711 > active1
length > isNatList > U51 > U52 > tt > U72 > mark1 > s > active1
length > isNatList > U51 > U52 > tt > nil > mark1 > cons > active1
length > isNatList > U51 > U52 > tt > nil > mark1 > U711 > active1
length > isNatList > U51 > U52 > tt > nil > mark1 > s > active1
length > isNatList > U51 > U52 > tt > U93 > mark1 > cons > active1
length > isNatList > U51 > U52 > tt > U93 > mark1 > U711 > active1
length > isNatList > U51 > U52 > tt > U93 > mark1 > s > active1
length > isNatList > U61 > U62 > tt > U72 > mark1 > cons > active1
length > isNatList > U61 > U62 > tt > U72 > mark1 > U711 > active1
length > isNatList > U61 > U62 > tt > U72 > mark1 > s > active1
length > isNatList > U61 > U62 > tt > nil > mark1 > cons > active1
length > isNatList > U61 > U62 > tt > nil > mark1 > U711 > active1
length > isNatList > U61 > U62 > tt > nil > mark1 > s > active1
length > isNatList > U61 > U62 > tt > U93 > mark1 > cons > active1
length > isNatList > U61 > U62 > tt > U93 > mark1 > U711 > active1
length > isNatList > U61 > U62 > tt > U93 > mark1 > s > active1
take > U81 > nil > mark1 > cons > active1
take > U81 > nil > mark1 > U711 > active1
take > U81 > nil > mark1 > s > active1
take > U91 > U92 > U93 > mark1 > cons > active1
take > U91 > U92 > U93 > mark1 > U711 > active1
take > U91 > U92 > U93 > mark1 > s > active1

Status:
U91^11: multiset
active1: multiset
zeros: multiset
mark1: [1]
cons: multiset
0: multiset
U11: multiset
tt: multiset
U21: multiset
U31: []
U41: multiset
U42: multiset
isNatIList1: [1]
U51: multiset
U52: []
isNatList: []
U61: []
U62: []
U711: multiset
U72: []
isNat: multiset
s: multiset
length: multiset
U81: multiset
nil: multiset
U91: multiset
U92: []
U93: []
take: multiset

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)

(44) Obligation:

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

U911(X1, active(X2), 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.

(45) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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)  =  U911(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
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

Recursive path order with status [RPO].
Precedence:
zeros > mark1 > active1 > s
zeros > cons > active1 > s
zeros > tt > active1 > s
isNatIList1 > U31 > mark1 > active1 > s
isNatIList1 > U31 > tt > active1 > s
isNatIList1 > U41 > U42 > mark1 > active1 > s
isNatIList1 > U41 > U42 > tt > active1 > s
isNatIList1 > isNat > U11 > mark1 > active1 > s
isNatIList1 > isNat > U11 > tt > active1 > s
isNatIList1 > isNat > U21 > mark1 > active1 > s
isNatIList1 > isNat > U21 > tt > active1 > s
isNatList1 > U51 > U52 > mark1 > active1 > s
isNatList1 > U51 > U52 > tt > active1 > s
isNatList1 > U61 > U62 > mark1 > active1 > s
isNatList1 > U61 > U62 > tt > active1 > s
isNatList1 > isNat > U11 > mark1 > active1 > s
isNatList1 > isNat > U11 > tt > active1 > s
isNatList1 > isNat > U21 > mark1 > active1 > s
isNatList1 > isNat > U21 > tt > active1 > s
length > 0 > mark1 > active1 > s
length > 0 > tt > active1 > s
length > U71 > U72 > mark1 > active1 > s
length > U71 > isNat > U11 > mark1 > active1 > s
length > U71 > isNat > U11 > tt > active1 > s
length > U71 > isNat > U21 > mark1 > active1 > s
length > U71 > isNat > U21 > tt > active1 > s
take > U61 > U62 > mark1 > active1 > s
take > U61 > U62 > tt > active1 > s
take > U81 > nil > 0 > mark1 > active1 > s
take > U81 > nil > 0 > tt > active1 > s
take > U91 > isNat > U11 > mark1 > active1 > s
take > U91 > isNat > U11 > tt > active1 > s
take > U91 > isNat > U21 > mark1 > active1 > s
take > U91 > isNat > U21 > tt > active1 > s
take > U91 > U92 > U93 > mark1 > active1 > s
take > U91 > U92 > U93 > cons > active1 > s

Status:
U91^11: multiset
active1: multiset
zeros: multiset
mark1: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: multiset
U31: []
U41: []
U42: []
isNatIList1: multiset
U51: []
U52: multiset
isNatList1: [1]
U61: multiset
U62: multiset
U71: []
U72: []
isNat: multiset
s: multiset
length: []
U81: []
nil: multiset
U91: multiset
U92: []
U93: []
take: []

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)

(46) Obligation:

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

U911(X1, active(X2), 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.

(47) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U911(X1, active(X2), 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)
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
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

Recursive path order with status [RPO].
Precedence:
zeros > 0 > tt > mark1 > active1 > nil
zeros > 0 > tt > cons > active1 > nil
zeros > 0 > U81 > mark1 > active1 > nil
isNatIList1 > U31 > tt > mark1 > active1 > nil
isNatIList1 > U31 > tt > cons > active1 > nil
isNatIList1 > U41 > U42 > tt > mark1 > active1 > nil
isNatIList1 > U41 > U42 > tt > cons > active1 > nil
isNatIList1 > isNat > U11 > tt > mark1 > active1 > nil
isNatIList1 > isNat > U11 > tt > cons > active1 > nil
isNatIList1 > isNat > U21 > tt > mark1 > active1 > nil
isNatIList1 > isNat > U21 > tt > cons > active1 > nil
isNatList1 > U51 > U52 > tt > mark1 > active1 > nil
isNatList1 > U51 > U52 > tt > cons > active1 > nil
isNatList1 > U61 > U62 > tt > mark1 > active1 > nil
isNatList1 > U61 > U62 > tt > cons > active1 > nil
isNatList1 > isNat > U11 > tt > mark1 > active1 > nil
isNatList1 > isNat > U11 > tt > cons > active1 > nil
isNatList1 > isNat > U21 > tt > mark1 > active1 > nil
isNatList1 > isNat > U21 > tt > cons > active1 > nil
length > 0 > tt > mark1 > active1 > nil
length > 0 > tt > cons > active1 > nil
length > 0 > U81 > mark1 > active1 > nil
length > U71 > U72 > s > isNat > U11 > tt > mark1 > active1 > nil
length > U71 > U72 > s > isNat > U11 > tt > cons > active1 > nil
length > U71 > U72 > s > isNat > U21 > tt > mark1 > active1 > nil
length > U71 > U72 > s > isNat > U21 > tt > cons > active1 > nil
take > U61 > U62 > tt > mark1 > active1 > nil
take > U61 > U62 > tt > cons > active1 > nil
take > U81 > mark1 > active1 > nil
take > U91 > U92 > isNat > U11 > tt > mark1 > active1 > nil
take > U91 > U92 > isNat > U11 > tt > cons > active1 > nil
take > U91 > U92 > isNat > U21 > tt > mark1 > active1 > nil
take > U91 > U92 > isNat > U21 > tt > cons > active1 > nil
take > U91 > U92 > U93 > mark1 > active1 > nil
take > U91 > U92 > U93 > cons > active1 > nil

Status:
U91^14: multiset
active1: multiset
zeros: multiset
mark1: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: []
U31: []
U41: []
U42: multiset
isNatIList1: multiset
U51: []
U52: []
isNatList1: [1]
U61: []
U62: []
U71: []
U72: []
isNat: multiset
s: multiset
length: []
U81: []
nil: multiset
U91: []
U92: []
U93: []
take: []

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)

(48) 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.

(49) PisEmptyProof (EQUIVALENT transformation)

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

(50) TRUE

(51) 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.

(52) 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)
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)  =  active(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
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

Recursive path order with status [RPO].
Precedence:
U81^11 > active1
zeros > cons > isNatIList > U31 > tt > s > mark1 > active1
zeros > cons > isNatIList > U41 > U42 > tt > s > mark1 > active1
zeros > 0 > isNatIList > U31 > tt > s > mark1 > active1
zeros > 0 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > 0 > isNatIList > U31 > tt > s > mark1 > active1
length > 0 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > isNatList1 > U51 > U52 > tt > s > mark1 > active1
length > isNatList1 > U61 > isNatIList > U31 > tt > s > mark1 > active1
length > isNatList1 > U61 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > isNatList1 > U61 > U62 > tt > s > mark1 > active1
length > isNatList1 > isNat > U11 > tt > s > mark1 > active1
length > isNatList1 > isNat > U21 > tt > s > mark1 > active1
length > U71 > U72 > s > mark1 > active1
length > U71 > isNat > U11 > tt > s > mark1 > active1
length > U71 > isNat > U21 > tt > s > mark1 > active1
take > U81 > nil > mark1 > active1
take > U91 > U92 > isNat > U11 > tt > s > mark1 > active1
take > U91 > U92 > isNat > U21 > tt > s > mark1 > active1
take > U91 > U92 > U93 > cons > isNatIList > U31 > tt > s > mark1 > active1
take > U91 > U92 > U93 > cons > isNatIList > U41 > U42 > tt > s > mark1 > active1

Status:
U81^11: multiset
active1: multiset
mark1: multiset
zeros: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: multiset
U31: []
U41: []
U42: multiset
isNatIList: multiset
U51: multiset
U52: multiset
isNatList1: multiset
U61: multiset
U62: []
U71: []
U72: []
isNat: multiset
s: multiset
length: multiset
U81: multiset
nil: multiset
U91: []
U92: multiset
U93: multiset
take: []

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)

(53) 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.

(54) PisEmptyProof (EQUIVALENT transformation)

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

(55) TRUE

(56) 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.

(57) 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)
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)  =  active(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
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

Recursive path order with status [RPO].
Precedence:
LENGTH1 > active1
zeros > cons > isNatIList > U31 > tt > s > mark1 > active1
zeros > cons > isNatIList > U41 > U42 > tt > s > mark1 > active1
zeros > 0 > isNatIList > U31 > tt > s > mark1 > active1
zeros > 0 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > 0 > isNatIList > U31 > tt > s > mark1 > active1
length > 0 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > isNatList1 > U51 > U52 > tt > s > mark1 > active1
length > isNatList1 > U61 > isNatIList > U31 > tt > s > mark1 > active1
length > isNatList1 > U61 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > isNatList1 > U61 > U62 > tt > s > mark1 > active1
length > isNatList1 > isNat > U11 > tt > s > mark1 > active1
length > isNatList1 > isNat > U21 > tt > s > mark1 > active1
length > U71 > U72 > s > mark1 > active1
length > U71 > isNat > U11 > tt > s > mark1 > active1
length > U71 > isNat > U21 > tt > s > mark1 > active1
take > U81 > nil > mark1 > active1
take > U91 > U92 > isNat > U11 > tt > s > mark1 > active1
take > U91 > U92 > isNat > U21 > tt > s > mark1 > active1
take > U91 > U92 > U93 > cons > isNatIList > U31 > tt > s > mark1 > active1
take > U91 > U92 > U93 > cons > isNatIList > U41 > U42 > tt > s > mark1 > active1

Status:
LENGTH1: multiset
active1: multiset
mark1: multiset
zeros: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: multiset
U31: []
U41: []
U42: multiset
isNatIList: multiset
U51: multiset
U52: multiset
isNatList1: multiset
U61: multiset
U62: []
U71: []
U72: []
isNat: multiset
s: multiset
length: multiset
U81: multiset
nil: multiset
U91: []
U92: multiset
U93: multiset
take: []

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)

(58) 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.

(59) PisEmptyProof (EQUIVALENT transformation)

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

(60) TRUE

(61) 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.

(62) 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)
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)  =  active(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
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

Recursive path order with status [RPO].
Precedence:
S1 > active1
zeros > cons > isNatIList > U31 > tt > s > mark1 > active1
zeros > cons > isNatIList > U41 > U42 > tt > s > mark1 > active1
zeros > 0 > isNatIList > U31 > tt > s > mark1 > active1
zeros > 0 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > 0 > isNatIList > U31 > tt > s > mark1 > active1
length > 0 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > isNatList1 > U51 > U52 > tt > s > mark1 > active1
length > isNatList1 > U61 > isNatIList > U31 > tt > s > mark1 > active1
length > isNatList1 > U61 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > isNatList1 > U61 > U62 > tt > s > mark1 > active1
length > isNatList1 > isNat > U11 > tt > s > mark1 > active1
length > isNatList1 > isNat > U21 > tt > s > mark1 > active1
length > U71 > U72 > s > mark1 > active1
length > U71 > isNat > U11 > tt > s > mark1 > active1
length > U71 > isNat > U21 > tt > s > mark1 > active1
take > U81 > nil > mark1 > active1
take > U91 > U92 > isNat > U11 > tt > s > mark1 > active1
take > U91 > U92 > isNat > U21 > tt > s > mark1 > active1
take > U91 > U92 > U93 > cons > isNatIList > U31 > tt > s > mark1 > active1
take > U91 > U92 > U93 > cons > isNatIList > U41 > U42 > tt > s > mark1 > active1

Status:
S1: multiset
active1: multiset
mark1: multiset
zeros: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: multiset
U31: []
U41: []
U42: multiset
isNatIList: multiset
U51: multiset
U52: multiset
isNatList1: multiset
U61: multiset
U62: []
U71: []
U72: []
isNat: multiset
s: multiset
length: multiset
U81: multiset
nil: multiset
U91: []
U92: multiset
U93: multiset
take: []

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)

(63) 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.

(64) PisEmptyProof (EQUIVALENT transformation)

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

(65) TRUE

(66) 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.

(67) 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)
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)  =  active(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
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

Recursive path order with status [RPO].
Precedence:
ISNAT1 > active1
zeros > cons > isNatIList > U31 > tt > s > mark1 > active1
zeros > cons > isNatIList > U41 > U42 > tt > s > mark1 > active1
zeros > 0 > isNatIList > U31 > tt > s > mark1 > active1
zeros > 0 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > 0 > isNatIList > U31 > tt > s > mark1 > active1
length > 0 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > isNatList1 > U51 > U52 > tt > s > mark1 > active1
length > isNatList1 > U61 > isNatIList > U31 > tt > s > mark1 > active1
length > isNatList1 > U61 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > isNatList1 > U61 > U62 > tt > s > mark1 > active1
length > isNatList1 > isNat > U11 > tt > s > mark1 > active1
length > isNatList1 > isNat > U21 > tt > s > mark1 > active1
length > U71 > U72 > s > mark1 > active1
length > U71 > isNat > U11 > tt > s > mark1 > active1
length > U71 > isNat > U21 > tt > s > mark1 > active1
take > U81 > nil > mark1 > active1
take > U91 > U92 > isNat > U11 > tt > s > mark1 > active1
take > U91 > U92 > isNat > U21 > tt > s > mark1 > active1
take > U91 > U92 > U93 > cons > isNatIList > U31 > tt > s > mark1 > active1
take > U91 > U92 > U93 > cons > isNatIList > U41 > U42 > tt > s > mark1 > active1

Status:
ISNAT1: multiset
active1: multiset
mark1: multiset
zeros: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: multiset
U31: []
U41: []
U42: multiset
isNatIList: multiset
U51: multiset
U52: multiset
isNatList1: multiset
U61: multiset
U62: []
U71: []
U72: []
isNat: multiset
s: multiset
length: multiset
U81: multiset
nil: multiset
U91: []
U92: multiset
U93: multiset
take: []

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)

(68) 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.

(69) PisEmptyProof (EQUIVALENT transformation)

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

(70) TRUE

(71) Obligation:

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

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.

(72) 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)
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
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)  =  x1

Recursive path order with status [RPO].
Precedence:
zeros > 0 > isNatIList > U31 > tt > U72 > s > U91 > U92 > U93 > mark1
zeros > 0 > isNatIList > U31 > tt > U72 > s > U91 > U92 > U93 > cons
zeros > 0 > isNatIList > U31 > tt > nil > mark1
zeros > 0 > isNatIList > U41 > U42 > tt > U72 > s > U91 > U92 > U93 > mark1
zeros > 0 > isNatIList > U41 > U42 > tt > U72 > s > U91 > U92 > U93 > cons
zeros > 0 > isNatIList > U41 > U42 > tt > nil > mark1
zeros > 0 > isNatIList > isNat1 > U21 > tt > U72 > s > U91 > U92 > U93 > mark1
zeros > 0 > isNatIList > isNat1 > U21 > tt > U72 > s > U91 > U92 > U93 > cons
zeros > 0 > isNatIList > isNat1 > U21 > tt > nil > mark1
zeros > 0 > U81 > nil > mark1
isNatList > U51 > U52 > tt > U72 > s > U91 > U92 > U93 > mark1
isNatList > U51 > U52 > tt > U72 > s > U91 > U92 > U93 > cons
isNatList > U51 > U52 > tt > nil > mark1
isNatList > U61 > isNatIList > U31 > tt > U72 > s > U91 > U92 > U93 > mark1
isNatList > U61 > isNatIList > U31 > tt > U72 > s > U91 > U92 > U93 > cons
isNatList > U61 > isNatIList > U31 > tt > nil > mark1
isNatList > U61 > isNatIList > U41 > U42 > tt > U72 > s > U91 > U92 > U93 > mark1
isNatList > U61 > isNatIList > U41 > U42 > tt > U72 > s > U91 > U92 > U93 > cons
isNatList > U61 > isNatIList > U41 > U42 > tt > nil > mark1
isNatList > U61 > isNatIList > isNat1 > U21 > tt > U72 > s > U91 > U92 > U93 > mark1
isNatList > U61 > isNatIList > isNat1 > U21 > tt > U72 > s > U91 > U92 > U93 > cons
isNatList > U61 > isNatIList > isNat1 > U21 > tt > nil > mark1
isNatList > U61 > U62 > tt > U72 > s > U91 > U92 > U93 > mark1
isNatList > U61 > U62 > tt > U72 > s > U91 > U92 > U93 > cons
isNatList > U61 > U62 > tt > nil > mark1
length > 0 > isNatIList > U31 > tt > U72 > s > U91 > U92 > U93 > mark1
length > 0 > isNatIList > U31 > tt > U72 > s > U91 > U92 > U93 > cons
length > 0 > isNatIList > U31 > tt > nil > mark1
length > 0 > isNatIList > U41 > U42 > tt > U72 > s > U91 > U92 > U93 > mark1
length > 0 > isNatIList > U41 > U42 > tt > U72 > s > U91 > U92 > U93 > cons
length > 0 > isNatIList > U41 > U42 > tt > nil > mark1
length > 0 > isNatIList > isNat1 > U21 > tt > U72 > s > U91 > U92 > U93 > mark1
length > 0 > isNatIList > isNat1 > U21 > tt > U72 > s > U91 > U92 > U93 > cons
length > 0 > isNatIList > isNat1 > U21 > tt > nil > mark1
length > 0 > U81 > nil > mark1
length > U11 > tt > U72 > s > U91 > U92 > U93 > mark1
length > U11 > tt > U72 > s > U91 > U92 > U93 > cons
length > U11 > tt > nil > mark1
length > U71 > isNat1 > U21 > tt > U72 > s > U91 > U92 > U93 > mark1
length > U71 > isNat1 > U21 > tt > U72 > s > U91 > U92 > U93 > cons
length > U71 > isNat1 > U21 > tt > nil > mark1

Status:
U72^12: [2,1]
mark1: [1]
zeros: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: []
U31: multiset
U41: multiset
U42: multiset
isNatIList: multiset
U51: multiset
U52: multiset
isNatList: multiset
U61: multiset
U62: []
U71: multiset
U72: []
isNat1: multiset
s: multiset
length: []
U81: multiset
nil: multiset
U91: []
U92: multiset
U93: multiset

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(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.

(74) 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)
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)  =  U721(x1, 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
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(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

Recursive path order with status [RPO].
Precedence:
zeros > cons > U41 > U42 > tt > U72 > mark1 > U711 > active1 > U72^12
zeros > cons > U41 > U42 > tt > U72 > mark1 > s > active1 > U72^12
zeros > cons > U41 > U42 > tt > U72 > mark1 > length1 > active1 > U72^12
zeros > cons > isNatList > U51 > U52 > tt > U72 > mark1 > U711 > active1 > U72^12
zeros > cons > isNatList > U51 > U52 > tt > U72 > mark1 > s > active1 > U72^12
zeros > cons > isNatList > U51 > U52 > tt > U72 > mark1 > length1 > active1 > U72^12
zeros > cons > isNatList > U61 > U62 > tt > U72 > mark1 > U711 > active1 > U72^12
zeros > cons > isNatList > U61 > U62 > tt > U72 > mark1 > s > active1 > U72^12
zeros > cons > isNatList > U61 > U62 > tt > U72 > mark1 > length1 > active1 > U72^12
zeros > 0 > tt > U72 > mark1 > U711 > active1 > U72^12
zeros > 0 > tt > U72 > mark1 > s > active1 > U72^12
zeros > 0 > tt > U72 > mark1 > length1 > active1 > U72^12
take > isNatIList > U31 > tt > U72 > mark1 > U711 > active1 > U72^12
take > isNatIList > U31 > tt > U72 > mark1 > s > active1 > U72^12
take > isNatIList > U31 > tt > U72 > mark1 > length1 > active1 > U72^12
take > isNatIList > U41 > U42 > tt > U72 > mark1 > U711 > active1 > U72^12
take > isNatIList > U41 > U42 > tt > U72 > mark1 > s > active1 > U72^12
take > isNatIList > U41 > U42 > tt > U72 > mark1 > length1 > active1 > U72^12
take > isNatIList > isNatList > U51 > U52 > tt > U72 > mark1 > U711 > active1 > U72^12
take > isNatIList > isNatList > U51 > U52 > tt > U72 > mark1 > s > active1 > U72^12
take > isNatIList > isNatList > U51 > U52 > tt > U72 > mark1 > length1 > active1 > U72^12
take > isNatIList > isNatList > U61 > U62 > tt > U72 > mark1 > U711 > active1 > U72^12
take > isNatIList > isNatList > U61 > U62 > tt > U72 > mark1 > s > active1 > U72^12
take > isNatIList > isNatList > U61 > U62 > tt > U72 > mark1 > length1 > active1 > U72^12
take > isNat1 > U11 > tt > U72 > mark1 > U711 > active1 > U72^12
take > isNat1 > U11 > tt > U72 > mark1 > s > active1 > U72^12
take > isNat1 > U11 > tt > U72 > mark1 > length1 > active1 > U72^12
take > isNat1 > U21 > tt > U72 > mark1 > U711 > active1 > U72^12
take > isNat1 > U21 > tt > U72 > mark1 > s > active1 > U72^12
take > isNat1 > U21 > tt > U72 > mark1 > length1 > active1 > U72^12
take > isNat1 > isNatList > U51 > U52 > tt > U72 > mark1 > U711 > active1 > U72^12
take > isNat1 > isNatList > U51 > U52 > tt > U72 > mark1 > s > active1 > U72^12
take > isNat1 > isNatList > U51 > U52 > tt > U72 > mark1 > length1 > active1 > U72^12
take > isNat1 > isNatList > U61 > U62 > tt > U72 > mark1 > U711 > active1 > U72^12
take > isNat1 > isNatList > U61 > U62 > tt > U72 > mark1 > s > active1 > U72^12
take > isNat1 > isNatList > U61 > U62 > tt > U72 > mark1 > length1 > active1 > U72^12
take > U81 > nil > 0 > tt > U72 > mark1 > U711 > active1 > U72^12
take > U81 > nil > 0 > tt > U72 > mark1 > s > active1 > U72^12
take > U81 > nil > 0 > tt > U72 > mark1 > length1 > active1 > U72^12
take > U91 > U92 > U93 > cons > U41 > U42 > tt > U72 > mark1 > U711 > active1 > U72^12
take > U91 > U92 > U93 > cons > U41 > U42 > tt > U72 > mark1 > s > active1 > U72^12
take > U91 > U92 > U93 > cons > U41 > U42 > tt > U72 > mark1 > length1 > active1 > U72^12
take > U91 > U92 > U93 > cons > isNatList > U51 > U52 > tt > U72 > mark1 > U711 > active1 > U72^12
take > U91 > U92 > U93 > cons > isNatList > U51 > U52 > tt > U72 > mark1 > s > active1 > U72^12
take > U91 > U92 > U93 > cons > isNatList > U51 > U52 > tt > U72 > mark1 > length1 > active1 > U72^12
take > U91 > U92 > U93 > cons > isNatList > U61 > U62 > tt > U72 > mark1 > U711 > active1 > U72^12
take > U91 > U92 > U93 > cons > isNatList > U61 > U62 > tt > U72 > mark1 > s > active1 > U72^12
take > U91 > U92 > U93 > cons > isNatList > U61 > U62 > tt > U72 > mark1 > length1 > active1 > U72^12

Status:
U72^12: multiset
active1: multiset
zeros: multiset
mark1: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: []
U31: multiset
U41: multiset
U42: multiset
isNatIList: multiset
U51: []
U52: multiset
isNatList: multiset
U61: multiset
U62: []
U711: [1]
U72: []
isNat1: [1]
s: []
length1: [1]
U81: []
nil: multiset
U91: []
U92: multiset
U93: multiset
take: []

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:
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.

(76) PisEmptyProof (EQUIVALENT transformation)

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

(77) TRUE

(78) Obligation:

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

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.

(79) 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)
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)  =  U711(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(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)  =  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

Recursive path order with status [RPO].
Precedence:
U71^11 > isNatList1
zeros > cons > U41 > U42 > mark1 > active1 > tt > isNatList1
zeros > cons > isNatIList1 > U31 > mark1 > active1 > tt > isNatList1
zeros > cons > U51 > U52 > mark1 > active1 > tt > isNatList1
zeros > 0 > isNatIList1 > U31 > mark1 > active1 > tt > isNatList1
zeros > 0 > U81 > nil > mark1 > active1 > tt > isNatList1
length > 0 > isNatIList1 > U31 > mark1 > active1 > tt > isNatList1
length > 0 > U81 > nil > mark1 > active1 > tt > isNatList1
length > U11 > mark1 > active1 > tt > isNatList1
length > U71 > U72 > s > U21 > mark1 > active1 > tt > isNatList1
length > U71 > U72 > s > isNatIList1 > U31 > mark1 > active1 > tt > isNatList1
take > U61 > isNatIList1 > U31 > mark1 > active1 > tt > isNatList1
take > U61 > U62 > mark1 > active1 > tt > isNatList1
take > U81 > nil > mark1 > active1 > tt > isNatList1
take > U91 > U92 > U93 > cons > U41 > U42 > mark1 > active1 > tt > isNatList1
take > U91 > U92 > U93 > cons > isNatIList1 > U31 > mark1 > active1 > tt > isNatList1
take > U91 > U92 > U93 > cons > U51 > U52 > mark1 > active1 > tt > isNatList1

Status:
U71^11: [1]
mark1: multiset
active1: multiset
zeros: multiset
cons: []
0: multiset
U11: multiset
tt: multiset
U21: multiset
U31: multiset
U41: multiset
U42: multiset
isNatIList1: [1]
U51: []
U52: multiset
isNatList1: multiset
U61: []
U62: multiset
U71: []
U72: multiset
s: []
length: []
U81: multiset
nil: multiset
U91: multiset
U92: multiset
U93: []
take: []

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)

(80) Obligation:

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

U711(X1, mark(X2), X3) → U711(X1, X2, X3)
U711(X1, X2, mark(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, 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)  =  U711(x1, 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(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

Recursive path order with status [RPO].
Precedence:
U71^12 > active1
zeros > cons > isNatIList > U31 > mark1 > 0 > active1
zeros > cons > isNatIList > U31 > mark1 > isNatList1 > tt > active1
zeros > cons > isNatIList > U41 > U42 > mark1 > 0 > active1
zeros > cons > isNatIList > U41 > U42 > mark1 > isNatList1 > tt > active1
zeros > cons > U51 > U52 > mark1 > 0 > active1
zeros > cons > U51 > U52 > mark1 > isNatList1 > tt > active1
zeros > cons > isNat > U11 > mark1 > 0 > active1
zeros > cons > isNat > U11 > mark1 > isNatList1 > tt > active1
zeros > cons > isNat > U21 > mark1 > 0 > active1
zeros > cons > isNat > U21 > mark1 > isNatList1 > tt > active1
length > U71 > U72 > s > U21 > mark1 > 0 > active1
length > U71 > U72 > s > U21 > mark1 > isNatList1 > tt > active1
length > U71 > isNat > U11 > mark1 > 0 > active1
length > U71 > isNat > U11 > mark1 > isNatList1 > tt > active1
length > U71 > isNat > U21 > mark1 > 0 > active1
length > U71 > isNat > U21 > mark1 > isNatList1 > tt > active1
take > U61 > U62 > mark1 > 0 > active1
take > U61 > U62 > mark1 > isNatList1 > tt > active1
take > U81 > nil > mark1 > 0 > active1
take > U81 > nil > mark1 > isNatList1 > tt > active1
take > U91 > U92 > U93 > cons > isNatIList > U31 > mark1 > 0 > active1
take > U91 > U92 > U93 > cons > isNatIList > U31 > mark1 > isNatList1 > tt > active1
take > U91 > U92 > U93 > cons > isNatIList > U41 > U42 > mark1 > 0 > active1
take > U91 > U92 > U93 > cons > isNatIList > U41 > U42 > mark1 > isNatList1 > tt > active1
take > U91 > U92 > U93 > cons > U51 > U52 > mark1 > 0 > active1
take > U91 > U92 > U93 > cons > U51 > U52 > mark1 > isNatList1 > tt > active1
take > U91 > U92 > U93 > cons > isNat > U11 > mark1 > 0 > active1
take > U91 > U92 > U93 > cons > isNat > U11 > mark1 > isNatList1 > tt > active1
take > U91 > U92 > U93 > cons > isNat > U21 > mark1 > 0 > active1
take > U91 > U92 > U93 > cons > isNat > U21 > mark1 > isNatList1 > tt > active1

Status:
U71^12: [1,2]
mark1: multiset
active1: multiset
zeros: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: multiset
U31: multiset
U41: []
U42: multiset
isNatIList: []
U51: []
U52: multiset
isNatList1: multiset
U61: []
U62: []
U71: []
U72: []
isNat: multiset
s: []
length: []
U81: multiset
nil: multiset
U91: []
U92: multiset
U93: []
take: []

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

(83) 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)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U711(x1, x2, x3)  =  U711(x1, x2, x3)
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(x1)
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)  =  x1
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

Recursive path order with status [RPO].
Precedence:
zeros > cons > U51 > U52 > tt > mark1
zeros > cons > U51 > U52 > tt > s
zeros > cons > U71 > isNat > U11 > tt > mark1
zeros > cons > U71 > isNat > U11 > tt > s
zeros > cons > U71 > isNat > U21 > tt > mark1
zeros > cons > U71 > isNat > U21 > tt > s
zeros > 0 > mark1
isNatIList1 > U31 > tt > mark1
isNatIList1 > U31 > tt > s
isNatIList1 > U41 > U42 > tt > mark1
isNatIList1 > U41 > U42 > tt > s
isNatIList1 > isNat > U11 > tt > mark1
isNatIList1 > isNat > U11 > tt > s
isNatIList1 > isNat > U21 > tt > mark1
isNatIList1 > isNat > U21 > tt > s
take > U61 > U62 > tt > mark1
take > U61 > U62 > tt > s
take > U81 > nil > 0 > mark1
take > U81 > nil > tt > mark1
take > U81 > nil > tt > s
take > U91 > U92 > U93 > cons > U51 > U52 > tt > mark1
take > U91 > U92 > U93 > cons > U51 > U52 > tt > s
take > U91 > U92 > U93 > cons > U71 > isNat > U11 > tt > mark1
take > U91 > U92 > U93 > cons > U71 > isNat > U11 > tt > s
take > U91 > U92 > U93 > cons > U71 > isNat > U21 > tt > mark1
take > U91 > U92 > U93 > cons > U71 > isNat > U21 > tt > s

Status:
U71^13: [3,1,2]
mark1: multiset
zeros: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: []
U31: multiset
U41: []
U42: []
isNatIList1: multiset
U51: []
U52: multiset
U61: []
U62: []
U71: []
isNat: multiset
s: []
U81: multiset
nil: multiset
U91: []
U92: []
U93: multiset
take: []

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(X1, active(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(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
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(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(x1, x2)

Recursive path order with status [RPO].
Precedence:
zeros > 0 > U81 > mark1 > tt > cons > active1
zeros > 0 > U81 > mark1 > tt > nil > active1
zeros > 0 > U81 > mark1 > tt > take2 > active1
isNatIList1 > U31 > mark1 > tt > cons > active1
isNatIList1 > U31 > mark1 > tt > nil > active1
isNatIList1 > U31 > mark1 > tt > take2 > active1
isNatIList1 > U41 > U42 > mark1 > tt > cons > active1
isNatIList1 > U41 > U42 > mark1 > tt > nil > active1
isNatIList1 > U41 > U42 > mark1 > tt > take2 > active1
isNatIList1 > isNat1 > U11 > mark1 > tt > cons > active1
isNatIList1 > isNat1 > U11 > mark1 > tt > nil > active1
isNatIList1 > isNat1 > U11 > mark1 > tt > take2 > active1
isNatIList1 > isNat1 > U21 > mark1 > tt > cons > active1
isNatIList1 > isNat1 > U21 > mark1 > tt > nil > active1
isNatIList1 > isNat1 > U21 > mark1 > tt > take2 > active1
isNatList1 > U51 > U52 > mark1 > tt > cons > active1
isNatList1 > U51 > U52 > mark1 > tt > nil > active1
isNatList1 > U51 > U52 > mark1 > tt > take2 > active1
isNatList1 > U61 > U62 > mark1 > tt > cons > active1
isNatList1 > U61 > U62 > mark1 > tt > nil > active1
isNatList1 > U61 > U62 > mark1 > tt > take2 > active1
isNatList1 > isNat1 > U11 > mark1 > tt > cons > active1
isNatList1 > isNat1 > U11 > mark1 > tt > nil > active1
isNatList1 > isNat1 > U11 > mark1 > tt > take2 > active1
isNatList1 > isNat1 > U21 > mark1 > tt > cons > active1
isNatList1 > isNat1 > U21 > mark1 > tt > nil > active1
isNatList1 > isNat1 > U21 > mark1 > tt > take2 > active1
length > 0 > U81 > mark1 > tt > cons > active1
length > 0 > U81 > mark1 > tt > nil > active1
length > 0 > U81 > mark1 > tt > take2 > active1
length > U11 > mark1 > tt > cons > active1
length > U11 > mark1 > tt > nil > active1
length > U11 > mark1 > tt > take2 > active1
length > U71 > U72 > s > U21 > mark1 > tt > cons > active1
length > U71 > U72 > s > U21 > mark1 > tt > nil > active1
length > U71 > U72 > s > U21 > mark1 > tt > take2 > active1
length > U71 > U72 > s > U91 > U92 > U93 > mark1 > tt > cons > active1
length > U71 > U72 > s > U91 > U92 > U93 > mark1 > tt > nil > active1
length > U71 > U72 > s > U91 > U92 > U93 > mark1 > tt > take2 > active1

Status:
active1: multiset
zeros: multiset
mark1: [1]
cons: multiset
0: multiset
U11: []
tt: multiset
U21: []
U31: []
U41: []
U42: multiset
isNatIList1: [1]
U51: []
U52: multiset
isNatList1: [1]
U61: []
U62: multiset
U71: []
U72: multiset
isNat1: multiset
s: []
length: multiset
U81: multiset
nil: multiset
U91: []
U92: multiset
U93: []
take2: [1,2]

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:
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.

(87) PisEmptyProof (EQUIVALENT transformation)

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

(88) TRUE

(89) 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.

(90) 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)
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)  =  active(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
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

Recursive path order with status [RPO].
Precedence:
U62^11 > active1
zeros > cons > isNatIList > U31 > tt > s > mark1 > active1
zeros > cons > isNatIList > U41 > U42 > tt > s > mark1 > active1
zeros > 0 > isNatIList > U31 > tt > s > mark1 > active1
zeros > 0 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > 0 > isNatIList > U31 > tt > s > mark1 > active1
length > 0 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > isNatList1 > U51 > U52 > tt > s > mark1 > active1
length > isNatList1 > U61 > isNatIList > U31 > tt > s > mark1 > active1
length > isNatList1 > U61 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > isNatList1 > U61 > U62 > tt > s > mark1 > active1
length > isNatList1 > isNat > U11 > tt > s > mark1 > active1
length > isNatList1 > isNat > U21 > tt > s > mark1 > active1
length > U71 > U72 > s > mark1 > active1
length > U71 > isNat > U11 > tt > s > mark1 > active1
length > U71 > isNat > U21 > tt > s > mark1 > active1
take > U81 > nil > mark1 > active1
take > U91 > U92 > isNat > U11 > tt > s > mark1 > active1
take > U91 > U92 > isNat > U21 > tt > s > mark1 > active1
take > U91 > U92 > U93 > cons > isNatIList > U31 > tt > s > mark1 > active1
take > U91 > U92 > U93 > cons > isNatIList > U41 > U42 > tt > s > mark1 > active1

Status:
U62^11: multiset
active1: multiset
mark1: multiset
zeros: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: multiset
U31: []
U41: []
U42: multiset
isNatIList: multiset
U51: multiset
U52: multiset
isNatList1: multiset
U61: multiset
U62: []
U71: []
U72: []
isNat: multiset
s: multiset
length: multiset
U81: multiset
nil: multiset
U91: []
U92: multiset
U93: multiset
take: []

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)

(91) 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.

(92) PisEmptyProof (EQUIVALENT transformation)

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

(93) TRUE

(94) Obligation:

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

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.

(95) 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)
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)  =  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

Recursive path order with status [RPO].
Precedence:
zeros > cons > U41 > U42 > mark1 > tt
zeros > cons > U41 > U42 > mark1 > s
zeros > cons > isNat > U11 > mark1 > tt
zeros > cons > isNat > U11 > mark1 > s
zeros > cons > isNat > U21 > mark1 > tt
zeros > cons > isNat > U21 > mark1 > s
zeros > 0 > mark1 > tt
zeros > 0 > mark1 > s
isNatList > U51 > U52 > mark1 > tt
isNatList > U51 > U52 > mark1 > s
isNatList > U61 > isNatIList > U31 > mark1 > tt
isNatList > U61 > isNatIList > U31 > mark1 > s
isNatList > U61 > isNatIList > U41 > U42 > mark1 > tt
isNatList > U61 > isNatIList > U41 > U42 > mark1 > s
isNatList > U61 > isNatIList > isNat > U11 > mark1 > tt
isNatList > U61 > isNatIList > isNat > U11 > mark1 > s
isNatList > U61 > isNatIList > isNat > U21 > mark1 > tt
isNatList > U61 > isNatIList > isNat > U21 > mark1 > s
isNatList > U61 > U62 > mark1 > tt
isNatList > U61 > U62 > mark1 > s
length > 0 > mark1 > tt
length > 0 > mark1 > s
length > U71 > U72 > mark1 > tt
length > U71 > U72 > mark1 > s
length > U71 > isNat > U11 > mark1 > tt
length > U71 > isNat > U11 > mark1 > s
length > U71 > isNat > U21 > mark1 > tt
length > U71 > isNat > U21 > mark1 > s
take > U61 > isNatIList > U31 > mark1 > tt
take > U61 > isNatIList > U31 > mark1 > s
take > U61 > isNatIList > U41 > U42 > mark1 > tt
take > U61 > isNatIList > U41 > U42 > mark1 > s
take > U61 > isNatIList > isNat > U11 > mark1 > tt
take > U61 > isNatIList > isNat > U11 > mark1 > s
take > U61 > isNatIList > isNat > U21 > mark1 > tt
take > U61 > isNatIList > isNat > U21 > mark1 > s
take > U61 > U62 > mark1 > tt
take > U61 > U62 > mark1 > s
take > U81 > nil > 0 > mark1 > tt
take > U81 > nil > 0 > mark1 > s
take > U91 > U92 > U93 > cons > U41 > U42 > mark1 > tt
take > U91 > U92 > U93 > cons > U41 > U42 > mark1 > s
take > U91 > U92 > U93 > cons > isNat > U11 > mark1 > tt
take > U91 > U92 > U93 > cons > isNat > U11 > mark1 > s
take > U91 > U92 > U93 > cons > isNat > U21 > mark1 > tt
take > U91 > U92 > U93 > cons > isNat > U21 > mark1 > s

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

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)

(96) 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)
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.

(97) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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(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(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(x1, x2)

Recursive path order with status [RPO].
Precedence:
zeros > cons > mark1 > take2 > active1 > nil
isNatIList1 > U31 > tt > U92 > U93 > cons > mark1 > take2 > active1 > nil
isNatIList1 > U41 > U42 > tt > U92 > U93 > cons > mark1 > take2 > active1 > nil
isNatIList1 > isNat1 > U11 > tt > U92 > U93 > cons > mark1 > take2 > active1 > nil
isNatIList1 > isNat1 > U21 > tt > U92 > U93 > cons > mark1 > take2 > active1 > nil
isNatList > U51 > U52 > tt > U92 > U93 > cons > mark1 > take2 > active1 > nil
isNatList > U61 > U62 > tt > U92 > U93 > cons > mark1 > take2 > active1 > nil
isNatList > isNat1 > U11 > tt > U92 > U93 > cons > mark1 > take2 > active1 > nil
isNatList > isNat1 > U21 > tt > U92 > U93 > cons > mark1 > take2 > active1 > nil
length > 0 > U81 > mark1 > take2 > active1 > nil
length > U11 > tt > U92 > U93 > cons > mark1 > take2 > active1 > nil
length > U71 > U72 > s > U91 > U92 > U93 > cons > mark1 > take2 > active1 > nil

Status:
mark1: [1]
active1: multiset
zeros: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: []
U31: multiset
U41: []
U42: multiset
isNatIList1: multiset
U51: multiset
U52: multiset
isNatList: multiset
U61: []
U62: multiset
U71: []
U72: multiset
isNat1: multiset
s: multiset
length: multiset
U81: []
nil: multiset
U91: []
U92: []
U93: multiset
take2: [1,2]

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)

(98) 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.

(99) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U611(mark(X1), X2) → U611(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U611(x1, x2)  =  U611(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
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

Recursive path order with status [RPO].
Precedence:
zeros > cons > U41 > U42 > mark1 > U61^11
zeros > cons > U41 > U42 > mark1 > tt
zeros > cons > U41 > U42 > mark1 > s
zeros > cons > U51 > U52 > mark1 > U61^11
zeros > cons > U51 > U52 > mark1 > tt
zeros > cons > U51 > U52 > mark1 > s
zeros > cons > U71 > U72 > mark1 > U61^11
zeros > cons > U71 > U72 > mark1 > tt
zeros > cons > U71 > U72 > mark1 > s
zeros > 0 > mark1 > U61^11
zeros > 0 > mark1 > tt
zeros > 0 > mark1 > s
isNatList > U51 > U52 > mark1 > U61^11
isNatList > U51 > U52 > mark1 > tt
isNatList > U51 > U52 > mark1 > s
isNatList > U61 > isNatIList > U31 > mark1 > U61^11
isNatList > U61 > isNatIList > U31 > mark1 > tt
isNatList > U61 > isNatIList > U31 > mark1 > s
isNatList > U61 > isNatIList > U41 > U42 > mark1 > U61^11
isNatList > U61 > isNatIList > U41 > U42 > mark1 > tt
isNatList > U61 > isNatIList > U41 > U42 > mark1 > s
isNatList > U61 > isNatIList > isNat > U11 > mark1 > U61^11
isNatList > U61 > isNatIList > isNat > U11 > mark1 > tt
isNatList > U61 > isNatIList > isNat > U11 > mark1 > s
isNatList > U61 > isNatIList > isNat > U21 > mark1 > U61^11
isNatList > U61 > isNatIList > isNat > U21 > mark1 > tt
isNatList > U61 > isNatIList > isNat > U21 > mark1 > s
isNatList > U61 > U62 > mark1 > U61^11
isNatList > U61 > U62 > mark1 > tt
isNatList > U61 > U62 > mark1 > s
length > 0 > mark1 > U61^11
length > 0 > mark1 > tt
length > 0 > mark1 > s
length > U11 > mark1 > U61^11
length > U11 > mark1 > tt
length > U11 > mark1 > s
length > U71 > U72 > mark1 > U61^11
length > U71 > U72 > mark1 > tt
length > U71 > U72 > mark1 > s
take > U61 > isNatIList > U31 > mark1 > U61^11
take > U61 > isNatIList > U31 > mark1 > tt
take > U61 > isNatIList > U31 > mark1 > s
take > U61 > isNatIList > U41 > U42 > mark1 > U61^11
take > U61 > isNatIList > U41 > U42 > mark1 > tt
take > U61 > isNatIList > U41 > U42 > mark1 > s
take > U61 > isNatIList > isNat > U11 > mark1 > U61^11
take > U61 > isNatIList > isNat > U11 > mark1 > tt
take > U61 > isNatIList > isNat > U11 > mark1 > s
take > U61 > isNatIList > isNat > U21 > mark1 > U61^11
take > U61 > isNatIList > isNat > U21 > mark1 > tt
take > U61 > isNatIList > isNat > U21 > mark1 > s
take > U61 > U62 > mark1 > U61^11
take > U61 > U62 > mark1 > tt
take > U61 > U62 > mark1 > s
take > U81 > nil > mark1 > U61^11
take > U81 > nil > mark1 > tt
take > U81 > nil > mark1 > s
take > U91 > U92 > isNat > U11 > mark1 > U61^11
take > U91 > U92 > isNat > U11 > mark1 > tt
take > U91 > U92 > isNat > U11 > mark1 > s
take > U91 > U92 > isNat > U21 > mark1 > U61^11
take > U91 > U92 > isNat > U21 > mark1 > tt
take > U91 > U92 > isNat > U21 > mark1 > s
take > U91 > U92 > U93 > cons > U41 > U42 > mark1 > U61^11
take > U91 > U92 > U93 > cons > U41 > U42 > mark1 > tt
take > U91 > U92 > U93 > cons > U41 > U42 > mark1 > s
take > U91 > U92 > U93 > cons > U51 > U52 > mark1 > U61^11
take > U91 > U92 > U93 > cons > U51 > U52 > mark1 > tt
take > U91 > U92 > U93 > cons > U51 > U52 > mark1 > s
take > U91 > U92 > U93 > cons > U71 > U72 > mark1 > U61^11
take > U91 > U92 > U93 > cons > U71 > U72 > mark1 > tt
take > U91 > U92 > U93 > cons > U71 > U72 > mark1 > s

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

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(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(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, 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
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

Recursive path order with status [RPO].
Precedence:
U61^12 > active1
zeros > cons > U41 > U42 > tt > mark1 > isNat1 > active1
zeros > cons > U41 > U42 > tt > mark1 > nil > active1
zeros > 0 > tt > mark1 > isNat1 > active1
zeros > 0 > tt > mark1 > nil > active1
zeros > 0 > U81 > mark1 > isNat1 > active1
zeros > 0 > U81 > mark1 > nil > active1
length > 0 > tt > mark1 > isNat1 > active1
length > 0 > tt > mark1 > nil > active1
length > 0 > U81 > mark1 > isNat1 > active1
length > 0 > U81 > mark1 > nil > active1
length > U11 > tt > mark1 > isNat1 > active1
length > U11 > tt > mark1 > nil > active1
length > isNatList > U51 > U52 > tt > mark1 > isNat1 > active1
length > isNatList > U51 > U52 > tt > mark1 > nil > active1
length > isNatList > U61 > isNatIList > U31 > tt > mark1 > isNat1 > active1
length > isNatList > U61 > isNatIList > U31 > tt > mark1 > nil > active1
length > isNatList > U61 > isNatIList > U41 > U42 > tt > mark1 > isNat1 > active1
length > isNatList > U61 > isNatIList > U41 > U42 > tt > mark1 > nil > active1
length > isNatList > U61 > U62 > tt > mark1 > isNat1 > active1
length > isNatList > U61 > U62 > tt > mark1 > nil > active1
length > U71 > U72 > s > U21 > tt > mark1 > isNat1 > active1
length > U71 > U72 > s > U21 > tt > mark1 > nil > active1
length > U71 > U72 > s > isNatIList > U31 > tt > mark1 > isNat1 > active1
length > U71 > U72 > s > isNatIList > U31 > tt > mark1 > nil > active1
length > U71 > U72 > s > isNatIList > U41 > U42 > tt > mark1 > isNat1 > active1
length > U71 > U72 > s > isNatIList > U41 > U42 > tt > mark1 > nil > active1
take > U61 > isNatIList > U31 > tt > mark1 > isNat1 > active1
take > U61 > isNatIList > U31 > tt > mark1 > nil > active1
take > U61 > isNatIList > U41 > U42 > tt > mark1 > isNat1 > active1
take > U61 > isNatIList > U41 > U42 > tt > mark1 > nil > active1
take > U61 > U62 > tt > mark1 > isNat1 > active1
take > U61 > U62 > tt > mark1 > nil > active1
take > U81 > mark1 > isNat1 > active1
take > U81 > mark1 > nil > active1
take > U91 > U92 > U93 > cons > U41 > U42 > tt > mark1 > isNat1 > active1
take > U91 > U92 > U93 > cons > U41 > U42 > tt > mark1 > nil > active1

Status:
U61^12: [2,1]
active1: multiset
zeros: multiset
mark1: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: []
U31: []
U41: multiset
U42: multiset
isNatIList: multiset
U51: []
U52: multiset
isNatList: multiset
U61: multiset
U62: multiset
U71: []
U72: []
isNat1: multiset
s: multiset
length: []
U81: multiset
nil: multiset
U91: []
U92: []
U93: []
take: []

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(active(X)) → ISNATLIST(X)
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)  =  active(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
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

Recursive path order with status [RPO].
Precedence:
ISNATLIST1 > active1
zeros > cons > isNatIList > U31 > tt > s > mark1 > active1
zeros > cons > isNatIList > U41 > U42 > tt > s > mark1 > active1
zeros > 0 > isNatIList > U31 > tt > s > mark1 > active1
zeros > 0 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > 0 > isNatIList > U31 > tt > s > mark1 > active1
length > 0 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > isNatList1 > U51 > U52 > tt > s > mark1 > active1
length > isNatList1 > U61 > isNatIList > U31 > tt > s > mark1 > active1
length > isNatList1 > U61 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > isNatList1 > U61 > U62 > tt > s > mark1 > active1
length > isNatList1 > isNat > U11 > tt > s > mark1 > active1
length > isNatList1 > isNat > U21 > tt > s > mark1 > active1
length > U71 > U72 > s > mark1 > active1
length > U71 > isNat > U11 > tt > s > mark1 > active1
length > U71 > isNat > U21 > tt > s > mark1 > active1
take > U81 > nil > mark1 > active1
take > U91 > U92 > isNat > U11 > tt > s > mark1 > active1
take > U91 > U92 > isNat > U21 > tt > s > mark1 > active1
take > U91 > U92 > U93 > cons > isNatIList > U31 > tt > s > mark1 > active1
take > U91 > U92 > U93 > cons > isNatIList > U41 > U42 > tt > s > mark1 > active1

Status:
ISNATLIST1: multiset
active1: multiset
mark1: multiset
zeros: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: multiset
U31: []
U41: []
U42: multiset
isNatIList: multiset
U51: multiset
U52: multiset
isNatList1: multiset
U61: multiset
U62: []
U71: []
U72: []
isNat: multiset
s: multiset
length: multiset
U81: multiset
nil: multiset
U91: []
U92: multiset
U93: multiset
take: []

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:
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.

(108) PisEmptyProof (EQUIVALENT transformation)

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

(109) TRUE

(110) Obligation:

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

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.

(111) 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)
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)  =  active(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
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

Recursive path order with status [RPO].
Precedence:
U52^11 > active1
zeros > cons > isNatIList > U31 > tt > s > mark1 > active1
zeros > cons > isNatIList > U41 > U42 > tt > s > mark1 > active1
zeros > 0 > isNatIList > U31 > tt > s > mark1 > active1
zeros > 0 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > 0 > isNatIList > U31 > tt > s > mark1 > active1
length > 0 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > isNatList1 > U51 > U52 > tt > s > mark1 > active1
length > isNatList1 > U61 > isNatIList > U31 > tt > s > mark1 > active1
length > isNatList1 > U61 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > isNatList1 > U61 > U62 > tt > s > mark1 > active1
length > isNatList1 > isNat > U11 > tt > s > mark1 > active1
length > isNatList1 > isNat > U21 > tt > s > mark1 > active1
length > U71 > U72 > s > mark1 > active1
length > U71 > isNat > U11 > tt > s > mark1 > active1
length > U71 > isNat > U21 > tt > s > mark1 > active1
take > U81 > nil > mark1 > active1
take > U91 > U92 > isNat > U11 > tt > s > mark1 > active1
take > U91 > U92 > isNat > U21 > tt > s > mark1 > active1
take > U91 > U92 > U93 > cons > isNatIList > U31 > tt > s > mark1 > active1
take > U91 > U92 > U93 > cons > isNatIList > U41 > U42 > tt > s > mark1 > active1

Status:
U52^11: multiset
active1: multiset
mark1: multiset
zeros: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: multiset
U31: []
U41: []
U42: multiset
isNatIList: multiset
U51: multiset
U52: multiset
isNatList1: multiset
U61: multiset
U62: []
U71: []
U72: []
isNat: multiset
s: multiset
length: multiset
U81: multiset
nil: multiset
U91: []
U92: multiset
U93: multiset
take: []

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)

(112) 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.

(113) PisEmptyProof (EQUIVALENT transformation)

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

(114) TRUE

(115) 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.

(116) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U511(X1, mark(X2)) → U511(X1, X2)
U511(X1, active(X2)) → U511(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U511(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
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(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

Recursive path order with status [RPO].
Precedence:
zeros > cons > U51 > U52 > tt > mark1 > s > active1
length > 0 > isNatIList > U31 > tt > mark1 > s > active1
length > 0 > isNatIList > U41 > U42 > tt > mark1 > s > active1
length > 0 > U81 > nil > tt > mark1 > s > active1
length > U71 > U72 > mark1 > s > active1
length > U71 > isNat1 > U11 > tt > mark1 > s > active1
length > U71 > isNat1 > U21 > tt > mark1 > s > active1
take > isNatIList > U31 > tt > mark1 > s > active1
take > isNatIList > U41 > U42 > tt > mark1 > s > active1
take > U61 > U62 > tt > mark1 > s > active1
take > U81 > nil > tt > mark1 > s > active1
take > U91 > U92 > U93 > cons > U51 > U52 > tt > mark1 > s > active1

Status:
mark1: multiset
active1: multiset
zeros: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: []
U31: []
U41: []
U42: multiset
isNatIList: multiset
U51: multiset
U52: multiset
U61: multiset
U62: []
U71: []
U72: multiset
isNat1: multiset
s: multiset
length: multiset
U81: multiset
nil: multiset
U91: []
U92: []
U93: []
take: []

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)

(117) Obligation:

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(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.

(118) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U511(mark(X1), X2) → U511(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U511(x1, x2)  =  U511(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)  =  length
U81(x1)  =  U81
nil  =  nil
U91(x1, x2, x3, x4)  =  x1
U92(x1, x2, x3, x4)  =  U92
U93(x1, x2, x3, x4)  =  U93
take(x1, x2)  =  x1

Recursive path order with status [RPO].
Precedence:
U51^11 > cons
zeros > tt > U92 > U93 > mark1 > cons
length > 0 > isNatIList > U31 > tt > U92 > U93 > mark1 > cons
length > 0 > isNatIList > U41 > U42 > tt > U92 > U93 > mark1 > cons
length > 0 > isNatIList > isNat > U11 > tt > U92 > U93 > mark1 > cons
length > 0 > isNatIList > isNat > U21 > tt > U92 > U93 > mark1 > cons
length > 0 > U81 > nil > tt > U92 > U93 > mark1 > cons
length > isNatList1 > U51 > U52 > tt > U92 > U93 > mark1 > cons
length > isNatList1 > U61 > U62 > tt > U92 > U93 > mark1 > cons
length > isNatList1 > isNat > U11 > tt > U92 > U93 > mark1 > cons
length > isNatList1 > isNat > U21 > tt > U92 > U93 > mark1 > cons
length > U71 > U72 > s > isNatIList > U31 > tt > U92 > U93 > mark1 > cons
length > U71 > U72 > s > isNatIList > U41 > U42 > tt > U92 > U93 > mark1 > cons
length > U71 > U72 > s > isNatIList > isNat > U11 > tt > U92 > U93 > mark1 > cons
length > U71 > U72 > s > isNatIList > isNat > U21 > tt > U92 > U93 > mark1 > cons

Status:
U51^11: multiset
mark1: multiset
zeros: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: []
U31: multiset
U41: []
U42: multiset
isNatIList: multiset
U51: []
U52: []
isNatList1: multiset
U61: multiset
U62: multiset
U71: multiset
U72: multiset
isNat: multiset
s: multiset
length: multiset
U81: []
nil: multiset
U92: multiset
U93: multiset

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)

(119) Obligation:

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

U511(active(X1), X2) → U511(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(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(active(X1), X2) → U511(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U511(x1, x2)  =  U511(x1, 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
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(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

Recursive path order with status [RPO].
Precedence:
zeros > cons > U71 > U72 > mark1 > tt > length1 > active1 > 0
zeros > cons > U71 > U72 > s > active1 > 0
isNat1 > U11 > mark1 > tt > length1 > active1 > 0
isNat1 > U21 > mark1 > tt > length1 > active1 > 0
isNat1 > isNatList1 > U51 > U52 > mark1 > tt > length1 > active1 > 0
take > isNatIList > U31 > mark1 > tt > length1 > active1 > 0
take > isNatIList > U41 > U42 > mark1 > tt > length1 > active1 > 0
take > U61 > U62 > mark1 > tt > length1 > active1 > 0
take > U81 > nil > mark1 > tt > length1 > active1 > 0
take > U91 > U92 > U93 > cons > U71 > U72 > mark1 > tt > length1 > active1 > 0
take > U91 > U92 > U93 > cons > U71 > U72 > s > active1 > 0

Status:
U51^12: [1,2]
active1: multiset
zeros: multiset
mark1: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: []
U31: multiset
U41: []
U42: []
isNatIList: multiset
U51: multiset
U52: multiset
isNatList1: multiset
U61: []
U62: multiset
U71: []
U72: multiset
isNat1: [1]
s: multiset
length1: [1]
U81: []
nil: multiset
U91: []
U92: []
U93: []
take: []

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(active(X)) → ISNATILIST(X)
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)  =  active(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
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

Recursive path order with status [RPO].
Precedence:
ISNATILIST1 > active1
zeros > cons > isNatIList > U31 > tt > s > mark1 > active1
zeros > cons > isNatIList > U41 > U42 > tt > s > mark1 > active1
zeros > 0 > isNatIList > U31 > tt > s > mark1 > active1
zeros > 0 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > 0 > isNatIList > U31 > tt > s > mark1 > active1
length > 0 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > isNatList1 > U51 > U52 > tt > s > mark1 > active1
length > isNatList1 > U61 > isNatIList > U31 > tt > s > mark1 > active1
length > isNatList1 > U61 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > isNatList1 > U61 > U62 > tt > s > mark1 > active1
length > isNatList1 > isNat > U11 > tt > s > mark1 > active1
length > isNatList1 > isNat > U21 > tt > s > mark1 > active1
length > U71 > U72 > s > mark1 > active1
length > U71 > isNat > U11 > tt > s > mark1 > active1
length > U71 > isNat > U21 > tt > s > mark1 > active1
take > U81 > nil > mark1 > active1
take > U91 > U92 > isNat > U11 > tt > s > mark1 > active1
take > U91 > U92 > isNat > U21 > tt > s > mark1 > active1
take > U91 > U92 > U93 > cons > isNatIList > U31 > tt > s > mark1 > active1
take > U91 > U92 > U93 > cons > isNatIList > U41 > U42 > tt > s > mark1 > active1

Status:
ISNATILIST1: multiset
active1: multiset
mark1: multiset
zeros: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: multiset
U31: []
U41: []
U42: multiset
isNatIList: multiset
U51: multiset
U52: multiset
isNatList1: multiset
U61: multiset
U62: []
U71: []
U72: []
isNat: multiset
s: multiset
length: multiset
U81: multiset
nil: multiset
U91: []
U92: multiset
U93: multiset
take: []

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:
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.

(127) PisEmptyProof (EQUIVALENT transformation)

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

(128) TRUE

(129) 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.

(130) 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)
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)  =  active(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
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

Recursive path order with status [RPO].
Precedence:
U42^11 > active1
zeros > cons > isNatIList > U31 > tt > s > mark1 > active1
zeros > cons > isNatIList > U41 > U42 > tt > s > mark1 > active1
zeros > 0 > isNatIList > U31 > tt > s > mark1 > active1
zeros > 0 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > 0 > isNatIList > U31 > tt > s > mark1 > active1
length > 0 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > isNatList1 > U51 > U52 > tt > s > mark1 > active1
length > isNatList1 > U61 > isNatIList > U31 > tt > s > mark1 > active1
length > isNatList1 > U61 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > isNatList1 > U61 > U62 > tt > s > mark1 > active1
length > isNatList1 > isNat > U11 > tt > s > mark1 > active1
length > isNatList1 > isNat > U21 > tt > s > mark1 > active1
length > U71 > U72 > s > mark1 > active1
length > U71 > isNat > U11 > tt > s > mark1 > active1
length > U71 > isNat > U21 > tt > s > mark1 > active1
take > U81 > nil > mark1 > active1
take > U91 > U92 > isNat > U11 > tt > s > mark1 > active1
take > U91 > U92 > isNat > U21 > tt > s > mark1 > active1
take > U91 > U92 > U93 > cons > isNatIList > U31 > tt > s > mark1 > active1
take > U91 > U92 > U93 > cons > isNatIList > U41 > U42 > tt > s > mark1 > active1

Status:
U42^11: multiset
active1: multiset
mark1: multiset
zeros: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: multiset
U31: []
U41: []
U42: multiset
isNatIList: multiset
U51: multiset
U52: multiset
isNatList1: multiset
U61: multiset
U62: []
U71: []
U72: []
isNat: multiset
s: multiset
length: multiset
U81: multiset
nil: multiset
U91: []
U92: multiset
U93: multiset
take: []

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)

(131) 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.

(132) PisEmptyProof (EQUIVALENT transformation)

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

(133) TRUE

(134) 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.

(135) 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)
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)  =  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)  =  x1
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)  =  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

Recursive path order with status [RPO].
Precedence:
zeros > cons > U51 > U52 > tt > mark1 > U721 > s
zeros > cons > U71 > isNat > U11 > tt > mark1 > U721 > s
zeros > cons > U71 > isNat > U21 > tt > mark1 > U721 > s
take > isNatIList1 > U31 > tt > mark1 > U721 > s
take > isNatIList1 > U41 > U42 > tt > mark1 > U721 > s
take > isNatIList1 > isNat > U11 > tt > mark1 > U721 > s
take > isNatIList1 > isNat > U21 > tt > mark1 > U721 > s
take > U61 > U62 > tt > mark1 > U721 > s
take > U81 > nil > 0 > tt > mark1 > U721 > s
take > U91 > U92 > U93 > cons > U51 > U52 > tt > mark1 > U721 > s
take > U91 > U92 > U93 > cons > U71 > isNat > U11 > tt > mark1 > U721 > s
take > U91 > U92 > U93 > cons > U71 > isNat > U21 > tt > mark1 > U721 > s

Status:
U41^12: [1,2]
mark1: [1]
zeros: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: multiset
U31: []
U41: multiset
U42: multiset
isNatIList1: multiset
U51: multiset
U52: multiset
U61: []
U62: []
U71: []
U721: [1]
isNat: multiset
s: []
U81: []
nil: multiset
U91: []
U92: []
U93: []
take: []

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)

(136) Obligation:

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

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.

(137) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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(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
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

Recursive path order with status [RPO].
Precedence:
zeros > mark1 > active1 > 0
zeros > mark1 > active1 > tt > cons
zeros > mark1 > active1 > tt > nil
zeros > mark1 > s
length > U11 > mark1 > active1 > 0
length > U11 > mark1 > active1 > tt > cons
length > U11 > mark1 > active1 > tt > nil
length > U11 > mark1 > s
length > isNatList1 > U51 > U52 > mark1 > active1 > 0
length > isNatList1 > U51 > U52 > mark1 > active1 > tt > cons
length > isNatList1 > U51 > U52 > mark1 > active1 > tt > nil
length > isNatList1 > U51 > U52 > mark1 > s
length > U71 > U72 > mark1 > active1 > 0
length > U71 > U72 > mark1 > active1 > tt > cons
length > U71 > U72 > mark1 > active1 > tt > nil
length > U71 > U72 > mark1 > s
take > isNatIList > U31 > mark1 > active1 > 0
take > isNatIList > U31 > mark1 > active1 > tt > cons
take > isNatIList > U31 > mark1 > active1 > tt > nil
take > isNatIList > U31 > mark1 > s
take > isNatIList > U41 > U42 > mark1 > active1 > 0
take > isNatIList > U41 > U42 > mark1 > active1 > tt > cons
take > isNatIList > U41 > U42 > mark1 > active1 > tt > nil
take > isNatIList > U41 > U42 > mark1 > s
take > isNatIList > isNatList1 > U51 > U52 > mark1 > active1 > 0
take > isNatIList > isNatList1 > U51 > U52 > mark1 > active1 > tt > cons
take > isNatIList > isNatList1 > U51 > U52 > mark1 > active1 > tt > nil
take > isNatIList > isNatList1 > U51 > U52 > mark1 > s
take > U61 > U62 > mark1 > active1 > 0
take > U61 > U62 > mark1 > active1 > tt > cons
take > U61 > U62 > mark1 > active1 > tt > nil
take > U61 > U62 > mark1 > s
take > isNat1 > U11 > mark1 > active1 > 0
take > isNat1 > U11 > mark1 > active1 > tt > cons
take > isNat1 > U11 > mark1 > active1 > tt > nil
take > isNat1 > U11 > mark1 > s
take > isNat1 > U21 > mark1 > active1 > 0
take > isNat1 > U21 > mark1 > active1 > tt > cons
take > isNat1 > U21 > mark1 > active1 > tt > nil
take > isNat1 > U21 > mark1 > s
take > isNat1 > isNatList1 > U51 > U52 > mark1 > active1 > 0
take > isNat1 > isNatList1 > U51 > U52 > mark1 > active1 > tt > cons
take > isNat1 > isNatList1 > U51 > U52 > mark1 > active1 > tt > nil
take > isNat1 > isNatList1 > U51 > U52 > mark1 > s
take > U81 > mark1 > active1 > 0
take > U81 > mark1 > active1 > tt > cons
take > U81 > mark1 > active1 > tt > nil
take > U81 > mark1 > s
take > U91 > U92 > U93 > mark1 > active1 > 0
take > U91 > U92 > U93 > mark1 > active1 > tt > cons
take > U91 > U92 > U93 > mark1 > active1 > tt > nil
take > U91 > U92 > U93 > mark1 > s

Status:
U41^11: [1]
active1: multiset
zeros: multiset
mark1: multiset
cons: multiset
0: multiset
U11: []
tt: multiset
U21: []
U31: multiset
U41: multiset
U42: []
isNatIList: multiset
U51: multiset
U52: multiset
isNatList1: [1]
U61: multiset
U62: multiset
U71: []
U72: multiset
isNat1: multiset
s: multiset
length: multiset
U81: []
nil: multiset
U91: []
U92: []
U93: []
take: []

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)

(138) Obligation:

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

U411(active(X1), 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(active(X1), 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)
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(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

Recursive path order with status [RPO].
Precedence:
U41^12 > active1
zeros > cons > U51 > U52 > tt > U72 > mark1 > active1
zeros > cons > U51 > U52 > tt > U72 > s > active1
zeros > cons > U51 > U52 > tt > nil > mark1 > active1
zeros > cons > U51 > isNatList1 > tt > U72 > mark1 > active1
zeros > cons > U51 > isNatList1 > tt > U72 > s > active1
zeros > cons > U51 > isNatList1 > tt > nil > mark1 > active1
zeros > 0 > tt > U72 > mark1 > active1
zeros > 0 > tt > U72 > s > active1
zeros > 0 > tt > nil > mark1 > active1
zeros > 0 > U81 > nil > mark1 > active1
isNat > U11 > tt > U72 > mark1 > active1
isNat > U11 > tt > U72 > s > active1
isNat > U11 > tt > nil > mark1 > active1
isNat > U21 > tt > U72 > mark1 > active1
isNat > U21 > tt > U72 > s > active1
isNat > U21 > tt > nil > mark1 > active1
isNat > isNatList1 > tt > U72 > mark1 > active1
isNat > isNatList1 > tt > U72 > s > active1
isNat > isNatList1 > tt > nil > mark1 > active1
length > 0 > tt > U72 > mark1 > active1
length > 0 > tt > U72 > s > active1
length > 0 > tt > nil > mark1 > active1
length > 0 > U81 > nil > mark1 > active1
length > U11 > tt > U72 > mark1 > active1
length > U11 > tt > U72 > s > active1
length > U11 > tt > nil > mark1 > active1
length > isNatList1 > tt > U72 > mark1 > active1
length > isNatList1 > tt > U72 > s > active1
length > isNatList1 > tt > nil > mark1 > active1
length > U71 > U72 > mark1 > active1
length > U71 > U72 > s > active1
take > U61 > isNatIList > U31 > tt > U72 > mark1 > active1
take > U61 > isNatIList > U31 > tt > U72 > s > active1
take > U61 > isNatIList > U31 > tt > nil > mark1 > active1
take > U61 > isNatIList > U41 > U42 > tt > U72 > mark1 > active1
take > U61 > isNatIList > U41 > U42 > tt > U72 > s > active1
take > U61 > isNatIList > U41 > U42 > tt > nil > mark1 > active1
take > U61 > isNatIList > isNatList1 > tt > U72 > mark1 > active1
take > U61 > isNatIList > isNatList1 > tt > U72 > s > active1
take > U61 > isNatIList > isNatList1 > tt > nil > mark1 > active1
take > U61 > U62 > tt > U72 > mark1 > active1
take > U61 > U62 > tt > U72 > s > active1
take > U61 > U62 > tt > nil > mark1 > active1
take > U81 > nil > mark1 > active1
take > U91 > U92 > U93 > cons > U51 > U52 > tt > U72 > mark1 > active1
take > U91 > U92 > U93 > cons > U51 > U52 > tt > U72 > s > active1
take > U91 > U92 > U93 > cons > U51 > U52 > tt > nil > mark1 > active1
take > U91 > U92 > U93 > cons > U51 > isNatList1 > tt > U72 > mark1 > active1
take > U91 > U92 > U93 > cons > U51 > isNatList1 > tt > U72 > s > active1
take > U91 > U92 > U93 > cons > U51 > isNatList1 > tt > nil > mark1 > active1

Status:
U41^12: multiset
active1: multiset
zeros: multiset
mark1: [1]
cons: []
0: multiset
U11: []
tt: multiset
U21: []
U31: []
U41: []
U42: []
isNatIList: multiset
U51: []
U52: []
isNatList1: multiset
U61: multiset
U62: []
U71: []
U72: multiset
isNat: multiset
s: multiset
length: multiset
U81: []
nil: multiset
U91: []
U92: multiset
U93: []
take: []

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(active(X)) → U311(X)
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)  =  active(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
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

Recursive path order with status [RPO].
Precedence:
U31^11 > active1
zeros > cons > isNatIList > U31 > tt > s > mark1 > active1
zeros > cons > isNatIList > U41 > U42 > tt > s > mark1 > active1
zeros > 0 > isNatIList > U31 > tt > s > mark1 > active1
zeros > 0 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > 0 > isNatIList > U31 > tt > s > mark1 > active1
length > 0 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > isNatList1 > U51 > U52 > tt > s > mark1 > active1
length > isNatList1 > U61 > isNatIList > U31 > tt > s > mark1 > active1
length > isNatList1 > U61 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > isNatList1 > U61 > U62 > tt > s > mark1 > active1
length > isNatList1 > isNat > U11 > tt > s > mark1 > active1
length > isNatList1 > isNat > U21 > tt > s > mark1 > active1
length > U71 > U72 > s > mark1 > active1
length > U71 > isNat > U11 > tt > s > mark1 > active1
length > U71 > isNat > U21 > tt > s > mark1 > active1
take > U81 > nil > mark1 > active1
take > U91 > U92 > isNat > U11 > tt > s > mark1 > active1
take > U91 > U92 > isNat > U21 > tt > s > mark1 > active1
take > U91 > U92 > U93 > cons > isNatIList > U31 > tt > s > mark1 > active1
take > U91 > U92 > U93 > cons > isNatIList > U41 > U42 > tt > s > mark1 > active1

Status:
U31^11: multiset
active1: multiset
mark1: multiset
zeros: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: multiset
U31: []
U41: []
U42: multiset
isNatIList: multiset
U51: multiset
U52: multiset
isNatList1: multiset
U61: multiset
U62: []
U71: []
U72: []
isNat: multiset
s: multiset
length: multiset
U81: multiset
nil: multiset
U91: []
U92: multiset
U93: multiset
take: []

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:
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.

(146) PisEmptyProof (EQUIVALENT transformation)

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

(147) TRUE

(148) 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.

(149) 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)
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)  =  active(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
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

Recursive path order with status [RPO].
Precedence:
U21^11 > active1
zeros > cons > isNatIList > U31 > tt > s > mark1 > active1
zeros > cons > isNatIList > U41 > U42 > tt > s > mark1 > active1
zeros > 0 > isNatIList > U31 > tt > s > mark1 > active1
zeros > 0 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > 0 > isNatIList > U31 > tt > s > mark1 > active1
length > 0 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > isNatList1 > U51 > U52 > tt > s > mark1 > active1
length > isNatList1 > U61 > isNatIList > U31 > tt > s > mark1 > active1
length > isNatList1 > U61 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > isNatList1 > U61 > U62 > tt > s > mark1 > active1
length > isNatList1 > isNat > U11 > tt > s > mark1 > active1
length > isNatList1 > isNat > U21 > tt > s > mark1 > active1
length > U71 > U72 > s > mark1 > active1
length > U71 > isNat > U11 > tt > s > mark1 > active1
length > U71 > isNat > U21 > tt > s > mark1 > active1
take > U81 > nil > mark1 > active1
take > U91 > U92 > isNat > U11 > tt > s > mark1 > active1
take > U91 > U92 > isNat > U21 > tt > s > mark1 > active1
take > U91 > U92 > U93 > cons > isNatIList > U31 > tt > s > mark1 > active1
take > U91 > U92 > U93 > cons > isNatIList > U41 > U42 > tt > s > mark1 > active1

Status:
U21^11: multiset
active1: multiset
mark1: multiset
zeros: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: multiset
U31: []
U41: []
U42: multiset
isNatIList: multiset
U51: multiset
U52: multiset
isNatList1: multiset
U61: multiset
U62: []
U71: []
U72: []
isNat: multiset
s: multiset
length: multiset
U81: multiset
nil: multiset
U91: []
U92: multiset
U93: multiset
take: []

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)

(150) 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.

(151) PisEmptyProof (EQUIVALENT transformation)

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

(152) TRUE

(153) 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.

(154) 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)
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)  =  active(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
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

Recursive path order with status [RPO].
Precedence:
U11^11 > active1
zeros > cons > isNatIList > U31 > tt > s > mark1 > active1
zeros > cons > isNatIList > U41 > U42 > tt > s > mark1 > active1
zeros > 0 > isNatIList > U31 > tt > s > mark1 > active1
zeros > 0 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > 0 > isNatIList > U31 > tt > s > mark1 > active1
length > 0 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > isNatList1 > U51 > U52 > tt > s > mark1 > active1
length > isNatList1 > U61 > isNatIList > U31 > tt > s > mark1 > active1
length > isNatList1 > U61 > isNatIList > U41 > U42 > tt > s > mark1 > active1
length > isNatList1 > U61 > U62 > tt > s > mark1 > active1
length > isNatList1 > isNat > U11 > tt > s > mark1 > active1
length > isNatList1 > isNat > U21 > tt > s > mark1 > active1
length > U71 > U72 > s > mark1 > active1
length > U71 > isNat > U11 > tt > s > mark1 > active1
length > U71 > isNat > U21 > tt > s > mark1 > active1
take > U81 > nil > mark1 > active1
take > U91 > U92 > isNat > U11 > tt > s > mark1 > active1
take > U91 > U92 > isNat > U21 > tt > s > mark1 > active1
take > U91 > U92 > U93 > cons > isNatIList > U31 > tt > s > mark1 > active1
take > U91 > U92 > U93 > cons > isNatIList > U41 > U42 > tt > s > mark1 > active1

Status:
U11^11: multiset
active1: multiset
mark1: multiset
zeros: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: multiset
U31: []
U41: []
U42: multiset
isNatIList: multiset
U51: multiset
U52: multiset
isNatList1: multiset
U61: multiset
U62: []
U71: []
U72: []
isNat: multiset
s: multiset
length: multiset
U81: multiset
nil: multiset
U91: []
U92: multiset
U93: multiset
take: []

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)

(155) 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.

(156) PisEmptyProof (EQUIVALENT transformation)

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

(157) TRUE

(158) 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.

(159) 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)  =  CONS(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)  =  x1
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  length
U81(x1)  =  U81
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

Recursive path order with status [RPO].
Precedence:
zeros > tt > s > mark1 > CONS1
zeros > tt > s > mark1 > cons
zeros > tt > s > mark1 > 0
zeros > tt > s > mark1 > U911
zeros > tt > U92 > U93 > mark1 > CONS1
zeros > tt > U92 > U93 > mark1 > cons
zeros > tt > U92 > U93 > mark1 > 0
zeros > tt > U92 > U93 > mark1 > U911
length > isNatList > U51 > U52 > tt > s > mark1 > CONS1
length > isNatList > U51 > U52 > tt > s > mark1 > cons
length > isNatList > U51 > U52 > tt > s > mark1 > 0
length > isNatList > U51 > U52 > tt > s > mark1 > U911
length > isNatList > U51 > U52 > tt > U92 > U93 > mark1 > CONS1
length > isNatList > U51 > U52 > tt > U92 > U93 > mark1 > cons
length > isNatList > U51 > U52 > tt > U92 > U93 > mark1 > 0
length > isNatList > U51 > U52 > tt > U92 > U93 > mark1 > U911
length > isNatList > U61 > U62 > tt > s > mark1 > CONS1
length > isNatList > U61 > U62 > tt > s > mark1 > cons
length > isNatList > U61 > U62 > tt > s > mark1 > 0
length > isNatList > U61 > U62 > tt > s > mark1 > U911
length > isNatList > U61 > U62 > tt > U92 > U93 > mark1 > CONS1
length > isNatList > U61 > U62 > tt > U92 > U93 > mark1 > cons
length > isNatList > U61 > U62 > tt > U92 > U93 > mark1 > 0
length > isNatList > U61 > U62 > tt > U92 > U93 > mark1 > U911
length > isNatList > isNat > U11 > tt > s > mark1 > CONS1
length > isNatList > isNat > U11 > tt > s > mark1 > cons
length > isNatList > isNat > U11 > tt > s > mark1 > 0
length > isNatList > isNat > U11 > tt > s > mark1 > U911
length > isNatList > isNat > U11 > tt > U92 > U93 > mark1 > CONS1
length > isNatList > isNat > U11 > tt > U92 > U93 > mark1 > cons
length > isNatList > isNat > U11 > tt > U92 > U93 > mark1 > 0
length > isNatList > isNat > U11 > tt > U92 > U93 > mark1 > U911
length > isNatList > isNat > U21 > tt > s > mark1 > CONS1
length > isNatList > isNat > U21 > tt > s > mark1 > cons
length > isNatList > isNat > U21 > tt > s > mark1 > 0
length > isNatList > isNat > U21 > tt > s > mark1 > U911
length > isNatList > isNat > U21 > tt > U92 > U93 > mark1 > CONS1
length > isNatList > isNat > U21 > tt > U92 > U93 > mark1 > cons
length > isNatList > isNat > U21 > tt > U92 > U93 > mark1 > 0
length > isNatList > isNat > U21 > tt > U92 > U93 > mark1 > U911
length > U71 > isNat > U11 > tt > s > mark1 > CONS1
length > U71 > isNat > U11 > tt > s > mark1 > cons
length > U71 > isNat > U11 > tt > s > mark1 > 0
length > U71 > isNat > U11 > tt > s > mark1 > U911
length > U71 > isNat > U11 > tt > U92 > U93 > mark1 > CONS1
length > U71 > isNat > U11 > tt > U92 > U93 > mark1 > cons
length > U71 > isNat > U11 > tt > U92 > U93 > mark1 > 0
length > U71 > isNat > U11 > tt > U92 > U93 > mark1 > U911
length > U71 > isNat > U21 > tt > s > mark1 > CONS1
length > U71 > isNat > U21 > tt > s > mark1 > cons
length > U71 > isNat > U21 > tt > s > mark1 > 0
length > U71 > isNat > U21 > tt > s > mark1 > U911
length > U71 > isNat > U21 > tt > U92 > U93 > mark1 > CONS1
length > U71 > isNat > U21 > tt > U92 > U93 > mark1 > cons
length > U71 > isNat > U21 > tt > U92 > U93 > mark1 > 0
length > U71 > isNat > U21 > tt > U92 > U93 > mark1 > U911
take > isNatIList > U31 > tt > s > mark1 > CONS1
take > isNatIList > U31 > tt > s > mark1 > cons
take > isNatIList > U31 > tt > s > mark1 > 0
take > isNatIList > U31 > tt > s > mark1 > U911
take > isNatIList > U31 > tt > U92 > U93 > mark1 > CONS1
take > isNatIList > U31 > tt > U92 > U93 > mark1 > cons
take > isNatIList > U31 > tt > U92 > U93 > mark1 > 0
take > isNatIList > U31 > tt > U92 > U93 > mark1 > U911
take > isNatIList > U41 > U42 > tt > s > mark1 > CONS1
take > isNatIList > U41 > U42 > tt > s > mark1 > cons
take > isNatIList > U41 > U42 > tt > s > mark1 > 0
take > isNatIList > U41 > U42 > tt > s > mark1 > U911
take > isNatIList > U41 > U42 > tt > U92 > U93 > mark1 > CONS1
take > isNatIList > U41 > U42 > tt > U92 > U93 > mark1 > cons
take > isNatIList > U41 > U42 > tt > U92 > U93 > mark1 > 0
take > isNatIList > U41 > U42 > tt > U92 > U93 > mark1 > U911
take > isNatIList > isNatList > U51 > U52 > tt > s > mark1 > CONS1
take > isNatIList > isNatList > U51 > U52 > tt > s > mark1 > cons
take > isNatIList > isNatList > U51 > U52 > tt > s > mark1 > 0
take > isNatIList > isNatList > U51 > U52 > tt > s > mark1 > U911
take > isNatIList > isNatList > U51 > U52 > tt > U92 > U93 > mark1 > CONS1
take > isNatIList > isNatList > U51 > U52 > tt > U92 > U93 > mark1 > cons
take > isNatIList > isNatList > U51 > U52 > tt > U92 > U93 > mark1 > 0
take > isNatIList > isNatList > U51 > U52 > tt > U92 > U93 > mark1 > U911
take > isNatIList > isNatList > U61 > U62 > tt > s > mark1 > CONS1
take > isNatIList > isNatList > U61 > U62 > tt > s > mark1 > cons
take > isNatIList > isNatList > U61 > U62 > tt > s > mark1 > 0
take > isNatIList > isNatList > U61 > U62 > tt > s > mark1 > U911
take > isNatIList > isNatList > U61 > U62 > tt > U92 > U93 > mark1 > CONS1
take > isNatIList > isNatList > U61 > U62 > tt > U92 > U93 > mark1 > cons
take > isNatIList > isNatList > U61 > U62 > tt > U92 > U93 > mark1 > 0
take > isNatIList > isNatList > U61 > U62 > tt > U92 > U93 > mark1 > U911
take > isNatIList > isNatList > isNat > U11 > tt > s > mark1 > CONS1
take > isNatIList > isNatList > isNat > U11 > tt > s > mark1 > cons
take > isNatIList > isNatList > isNat > U11 > tt > s > mark1 > 0
take > isNatIList > isNatList > isNat > U11 > tt > s > mark1 > U911
take > isNatIList > isNatList > isNat > U11 > tt > U92 > U93 > mark1 > CONS1
take > isNatIList > isNatList > isNat > U11 > tt > U92 > U93 > mark1 > cons
take > isNatIList > isNatList > isNat > U11 > tt > U92 > U93 > mark1 > 0
take > isNatIList > isNatList > isNat > U11 > tt > U92 > U93 > mark1 > U911
take > isNatIList > isNatList > isNat > U21 > tt > s > mark1 > CONS1
take > isNatIList > isNatList > isNat > U21 > tt > s > mark1 > cons
take > isNatIList > isNatList > isNat > U21 > tt > s > mark1 > 0
take > isNatIList > isNatList > isNat > U21 > tt > s > mark1 > U911
take > isNatIList > isNatList > isNat > U21 > tt > U92 > U93 > mark1 > CONS1
take > isNatIList > isNatList > isNat > U21 > tt > U92 > U93 > mark1 > cons
take > isNatIList > isNatList > isNat > U21 > tt > U92 > U93 > mark1 > 0
take > isNatIList > isNatList > isNat > U21 > tt > U92 > U93 > mark1 > U911
take > U81 > nil > tt > s > mark1 > CONS1
take > U81 > nil > tt > s > mark1 > cons
take > U81 > nil > tt > s > mark1 > 0
take > U81 > nil > tt > s > mark1 > U911
take > U81 > nil > tt > U92 > U93 > mark1 > CONS1
take > U81 > nil > tt > U92 > U93 > mark1 > cons
take > U81 > nil > tt > U92 > U93 > mark1 > 0
take > U81 > nil > tt > U92 > U93 > mark1 > U911

Status:
CONS1: [1]
mark1: [1]
zeros: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: []
U31: []
U41: multiset
U42: []
isNatIList: multiset
U51: []
U52: multiset
isNatList: multiset
U61: multiset
U62: []
U71: []
isNat: []
s: multiset
length: []
U81: []
nil: multiset
U911: multiset
U92: multiset
U93: multiset
take: []

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)

(160) 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.

(161) 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(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)  =  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(x1, x2)

Recursive path order with status [RPO].
Precedence:
zeros > mark1 > tt > take2 > CONS2
zeros > cons > CONS2
isNatList1 > U51 > U52 > mark1 > tt > take2 > CONS2
isNatList1 > U61 > isNatIList1 > U31 > mark1 > tt > take2 > CONS2
isNatList1 > U61 > isNatIList1 > U41 > U42 > mark1 > tt > take2 > CONS2
isNatList1 > U61 > U62 > mark1 > tt > take2 > CONS2
length > 0 > isNatIList1 > U31 > mark1 > tt > take2 > CONS2
length > 0 > isNatIList1 > U41 > U42 > mark1 > tt > take2 > CONS2
length > 0 > U81 > nil > mark1 > tt > take2 > CONS2
length > U11 > mark1 > tt > take2 > CONS2
length > U71 > U72 > s > U21 > mark1 > tt > take2 > CONS2
length > U71 > U72 > s > isNatIList1 > U31 > mark1 > tt > take2 > CONS2
length > U71 > U72 > s > isNatIList1 > U41 > U42 > mark1 > tt > take2 > CONS2
length > U71 > U72 > s > U91 > U92 > U93 > mark1 > tt > take2 > CONS2
length > U71 > U72 > s > U91 > U92 > U93 > cons > CONS2

Status:
CONS2: [2,1]
mark1: [1]
zeros: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: []
U31: multiset
U41: multiset
U42: multiset
isNatIList1: [1]
U51: multiset
U52: multiset
isNatList1: multiset
U61: []
U62: []
U71: multiset
U72: []
s: multiset
length: []
U81: []
nil: multiset
U91: []
U92: multiset
U93: []
take2: [1,2]

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)

(162) 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.

(163) 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
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

Recursive path order with status [RPO].
Precedence:
zeros > mark1 > active1 > tt > nil > 0
zeros > cons > active1 > tt > nil > 0
isNatList1 > U51 > U52 > mark1 > active1 > tt > nil > 0
isNatList1 > U61 > U62 > mark1 > active1 > tt > nil > 0
isNatList1 > isNat > U11 > mark1 > active1 > tt > nil > 0
isNatList1 > isNat > U21 > mark1 > active1 > tt > nil > 0
length > U71 > U72 > mark1 > active1 > tt > nil > 0
length > U71 > U72 > s > active1 > tt > nil > 0
length > U71 > isNat > U11 > mark1 > active1 > tt > nil > 0
length > U71 > isNat > U21 > mark1 > active1 > tt > nil > 0
take > isNatIList > U31 > mark1 > active1 > tt > nil > 0
take > isNatIList > U41 > U42 > mark1 > active1 > tt > nil > 0
take > isNatIList > isNat > U11 > mark1 > active1 > tt > nil > 0
take > isNatIList > isNat > U21 > mark1 > active1 > tt > nil > 0
take > U61 > U62 > mark1 > active1 > tt > nil > 0
take > U81 > mark1 > active1 > tt > nil > 0
take > U91 > isNat > U11 > mark1 > active1 > tt > nil > 0
take > U91 > isNat > U21 > mark1 > active1 > tt > nil > 0
take > U91 > U92 > U93 > mark1 > active1 > tt > nil > 0
take > U91 > U92 > U93 > cons > active1 > tt > nil > 0

Status:
active1: multiset
zeros: multiset
mark1: multiset
cons: []
0: multiset
U11: []
tt: multiset
U21: []
U31: []
U41: multiset
U42: multiset
isNatIList: multiset
U51: multiset
U52: multiset
isNatList1: [1]
U61: []
U62: []
U71: []
U72: []
isNat: multiset
s: []
length: []
U81: []
nil: multiset
U91: []
U92: []
U93: []
take: []

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)

(164) Obligation:

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

CONS(active(X1), X2) → CONS(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(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(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(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

Recursive path order with status [RPO].
Precedence:
zeros > cons > active1 > s
zeros > 0 > mark1 > nil > active1 > s
zeros > 0 > tt > nil > active1 > s
length > 0 > mark1 > nil > active1 > s
length > 0 > tt > nil > active1 > s
length > U11 > mark1 > nil > active1 > s
length > U11 > tt > nil > active1 > s
length > U71 > U72 > mark1 > nil > active1 > s
take > isNatIList > U31 > mark1 > nil > active1 > s
take > isNatIList > U31 > tt > nil > active1 > s
take > isNatIList > U41 > U42 > mark1 > nil > active1 > s
take > isNatIList > U41 > U42 > tt > nil > active1 > s
take > isNatIList > isNatList1 > U51 > U52 > mark1 > nil > active1 > s
take > isNatIList > isNatList1 > U51 > U52 > tt > nil > active1 > s
take > isNatIList > isNatList1 > U61 > U62 > mark1 > nil > active1 > s
take > isNatIList > isNatList1 > U61 > U62 > tt > nil > active1 > s
take > isNatIList > isNat > U11 > mark1 > nil > active1 > s
take > isNatIList > isNat > U11 > tt > nil > active1 > s
take > isNatIList > isNat > U21 > mark1 > nil > active1 > s
take > isNatIList > isNat > U21 > tt > nil > active1 > s
take > U81 > mark1 > nil > active1 > s
take > U91 > isNat > U11 > mark1 > nil > active1 > s
take > U91 > isNat > U11 > tt > nil > active1 > s
take > U91 > isNat > U21 > mark1 > nil > active1 > s
take > U91 > isNat > U21 > tt > nil > active1 > s
take > U91 > U92 > U93 > mark1 > nil > active1 > s
take > U91 > U92 > U93 > cons > active1 > s

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

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:
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.

(167) PisEmptyProof (EQUIVALENT transformation)

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

(168) TRUE

(169) 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.